Quantum Probability And Infinite Dimensional Analysis: From Foundations To Applications Krupp-Kolleg Greifswald, Germany 22-28 June 2003 (Qp-Pq Quantum Probability and White Noise Analysis)

QP-PQ: Quantum Probability and White Noise Analysis Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy QP-PQ: Quantum Probability and White Noise Analysis VOl. 18:

Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds. M. Schurmann and U. Franz

Vol. 17:

Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki

Vol. 16:

Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hora

Vol. 15: Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg Vol. 14:

Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola

Vol. 13:

Foundations of Probability and Physics ed. A. Khrennikov

QP-PQ VOl. 10:

Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay

VOl. 9:

Quantum Probability and Related Topics ed. L. Accardi

VOl. 8:

Quantum Probability and Related Topics ed. L. Accardi

VOl. 7:

Quantum Probability and Related Topics ed. L. Accardi

Vol. 6:

Quantum Probability and Related Topics ed. L. Accardi

Qp-pQ Quantum Probability and White Noise Analysis Volume XVIII

From Foundations to Applications Krupp-Kolleg Greifswald, Germany

22 - 28 June 2003

Editors

Michael Schurmann Uwe Franz University of Greifswald, Germany

NEW JERSEY

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EeWorld Scientific LONDON * SINGAPORE

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QP-PQ: Quantum Probability and White Noise Analysis Vol. XVIII QUANTUM PROBABILITY AND INFINITE-DIMENSIONAL ANALYSIS From Foundations to Applications Copyright 0 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof;may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying,recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

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INTRODUCTION

In Spring 2003, we organized the Special Semester “Quantum Probability: From Foundations to Applications” in Greifswald, Germany. The main events were 0

0

0

a school “Quantum Independent Increment Processes: Structure and Applications to Physics,” two workshops on “Non-commutative Martingales and Free Probability” and “Dilations, Endomorphism Semigroups, and their Classification by Product Systems,” and a n international conference “Quantum Probability and Infinite Dimensional Analysis.”

Furthermore, in Fall 2003 a workshop on “Quantum Independent Increment Processes” was held in Levico Terme, Italy, as a continuation of the Greifswald School. The goal of this Special Semester was to bring together scientists with different backgrounds who work in quantum probability and related fields, and to stimulate new collaborations. We would like t o thank all participants and lecturers for their hard work and for helping t o create a very stimulating atmosphere. The research papers collected in this volume represent the topics discussed during the Special Semester and reflect the active developments ranging from the foundations of quantum probability to its applications. The lecture notes of the school on “Quantum Independent Increment Processes” will be published in the Springer series Lecture Notes in Mathematics. Many people have been involved in the organization of the Special Semester, it would be impossible to name them all. We are particularly indebted t o the Volkswagen Foundation and the DFG, who supported the school on “Quantum Independent Increment Processes” and the international conference ‘Quantum Probability and Infinite Dimensional Analysis,” and thereby made the Special Semester possible. We also acknowledge the support by the European Community for the Research Training Network

V

vi

“QP-Applications: Quantum Probability with Applications to Physics, Information Theory and Biology” under contract HPRN-CT-2002-00279. Special thanks go to Mrs. Zeidler for taking care of the logistics of the Special Semester and for her help with the editing of these proceedings.

Michael Schiirmann Uwe Franz Greifswald, October 2004

CONTENTS

Introduction

V

Probability Measures in Terms of Creation, Annihilation, and Neutral Operators Luigi Accardi, Hui-Hsiung Kuo and Aurel Stan

1

Semi Groupes Associes B l’operateur de Laplace-Levy Luigi Accardi and Habib Ouerdiane

12

Stochastic Golden Rule for a System Interacting with a Fermi Field Luigi Accardi, R.A. Roschin and I.V. Volovich

28

Generating Function Method for Orthogonal Polynomials and Jacobi-Szego Parameters Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo

42

Low Temperature Superconductivity and the Stochastic Limit Fabio Bagarello Multiquantum Markov Semigroups, Interacting Branching Processes and Nonlinear Kinetic Equations. Finite Dimensional Case V.P. Belavkin and C.R. Williams A Note on Vacuum-Adapted Semimartingales and Monotone Independence Alexander C.R. Belton Quantum Stochastic Processes and Applications Mohamed Ben Chrouda, Mohamed El Oued and Habib Ouerdiane Regular Quantum Stochastic Cocycles have Exponential Product Systems B.V. Rajarama Bhat and J. Martin Lindsay

vii

56

67

105 115

126

viii

Evolution of the Atom-Field System in Interacting Fock Space P.K . Das Quantum Mechanics on the Circle through Hopf q-Deformations of the Kinematical Algebra with Possible Applications to LQvyProcesses V.K. Dobrev, H.-D. Doebner and R. Twarock

141

153

On Algebraic and Quantum Random Walks Demosthenes Ellinas

174

Dual Representations for the Schrodinger Algebra Philip Feinsilver and Rent! Schott

201

Harmonic Analysis on Non- Amenable Coxeter Groups Gero Fendler

216

A Limit Theorem for Conditionally Independent Beam Splittings K.H. Fichtner, Volkmar Liebscher and Masanori Ohya

227

On Factors Associated with Quantum Markov States Corresponding to Nearest Neighbor Models on a Cayley Tree Francesco Fidaleo and Farruh Mukhamedov

237

On Quantum Logical Gates on a General Fock Space Wolfgang Freudenberg, Masanori Ohya and Noburo Watanabe

252

The Chaotic Chameleon Richard D. Gill

269

On an Argument of David Deutsch Richard D. Gill

277

Volterra Representations of Gaussian Processes with an Infinite-Dimensional Orthogonal Complement Yuji Hibino and Hiroshi Muraoka

293

The Method of Double Product Integrals in Quantisation of Lie Bialgebras Robin L. Hudson

303

ix

On Noncommutative Independence Romuald Lenczewski

320

Lkvy Processes and Jacobi Fields Eugene Lytvynov

337

Ic-Decomposabilityof Positive Maps Wtadystaw A. Majewski and Marcin Marciniak

362

An Introduction to LBvy Processes in Lie Groups Ming Liao

375

Duality of W*-Correspondences and Applications Paul S. Muhly and Baruch Solel

396

Extendability of Generalized Quantum Markov States Hiromichi Ohno

415

White Noise Approach to the Low Density Limit Alexander N . Pechen

428

Asymptotics of Large Truncated Haar Unitary Matrices Jlilia Riffy

448

Quantum Optical Scenarios for Stochastic Resonance Vyacheslav Shatokhin, Thomas Wellens and Andreas Buchleitner

457

On a Classical Scheme in a Noncommutative Multiparameter Ergodic Theory Adam G. Skalski

473

LBvy Processes and Tensor Product Systems of Hilbert Modules Michael Skeide

492

Three Ways to Representations of B A ( E ) Michael Skeide

504

The Hamiltonian of a Simple Pure Number Process Wilhelm von Waldenfels

518

On Topological Entropy of Quotients and Extensions Joachim Zacharias

525

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PROBABILITY MEASURES IN TERMS OF CREATION, ANNIHILATION, AND NEUTRAL OPERATORS

LUIGI ACCARDI Centro Vito Volterm Facoltb di Economia Uniuersitd di Roma “Tor Vergata” 00133 Roma, Italy E-mail: accardi0uolterra.mat.uniroma2.it HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: kuo0math.1su.edu AUREL STAN Department of Mathematics University of Rochester Rochester, N Y 14627, USA E-mail: astan0math.rochester.edu Let p be a probability measure on Rd with finite moments of all orders. Then we can define the creation operator a + ( j ) , the annihilation operator a - ( j ) , and the neutral operator a o ( j )for each coordinate 1 5 j 5 d . We use the neutral operators ao(i) and the commutators [a-( j ) ,a + ( k ) ] to characterize polynomially symmetric, polynomially factorizable, and moment-equal probability measures. We also present some results for probability measures on the real line with finite support, infinite support, and compact support.

1. Creation, annihilation, and neutral operators

Let /A be a probability measure on Rd with finite moments of all orders, namely, for any nonnegative integers i l , i2,. . . ,id,

1

2

where x = (21,x2,.. . ,xd) E Pd.Let Fo = P and for n 2 1let F, be the vector space of all polynomials in x1,22,. . . , z d of degree 5 n. Then we have the inclusion chain

Fo C FI C ... C Fn C ... C L2(p)Next, define Go = R and for n 2 1 define G, to be the orthogonal complement of F,-1 in F,. Then the spaces G,, n 2 0 , are orthogonal. Define a real Hilbert space 3t by 00

3t = @ G,

(orthogonal direct sum).

n=O

For each n 2 0, let P, denote the orthogonal projection of 3t onto G,. Let X j , 1 5 j 5 d, be the multiplication operator by xj. Accardi and Nahni5 have recently observed that for any 1 5 j 5 d and n 2 0

XjG, I G k , Q k f n - l , n , n + l , where G-1 = ( 0 ) by convention. Then they used this fact to obtain the following fundamental recursion equality

XjPa = Pn+lXjPn

+ PnXjPn + Pn-1XjPn,

1 <_ j

5 d, n >_ 0 ,

(1)

where P-1 = 0 by convention. When d = 1, this equality reduces to the well-known recursion formula

xPn(x) = Pa+l(x) + anPn(z)+ wnPn-l(z),

(2)

where P,(z)’s are orthogonal polynomials with respect to p , Pn(x)is a polynomial of degree n with leading coefficient 1, and {an,wn}’S are the Jacobi-Szego parameters of p . Now, for each n 2 0 and 1 5 j 5 d, define three operators by

D i ( j ) = Pn+lXjPn

:

Gn +Gn+lr

D,(j) = Pn-lXjPn : Gn +Gn-1, DO,(j) = P,XjP,

:

G,

+Gn.

Using these operators, we can define for each 1 5 j 5 d three densely defined linear operators a + ( j ) , u - ( j ) , and a o ( j )from 3c into itself by n

2 0,

= D,(j),

n

2 0,

aO(j)lcn= D%),

n

2 0.

a+(j)lc, = DRW, a-(j)Ic,

3

The operators a + ( j ) , u - ( j ) , and u o ( j )are called creation, annihilation, and neutral operators, respectively. It can be checked that u - ( j ) = a+(j)* and u o ( j ) = u o ( j ) * for each 1 5 j 5 d. The collection I1 I j I d )

{X,a+W, a - ( A , &)

is called the interacting Fock space of the probability measure p . For convenience, we will use the term “CAN operators” to call the creation, annihilation, and neutral operators. By using the multiplication and CAN operators, we can rewrite the fundamental recursion equality in Equation (1) as the equality in the next theorem. Theorem 1.1. For each 1 xj

j

5 d , the following equality holds

= u+(j)

+ 2 ( j )+ u - ( j ) .

(3)

We can use the equality in Equation (3) to extend the Accardi-Boiejko unitarity theorem’ to the multi-dimensional case. In this paper we will present some results to answer the following question. Question: What properties of p are determined by the associated CAN operators? 2. Polynomially symmetric measures Definition 2.1. A probability measure p on Rd is said to be polynomially symmetric if

for all nonnegative integers i l , i2,. . .id with il integer.

+ i2 + . . . + id being an odd

Note that if p is a symmetric measure with finite moments of all orders, then it is polynomially symmetric. But the converse is not true. Consider the function e(z) = e - ( ’ n ~ sin(2n )~ In $1,

z

> 0.

(4)

It is well-known that P“

zV(z) dz = 0, 10

V n = 0, 1,2, . . .

(5)

4

Define a function

f ( x )=

{

c8+(x), 0, c8-(-x),

if x > 0, if x = 0, if x

< 0,

where 8+ and 8- are the positive and negative parts of 8, respectively, and the constant c is chosen such that Jw - m f (x)d x = 1. By using Equation (5) one can easily check that the probability measure

dP(Z) = f dx is polynomially symmetric. Obviously, p is not symmetric. The next theorem has been proved in our paper2. Theorem 2.1. A probability measure p o n Rd with finite moments of all orders is polynomially symmetric if and only i f u o ( j ) = 0 f o r all j = 1,2,...,d. 3. Polynomially factorizable measures Definition 3.1. A probability measure p on Rd is said to be polynomially factorizable if

for all nonnegative integers il ,i 2 , . . .id. Obviously, if p is a product measure with finite moments of all orders, then it is polynomially factorizable. However, the converse is not true. Consider two modified functions of the function 8 in Equation (4):

el (x)= e-Qn e2(2) = e-('" ')'

2 ) '

[I

+ sin(2r In x ) ],

x > 0,

[I - sin(2r In x)], x > 0.

Define a function g ( x , y ) on R2 by

(0'

k[el(x)sin2 y

dX,Y) =

+ &(x) cos2y l e - ~ ,

if x

> 0, y > 0 ,

elsewhere,

where the constant k is chosen so that probability measure

d&,

JR2

g(x,y) d x d y = 1. Then the

Y ) = g(x,Y ) dXdY

5 can be shown to be polynomially factorizable, but not a product measure. The next theorem follows from Theorem 4.10 in our paper2. Theorem 3.1. A probability measure p on Rd with finite moments of all orders is polynomially factofizable if and only if for any i # j ,

the operators in {a+(i),a-(i), a o ( i ) } commute with the operators in {.+(A, a - w , a O ( j ) } . 4. Probability measures by means of the CAN operators

Let p be a probability measure on Rd with finite moments of all orders. We have the associated CAN operators a + ( j ) , u - ( j ) , and u o ( j ) . Define a d x 1 matrix A t and a d x d matrix A;>+ by

where [a,b] = ub - ba, the commutator of a and b. Definition 4.1. Two probability measures p and u with finite moments of all orders are said to be moment-equal if

for all monomial functions m ( x ) . The next two theorems have been proved in our paper3. Theorem 4.1. Two probability measures p and u on Rd with finite moments of all orders are moment-equal if and only if A t = A: and A;,+ = & I + .

6

Theorem 4.2. A probability measzlre p o n Rd with finite moments of all orders is the standard Gaussian measure o n Rd i f and only i f A: = Od and A;'+ = I d , namely, ao(i)= 0 f o r all 1 5 i 5 d and [a-(j),a+(k)]= d j , k I f o r all 1 5 j , k 5 d. The above discussion leads to the following problem for specifying a probability measure /I in terms of the matrices A: and A;*+.

Problem: Let V be the vector space of all polynomials on Rd. Let ai and a j , k be linear operators on V for 1 5 i , j , k 5 d. Find conditions on {~i}f and =~ {aj,k};,k=l so that there exists a probability measure p on Rd satisfying ai = ao(i)and a j , k = [ u - ( j ) , a + ( k ) ]for all 1 5 i , j , k 5 d. In the next section we will give some results on the solution to the above problem for the case when d = 1.

5. Probability measures on the real line Let p be a probability measure on R with finite moments of all orders. Let V be the vector space of all polynomials in 2 and let V,, be its subspace consisting of all polynomials of degree 5 n. Let F,, = Vn/-. Here the equivalence relation is given by p-almost everywhere, namely, f g if f = g holds p-a.e.

-

-

Assumption. In this section all linear operators T : V + V are assumed to satisfy the condition that T(V,) C V, for all n 2 0, namely, all subspaces Vn,n 2 0, are invariant under T . 5.1. Probability measures o n R with finite support Observe that if a probability measure p on R is supported by m distinct points, then

Fj=V,,

,...,m - 1 , j = m , m + l , ....

j=O,l,2

Fj =Vm--l,

The following theorem can be easily verified.

Theorem 5.1. Suppose p is a probability measure o n R supported by m distinct points. T h e n the following equalities hold: (1)

D(u:) , I

(2) TI-( [.;,a:]

= TI-(a:

), I

=

lvm-,)

f o r all k 2 m - 1.

o f o r ail k 2 m - 1.

7

Definition 5.1. Two linear operators S and T from V into itself are called trace equivalent on V , denoted by S T on V , if n-(slvb)

=Tr(TIVb))

vk_>o'

They are called trace equivalent on Vn, denoted by S

T on Vn,if

VOIkSn.

n ( s ( V ,= ) n(Tlvb),

The next theorem is from our paper3. It characterizes those measures supported by finitely many points in R in terms of the CAN operators.

Theorem 5.2. Let m 2 1 be afixed integer. Let a' and a-*+ be two linear operators f r o m V,-l into itself. Then there exists a probability measure p t o n R supported by m distinct points such that a' A a t and a-i+ [a;,.:] o n V,-l if and only if the following conditions hold:

-

(1) The spaces v k , 0 5 k

I,) (3) Q(a-*+ IvJ (2) Tr(a-9'-

5 m - 2, are invariant under a' and a->+. > 0 for all 0 5 k 5 m - 2. = 0,

5.2. Probability measums on R with infinite support Let p be a probability measure on R with infinite support, namely, the support of p contains infinitely many points. In this case, we have

Fn =Vn,

Vn_>O.

The next theorem has been proved in our paper3.

Theorem 5.3. Let a0 and a-i+ be two linear operators f r o m V into itself. Then there exists a probability measure p o n R with infinite support such that a' a! and a->+ [ a ; , a i ] o n V i f and only i f the following conditions hold: (1) The spaces Vn,n 2 0, are invariant under a' and a-i+. (2) Tr(a-i+

I,)

> o for all n 2 0.

Let E denote the set of all trace equivalent classes of ordered pairs (ao,a-i+) of linear operators from V into itself satisfying either one of the following conditions (a) and (b): (a) Q( a-$+

l v n)

> 0, V n 2 0.

8

(b) There exists m such that (1) n ( a OIVJ = n ( a Olvm-l), V k

(2)

n(a-,+

(3)

Tr(a-l+

lvk) lvk)

2m-

1,

> 0, vo 5 k 5 m - 2, = 0, V k 2 m - 1.

Theorem 5.4. There is a one-to-one correspondence between the set B and the set of all probability measures o n R with finite moments of all orders. 5.3. Probability measures on R with compact support

The Paley-Wiener type problem is to characterize probability measures with compact support. We have the next theorem from our paper3.

Theorem 5.5. A probability measure p o n R with finite moments of all orders has compact support if and only if the following two sequences of real numbers are bounded: (1) m

(2)

( 4), I

-*(a;

n ( q +I F = ) ,

n> - 1.

F l J7

n 2 1.

6. Classical probability measures on the real line Let p be a probability measure on R with finite moments of all orders. We have the associated orthogonal polynomials { Pn} and the Jacobi-Szego parameters {an,un}as given in Equation (2). The corresponding CAN operators are given by

UEP,

= pn+l,

aGPn = WnPn-1,

U:P~

= CX,P,, n 2 0,

where P-l = 0 by convention. Therefore, the commutator a;'+ is given by

= [a;,

]a :

Consider the following classical probability measures on the real line: 1 (1) Gaussian: dp(x) = -e - S dx, x E R (g > 0). &a

.=

(2) Poisson: p ( { l ~ }= ) e-'-,

Ak

k!

k = 0,1,2,. . . (A > 0).

9

(3) gamma:

Q

> 0. 1

d p ( x ) = -x a - l e -3 d x , x J3a) (4) Pascal: T > 0, 0 < p < 1.

> 0.

1

( 5 ) uniform: dp(x) = - d x , - 1 5 x 5 I. 2 1 1 (6) arcsine: d p ( x ) = - ___ d x , - 1 < x < 1.

di=?

(7) semi-circle: d,u(x) = Z d S d x , lr

(8) beta-type: j? > -1/2,

-1

1x 5 1.

,8 # 0.

For the above probability measures, the Jacobi-Szego parameters are given in the next table. By convention, wo = 1.

measure u Gaussian Poisson

+ n - 1)

Pascal

+ 2n ( 2 - p)n + ~ ( 1 p- ) P

p”

uniform

0

n2 (2n 4-1)(2n- 1)

arcsine

0

semi-circle

0

-

beta-type

0

n(n - 1 28) 4(n P)(n - 1 + P )

gamma

a!

n(Q n(n

+ T - 1)(1 - P)

-

ifn=l

1

4

+

+

10

Furthermore, the operator a: and the commutator a;’+ = [a,, a]: given in the following table.

are

measure p Gaussian

D 2Pn

Poisson gamma

APn (a!

+ 2n)P,

(a!

+ 2n)P,

Pascal uniform ifn=O arcsine

0

--PI,

(0,

semi-circle

0

beta-type

0

if n = 1 ifn>2

It is interesting to compare the above table with Theorems 2.1, 5.1, 5.2, 5.3, and 5.5.

Acknowledgments This research was initiated in May-June 2002 during the visits of H.-H. Kuo and A. Stan to the Centro Vito Volterra (CVV), Universita di Roma “Tor Vergata”. They are grateful to the CVV for financial support and would like to thank Professor L. Accardi and his staff members for the warm hospitality during the visits.

References 1. Accardi, L. and Boiejko, M.: Interacting Fock space and Gaussianization of probability measures; Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics 1 (1998) 663-670 2. Accardi, L., Kuo, H.-H., and Stan, A.: Characterization of certain probability measures by creation, annihilation, and neutral operators; Infinite Dimensional Analysis, Quantum Probability and Related Topics (to appear) 3. Accardi, L., Kuo, H.-H., and Stan, A.: Moments and commutators of probability measures; Preprint (2003) 4. Accardi, L., Lu, Y. G., and Volovich, I.: The QED Hilbert module and interacting Fock spaces; I I A S Reports No. 1997-008 (1997) International Institute for Advanced Studies, Kyoto 5. Accardi, L. and Nahni, M.: Interacting Fock space and orthogonal polynomials in several variables; Preprint (2002) 6. Chihara, T. S.: A n Introduction t o Orthogonal Polynomials. Gordon and Breach, 1978 7. Kubo, I.: Generating functions of exponential type for orthogonal polynomials; Inifinite Dimensional Analysis, Q u a n t u m Probability and Related Topics 7 (2004) 155-159 8. Szego, M.: Orthogonal Polynomials. Coll. Publ. 23,Amer. Math. SOC.,1975

SEMI GROUPES ASSOCIES A L’OPERATEUR DE LAPLACE-LEVY

LUIGI ACCARDI Centro Vito Volterra Facoltb d i Economia Universitd di Roma “Tor Vergata” 00133 Roma, Italy E-mail: [email protected] HABIB OUERDIANE Dipartement de Mathimatiques FacultC des sciences de Tunis i060 Tunas, Tunisia

1. Introduction et prkliminaires

Soit H un espace de Hilbert skpkrable rkel et A un opkrateur positif auto adjoint sur H tel que son inverse A-’ soit de type Hilbert-Schmidt. Alors il existe une suite de nombres rkels positifs 0 6 A1 _< A2 5 ... 5 A,. . . et c Dom(A) tel que: une suite de vecteurs

La suite (en)n21 forme une base orthonormke de H . Pour tout rkel p E R posons:

n=l

oh (., -) est le produit scalaire dans H et I. 10 est la norme Hilbertienne sur H . Pour tout pour p 2 0, Ep = {[ E H , [Elp < oo} est un espace de Hilbert

muni de la norme [
D-h 00

npz0Ep car l’opkrateur A est de

type Hilbert-Schmidt. D’oh le triplet de Gelfand nuclkaire suivant:

E

C

H

11

H’ C E*

12

(1)

13

oil E* = Lim ind E-, =

Up>_o E-, est le dual topologique fort de E.

La

P++W

forme bilinkaire canonique sur E* x E est notde par < ., . > . L’ensemble typique de triplet de type (1) qu’on considbre en analyse du Bruit blanc, voir 191, [13] et [14] est le triplet:

E I S(R) c L2(R,dx)c E* = S(R) oh S(R) est l’espace de Schwartz des fonctions

B ddcroissance rapide sur R, L2(R,dx) est l’espace des fonctions de carrk intdgrable par rapport B la mesure de Lebesgue dx sur R et S ( R )l’espace des distributions tempdrkes. Soit l’op6rateur

et (e,),>lune d’HermiG

base orthonorm6e de L2(R,dx) formke par les fondions

Oh

+

est le polyndme d’Hermite de degr6 n. Alors Ae, = 2 ( n l)en, i. e. An = 2 n + 2, n = 0 ’ 1 ’ 2 , . . . sont des valeurs propres de l’opdrateur A et de plus W

1

< 00 si p > 1 / 2

n=O

Pour tout p 1 0, on d6finit lfl, = lAPflo oh de L2(R,dx). Alors l f l p est donnke par:

[.lo

est la norme Hilbrtienne

\n=o

If/,

Soit l’espace de Hilbert S,(R) = {j E L2(R,dx), < co} alors S(R) = nploS,(R) et S(R) est un espace de F’r6chet nucleaire. Soit SL(R) = S-,(R) le dual topologique de S,(R) muni de la norme Hilbertienne Ifl-, = IA-Pflo . On obtient alors les inclusions suivantes

S(R) C S,(R) C L2(R,dx) C S-,(R) C S ( R )= U,?oS-,(R)

14

1.1. Fonctions differentiables

Une fonctions F : E + R (ou C ) est dite de classe C 2 au point [ E E s’il existe un klkment F’([) E E* et PI’(<)E L ( E ,E * ) telle que:

avec

o(trl) = 0. lim t2

t+O

<

Si de plus on suppose que les applications: 3 F’(J)et [ + F”(<) sont continues, alors puisque E est nuclkaire complet voir [19] et [21] on a:

L(E,E * ) fl ( E @ E)* N B ( E ,E ) et donc (F”(<)q,q)= (F”(E),q63 7) = D,D,F(<)= F”([)(q,r))oh D, est la derivke au sens de Frkchet dans la direction de 11, i.e.,

(D,F)([) = lim

w+

X+O

x

-F(I)

Posons

Alors I’action de l’opkrateur de Laplace- Lkvy AL sur toute fonction F de V est dkfinie par .

N

1.2. Fonctions e n t i h s b croissance exponentielle

Soit 8 : R+ 4 R+ une fonction continue, convexe, strictment croissante et v6rifiant les conditions:

e(z) = +oo lim -

n++m

5

et O(O> = o

Une telle fonction 8 s’appelle fonction de Young c’est A, dire la fonction de Young suivante: 8*(2) = sup(t2 - 8 ( t ) ) t 20

.

(6)

Soit 8* la polaire de 8

15

On montre facilement que 8' = 8. En complexifiant le triplet reel (1) on obtient le triplet nuclkaire complexe suivant

E,

c H, IIHA c E,*

+ i E , H , = H + iH et E,* est le dual topologique fort de E,. Soit ( B ,11.11) un espace Banach complexe. Pour toute fonction de Young 8 et m > 0 notons par Ezp(B,e,m) l'ensemble de toutes les fonctions complexes f qui sont entibres sur B telles que oh E, = E

~ l f l l ~=, sup ~ If(<)le-e(mll~ll) < 00 tEB

(7)

Alors l'espace de fonctions entibres sur E,*, d'ordre de croissance 8-exponentiel de type minimal est defini par

.WE,*) =

n

Ezp(E,,-,,e,m)

(8)

p>O,m>O

qu'on munit de sa topologie limite projective. De m&mel'espace

G, (E,):=

(J

EZP(E,,,, 8, m )

(9)

p>O,m>O

muni de sa topologie limite inductive est appeld espace de fonctions entibres sur E, d'ordre de croissace 8-exponentiel de type arbitraire. Par dCfinition f E T,g(EE)et g E Ge(E,) admettent comme developpement en sdrie deTaylor B l'origine

f(z) = C(z"",fn,,z

E E,*

(10)

n>O

n>O

oh (., .) est la forme canonique de dualit4 entre (E,*)Bnx E,On et oii E,On

d4signe le produit tensoriel symbtrique d'ordre n de E,. En vue de caret Gg(E,) en fonction de leurs sdries formelles actkriser les espaces Fg(E,*) de Taylor, on intoduit les espaces de Fock interactifs suivants:

oh pour tout entier n

16

np>O,m>O Fe,m(Ec,p)muni de la topologie limite pro-

Soit alors Fo(E,) = jective. De mGme on pose

Ge(E,*)=

u

Gs,m(Ec,-p)

p>O,m>O

muni de la topologie limite inductive oh pour tout m

Gs,m(E,*,-p)=

{ 6 = (@n)n>o :

@n

> 0 et p > 0

E (Ec,-p)o",l1611e,m,-p < m}

avec n>O

I1 est facile de verifier que Fe(E,) est un espace de F'rCchet nucl6aire et que les espaces Fg(E,) et Ge(E,*)sont en dualit6, auterement dit que le dual fort de Fe(E,) est identifik B l'espace Ge(E,*).La forme de dualit6 entre Gs(E,*)et Fs(E,) est donn6e par m

n=O

L'application Skrie de Taylor 7 (A l'origine) associe B toute fonction entibre f(z) = Cn>o(z@p",fn) E Fe(E,*)ses coefficients de Taylor = (fn)n2~. Il est demo&r6 dans [ll] que l'application 7 Btablit deux isomorphismes topologiques:

f

1.3. Transformation de Laplace Soit FT*, (E:) le dual topologique fort de 3 0 (E,').La dualitk entre FT*, (E,*)et Gp(E,) est d6finie par la forme de dualit6 (11) entre les espaces de sdries formelles associks Fe (E,) et Gs (E:): 00

w E F:(E,*)et

f E F ~ ( E , * )i

((4,f)):= @,fj= C.!(dn,fn) n=O

De la condition

on dkduit que pour tout E E, la fonction exponentielle et : E,' + C , qui A tout z E E,* associe ec(x) = e("iE) est un Clement de Fs(E,*).Donc pour

17

toute fonctionelle

+ E 3:(E,*)sa transformke de Laplace est dkfinie par:

.~(+)(t) =

= (6, e') =

Pn = C(+n, C n!(+n,7) Pn) n.

(15)

On utilisea dans la suite un thkorbme de dualitk obtenu dans [ll]

Theorem 1.1. La transformation de Laplace L e'tablit un isomorphisme topologique

Fi(EE)+ Go* (Ec) 1.4. Reprdsentation intdgmle de distributions positives

Une fonction test cp de Fo(E,*) est dite positive ( et on note cp 10), si pour tout x E E , cp(x io) 1 0 . Une distribution 4 E Fi(E*)est dite positive si pour toute fonction test cp positive de Fo(EE)

+

(4,cp) 1 0 On dksigne dans la suite par Fi(E*)+l'ensemble de toute les distributions positives.

Theorem 1.2. [21] Pour toute distribution 4 E Fi(E,*)+, il existe une unique mesure de Radon p sur l'espace ( E * ,23) muni de sa tribu Borelienne telle que

vf

(4, f ) =

E Fe(E,*)

/

E '

f(x + io)dp(x)

(16)

De plus la transforme'e de Fourier de p est donne'e par:

fi(<) =

/

ei(nsc)dp(x)= ( 4 , e i t )

E'

Une sorte de rkciproque du thkorkme prkckdent est donnke par la caractkrisation de telle mesures:

Theorem 1.3. [24] Soit p une mesure positive et finie sur E* muni de sa tribu Borelienne BE.. Alors la mesure p repre'sente une distributions 4 E Fi(E,*)+ si et seulement si elle ve'rifie Ees deux conditions suivantes: 1) Il existe q > 0 telle que p soit porte'e par l'espace E - q , i.e. P ( E - ~= )1 2)3m > 0 tel que

18

2. Laplacien de LQvyagissant sur les distributions

Dans cette section, on va etudier l’action de I’opCrateur de L6vy A L sur l’espace des fonctions gCnCralis6es 7;(EZ). Notons par TLla rkstriction de I’opCrateur de LCvy d6fini dans (5) B l’espace 0 0 . (Ec).Comme d’aprbs le thCorkme 1 de dualit6 , la transformee de Laplace de toute distribution q5 E 7;(E:) est une fonction de I’espace ( E J , on dCfinit le domaine de l’opdrateur A L :

et oij (en)est une base orthonorm6e de H . Soit a = C,”=, a,e, E E*, alors en posant pour tout entier n, a, = ( a , e , ) , on a:

et cette limite peut 6tre finie ou infinie. On introduit alors l’espace suivant: 2 E E* telle

que lla11i = lim N++w

l

N

N

n=l

- ~ ( x , e , ) existe ’ et soit finie

il est clair que E c H c E i , , c E*, et que ces inclusions sont strictes, voir [7]. Consid6rons maintenant un sous espace vectoriel V de E* verifiant V c ET,,,. Comme pour tout p > 1, V c E-, alors le complet6 ? de V par B rapport B la norme I . I-, est un espace de Hilbert. On notera par la suite cet espace par El. D’oh les inclusions suivantes:

H c 9 = E,* c E-, c E* En tranposant ces injections canoniques on oblient le triplet:

E

c E, c Eo c H

M

H’ v E,* c E-, c E*

Theorem 2.1. [7] Pour tout a € E * , tel que lla11;= limN,, soit finie, on a:

ALea = ( ( u ( ( i e a

(19) I

N

(a,en)’

(20)

Soit E = Vect{ea,a E E*,11~11; < m} l’espace vectoriel engendr6 par ces fonctions exponentielles. I1 est clair E c Be* ( E )et que griice B 1’6galit6 (20)’ l’op6rateur A , est lin6aire continu sur E . On va dans la suite Ctudier

19

l'action de AL sur l'espace de fonctions gheraliskes F;(E,*). En utilisant le le thkorkme 1 on a:

Proposition 2.1. La transformation de Laplace C e'tablit un isomorphisme topologique:

Lemma 2.1. (Convergence des suites de fonctions) Soit (F,), u n e suite de G,+,(Ec).Alors (F,) converge dans G,(E,) si e t seulment s i (F,) est borne'e dans G,+,(Ec) et converge simplement. Proof. La preuve de ce lemme est basde sur la nucl6aritd des espaces F;(E,*) et Gp(E,). Une demonstration similaire de ce lemme a et6 6tablie 0 avec plus de details dans [16] et [24]. Proposition 2.2. P o u r tout

4 de F;(E,*);

est bien defini et est un e'lement de F:(E,*) Proof. Soit 4 E Fo(E,*) alors d'aprks la proposition (5) , sa transformke de Laplace $(<) E GO* (Eo) . En utilisant la densit6 des fonctions exponentielles dans l'espace G p (Eo), il existe une suite (F,), E E = V e c t { e a , a E E,*}telle que limn+oo F, = $ En utilisant d'une part la stabilitk et la continuitd de AL sur l'espace E et d'autre part le crithe de convergence des suites dans 66. ( E o )donn4 dans le lemme 6, on deduit que

,

ALF,+AL$

n++m

ceci pouve que AL$ E G p (Eo)et donc d'aprbs le thdorbme de dualitd ,L-'(AL$)

:= A

L E~F;(E:)

0

Theorem 2.2. L e s e m i groupe Pt associe' d l'ope'rateur de Ldvy-Laplace est Markovien. D e plus pour t o u t 4 de F;(E,*)+

04 d p ( x ) est la mesure qui repre'sente la distribution positive

4.

20 Proof. Soit 4 m e distribution positive (4 E .Ti(E,*)+).Il existe d'aprks le thkorkme 2 une mesure de Radon p sur ( E * ,B):

(4'cp) =

1

cp(Z)dP(Z)

E'

;

Vcp E .Ti(-%)

Or d'aprks la proposition 7,

A L =~L-'(ALJ) Donc on va d'abord calculer

A L ~En . effet on a:

d'ou:

et par conskquent si Pt est le semi groupe associk A AL il vient:

=

S,.

etllz112e(z>t)dp(z)

Pour montrer la convergence de cette dernikre intkgrale on utilise la condition (17) d'integrabilitk verifike par la mesure p associke a la distribution positive 4, i.e., 3 q > 0 , m > 0:

en effet

en posant alors

21

on a

e@”)dv(z)= (,cY)(<)= o(<) En utilisant de nouveau le thkorkme 1 de dualitk prkcedente:

, on dkduit

de l’kgalitk

/ P ( P t J ):= Ptq5 = P ( , C v )= v d’oii = y = etllzll”p(z)

comme p est une mesure positive, alors Ptq5 = v l’est aussi. Pour finir la preuve il est clair d’aprhs (22) que le semi groupe Pt transforme l’identitk en elle m&me,i.e., PtSo = So. 0 Remark 2.1. Pt transforme les distributions positives en des distributions positives. E n particulier les mesures positives en mesures positives. Dans [l]l’espace M i ( E * )des mesures SUL’ E*, invariantes par l’opkrateur (en) schift S dkfini par S(e,) = en+l a 6t6 introduit. Alors une application directe du thkorkme prkckdent permet de retrouver le thkorkme obtenu par Luigi Accardi dans [2] :

Si 4 = p E M i ( E * ) et FO = ji sa transforme‘ de Fourier, alors le problkme de Cauchy suivant:

Theorem 2.3.

admet l'unique solution donne par f

avec

3. Semi groupe de LQvyagissant sur l’espace des fonctions

On a donnk dans le paragraphe pr6ckdent une formule explicite du semigroupe de Lkvy Pt agissant sur les distributions positives (et donc en particulier sur certaines mesures de Radon sur E l ) . Soit ‘p une function de Young donnke. Le but de ce paragraphe est de dkfinir et de donner explicitement le semi- groupe associk au Laplacien de Lkvy defini sur l’espace G,(Eo,~)des fonctions g : + C qui sont entihres sur A croissance

22 9-exponentiel d'ordre quelconque. Comme E o , est ~ le complexifid de Eo, la restriction de tout element g E G,(Eo,~)B EO est definie par:

gIEo := g(Z

+ io), 2 E EO.

Definition 3.1. Soit f : E + C . On dit que f est une fonction cmactdristique si f est continue, ddfinie positive et verifiant f(0)= 1 Posons alors G:(E0) l'ensemble de toute les fonctions g de G,(Eo,~) dont la restriction glEo est une fonction caratdristique.

Proposition 3.1. Soit g un e'lement de G;(Eo). Alors la mesure de probabilite' qua lui est associe'e, pg ve'rijie la conditions d'inte'grabilite' suivante: 3m>0 etpEN*,

I

e'P'(ml"I-P)dpg(Z)

< co

%--P

D e plus la transforme'e de Fourier F ( p g ) de pg est donne'e par: F'(LLg)(t)

= 35) =

J

ei("9%g(z)

i

v t E Eo

EO,-p

Proof. En effet en utilisant le thdorhme de Bochner-Minlos, voir [14], il existe une unique mesure de probabilite pg sur le dual fort de Eo et qui est associke B la fonction caractdristique S I X , . Plus prbcisement, comme g E G,(Eo)alors

Alors on sait qu'il existe un entier q > p tel que I'injection i, : Eo,q + Eo,p est de type Hilbert-Schmidt et , L L ~ ( E O= , - 1. ~ )Pour montrer la condition d'intkgrabilitb vdrifide par la mesure pg on a besoin d'avoir une estimation et n E N des moments de pg. En effet on a V < E

En utilisant maintenant la formule de Cauchy, alors pour tout

,W)

en posant 9 k = infrBo7 , k E N on a:

T

>0:

23 En utilisant maintenant la formule d'identitk de polarisation et l'indgalitd (23) on obtient

1

lzlzqdpu,(z) 5 (11911q,p,mcpzn(2n)!)112( h m l l i q p l l H S ) n , \ ~ EnN

(25)

oh ~ ~ i P est qla ~ norme ~ ~Hilbert-Schmidt s de l'injection i p q .Pour compldter

la preuve de cette proposition on a pour tout t 2 0 et

E

> 0:

D'autre part on peut montrer facilement que 1

\Jt2 0 , e-q(t)tn < -(Pn

D'oh:

En utilisant maintenant l'estimation (24) des moments de la mesure puset le fait que: &12npn

7

\Jn€N:

on a

comme

d'oh en choissant 44mlliq,pllHS

<1

alors la serie (25) converge et on a:

I

EO,

-*

ep*(EIZI-q)dpg(2)<

00

Ceci provient du fait que sup{etElzl-q-q(t) = exp(sup{tElzl-, - cp(t)}= exp cp* ( ~ l z l - ~ ) t>o

t

par ddfinition meme de la fonction polaire cp* associde

cp.

0

24

Theorem 3.1. Le semi-groupe Pt associe‘ d l’ope‘rateur de Le‘vy-Laplace agissant sur l’espace des fonctions G;(Eo)= ( 9 E G,(Eo),glx0 soit definie positive} est Markovien. De plus o n a la formule explicite du semi-groupe

9: V g E G$(Eo) , ( k t g ) ( t )= ( e t A L g ) ( t )=

/

e-tll”lls,i(.~r)dpg(2)

(27)

E,.

04 (mg)(t)

= 9(t)

Proof. Soit g une fonction appartenant ? l’espace i G;(Eo). Alors d’aprb la proposition (12)’ on sait qu’B tout 6lkment g de B: (Eo),il lui est associCe une unique mesure de Radon pg telle que:

d’ob

D’aprCs l’Cgalit6 (20) on dgduit que

d’oh:

Pour montrer enfin la convergence de cette expression on utilise le fait que la mesure p 9 vgrifie d’aprgs la proposition (12) la condition d’intggrabilitk:

3m > 0 e t p > 0,

I

ev*(mlzl-p)dpg(2)< 00

E O F P

On en d6duit alors que: 3 c > 0, m > 0 et p

> 0 tel que

e,*(mlzl-p)dpg(x) < o o , ~ Et

I(Ptg)(t)l< c / E0r-P

EO

25 et c = c(t, E). Finalement pour montrer que le Processus Pt est Markovien, il suffit de montrer que si g = 1, i.e., g(() = 1, VE, alors pg = So. En appliquant la formule explicite (27) du semi-groupe Ft,on a

d’autre part comme pour tout g dans G$(Eo):

En posant dvg(x):= e-tllzlltdpg(x) , i.e., dv9W = e-t11211:

4%(x) oiI

dP&l(X)

est la d6rivde au sens de Radon-Nikodym. Alors on a: ($‘g)(() =

]

ei(”z[)dv(z)= .T(vg)(E)

E,*

qui est par d6finition une fonction caract6ristique’ et donc le semi-groupe Pt transforme les fonctions definies positives en fonctions definies positives, i.e.:

Pt : G p o )

-+ G,+(Eo)

(28)

D’autre part si P t g = P t h alors 7 v g = Fvh, et comme la transformation de Fourier est injective ceci donne: vg = V h et donc e-tIIzI12dpg (x)= e-tIIzII:dph (z)

d’oh p g = p h et par suite F 1 ( p g ) = T-l(ph), donc g = h. Donc l’application ddfinie par (27) est injective. Montrons qu’elle est aussi surjective. En effet d’aprks la proposition prdcddente, pour tout G de G$ (Eo) alors il existe une mesure p telle que 3 ( p ~ =) G. Cherchons alors s’il existe un antbcedent g (3 G$(Eo)de G , i.e.,

P t g = G equivaut B - T ( ~ G=) Ptg = 3 ( u g ) d’oh p~ =

or

dug- - e-tllz112 - dPG dP9

dP9

26

d’oii puse t donc g par Laplace inverse.

0

Corollary 3.1. En combinant les ope‘rateursPt agissant sur Ees distributions positives et agissant sur les fonctions @(Eo) on a Ee diagramme suivant:

m%)+ -5Wa+

T

7-l

oi F est la transformation de Fourier. D’o% la relation :

References 1. L. Accardi, P. Gibilisco and I. V. Volovich: The Ldvy Laplacian and the Yang-Mills equations, Rendiconti dell’Accademia dei Lincei,( 1993). 2. L. Accardi :Yang Mills equations and LCvy-Laplacians, Dirichlet Forms and Stochastics Processes,(l995), 1-23, Eds.: Ma/Roekner/Yan.Walter de Gruyter and Co., New York . 3. L. Accardi and V. I. Bogachev: The Ornstein-Uhlenbeck process associated with the Lkvy Laplacian, and its Direchlet form, Probability and Math. Statistics , Vo1.17, Fasc.1 (1997), 95-114. 4. L. Accardi and 0. G. Smolyanov: On the Laplacians and traces, Rend. Sem. Mat. Bari, Ottobre 1993, Preprint- Volterra. 5. L. Accardi, P. Roselli and O.G. Smolyanov: The Brownian Motion Generated by the Ldvy-Laplacian, Mat.Zamatki, 54 (1993), 144-148 6. L. Accardi and N. Obata: Derivation Property of the Ldvy-Laplacian, White noise analysis and quantum probability, (N. Obata, ed.) RIMS Kokyuroku 874, Publ. Res. Inst. Math. Sci.(1994), 8-19. 7. L. Accardi and H. Ouerdiane: Fonctionelles analytiques assocides B I’opCrateur de LaplaceLdvy, Preprint No. 477 (2001), Centro Vito Volterra. 8. S. Albeverio, Y. L. Daletsky, Y. G. Kondratiev and L. Streit: Non-Gaussian infinite dimensional analysis. Journal of functional analysis 138 (1996), 311-350. 9. D. M. Chung U. C. Ji. and K. SaitB: Cauchy problems associated with the Ldvy Laplacian on white noise analysis. Infinite Dim. Analysis, Quantum Prob. and Related Topics. Vol. 2, No. l(1999). 10. W. G. Cochran., H-H. Kuo and A. Sengupta: A New Class of White noise Generalized Functions. Infinite Dim. Analysis, Quantum Prob. and Related Topics. Vol.1, No. 1(1998), 43-67. 11. R.Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui: Un thdorhme de dualitd entre espaces de fonctions holomorphes B croissance exponentielle. J. Funct. Anal., Vo1.171, No.1 (2000), 1-14.

,

27

12. R. Gannoun, R. Hachaichi, P. Kree and H. Ouerdiane: Division dc fonctions holomorphes ti croissance 8- exponentielle. Preprint, BiBos NO : 00-01-04. (2000). 13. I. M. Gel’fand and N. Ya. Vilenkin: Generalized Functions, volume 4, Academic Press. New York and London (1964). 14. T. Hida: Brownian Motion, Springer Verlag, Berlin-Heidelberg-New York, 1980. 15. T. Hida, H- H. Kuo, J. Potthoff and L. Streit: White noise, An infinitedimensional calculus, Kluwer Academic Publishers Group,l993. 16. P. Krbe and H. Ouerdiane: Holomorphy and Gaussian analysis. Prkpublication de 1’Institut de mathkmatique de Jussieu. C.N.R.S. Univ. Paris 6 (1995). 17. H- H. Kuo, N. Obata and K. SaitB: Lkvy Laplacian of generalized functions on a nuclear space, 3. Funct. Anal. 94 (1990) 74-92. 18. P. Lbvy: Leqons d’analyse fonctionnelle, Gauthier-Villars, Paris (1922). 19. N. Obata: White noise calculus and Fock space. L. N. Math. 1577 (1994). 20. H. Ouerdiane: Fonctionnelles analytiques avec conditions de croissance et application B l’analyse gaussienne. Japanese Journal of Math. Vol. 20, No. 1, (1994), 187-198 . 21. H. Ouerdiane: Noyaux et symboles d’opkrateurs sur des fonctionnelles analytiques gaussiennes. Japanese Journal of Math. Vol21. No.1. (1995), 223-234 22. H. Ouerdiane: Alghbre nuclkaires et kquations aux dkriv6es partielles stochastiques. Nagoya Math. Journal Vol. 151 (1998), 107-127. 23. H. Ouerdiane :Distributions gaussiennes et applications aux Bquations aux dkrivkes partielles stochastiques in Proc. International conference on Mathematical Physics and stochastics Analysis ( in honor of L. Streit’s 60 th birthday ) S. Albeverio et al.(eds.) World Scientific (2000). 24. H. Ouerdiane and A. Rezgui: Reprksentations integrales de fonctionnelles analytiques. Stochastic Process, Physic and Geometry : New interplays. A volume in honor of S.Albeverio. Canadian Math. Society. Conference Proceedings Series. Vol. 28 (2000), 283-290. 25. K. Sait6 and A. Tsoi: The Lkvy Laplacian as a self-adjoint operator, Proceeding of the first Int. Conf. Quantum Information. World Scientific (1999). 26. L. Schwartz: Radon measures on arbitrary topological spaces and cylindrical measures, Oxford Univ. Pr.,1973.

STOCHASTIC GOLDEN RULE FOR A SYSTEM INTERACTING WITH A FERMI FIELD

L. ACCARDI Centro Vito Volterra Universitd d i Roma "Tor Vergata" Via Columbia, 2, 00133, Roma, Italy E-mail: accardiQvolterra.mat.uniroma2.it R.A. ROSCHIN Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 117966, GSP-1, Moscow, Russia E-mail: roschin0mi.ras.ru I.V. VOLOVICH Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 117966, GSP-1, Moscow, Russia E-mail: volovich0mi.ras.m We consider the (causally) normally ordered form of the quantum white noise equation for a Fermi white noise. We find a new form of the normally ordered equation for some class of Hamiltonians and we obtain the inner Langevin equation for such Hamiltonians.

1. Introduction

The stochastic limit approach is a powerful method that allows to study the quantum dynamics of an open system. This method proved to be very efficient in study of various systems, interacting with Bose fie1ds.l In the present paper we concentrate on the Fermi case. In this case the superselection rules restrict the class of fundamental Fermi Hamiltonians. This restriction is discussed in Sec. 2. The main goal of the stochastic limit approach is to study the quantum dynamics of open systems. Such a dynamics is given by an evolution operator U ( t ) ,which satisfies the evolution equation in the interaction rep28

29

resentation:

& U ( t ) = -iAK,,t(t)U(t), U ( 0 ) = 1

.

(1)

Here x,t(t) is the evolution of the interaction Hamiltonian under the free evolution. However, even in the simplest non-trivial cases, there are no exact solutions of Eq. (1). Therefore, we have to use some approximations. One can express the solution of Eq. (1) using the iterated series and Feynman diagrams. The Feynman diagrams are a graphical representation of the combinatorial procedure, which is called "normal ordering". Usually, only a few terms of these series can be explicitly calculated. The stochastic limit approach provides another, more efficient and elegant way of solving Eq. (1).

Consider the evolution of an open quantum system in the rescaled time: In the stochastic limit approach we are interested in the limits of the rescaled evolution operator and of the rescaled interaction Hamiltoniama

t

+ t/A2.

In many physically important cases, one can prove that Ut satisfy the (Bose) quantum white noise equation:

&Ut = -ihtUt, Uo = 1 ,

(3)

where the limit of the interaction Hamiltonian, hi, can be expressed in terms of (Bose) quantum white noises. Although Eq. (3) seems to be very similar to its predecessor, Eq. (l),it can be explicitly solved. Again, we are going to find the normally ordered form of Eq. (3). It is called causal normal order to emphasize the fact that the commutation relations used to bring to normal order the terms of the iterated series, only have a meaning for time ordered products of creation and annihilation operators. The causally normally ordered form of Eq. (3) can be found explicitly. The technique that allows to find and bring to causal normal order Eq. (3) is called the stochastic golden rule. The aim of the present paper is to develop the stochastic golden rule for the cases when the interaction in (1) is of dipole type (cf. (8) below) and driven by a Fermi field. aThe exact sense of these limits is the subject of Statement 1

30

2. Hamiltonian of the model

Let 31 be a Hilbert space of the form

(4)

?I!=?I!S@%!R

where 7-Is is a finite dimensional Hilbert space, and 3 1 is~ an infinite dimensional Hilbert space. Suppose H is a Hamiltonian of the form

H =Hs @ 1

+ 1 @ HR + XVjnt

7

(5)

where X is a positive real parameter. We denote the first two terms of this Hamiltonian by Ha, and refer HO as ”free Hamiltonian”. Then, the pair (31,H) describes an open quantum system. In other words, an open quantum system is a ”small”, discrete spectrum quantum system, interacting with some infinite dimensional environment, or field. The simplest model that gives the Fermi white noise equation in the stochastic limit is a two-level system, interacting with a Fermi field. Let us introduce this model, using the notation of Eqs. (4,5). Let the Denote the elements of the orthonormal system Hilbert space be 31s = basis for this space by ti), i = 1,2. Let the environment Hilbert space be the Fermi (antisymmetric) Fock space

a?.

M

n=O

Denote the Fermi creation and annihilation operators in this space by ul and U k , k E R3,respectively. They satisfy anticommutation relations:b

{al,uk’) = 6 ( k - k ’ ) , { a k , a k f } = o . Let the free Hamiltonian be:

HO = H s 4-H R = E 11)(11 @ 1 + 1 8

s

w(k)aiuk dk .

(7)

Here E is a positive real number, w ( k ) = k2+m2, m 2 0. Let the interaction Hamiltonian be:

the following, {,} denote anticommutator, {x,y} = xy +yx, and [,] denote commutator, [x,y]= zy - yx.

31

where D is a bounded operator, acting on the system space, and g(k) is a smooth complex function with finite support. In the following, we omit the tensor product in our formulae. The superselection rules (see Ref. 2, require that fundamental Hamiltonians should be even in the Fermi operators.c If the Hamiltonian is even, then D should be odd in the Fermi operators. Hence, D and a,at should anticommute.d We will see that this case leads to a stochastic golden rule similar to the one known in the Bose case. Effective (for example, non-relativistic) Hamiltonians may be odd in Fermi operators, in these cases D and a,at may commute. An example of such Hamiltonian was considered in Ref. 4. If D and a,at commute, then we get the new kind of the stochastic golden rule. We will study both cases; first the ”commutative”, then the ”anticommutative” one. As mentioned in the introduction, there are two main components in the stochastic golden rule: the existence of the limit and the causal normal order form of the white noise equation. The first part, the existence of the limit, is quite similar for Bose and Fermi cases. Both cases were studied in Ref. 1. Here we just formulate the result for Fermi case. Statement 1. The limit (2) of the evolution operator in the model defined by Eqs. (6-8), exists in the sense of correlators. Moreover, the limit of the rescaled evolution operator Ut satisfies the white noise equation atUt = -ihtUt, with the white noise Hamiltonian of the form ht = D’bt

(9)

+ bfD,

where bt, bi are Fermi Fock white noise creation and annihilation operators. This means that they are operator valued distributions, acting on the Fock space r = F ( L 2 ( R ) )and satisfying the following relations:

{ bf, b”’} = y-6+(t

- t’) ; {bt, bt’} = 0 .

(11)

where d+(t) is the causal &function (see Ref. 1) and

‘An operator is called even, if it commutes with the parity operator, and odd, if it anticommutes. The parity operator in this case can be defined as in Sec. 3. dAn obvious generalization of Eqs. (4-8) is needed to include this case.

32 The relation between D and b, bt is the same as the relation between D and a,at: they either commute, or anti-commute.

3. The "Commutative" case The causally normally ordered form of the Fermi white noise equation and the Langevin equation in the "commutative" case are the main results of the present paper. Let us formulate these results as a theorem. First of all, let us give some definitions. Consider a monomial of the form:

bt: biz

. ..bi," ,

were, bii denote either bt, or bj. Denote the vacuum state in I? by 90. There exists a unique operator 0 , called parity operator, such that for any monomial

Ob;ibf,2 ...b:,"Qo = (-l)nb;:b;,2.. Note, that O2= 1, hence 0-l = 0 = O*. I f A is an operator, then the map: A automorphism. We will use the notation:

+

A := O A O .

.b;,"@o.

(13)

C3AO-l = OAO is a

*-

(14)

Theorem 2. Keep the notation and the assumptions of Statement 1. Suppose the operator D is such that

[D,bt]= [D,bl] = 0 . A

h

and define Ut U := OUO. Then, (1) The following relations are satisfied:

(15)

33

(2) The causally normally ordered form of (9) is the system

+ Dbf Ut) - y-DtDUt = i (DtUtbt + D b f c t ) - y - D t D c t

&Ut = -i ( D t ctbt

{

@t

(17)

h

with the initial conditions Ut = Ut = 1. (3) Let X be an operator on the system space. Consider the following matrices:

T:= Then,

(Jij

(;:),

S:=

(:_4)

(19)

( X ) )satisfies the following inner Langevin equation:

at ( ~ ( x=)-7-) ( J ( x D ~ D ) -) 7: ( ~ ( o t ~ x ) ) (7- + y ? ) J i i ( D t x D ) (-7-+ 7 T ) J i 2 ( D t X D ) + ((-7: + y - ) J 2 i ( D t X D ) (7- + f ) J 2 2 ( D t X D ) ) + ibfST ( & ( D X ) ) - ibfT ( J & ( X D ) )S

+iS ( A j ( D t X ) )Tbt - i ( , 7 , j ( X D t ) )TSbt . (20) -

(4) The master equation for the partial expectation (U,*XUt), = X is:

&X

= -y-XDtD - y-DtDX

+ 2 (!J?y-) D t X D .

(21)

The proof of Theorem 2 is given in the next section. Remark. The causally normally ordered form of the Fermi white noise This is the new feature of the equation is a pair of equations for U and Fermi case.

c.

4. Proof of Theorem 2. 4.1. Quasi-commutation mles for bt and U t .

Let us express the evolution operator as Ut = lim U,( N ) , N+a, N

Ir

u:N)= z(-i)" dtl f ' n=O

0

0

dt2.. . fn-' dt,

(fi

0

j=1

(Dtbtj + D b l j ) )

34

The time-consecutive principle (see Ref. 2) states that expanding the product btUr according to (22), any term of the form btb;: b;,2 . . . b::

with t 2 tl 2

t2

,

(23)

2 . . . 2 tn7 ~i E { ,t}, is equal to (-l)nb,E:bL,2.. .bE,"bt

+ { b t , b::} b:,Z . ..biz

,

where the anti-commutation relation for the fields bt is given by (11). Using this, we obtain:

'&-i)n

l' l dtl

tl

d t 2 . ..

Ln-'

dt,(-l)n j=1

n=l

(fi

{ bt, bfl}

+ (-i)(-i)n-lD

(D'bt,

j=2

= 6:N)bt - i D y - ~ [ ~ , ~ l ( t ) U .: ~ - ' ) Here X[,,~I is the characteristic function of the given interval [ O , T ] . In the limit N -+ 00 we obtain Eq. (16.1) A

btUt = Utbt - i r - D U t , For btU,* we obtain:

btUjN)*=

?(i)" n=l

IT Jo" Jd'=-' dtl

dtg

. ..

(h

+ (i)(i)n-l

j=n

(Dtbtj

+ Db!,))

D { bt, bf,}

j=n

and in the limit we get Eq. (16.2):

btU,* = @bt

+ iy-U,*D.

Let A , B be some operators. Since 0' = 1, observe that

P ( A B ) = OABO = OAG2B0 = P ( A ) P ( B ) .

35

and

(PA)* = P ( A * ) . l?rom the general formula (13) we get

(25)

Pbf = -bf .

Pbt = -bt, Let us apply P operator to bt6t and have:

bt6,*,

(26) and use Eqs. (16.1,16.2). We

(-Utbt - ir-DUt)

bt6t = -P (btUt) = -P

= - (-U& - iT-DUt) = Utbt + i y - D 6 t . 6

Hence, we get Eq. (16.3):

btUt = Utbt bt6,* = -P (btU,*)= -P

+ iy-D6t.

(6,*bt + ir-U,*D = - (-U,*bt

1

+ ir-U,*D - 1 = U,*bt- i ~ - 6 , * D

Hence, we get Eq. (16.4): A

btU,* = U,*bt - ir-U,*D. Eqs. (16.1*)-(16.4*) are adjoint of Eqs. (16.1)-(16.4). In Appendix A we prove the relations (16) using the integral form of the evolution equation, as done in Ref. 1 for bosons. 4.2. Normally ordered equation f o r Ut

.

Let us rewrite the evolution equation (9),using (16.1).

atUt = -i (Dtbt

+ Dbf) Ut = -i

(Dt6tbt

+ DbfUt) - T-DtDUt.

Eq. (27) is not close because it involves both Ut and operator to it, and using (24-26), we obtain:

(27)

6t.e Applying the P

&6t = i (DiUtbt + DbtGt) - T - D t D 6 t .

(28)

The system of Eqs. (27,28) is closed and causally normally ordered in the sense that all the bf operators are on the left hand side and all the bt are on the right hand side of the Ut system. eThe operators U and 6 are dependent, hence if one substitutes the definition of 6into (27), then the result will be closed. R.R. is grateful to Prof. Y.G.Lu for pointing this out.

36 4.3. Inner Langevin equation.

The inner Langevin equation is a result of a direct computation. The idea of this computation is to apply Eqs. (27,28) to express &Ut, then to use Eq. (16) to find the causally normally ordered form of the terms. Let us compute a t J l l ( X ) ( t ) :

atgll(x)(t) = at (u,*xut) = atu,*xut + u,*xatvt

+ U,*Dtbt) - f U , * D t D ) XUt + U,*X (-i(Dt6tbt + bfDUt)- r-DtDUt) = (i(bi6,*D

= ibf6,*DXUt +iU,*DtXbtUt - f U , * D t D X U t

- iU,*XDtGtbt - iU,*bfXDUt - r-U,*XDtDUt Using

iU,*DtXbtUt = iU,*DtX6tbt

+ r-U,*DtXDUt

and

-iU,*bfXDUt we obtain:

Let us compute

Using

and

-ibfG,*XDUt

1

(X)(t):

+ rZU,*DtXDUt

37

we obtain:

Let uss compute

(X)(t):

Using

and

W e ob tain :

The computation of (X)(t) is simila and therefore omitted. Combining all terms in the matrix equation, we find:

Using the matrices T and S, defined in (19), one can rewrite the last two terms as (in the notation (18))

+

i b f ( S T J ( D X )- T J ( X D ) S ) i ( S , 7 ( D t X ) T - , 7 ( X D t ) T S ) b t . this can be checked directly by matrix multiplication.

(33)

4.4. Canonical form of the Jangevin equation.

We would like to represent the following inner Langevin equation: at

( & ( X ) ) = -7- (Zj(XDtD)) - 7' ( & ( D t D X ) )

+

(-7-+ 7')&2(DtXD) (-7' + r - ) & @ X D ) (7- + 7')&2(DtXD) + ibfST ( s j ( D X ) )- ibfT ( A j ( X D ) )S is ( z j ( D t X ) )Tbt - i ( z j ( X D t ) )TSbt

(y- 7 ' ) J l , ( D t X D )

)

+

(34)

in the canonical form at

(zj(x))=

(Jkl(@iij(x))

+ blJkl(8Lij(x))+ Jkl(8,,j(X))bt)

*

kl=1,2

(35) Let us find structure maps 8O, 8%. The map Zj is linear, hence 8O, 8% can be obtained by rewriting the matrix multiplications in (34) in the index form: ( a i j ) ( b j l )= (Cjaijbjl). Thus, where

and

where

nij

:=

(0 ) , -1

(nij) = T

s = (Sij) = oz = (::1).

S,

39 4.5. The Master equation.

The master equation is the vacuum expectation of the inner Langevin equation (32). The last two terms, containing bt and b i , vanish. It is enough to take vacuum expectation only for one matrix element. The most interesting is A I ( X ) . Denote by X the vacuum expectation (,711(X))0.We obtain:

8tX = - y - X D t D - y - D t D X + 2 8 y - D t X D .

(36)

5. "Anti-commutative"case Theorem 3. Keep the notation and assumptions of Statement 1. Suppose the operator D is such that

{D,bt} = { D , b i } = O .

(37)

Then, the following relation is satisfied

Proof. Let us express the evolution operator as:

Using the time-consecutive principle (see (23)), we obtain: bt U i N ) =

c(-i)" /' dtl [' N

=

n=l

0

dtg . . .

0

rn-'

(fi

dt,(-1)2n

{b,,bil}

+ b i j D ) ) bt

j=1

0

+ (-l)(-i)(-i)n-lD

(Dtbtj

(fi

(Dtblj

+ bl,D

j=2

= U:N)bt Taking the limit N

+ iDy-~[o,~](t)U$~-l) (39)

-+ oi, of Eq. (39) we obtain: bt Ut = Utbt

which is (38). The theorem is proved.

+ iy- DUt

40

The relation (38) between b and U is similar to the Bose case.’ In the case of Bose quantum white noise, the causally normally ordered form of the white noise equation (3) can be obtained using only the relation between Bose white noise and the evolution operator. Using the proof for the Bose case, one can easily prove that replacing Bose by Fermi quantum white noise operators, the causally normally ordered form of the white noise equations is the same.

6. Conclusions We studied the Fermi quantum white noise equations with a dipole-type interaction Hamiltonian (10). In the ”commutative” case we obtain a new form of causally normally ordered white noise equation (17) and of inner Langevin equation (20). In the ”anti-commutative’’ case we found the commutation relation (38). From Eq. (38) we get that in the ”anticommutative” case the causally normally ordered form of a Fermi white noise equations is the same as in the Bose case.

Acknowledgements This research was carried out while one of the authors (R.R.) was visiting at Centro Vito Volterra. R.R and I.V. were partially supported by the RFFI grant 02-01-01084 and the grant 1542.2003.1 for scientific schools.

Appendix A. Proof of the relations (16) using integral equation The evolution operator Ut satisfies the following integral equation

i?t

Ut = 1- i

I’

dt’(Dbi, + Dtbr)Uti .

Gt = 1+ i

1

dt’(Dbi, + Dtbtl)ctj

satisfies

t

Consider the iterated series U ( N )for the solution of the integral equation with the initial condition u(0)=

and relation

1

41

and the same series for 6 with the initial condition 6p)= 1. The limit of the series (which exists under our assumptions) is the solution of the integral equation

We want to prove that for t 2 T h

btUT = UTbt - ~ ~ - D X [ O , T ] ( ~ ) U T .

(-4.1)

Let us prove the following relation for the iterated series (for t

btU$N)= @"bt

-iy-D~[o,~](t)U$~-~).

This equation clearly holds for N = 1. Suppose it holds for all N Let us proof it for N = M :

btU$M)= bt (1 - i

Jd'

2r):

5M

- 1.

dt'(Dbf, + Dtbtt)U,(,M-')

= bt - i

I'

)

+

dt' bt(Dbf, Dtbtt)U{?-"

- ir-DX[o,T](t)UT (M-1)

= G!M)bt

+ 0 - iy-D~[o,~](t)U$~-~) . (A.2)

The second term is equal to 0, because t 2 r. Taking the limit N -+ 00, we obtain (A.1). Substituting t for T in ( A . l ) , we get (16.1). References 1. L. Accardi, Y.G. Lu and 1.V. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, 2003. 2. Bogoliubov, N. N., Logunov, A. A., Oksak, A. I., Todorov, 1. T., General Principles of Quantum Field Theory, Kluwer, Dordrecht 1990 3. R.L. Hudson and K.R. Parthasarathy, Unification of boson and f e n i o n stochastic calculus, Commun. Math. Phys., 104, 457-470 (1986). 4. A.F.Andreev, Mesoseopic superconductivity in superspace, JETP Lett. 68, 673 (1998)

GENERATING FUNCTION METHOD FOR ORTHOGONAL POLYNOMIALS AND JACOBI-SZEGO PARAMETERS

NOBUHIRO ASAI Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan E-mail: asaiQkurims. kyoto-u. ac.jp IZUMI KUBO Graduate School of Environmental Studies Hiroshima Institute of Technology Hiroshima 731-5193, Japan E-mail: kubo Qcc.it- hiros hima. ac.jp HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: kuoOmath.1su. edzl Let p be a probability measure on R with finite moments of all orders. Suppose p is not supported by a finite set of points. Then there exists a unique sequence {Pn(z)}rz0 of orthogonal polynomials such that P,(z) is a polynomial of degree n with leading coefficient 1 and the equality ( 2 - an)P,(z)= P,+i(z) +w,P,-i(z) holds. The numbers {anr u,}?=~ are called the Jacobi-Szego parameters of p . The family {Pn(z),an,u,}:=~ determines the interacting Fock space of p . In this paper we use the concept 'of generating function to give several methods for computing the orthogonal polynomials Pn(z)and the Jacobi-Szego parameters a, and wn. We also describe how to identify the orthogonal polynomials in terms of differential or difference operators.

1. Accardi-Boiejko unitary isomorphism

Let p be a probability measure on R with finite moments of all orders. Assume that p is not supported by a finite set of points and that the linear span of the monomials (2";n 2 0 ) is dense in the complex Hilbert space L 2 ( p ) .Then we can apply the Gram-Schmidt orthogonalization procedure

42

43

to the monomials {1,x,x2,.. . , x n,...}, in this order, to get orthogonal polynomials {Po(x),P1(x),. . . ,P,(x), . . .}. Here P,(x) is a polynomial of degree n with leading coefficient 1. It is well-known (see, e.g., the books by Chihara5 and by Szego7) that these orthogonal polynomials satisfy the recursion formula

(x - a n ) P n ( x ) = pn+l(x) + wnPn-l(x),

n >_ 0 ,

(1)

where a, E R, w, > 0 and by convention wo = 1, P-1 = 0. The numbers a, and w, are called the Jacobi-Szego parameters of p. associated with the measure p by Define a sequence A, = wow1 .-.w,,

n 2 0.

(2)

It can be easily checked that A, = JRIPn(x)12dp(z). Assume that the sequence satisfies the condition that infn>O - A" :/ > 0. Define a complex Hilbert space rp by

with norm

I\ . 1) given by

+

Let @, = (0,. . ., O , l , O , . . .) with 1 in the (n 1)st component. Define the creation, annihilation, and neutral operators a+, a-, and ao acting on F p , respectively, by

a+@, =

a-Gn = wn@,-l,

a'@, = anan, n 2 0 ,

where @.-I = 0 by convention and an's and w,'s are the Jacobi-Szego parameters of p. It can be easily shown that the operators a+ and a- are adjoint t o each other. The Hilbert space F p together with the operators {a+, a-, a o } is called the interacting Fock space associated with the measure p. It has been shown by Accardi and Bozejkol that there exists a unitary isomorphism U : F p + L 2 ( p )satisfying the conditions: (1) U@O = 1, ( 2 ) Ua+U*P, = Pn+1, (3) Ua-U*P, = WnPn-1, (4) U(a+ a- a0)u* = X,

+ +

44

where the polynomials Pn(z)’s are given in Equation (1) and X is the multiplication operator by z. Note that the Hilbert space is determined only by the numbers w,’s, while the numbers an’s and the polynomials P,(z)’s are related to the unitary operator U . It is natural to ask the following question: Question: Given a probability measure p on B!, how to compute the associated orthogonal polynomials and the Jacobi-Szego parameters {Pn, *n, wn}?

In Section 2 we will explain the generating function method to derive the orthogonal polynomials. In Section 3 we will describe two ways for the computation of the Jacobi-Szego parameters. In Section 4 we will discuss the computation of the Orthogonal polynomials by differential and difference operators. In Section 5 we will list some important classical examples from the viewpoint of generating functions. 2. Pre-generating and generating functions

Let p be a probability measure on R satisfying the conditions mentioned in Section 1. In a series of papers2>324we have introduced the generating function method to derive the associated orthogonal polynomials {Pn(z)} and the Jacobi-Szego parameters {an,wn). A pre-generating function is a function cp(t,x) which admits a power series expansion in t as follows:

n=O

where gn(z) is a polynomial of degree n and limsup,,, ~ ~ g n <~CCJ. ~ ~ & ~ A generating function for p is a pre-generating function of the form W

n=O

where P,(z)’sare the orthogonal polynomials associated with ,u as given in Equation (1). Note that a generating function for p is not unique because we can always replace t in Equation (3) with ct for a nonzero constant c # 1 to get a different function $ ( t , z ) . However, it is possible to have two essentially different generating functions for the same measure. For

45

example, the following functions

$Nt,x) =

(

2 ( 1 - 2tx

+ t2)(1- tx + J1-

)

2tx + t 2 )

are generating functions for the measure dp(x) = SJ-dx, 1x1 5 1. Suppose p(t,x) is a pre-generating function. Consider its multiplicative renormalization defined by

Theorem 2.1. The multiplicative renormalization $(t,x ) is a generating function for p i f and only if E,[$(t, -)+(s, -)I is a function of ts.

If we can check that E,[@(t,-)$(s, .)I is a function of ts, then by the above theorem $ ( t ,x) is a generating function. We can expand $(t,2) as a power series in t to get

n=O

where Qn(x)is a polynomial of degree n. Let a, be the leading coefficient of Qn(x) and let P,(x) = &,(%)/a,. Then the polynomials {P,(x)}are the orthogonal polynomials satisfying Equation (1) for the measure p. In the papers2i3s4 we have applied the generating function method to pre-generating functions of the form

d t , 2) = h(P(tb)9 where h(x) = ez or h(x) = (1- x)c and p(t) is a function to be derived so that the condition in Theorem 2.1 is satisfied. Case 1:

h ( z ) = ez measure Gaussian Poisson gamma negative binomial

I

polynomials Hermite Charlier Laguerre Meixner

46

Case 2: h ( z )= (1 - z)' measure

polynomials

uniform arcsine semi-circle beta-type

Legendre Chebyshev of 1st kind Chebyshev of 2nd kind Gegenbauer

The above polynomials are derived from the power series expansion of the resulting generating functions. Consequently they are expressed in terms of sums of monomials. In Section 4 we will use differential and difference operators to identify these polynomials. Recently all measures of exponential type have been derived in the paper6. In particular, the probability law of the L h y stochastic area is

in this class.

3. Computation of orthogonal polynomials and Jacobi-Szegii parameters Suppose $(t,z) is a generating function for p. The object is t o compute the orthogonal polynomials {Pn(z)} and the Jacobi-Szego parameters {a,, wn} from $(t,z). Recall that A, = wow1 .w,, n 2 0, as defined by Equation (2). The following theorem has been proved in our paper3.

Theorem 3.1. Let $(t,z) = C,"==, anPn(z)tnbe a generating function for p. Then we have t+O lim

+(t,

f) =

c 00

anzn,

n=O

n=O

n=O

where a-1 = 0 by convention.

Thus once we have found a generating function for p, then we can compute {P,,a,, w,} as follows:

47

Namely, first expand $(t,x) as a power series in t to get a, and P,(x),which is expressed as a sum of monomials. If we are not interested in finding P,, then we do not have to expand $(t,x) as a power series in t. We can simply use Equation (4)to find a,. Then we use Equation (5) to find A,, which can be used in turn to find w, since w, = An/An--l, n 2 1, wg = 1. Finally we can use Equation (6) to derive a,. In our paper3 we have used this method to compute (Pn,a,,w,} for those measures listed in Section 2. However, the computation is somewhat complicated due to the following difficulties:

Difficulties: There are several difficulties in applying Theorem 3.1. (1) It might be difficult to find the series expansion of $ ( t , x ) in t. (2) The computation of E , [ $ ( t , - ) 2 ]in Equation (5) might be very complicated. (3) The computation of E, [x$(t, -)2] in Equation (6) is even more involved. (4) The orthogonal polynomials are expressed as sums of monomials. How to find the close forms for them?

Resolution: Here are some ideas to overcome the above difficulties: (a) Find a computation method without having to use the power series expansion of $(t,x) in t. (b) Find a system of linear equations for the Jacobi-Szego parameters. (c) Determine P,(z) by a differential or difference operator. (d) Find a differential equation satisfied by P,(x). This equation is determined by the measure p. (e) Determine {a,,w,} from the eigenvalues of a differential operator.

We have made some progress regarding to Items (a), (b), and (c). The key idea is to use the series expansions of $ ( t , O ) and &$(t,x)Is=~ in t. Then we can avoid the difficulty (1). So, suppose cp(t,x) is a pre-generating function and assume that its multiplicative renormalization $(t,x) = cp(t,x)/E,cp(t, .) is a generating function for p. Define three functions A ( x ) , B ( t ) , and C ( t ) with their respective power series expansions by

00

B ( t ) = $(t,O) =

b,tn, n=O

The next theorem is from our paper4.

Theorem 3.2. Suppose $(t,x) = p(t,x)/Ep(p(t,.) is a generating function for p . Let a,, b,, c, be the numbers defined in Equations (7)-(9)- Assume that bncn-l # bn-lc,. Then the Jacobi-Szego parameters {an,w,} are the unique solution of the system of the linear equations:

Thus we can compute the Jacobi-Szego parameters as follows:

Now, consider the special case when the pre-generating function V ( t ,z) is of the form

where p ( t ) = C,“==, pntn is an analytic function near z = 0 with p1 # 0 and h(x) = C;=, h,zn is an analytic function near z = 0 with ho = 1, h, # 0 for all n 2 1 and there exists tl > 0 such that p(tz) is analytic in z on the support of p for all It1 < t l , and limsup,,, (1h.I I~PIILZ(~))”~ < 00. In this case the function A ( z ) in Equation (7) is given by

49

Therefore, a, = hnpr and so Equation (10) becomes

In particular, when p is symmetric, we have 01, = 0 for all n. Then Equation (12) yields the values of wn’s as follows:

4. Orthogonal polynomials in terms of differential or difference operators

Suppose $(t,x ) is a generating function for p. Then we can expand $(t,z) as a power series in t

n=O

where Pn(x)’saxe the orthogonal polynomials associated with p . As pointed out in Section 3, these polynomials are expressed as sums of monomials. Since it might be difficult to compute the power series expansion of $(t,z), it is desirable to determine or identify these polynomials in some other ways, namely, we raise the following Question: How to determine or identify the orthogonal polynomials {Pn(x)}without using the power series expansion of $ ( t ,x ) in t?

For absolutely continuous measures we have the next two theorems from our paper4. Recall that A ( z ) = Cr=,anzn from Equation (7) and that An = wow1 * * . w,. Theorem 4.1. Let p be a measure on ( a ,b) with Q smooth density function e(x) and assume that $ ( t , x ) is a generating function for p . Suppose qn(x) is a smooth function satisfying the conditions: 1 (a) -0,” [qn(xP(z)]E L2(p). O(X)

50

(b) 0,k[qn(x)B(x)]= 0 at x = a, b f o r all 0 5 k

< n.

Then the orthogonal polynomial P,(x) associated with p is given by 1 P,(x) = -0," [qn(x)o(x)] knB(x) with some constant k, if and only i f

[ ~ : + ( Xt ,I ] q n ( x ) d ~ x =) dntn with some constant d,. In this case, d, and k, are related by dn = (-l)"X,a,k,. For the special case when cp(t,x) is of the form in Equation (ll),the generating function is given by

In this case, we can make use of the function B ( t ) defined in Equation (8) and take q n ( x ) = e,(x)/e(x) in Theorem 4.1 to get the next theorem. Theorem 4.2. Let p be a measure on (a,b ) with a smooth density function e ( x ) and assume that

is a generating function f o r p . Suppose & ( x ) is a smooth function with support in [a,b] satisfying the conditions:

(b) D,k [&(x)]= 0 at x = a , b for all 0

5 k < n.

T h e n the orthogonal polynomial P n ( x ) associated with p is given by

with some constant k, i f and only i f

with some constant d,. In this case, d , and kn are given

51

For discrete measures on the set & = {0,1,2,. . .,n,. . .} we have the analogous results in the following two theorems from our paper4. Define the right- and left-hand difference operators A,+ and Ax- acting on functions defined on & by = f (x

Ax+f Ax-f

(2)

+ 1) - f (21,

= f (2) - f .( - 11,

where f (-1) = 0 by convention. We have A:+ f (x - n) = A:- f (x). Theorem 4.3. Let p be a measure o n & and let e(x) = p ( { x } ) , x E & . A s s u m e that +(t,x) is a generating function f o r p . Suppose qn(x) is a

function o n & satisfying the conditions:

(b) A!+ [q,(x)e(x)] = 0 at x = 0 , 00 f o r all 0

5 k < n.

T h e n the orthogonal polynomial P,(x)associated with p i s given by

with some constant k, i f and only i f m

2 [ ~ : + + (.)It , x=o

qn(x)e(x) = dntn

with some constant d,. In this case, d , and k, are related by d , = (-l)n Xn ~ n k n . The next theorem is analogous to Theorem 4.2. It is a special case of Theorem 4.3 for the multiplicative renormalization of the pre-generating function cp(t,x) = ep(t)z. Theorem 4.4. Let p be a measure o n & and let e(x) = p ( { x } ) , x E & .

A s s u m e that the multiplicative renormalization

pntn is analytic with i s a generating function f o r p, Here p ( t ) = C,"==, # 0. Suppose &(x) is a probability m a s s function o n & f o r each n satisfying the conditions:

p1

52

(8)

1

e(z)A:-

[en(x)]E L~(P).

(b) A:+ [&(x)]

5

0 at x = 0 , 00 for all 0 5 k < n.

Then the orthogonal polynomial P,(x) associated with p is given b y

with some constant k, if and only if

with some constant do. I n this case, d, and k, are given by dn = $,

kn = (-I),-.

n!

An

5. Classical examples In our paper3 we have used our method to derive generating functions $ ( t , x ) and {P,(z), ali, wn} for those measures listed in Section 2. But some computations, for example in identifying the polynomials P,(x),are rather complicated. In the paper4 the computations are simplified by using Equations (12) and (13) and Theorems 4.2 and 4.4. Recall that wg = 1.

Example 5.1. (Gaussian measure) e(x) = -&e-x2/2"2,

x

E

R

g ( t ,x) = etz--a2t/2, P,(x)= (-02)nez2/2"20," [e-z2/2"2] (Hermite polynomial), n20,

a, = 0 ,

w, =02n, n

2 1.

Example 5.2. (Gamma distribution) a > 0.

e(x) = r(1a7, p - 1 e --I , x > 0. Here

4(t,x) = (1 + t)-ae+z, n Pn(2) = (-1),x -a+l e zD, [ ~ ~ + ~ - - l e (Laguerre -~] polynomial), an = 2 n + a , n 2 0, w, = n(n a - l), n 2 1.

+

53

Example 5.3. (Beta-type distribution)

where the parameter /? > -l/2, 6 ,

# 0.

(Gegenbbauer polynomial),

We have two special cases. When p = 1/2, the measure p is the uniform measure on [-1,1] and Pn(z) is the Legendre polynomial. When ,B = 1, the measure p is the semi-circle distribution on [-l,l] and Pn(z) is the Chebyshev polynomial of the second kind, which can be verified to equal

Pn(z)=

+

1 sin [(n 1) c0s-l x] sin [cos-l , n>0.

4

Note that we have to exclude P = 0 in this example since the function in Equation (14)) is not a generating function when ,B = 0. The next example takes care of this case.

Example 5.4. (Arcsine distribution) e(z) = +-&J,

121

< 1.

(Chebyshev polynomial of the first kind),

w n = { 1 / 2 , if n = 1, 114, if n

> 2.

54

Example 5.5. (Poisson measure) O(x) = e - X s , x E &, X

> 0.

(Charlier polynomial), an=n+X, wn=Xn,

n10, n > 1.

Example 5.6. (Negative binomial measure)

where r

> 0 and 0 < p < 1.

(Meixner polynomial),

Acknowledgments N. Asai gives his deepest appreciation to Professor I. Ojima of RIMS, Kyoto University for challenging discussions and constant encouragement during his stay from April 2002 to March 2004. He is grateful for the postdoctoral fellowship from the program of the 21st century COE Kyoto Mathematics fellowships. H.-H. Kuo is grateful for research support from the Academic F'rontier in Science of Meijo University. He would like to thank Professors T. Hida, K. Nishi, and K. Sait6 for their warm hospitality during his visits in the past years.

55 References 1. Accardi, L. and Bozejko, M.: Interacting Fock space and Gaussianization

2. 3.

4.

5. 6.

7.

of probability measures; Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 (1998) 663-670 Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functions I.; Taiwanese Journal of Mathematics 7 (2003) 89-101 Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functions 11.; Taiwanese Journal of Mathematics 8 no. 4 (2004) Asai, N., Kubo, I., and Kuo, H.-H.: Generating functions of orthogonal polynomials and Szego-Jacobi parameters; Probability and Math. Stat. 23 (2003) 273-291 Chihara, T. S.: A n Introduction to Orthogonal Polynomials. Gordon and Breach, 1978 Kubo, I.: Generating functions of exponential type for orthogonal polynomials; Inifinite Dimensional Analysis, Quantum Probability and Related Topics 7 (2004) 155-159 Szego, M.: Orthogonal Polynomials. Coll. Publ. 23,Amer. Math. SOC.,1975

LOW TEMPERATURE SUPERCONDUCTIVITY AND THE STOCHASTIC LIMIT*

F. BAGARELLO Dipartimento d i Matematica ed Applicazioni, Facoltd d i Ingegneria, Universitd d i Palemno, V i d e delle Scienze, I-90128 Palemno, Italy e-mail: 6agarellOunapa.it

In this paper we review some recent results concerning the analysis of the Open BCS model as proposed by Buffet and Martin as well as some generalizations. Our attention is mainly focused on the computation of the critical temperature.

1. Introduction

In this paper we review two recent papers concerning the Open BCS model as first introduced in Ref. 1, 2 and some possible generalizations of this m0de1.~>~ The technique that we have adopted is that of the stochastic limit approach (SLA), whose details are described in the m ~ n o g r a p h .In ~ particular, we will show here that the SLA allows us to compute the same value of the critical temperature T,when we consider the model in Ref. 1 but in a significantly simpler way, and it allows a simple control of T, also for different models, like the ones discussed in Ref. 4. The paper is organized as follows: In the next section we review in some details the model analyzed in Ref. 3. In Section 111, after some physical justification, we discuss the role of a second reservoir in the open BCS model and we show how this reservoir may affect the value of T,,reviewing some of the results contained in Ref. 4. 2. The physicals model and its stochastic limit

Our model consists of two main ingredients, the system, which is described by spin variables, and the reservoir, which is given in terms of bosonic 'This paper is dedicated to my beloved parents, which are always close to me

56

57

operators. It is contained in a box of volume V = L 3 , with N lattice sites. We define, following Ref. 1, 2

where the Pauli matrices satisfy the following commutation rules

bi*

[a;,o-] 3 =&joy,

7

4

- T 2dij0f.

We will use the following realization of these matrices:

If we now define the following bounded operators, -

N

Hps)

can be simply written as = N(ZS& - gRN) and it is easy to check that [S%,RN]= [H P ” , R j v ] = [H$”),S%] = 0, for any given N > 0. It is also worth noticing that the commutators [ S ~ , C $go ] to zero in norm as & when N + 00, for all j , a and p. Our construction of the reservoir follows the same steps as in Ref. 2, but for the commutation rules. In particular, we introduce here as many bosonic modes a3,j as lattice sites are present in V . This means that j = 1,2, ..., N . p’ is the value of the momentum of the j-th boson which, if we impose periodic boundary condition on the wave functions of the bosons, has necessarily the form p’ = %fi, where n’ = (121,122,723) with nj E 2. These operators satisfy the following CCR,

[ap;i,ag,jl = [at. P,%.,at. 923.I = 0, and their free dynamics is given by

[ag,i,at. 4 > .I 3 = dij+g

(4)

N

-a 47r2(4+?4+4) where A N = {p’= %n’, n’ E Z 3 } and €3 = L2m = 2mL2 ‘ The form of the interaction between reservoir and system is assumed to be

N

+

~ $ 1= C ( a j + a j ( f ) h.c.1, j=1

(6)

58

where aj ( f ) = f (p3, f being a given test function which will be asked t o satisfy some extra conditions, see equation (15) below and the related discussion. We would like to stress that, in order to keep the notation simple, we will not use the tensor product symbol along this paper whenever the meaning of the symbols is clear. The finite volume open system is now described by the following Hamiltonian,

H N = H&

+ AH;),

where H L = H P s J

+Hge8)

(7)

and X is the coupling constant. The free evolution of the interaction Hamili H g V " ) t + -iHg'")t i H g e " ) t ( I ) ( t ) = eiHktH$)e-iHkt = EEI ( e tonian, H N e aj e

+

a j(f)e-iHge"'t h.c.), can be easily computed using a semiclassical approximation . Defining

(8)

where w = g,/(So)2 Q = 0 , f, we get

+ 4S+S-,

Y

= 22

+ gSo and va(p3 = v - e,j + o w ,

N j=1 c r = O , f

The next step in the SLA consists in computing the following quantity

and its limit for X going to zero. Here the state wtot is the following product state wtot = wsyswp, where waysis a state of the system, while wp is a KMS state corresponding to an inverse temperature ,b = It is convenient here t o use the so-called canonical representation of thermal state^.^ For that we introduce two sets of mutually commuting bosonic operators {#}, y = a , b, as follows:

&.

a5,j = Jmlj)cgj

+&'j$c~~~+,

(11)

where

mm = 4 a , j , j a ; , j ) = 1-

1 e-pep>

e-pec nm = wp(ap!,ja,j,j)= 1 - e-pep. (12)

59

The operators c $ ! satisfy the following commutation rules rC$,j (a),cLT)t] g,b = 6J.k 6p-q4

(13)

while all the other commutators are trivial. Now, if we introduce the vacuum of the operators c g j , @o, $!Go >J = 0, Vfl E AN, j = 1, ..N, a = a, b, then wg can be represented as a vector state: wp(X,) =< @o,X,@o> for any observable of the reservoir, X,. Finally, if we define fm@J = and fnm = we get

m f m

mfm,

cc N

H$)(t) =

{ph ( c ~ ) ( f m e i t ” -+) c ~ ) ’ ( f n e i t V a )+ ) hx}.

(14)

j=1 a=O,f Finally, if we require that the following integral exists finite:

where f,(jJis fm@J or fn@J and v,@J

is given above, we find that

(16) where the two complex quantities

(17) both exist because of the assumption (15). This suggest to define the following stochastic limit Hamiltonian N

H F ) ( t )=

c {d

(c&)(t)

j=1 a=O,f

+ c a j ( t )) + h.c } ,

(18)

where the operators ch‘j‘(t) are assumed to satisfy the following commutation rule,

[C$)(t),cg)t(tr)l= sj,

6,,6(t

-

t/)rp),

for t

> t’,

since, as it is easily checked, the following quantity

J ( t ) = (-i)2 [ d t l 0

[’ 0

dtzRtot(H~)(tl)H~‘)(t2))

(19)

60

coincides with I ( t ) . Here Otot = wsvsO = wsys < Qo, QO >, where QO is the vacuum of the operators c$)(t): c$)(t)Qo = 0 for all a , j , y and t , cf. Ref. 5 . Following the SLA5 we now use H g ) ( t )to compute the generator of the theory, which is found recalling that Otot(dtjt(X 81I,)) = O t o t ( j t ( L ( X ) ) ) , where j t ( X @ It,) = U j ( X @ Il,)Ut and Ut is the wave operator, which satisfies the following differential equation: at Ut = - i H g ) ( t ) U t . For all self-adjoint observables X we get

L ( X )=

+L2(X),

(20)

where

cf. Ref. 3. As discussed in Ref. 1, 2, we need to find the dynamics of S& and RN to get some insight about the phase structure of the model and, in particular, to compute the value of the critical temperature corresponding to a transition from a normal to a superconducting phase. These intensive operators are both self-adjoint, so that we can use equations (20) and (22). It is now a direct computation to deduce that

where it is necessary to introduce the F - strong topology instead of the uniform t ~ p o l o g y ,and ~ , ~we have defined

while

61

The phase structure of the model is now given by the right-hand sides of Equations (23) and (25),see Ref. 1, 2, and, in particular, from the zeros of the functions

where x = So and y = S+S-. In particular, the existence of a superconducting phase corresponds to the existence of a non trivial zero of fl and f 2 , cf. Ref. 1, 2, that is to a non trivial zero of the function h: h(xO,yo) = 0 with (x,,y,) # (0,O). In order to find such a solution, it is first necessary to obtain an explicit expression for the coefficients Rrp). This is easily done:

lf,rn~~s(~*rn),

Erg) = =

xrf) = r

C IMPI~S(Y*(P~). IXAN

$€AN

(27) It is now almost straightforward to recover the results of Ref. 1,2.Following Buffet and Martin’s original idea, we look for solutions corresponding to Y = 0. This means that x = -2Z/9, v+@j = w - €5, which is zero if and only if w = €5, and v- (p3 = -w - €5, which is never zero. For these reasons we deduce that Rr?) = 0, y = a,b, while the sums in (27) for Rry) are restricted to the smaller set, EN c AN, of those values of $such that, if 4‘ E EN then 6; = w. Therefore, recalling the expression of m(P3 and n(p3 in (12),we find

From Definition (24), finally, we get the following equation

or

which, as we have proven in Ref. 3, is equivalent to the one obtained in Ref. 1, 2, g tanh = w. For this reason, then, we deduce the existence of a critical temperature, T, := &, coinciding with that found by Martin and Buffet, such that, when T < T,, the system is in a superconducting phase. The values of the order parameters also are recovered.

% 0

62

3. Possible generalizations In this section we discuss some possible generalizations of the model which may produce higher values of T,. The idea is very simple and is well emphasized using the SLA: suppose that the free evolution of the annihilation operator of the reservoir, ag,i(t) = ap-, z.e--iept , is replaced, for some reason, which we will investigate later, by ag,i(t) = ag,,ie-iTept, y being some real constant less than one, y < 1. Then we have shown in Ref. 4 that Equation (29) is replaced by following one:

eDw/r = 9 +W 9-W’

which admits a non trivial solution in 10, g [ if gD/y - 2 > 0, that is under a new critical temperature = = which is larger than T, since y is smaller than 1 by construction. Therefore, this very easy mechanism makes the value of the critical temperature to increase. It is worth stressing that a similar conclusion was by no means evident in Ref. 1, 2. The main point, therefore, is to find a mechanism able to change the free evolution of the boson operators, possibly as shown above. For that, a possibility consists in switching on an interaction between the boson reservoir in Ref. 3, which we will call R1 and mother reservoir, Rz,which only communicates with R1 and not with the system S. This will be our point of view: we will discuss now a single model which generalizes the one discussed in Section I1 and the consequences of the presence of this second reservoir on the value of T,. More models are considered in Ref. 4. Let

‘TP)

5,

where H g Y S )is given in (l),H$) in ( 6 ) and

(32) Here both the reservoirs satisfy a bosonic statistic and they are mutually independent:

63

With these definitions it is clear that the free time evolution of H E ) , H g ) ( t ) = e i H g t H g ) e - i H s t ,depends on R2 only through its interaction with R1.We have, using the definitions in (8), e i H ~ t o ~ e - i H= ~ te i H g “ ) t

+e - i H g u ” t -eiutd oj

+ ei(u+w)tpj+ + e i ( u - w ) t

j

P-

>

while

e - i H & t - e i H $ e S ) t a,;ie- i H g e S ) t aP,Z,(t)= eiNa t U 6 , i - e-icpt [a,-,i -

cos(pt) - ib,j,i sin(pt)] ,

as it can be easily derived. If we now introduce the following function:

vapO=v-€$++QIW+Pp, + = O , f ,

P=&,

(34)

we get

+bj(feitva-)

+ h.c),

- bj(feitua+)]

(35) a- . - b - .

Let us now define the operators Ag,j = a5d&bb3j, B s , ~= p ~ l p 3 1 . The only non trivial commutation rules are [ A g , i , A > , . ] = [ B ~ , i , B i = , ~dijdcf, l and we can write H $ ) ( t ) as

which producess, as in the previous section

The only difference is in the coefficients defined as

Here we have introduced

and

which are now

64

and we recall that m ~ = oW A ( A ~ , ~ Am $ ,~~@ ) ,= 3w ~ ( B ~ , ~ B nA(p3 $ , ~=) , W A ( A $ , ~ Aand ~ , ~nB(p3 ) = w ~ ( B $ , ~ B c ,Wj A) . and WB are the KMS-states of respectively the A and the B-operators. The total state is wtot = wsYs @ WA

@WB.

Remark:- For all our results to be meaningful, we have to require that all these integrals exist finite. This gives a condition on f(p3, which extends the analogous one previous given in the previous section, due to the appearance of two different reservoirs. Due to the fact that I ( t ) in (37) coincides formally with the one in (16), it is easy to check that all the previous steps can be repeated and the conclusion is again the same: the system undergoes a phase transition, from a normal to a superconducting phase, if the function h(x,y) defined in analogy with (24) as

has a non trivial zero (20,yo). Find such a zero may be very hard, in general. First we observe that

i

= f &AN = f &AN sry)= ;

c

lf(P3l2(mA@36(Y+-(p3) + mB(p3d(Y++(p3)) lf(P3l2( m A ( p 3 6 ( Y - - ( p 3 ) + mB(p36(Y-+(p3)) > (41) (nA(p36(Y+-(p3) + nB(p36(Y++(p3)) 7

sr- -- 57r Cc,=,nN lf(p3I2 (nA(p36(V--(p3)+ nB(p36(v-+(p3)) (n)

7

which are much more complicated than the expressions in (27). We have discussed in Ref. 4 that choosing Y = 0 does not allow us to find any new result: the value of the critical temperature obtained in Ref. 1, 2 and Ref. 3 is recovered. For this reason we now assume that Y # 0 and, in particular, we look for solutions such that only u+-(PT assume, for some 3, the value 0, while u++(fl,u--(p3 and v+-+(p? are always different from zero. For such a solution t o exists it is enough that the following inequalities are all satisfied: Y

{

+ w + p < 0,

Y -

Y

w-p

< 0,

- w + p < 0,

v+w-p>o.

65

A trivial solution surely exists if we fix Y = p as far as w E ]2p,-2p[. This means, because 0 5 IwI 5 f i g , that the coumust be negative and less than - g g . pling constant p in However, it is not hard to check that with this choice =

%rim)

%r(”) - = 0, while %I$?

= I ~ ( J $ ~ ~ ~ ~ ( J $ ~ ( Y +and - ( J%I??) $) = If(p312n~(p36(~+-(P3). Now, as an easy consequence, we recover the same equation as in the previous section, epAw = g+w which implies 9 --w ?r

that the critical temperature is not affected in this case. More interesting is the situation when the System (42) holds true without having v = p. This is possible: the choice p = -w, v = is an example of this situation. If (42) is satisfied we deduce that

-:

In order t o check whether this equation admits non trivial solutions for some w €10, g[, we consider three different situations: (i) if Y = p then we go back to the usual ~ o n d i t i o n and , ~ to our previous analysis, and we deduce the existence of a critical temperature which coincides with the usual one. (ii) if v > p then the situation is different: since the function F ( w ) := e B A ( w + Y - p ) - @ is such that F ( 0 ) = e P A ( ” - p ) -1 > 0 and lim-w+g-F ( w ) = g--w -00, and since F ( w ) is continuous, then we surely have a solution F(w,-,) = 0 with wg ~]0,g[,for all the values of PA. This suggests the existence of a superconducting phase for all values of the temperature! (iii) if Y < p then F ( 0 ) = e f i A ( V - - I 1 ) - 1 < 0 and we cannot conclude that a non trivial solution of the equation F( w) = 0 does exist even in this case. A deeper investigation need to be carried out in this case but, since, it is not relevant for our present purposes, we will omit it here. The conclusion of this analysis is therefore that, at least for those values for which System (42) holds true (if Y # p), the equation h(z,g) = 0 has always a non trivial solution independently of the temperature. This seems quite promising and we hope to get a deeper understanding of this fact in a close future, since it suggests the possibility of using this model in a first approach to high temperature superconductivity. We also would like t o mention that more models, sharing with the one considered here a sort of control on the critical temperature depending on the value of the parameters, are discussed in Ref. 4. Remark:- As discussed in Ref. 3, the operators of both R1 and R2 are

66

in general unbounded. For this reason a special care is required to make everything rigorous. This can be done using the framework discussed in Ref. 7, but we will not do it here to avoid a useless complication of the procedure. References 1. E. Buffet, P.A. Martin, Dynamics of the Open BCS Model, J. Stat. Phys., 18,NO.6, 585-632, (1978) 2. P.A. Martin, ModBles en MCcanique Statistique des Processus IrrCversibles, Lecture Notes in Physics, 103, Springer-Verlag, Berlin, (1979) 3. F. Bagarello The stochastic limit in the analysis of the open BCS model, J. Phys. A, in press 4. F. Bagarello The role of a second reservoir in the open BCS model, submitted to IDAQP 5. L. Accardi, Y.G. Lu, I. Volovich, Quantum Theory and its Stochastic Limit, Springer (2002) 6. W.Thirring and A.Wehr1, On the Mathematical Structure of the B.C.S.Model, Commun.Math.Phys. 4, 303-314 (1967) 7. F. Bagarello, Applications of Topological *-Algebras of Unbounded Operators, J. Math. Phys., 39,6091-6105, (1998)

MULTIQUANTUM MARKOV SEMIGROUPS, INTERACTING BRANCHING PROCESSES AND NONLINEAR KINETIC EQUATIONS. FINITE DIMENSIONAL CASE.

V.P. BELAVKIN AND C.R. WILLIAMS School of Mathematical Sciences University of Nottingham University Park Nottingham NG?’ 2RD, U.K.

1. Introduction

This paper introduces an algebraic approach to the theory of systems consisting of random numbers of particles of the same type. In this paper we will consider those systems such that each particle is described as a finite dimensional quantum state, with an associated algebra of (finite dimensional) observables. The multi-particle systems are described by a decomposable algebra of observables compatible with the total number operator on Fock space. The equivalence is shown between multi-particle states and complex valued positive definite contractions defined on a unit ball of matrices and analytic within this ball. A similar correspondence is shown for transformations of multi-particle states, by extension to a family of matrix valued functions. We consider semi-groups of operators on an infinite dimensional space, which are represented as an infinite block matrix of transformations between the finite particle states. Such semi-groups are defined by a suitable family of matrix valued analytic functions of the unit ball of single particle observables which we call generating functions for the semi-group. We define a Markovian multi-particle process in a broad sense by introducing conditions on these matrix valued generating functions and determine necessary and sufficient conditions on the infinitesimal generator of these generating functions for the existence of a Markovian multi-particle process. We consider the class of all possible non-interacting systems and a particular class of interacting systems having an overall interaction strength. A family of generating functionals indexed by the positive real line is given 67

68 which describe the evolution of a multi-particle state for non-interacting systems. We introduce a pair of equations called a canonical pair of quantum kinetic equations and give, in the case of interacting systems having an overall interaction strength, a formal asymptotic solution to the family of state generating functionals in the case where the canonical pair have a solution satisfying a mixed boundary value condition, as the interaction strength tends to zero. At least in the case of pairwise interaction it is possible to show that the asymptotic solution tends to an actual solution in such a limit. 2. The algebraic theory of finite dimensional multi-particle

systems Let A be a C*-algebra of finite dimensionality a, which we will call the algebra of observables for a single particle system. For every such algebra there exists a Ic E N and a set

such that A has the structure k

A3A=@Ai i=l

for some collection of matrices Ai E M (nj). We therefore represent A as a *-subalgebra of M ni having the block diagonal form:

&, 0

‘M(n1) 0 * - . 0 M(n2).

0

..

0

, o

0M

(nk)

For each 1 5 i 5 k let {&,k : 1 5 k 5 n f } be an orthonormal basis for M (ni) with respect to the usual matrix inner product:

( A , B )= T r { A * B } For each 1 5 i 5 k and 1 5 k 5 nf let h i & E A be the matrix; k Ei,k

= @6i&,k j=1

(1)

69

and let {Ei : 1 5 i

I a } be the orthonormal basis set for A on M

(with respect to the inner product (1)) given by Uf=:=,{&,k

:1

x:=l 0 5k5 ni

n:}.

k We define by %(’) the Hilbert space @ ( l ) where d(1) = Ci=:=,ni. We define A’ to be the commutant of A, namely the algebra of all matrices in M (EL1 n . which commute with any single particle observable A E A. Specifically A’ is the algebra of diagonal matrices of the form

.)

‘AIInl

0

0 hIn2

.

.. . . ..

0

..

0

.. .

, o

0 L,In,,

where X I , . . .,A,, E C‘ and Inis the identity matrix on M (n). A single particle state is defined as a positive linear functional having norm at most one (we call such a state stable if it has norm one). Note that any linear functional p on A can be represented by a matrix e = : ei E M (ni) in the sense k i= 1

for any A = $f=lAi E A. It immediately follows that p 2 0 if and only if ei 2 0 for every 1 5 i 5 t. Of course the representation e = is not unique amongst all density matrices on M ni) , but it is the unique A valued element in the equivalence classes defined by the relation:

(‘&

el

N

ez iff Tr { A e l } = Tr { A Q ~VA } EA

Z:=l BT A } where { O BT is the transpose of B , which obviously has the same structure as A. Note Let dT be the transposed algebra B E M

ni

:

E

that positivity is invariant under matrix transposition. We define a pairing

( A ,B ) = Tr { A B T } between A and AT, so that any functional p is uniquely associated (via this paring) with an element in AT. We define a norm on dT as;

AT

3Q

I---)

Ilell, = SUP { ( A ,e> : A E A, IlAll I 11

70

where JJAJ1 is the usual operator norm, so that any single particle state is uniquely represented by a positive matrix in AT having trace at most one. We call AT the algebra of single particle states. is The algebra of n particle observables, which we will denote by defined as the symmetrical nthtensor power, defined as the algebra of finite linear combinations of operators;

with X@O = 1 for all X E A. Such an algebra has a representation in the finite dimensional algebra M (EfZl n,)"> with a basis given by the symmetric tensor products:

(

E ~ @ . . . @ E ? :~n -l +...+TI,=n can also be repreHowever, being finite dimensional, the algebra sented as an algebra of block diagonal matrices. Then any A(") E A(") has the form A!") for some Ic (n) E N, Ain) E M (mi"') for some

@z)

mjn) E N and we define d ( n ) = ~ ~ ~ ' r n { " We ) . define to be the Hilbert space of this representation. We denote by A(")' C M (d ( n ) ) the commutant of the algebra represented on C?(n). We define A:) as the transposed algebra to A(").We define n-particle ( states as the positive matrices en E A;) having trace at most one. We use the collection of algebras A(") : n E N to construct, for 6 > 0, (infinite dimensional) algebras .Ct as the Banach spaces of E summable sequences e = [en]:=o with entries en E A$) such that:

n=O

The multi-particle states are defined as those and sub-normalised in the sense that en =

e E ,C1 which are both positive

aand:

M

n=O

M

n=O

If e is normalised, llell = 1, then we say that the system has a random but finite number of particles with a probability distribution given by p , = Tr{en} for the events that the system consists of n particles. Thus a normalised n-particle state en has the physical interpretation of a multiparticle system which contains (with certainty) exactly n particles.

71

The more relaxed condition of sub-normalisation allows for the description of unstable systems having a non-zero probability 1 - llell of existing in a state having an infinite number of particles. The dual to the space C1, denoted by CT, is the space of all multiparticle observables, which are defined as sequences

satisfying the boundedness condition:

For a given e E Lc we can define a generating functional R (similar to those introduced in Ref. 1) on the unit ball Bt = { X E M : 1JXJI5 [} as the absolutely convergent sum:

The generating function R is an analytic function of a variables in The following proposition establishes the converse.

Proposition 2.1. If R : I3c e E Cg such that:

+ C! is analytic

c

Bt.

then there exists a sequence

W

R (X)=

Tr { $x@n}

n=O

Proof. Such maps R can be thought of as analytic functions in a variables. These agree with their Taylor series locally, giving W

nt=l

W

n,=l

where we denote a

i= 1

and:

The set of complex numbers

72

define a linear functional pn on the algebra d(n)via the linear extension of pn

(Epn'@. .@E?"-) = r (721,. . . ,na)

let en E A$) be the implementation of this form in the algebra A:), in the sense:

We note that the sums nl+...+n,z =n

can be written as ( X @ n , ~ ndefining )

R (X)=

C Tr { e:XNn> n=O

where: 00

C

n=O

{en) = R (61) <

0

Obviously if @ E Ct is a multi-particle state then the generating functional (2) is positive definite and R ( I ) 5 1, the following propositions shows that the converse is true. Proposition 2.2. If R : Bt + C is positive definite and analytic throughout BC then it is the generating functional of some positive @ € Cg. If furthermore 6 = 1 and R ( I ) 5 1 then R is the generating functional f o r a multi-particle state.

Proof. The previous proposition proves the existence of some p E Lt such that: M

n=O

Clearly if 6 = 1 and R ( I ) 5 1 then Cauchy integral formula we can write

llelll 5

1. Furthermore, using the

73

where C is any positively oriented simple closed contour around the origin. Hence we may write

Tr { e;xBn}

by approximating the integrals as the limit of summation, the positivity of Tr { e:X(,)} follows from the positive definiteness of the forms X(,) R. The positivity of the matrix e: in N(")follows from the relation

for any f = @f$)fi E following from that of.:Q

such that fi E P i ( nthe ) , positivity of That e = [ ~ n ] r = oE Lc follows from R ((I) <

en

We have shown a one to one correspondence between positive definite analytic functionals of Q variables and multi-particle states generated by the single particle algebra A. The following proposition is the simplest example of a result we will prove in the following section. Proposition 2.3. An analytic functional R : t3c if and only if

R ( X )=

+ C is positive definite

C C A, (i)* XBnAn (i) n=O i=l

for A, (i) : C + a(")satisfying a(") has dimension d (n).

C,"==, Ct)'

A, (i)* A, (i)

<

00

where

Proof. If R : Bc + C is analytic and positive definite then there exist, for each n E N,positive e: E d(,) such that R is the generating functional of Let A, (i) be the function defined by the column the sequence e = [en],"==o.

74 where R{ is the ( i , j ) t h entry in the square root R of

e:.

Since R ( I ) < 00 it follows that: 03

d"

n=O i=I

Furthermore: 00

n=O

n=O i=l

n=O

The converse follows from the equality

for any I

and ssets

3. Transitional transformations

In this section we consider generalised branching transformations of our : .4c)+ ,CE be a bounded linear transformation having algebra. Let components (@T):) : A$?) + A&m',such a transformation is called an n-particle branching transformation if (QT):) # 0 for at least one m # n. The conjugate transformation dn) : ,CT + d(n) are defined, for en E A?)

@g)

and X = [ X ( m )m=O O3 ] E ,CT by

can be used to construct an F(") : Bc 3 given by:

valued (analytic) generating function

m=O

: We define an n-particle branching evolution as any positive C:=, @g)* ( I @ m )5 Ign. The special case where n = 0 correspond to the multi-particle states. Of

A(") -+ L1 satisfying the contraction condition

75

particular interest will be the branching evolutions having components with completely positive conjugates. The following proposition generalises Proposition 2.1 from transitional transformations (which are simply the elements of C,) to general transformations of the described type.

Proposition 3.1. If F(n) : Be : dg) + C, szlch that:

+ d(n)is

analptic then there exists a

03

m=O

Proof. The proof is similar to the proof of Proposition 2.1. The analyticity gives a power series expansion M

nl=l

n.=l

where f (nl,. . . ,n,) E A(") are defined as the mixed derivatives

as before. We now define an A(") valued bounded linear operator @k) on for each m E N,by the continuous linear extension of the algebra d(m),

(Ef+"'@.. .@ E y " ) = f

+ +

. . . ,n,)

(n1,

where n1 . .. n, = m. Clearly F(n) is the generating function corresponding t o the sequence @$"I = ( @ T )( ,n ) ]O0 and ( e n ) E Cc for any

[

m=O

@g)

E dg).

0

@k)

Clearly if each is completely positive it follows that F(n) must be positive definite. The following proposition proves the converse.

Proposition 3.2. If F("): + A(") is positive definite and analytic : d?) + C, throughout t3c then it is the generating function of some having components with completely positive conjugates : d(m) + d(.).

@g)

@k)

Proof. The previous proposition proves the existence of some M(") + Cc such that: M

m=O

:

76

We again use the Cauchy integral formula, this time applying it to the derivatives g F ( n )( t X ) l to conclude that: t=o

Then for all N E N and for all qi, qj E Canand X i m ) ,X j m ) E M(") where 1 5 i,j 5 N we have

Egl

EM XiY""

for some X i E @, Ci = X i q j and = exp { -ie,} Y,. This proves and the complete the positive definiteness of the conjugate elements 0 positivity follows from the linearity of the operators. where Xj"'

=

2=1

. z ~ + ,= ~ exp {nie,} A&.<, and finally &+,L

@k)

We conclude this section by providing a dilation for the generating functions of the transformations d n ) . Proposition 3.3. An analytic function nite if and only i f M

F ( n )( X ) =

cc

F("): Bc

+

is positive defi-

k(") k(m) kjm)kj.")

@

m=O j=1 i=l

for

A:) (i,j , r)*X B m A k ) (i, j , r )

(4)

r(i,j)=l satisfying;

m=Oj=l i=l

,-(i,j)=l

where k:"), ky' E N for j = 1,... ,k(") and i = 1,.. . ,k(") are the dimensionalities of the block matrices in the block diagonal representations of d ( m )and respectively.

77

Proof. Obviously if F ( n )(X) takes the form (4) then it is positive definite, it remains to show the converse. For each rn E N we use the complete positivity of to construct a positive linear form d, on the Hilbert space 31(")* @ 31(") of d (m) x d (n)complex valued matrices such that

(5)

where

for some vectors

(9),xj ( h ) E dj for j = 1, . . . ,k(") and vectors (n)

[j

ai (9),pi( h ) E dl"' for i = 1,.. .,k(") such that g = 1,.. .,M , h = 1,.. . ,N . By linearity the must take the form operator @g)

where (for each i = l,...,k(") and j = 1,.. . ,k(n))cp(i,j) is a matrix whose entries are positive x valued matrices. Then (5) takes the form

Icy) Icy)

&"'

and each summand defines a positive sesquilinear form on Ckim)*8 which is implemented as the trace with respect to a positive matrix over &m)kjn) having square root R (i,j),i.e. (Sj

(9)7 Tr { ai (9)*Pi ( h )cp (4A } xj ( h ) )

78

where:

We define, for T (i,j ) = 1 , . . .,kim)k:n) the operators A k ) (i,j , T ) to be the

kim) x k p ) complex valued matrices;

where R (i, j ) : denotes the entry of R (i, j ) at the p t h row and qth column. It follows that

and hence that ( 5 ) takes the form

giving

for any ai,Pi E C k i m ) That . F ( n )( X ) takes the form 4 follows immediatela Proposition 2.3 is a corollary of Proposition 3.3 which gives a non linear extension of Stinespring's construction,2 in the finite dimensional case.

79

4. Markovian multi-particle processes

In this section we define a special class of generalised branching evolutions which we call Markovian multi-particle processes (in a broad sense as in Ref. 3). Of articular importance is the class of branching evolutions such that @k7 is completely positive. The previous section shows a one to one correspondence between such branching evolutions and the A(") valued positive definite analytic functions of a variables. We exploit this correspondence in the following definition. Definition 4.1. Let IIT ( t ) : t E J.R+ be a semi-group of branching evolutions IIT ( t ) = [IIcn) (t)];==,in the algebra L1 described by the oneparameter family of Ly valued analytic functions defined on the unit ball B1 as:

P (t,X ) =

[m( t , X ) ]

O0

n=O

We say that II (t)is a stationary continuous time Markovian multi-particle process if the following three conditions hold: (1) The transformation P ( t ) is positive definite (2) For all t > 0, P ( t ) is a contraction P ( t , l )5 I@ = (3) P ( t ,X ) + X @ = [x@~];=~ weakly as t + o for all

[I@n]z=O x

E B1.

We interpret such branching transformations as the partial description of a mechanism by which an n-particle component of a multi-particle state evolves in a system where interactions occur which do not necessarily preserve the number of particles. In particular the mechanism describes the averaged evolution of a system undergoing random branching interactions, which is the only possible description of the evolution of a multi-particle system if the individual branching interactions are not observed.

[@g)]

We say that a transformation 9~= O0 where n=O is the generator of a Markovian multi-particle system if:

:

+ L1

In particular we make no assumptions on the boundedness of the sequence 9 and will later see that some natural branching evolutions can only be described with an unbounded generator. Such a generator 9 is described by the unbounded sequence F = [F(n)],",oof A(") valued functions F(") (defined on and analytic within

80

the unit ball 23') which satisfy: d

F(") ( X ) = -P(") ( t , X ) dt Proposition 4.1. If P(")( t ) : B' + A(") is a positive definite function analytic in 23' describing a Markovian multi-particle system and F(") ( X ) is its generator

then F(") ( X ) is conditionally positive definite, satisfying for all M E 1 5 k,Z 5 M , X I E %(") and Xl E 23'

N,

M

k,l=l and F

Furthermore F

Proof. If

then

from which it follows that limt+o (P(") ( t , X ; X l ) - X;@'"X?") is conditionally positive definite. From Proposition 3.3 it follows that P(")( X ) * = P(")( X * ) and hence (P(") ( t , X ) - X@")* = P(") (t,X * ) - X * @ . Since P(")( t , I ) 5 I@" it follows that F(") ( I ) 5 0. 0

If the enerator F(") ( X ) is analytic also, then it determines transforA("),we now show that these must be conditionally mations ak) : positive definite. Proposition 4.2. If F(") ( X ) : B1 + A(") is analytic in B1 and condi: B1 -+ L1 tionally positive definite then there exists a transformation @$"I such that dn) is conditionally positive definite and:

c a): 00

F(") ( X ) =

m=O

(Xmm)

81

If furthemnore F(") ( X ) * = F(n)( X ' ) t h e n dn) (X)*= dn) (X') for any

x E LY. Proof. The existence of d")is guaranteed by Proposition 3.1. It follows immediately from F(") ( X ) * = F ( n )(X*) and the density in Cy of

(cx,xp : N E N,xi E Bl, xi E c i=l

that

for some 1

and

Such that

then

the positivity following from the conditional positive definiteness of Fcnh then a(") is conditionally completely positive throughout CT. Proposition 4.3. F(") (X) : B1 + A(") is analytic in B1 and conditionally ) = F(") (x*) i f and only zf positive definite and ~ ( n(x)*

for

satisfying:

m=Oj=l i=l r ( i , j ) = l

where k!"), k y ) E N for j = 1 , . . . ,k(n) and i = 1 , . . . ,k(") are the dimensionalities of the block matrices in the block diagonal representations of and A(") respectively and B(") E A(n). Proof. It is well known that linear w*-continuous maps A = A* between Von Neumann algebras A and B are conditionally positive definite with respect t o a w*-representation A : A + B if and only if;

A ( A ) = r ( A ) - A ( A )B - B*A( A )

82 for some completely positive I' : A -+ f3 and B E f3. We apply this result to the map conditionally positive definite map dn)satisfying dn) (X)*= dn) (X*) for any X E CT resulting from the previous proposition. The result then follows from Proposition 3.3. 0 The above proposition is a non-linear extension of the Lewis-Evans construction4 to non-linear conditionally completely positive analytic transformations with values in the finite dimensional algebra The Proposition 4.1 gives the necessary conditions for the generator of a Markovian multi-particle system. We now use Propositions 4.2 and 4.3 to show that these conditions are also sufficient. Theorem 4.1. Let F ( n ): B1 + A(")for n E N be a family of conditionally positive definite maps analytic in L?' such that F(n)( I ) 5 0 and F ( n )( X ) * = F(n)( X ) . Then there exists a Markovian multi-particle process described by a family P(n)( t ): B1 -+ A(") such that:

1

d F@)( X ) = -P(") (t,X ) dt t=O Proof. We seek to represent P(n)(t,X ) as the solution to the integral equation

where we use Proposition 4.3 to express:

Ak)

where (i:) 7-6; + '?are I( defined ") by an A g ) ( i , j ,T ) : Ckln)-+ Ckjm) and Cd(n)d m, AL (i)* A?) ( i )5 B(")- B(n)*. z=1 We are required to show that such a solution exists, satisfies P(n)(t,I ) 5 I for all t > 0, is positive definite, and furthermore is the generating function for a semi-group II(n): A(n)+ L1. Using the Du Hammel principle we look

83 for a solution t o the recurrent system

with starting point Po'") (t,X ) = exp { -tB(")*}XBL"exp { -tB(")}. Using the recurrence relation we can define (t,X ) as the summation

{

( t ,X ) = exp -tB(")*}~

8 exp "

{- t ~ ( n ) }

+k /...I g...c 1

d(n)d(ml)

w

mi=O

N=l

exp

{

{

-SIB(")*)

A?;*

exp s1Bcml)}A::

mn=O

(il) exp

(il) exp

{

d(mnr-l)d(mnr)

...

ii=l

C

iN=l

sI~(ml)*)

{-SIB(")}

dsl

. . .dsN

and so the family of approximations P,'") (t,X ) consists of completely positive members which are increasing (in a C.P. sense) with increasing i. We will now show that Pi"' ( t , X ) 5 I@" if t > 0 and X 5 I . First we note that

84

from which follows exp { -tB(n)*}exp { -tB(")} 5 Ian, Let us assume that X 5 I and Pjn)( t , X ) 5 IBn for all 1 5 i 5 j . Then:

p!") ( t , ~5 )e-tB(n)*e -tB(") z+ 1

5 Ian It follows that the increasing sequence must have a limit 0 5 Pg)( t ,X ) 5 Ian given by the infinite series

{

{

P(")(t,X)= exp - t ~ ( n ) * ~} 8 'exp " -t~(n)}

n

which, furthermore must satisfy the integral equation since

(7)

85 and so it follows;

5 P(n)( t ,X ) whence equality. By differentiating (6) it is clear that the constructed solution does indeed satisfy the equation for the generator:

dt

t=O

It remains only to show that the functions II(n)( t )defined by the derivatives

are a ssemi-group. We note that

+g J.. .J g . . . c~ ... c 00

N= 1 O<sl<...<SN
{

exp S I B ( ~ 'Ag! ) } ( i l )exp from which the semi-group property is clear.

mn--l=O

d(n)d(ml)

d(mN-l)d(m)

i1=l

SIB(^)}

dsl

iN=l

. . . dslv 0

The Markovian semi-group II ( t ) = [IIIcn)(t)] defined by the family P(") (t,X ) given by equation (7) gives a weak solution e ( t ) = e o II ( t ) to

86

the Cauchy problem d

-e(t) = @ O @ dt

is the initial state of the system and @ = .Ly + d(n)are transformations determined by:

where

Q

[dn)]r=O where dn) :

m=O

The solution is defined by the analytic generating functionals

n=O

5. Interaction free branching processes In this section we classify the Markovian branching evolutions corresponding to interaction free systems. We will begin by giving the simplest example of a multi-particle evolution, corresponding to a pure birth process of particles from the vacuum. Example 5.1. Let G ( X ) : B1 + C be analytic in the unit ball B1 and positive definite. Let c E iR+ be such that G ( I ) 5 c. We define a semigroup of analytic functions V ( t , X ) : B1 + C as the exponents

v ( t ,X ) = exp {t (G (W - 4) these have the interpretation as the generating functionals of some states u (t)E L1. Furthermore these states must satisfy M

+

from which follows un (t T ) = family of analytic functions;

C:=, um ( t )@un-m ( T ) .

We define a

P(n)(t,X ) = v (t,X ) X B n these have a family of generators F(n)( X ) = (G ( X ) - c ) X@'" which are necessarily conditionally positive definite, and satisfy F(") ( X ) * =

87

F ( n )(X*) , F ( n )(I)5 0. Hence P(n)( t ,X) describes a Markovian multiparticle process ll ( t ) satisfying: m2n l-Ik)*( t )= {ovmPn( t )ifotherwise

This process defines an evolution of the state e ( t )E L1 of a multi-particle system having initial state e E L1 and generating functional R ( t ,X) via the equation

c(en,dn) 03

R (4 X)=

(t,X)} = R (0, X ) V (t,X)

n=O

from which it follows that en (t) = C:=, em @ vn-m (t). Such a system has a clear interpretation as describing a multi-particle system whose dynamics consist of birth of particles from the vacuum. Conversely let us consider a system consisting only of the birth of particles from the vacuum, i.e. a system whose state at time t E IB+ is described by en ( t ) = em @v2),-m ( t )for some initial state e E L1 and where v ( t ) E L1 are, for each t E lit+, the state of birthed particles at time t. If we require our system to be Markovian, then it must be that ll ( t )defined by

c:=,

m=O

be a semi-group. From this it follows that n

n-m

m=O k=O

+

and hence v n (t T ) = vm ( t )@ vn-m ( T ) . Then the family of generating functionals of the state of the birthed particles at times t E IR+ is a positive definite contractive semi-group, hence it has a conditionally positive definite generator H (X) such that H ( I ) 5 0. Hence any Markovian system of birth from the vacuum is of the type described by Example 5.1. The second fundamental type of process without interaction is that involving processes on single particles in the system. This type of interaction will be described in the following example: Example 5.2. Let us define G (X) as a d(') valued positive definite analytic function. Furthermore let us assume that there exist some B E M ( I ) such that G ( I ) 5 B B*. Let the family QT (t,X) E d(') be analytic

+

88

throughout 23' and a solution to the non-linear inverse Heisenberg Kolmogorov equation defined by

d dr

-QT

( t , X ) = Q T (t,X ) B -

. m=O

+ B*Qr (t,X) '

A , (i)* QT (t,X)@'" A , (i)

i=l

for 0 5 r < t with boundary condition Qt ( t , X ) = X and where the matrices A , (i) are determined by the dilation of G in Proposition 3.3. We show that such a solution exists and is analytic for X E B1 using an iterative method on the integral equation

T (t,X ) = exp { -tB*} X exp { -tB*}

+

I,"

exp {-sB*} G (T (t - s,X)) exp (-sB}cls

similarly t o the proof of Theorem 4.1 where T (t - 7, X)= QT (t,X ) . The family T (t,X) necessarily has generator G ( X ) - X B - B*X and satisfies T (t,X ) 5 I for all X E B1. Now consider the family of analytic maps P(n)( t , X ) : 23l + M(") defined by: P(n)( t , X ) = T ( t , X ) @

These have unbounded generator F(")( X ) given by n

X@j-' @ (G (X)- X B - B ' X ) @ X@(n-i)

F(")(X)= j=1

which are obviously conditionally positive definite and satisfy dn) (I)5 0 , F(n)( X ) * = F(n)(X*).Hence P(n) t X ) describe a Markovian multi( t )= 6p and, for n > 1 particle process 'II( t ) which satisfies

(t)]Z0

where [ O k are defined by T (t,X) using Proposition 3.1. This process defines an evolution of the state e ( t )E ,C1 of a multi-particle system having initial state e E L1 and generating functional R (t,X ) via the equation

89

from which it follows that eo ( t )= ~0 and for n

c

> 1:

m

en ( t )=

m=l k l +

...+k,=n

( t )@ * * * @ @k,

(@kl

( t ) )e m

Such a system has a clear interpretation as describing a multi-particle system whose evolution consists of independent branching. We now show that evolutions of the type typified by the above example are the only Markovian multi-particle processes n(t): t 0 which correspond t o single particle independent branching processes. Due to the condition that the multi-particle states have n particle components in the symmetric algebra A(") it must be that any such process is described at time t 2 0 by a multi-particle state e ( t ) of the type eo ( t )= eo

>

m

m=l kl+

where @ k ( t ):

...+k , = n It follows that

-$

c

Ilk)( t )=

@k1

Ilk)= 6r and that;

( t )@ * * ' @ @ k , ( t )

kl+...+k,=m

and P("-)(t,X) = T ( t ,X)@'". By taking n = 1 we see that T ( t ,X ) is positive definite, and that T ( t ,I ) 5 I for t > 0 from which follows T ( t ,X) E B1 for all X E B 1 . The requirement that ( t )be a semi-group means that for all n E Z+ we have;

IIk)

m

n

.

=m k=O k l + . .+k, =k j=1

MI

from which it follows (by taking n = 1) that

and hence that:

T (t

+ T , X ) = T ( t ,T

(T, X

))

90 By considering the case n = 1 we see that ;di i T ( t , X ) l = F ( l ) ( X ) t=o

which is necessarily conditionally positive definite and satisfies F ( l )(X)*= F ( l ) (X*) hence there exist A , (i) and B such that:

c

00

d(m)d(l)

m=O

i=l

F(l)( X ) =

A , (i)* X B m A , (i) - X B - B * X

Hence we see that; d 1 -T (t,X) = lim - (T (st,T (t,X)) - T (0,T (t,X))) dt 6t-0 6t =

C C

A , (i)* T ( t , X ) B ' m A m ( i )

m=O i=l -T (t,X ) B

- B*T(t,X

)

showing that the above example does indeed characterise all Markovian multi-particle systems of independent branching.

Example 5.3. Let G ( X ) : B1 + C and N ( X ) : B1 + d(l)be analytic in the unit ball 23' and positive definite. Let c E I&.+ be such that G ( I ) 5 c , let B E be such that H ( I ) 5 B B*. Let Qr (t,X) : B1 + be analytic throughout B1 and a solution to the non-linear inverse Heisenberg Kolmogorov equation defined by d -Qr ( t , X ) = Qr ( t J ) B + BQr ( t , X ) dr

+

m=O

i=l

for 0 5 r < t with boundary condition Qt ( t , X ) = X and where the matrices A , (i) are determined by the dilation of H in Proposition 3.3. As before we define T (t,X ) to be the analytic map satisfying T (t - r ,X ) = Qr (t,X ) , which necessarily has generator H ( X ) XB B * X and satisfies T ( t ,X) 5 I for all X E 23'. Define V (t,X ) : B1 + U2 as the exponent:

+

+

)

v ( t , X )= e x p ( l G ( T ( s , X ) ) d s - c t Now consider the family of analytic maps P('l)(t,X ) : B1 -+ d('l)defined by:

P(")(t,X ) = T (t,X)@'lV (t,X )

91

These have unbounded generator F(n) ( X ) given by n

F(n)(X) = G (X) - cX

+ C X @ j - l @ ( H (X)- X B - B * X )@ X@("-j) j=1

which are obviously conditionally positive definite and satisfy F ( n )( I ) 5 0 , F(") (X)*= F(n)(X*).Hence P(n)( t , X ) describe a Markovian multiparticle process II (t)which satisfies m.

k=O k l + . . . + k , = k

(t)]EOare

defined by T ( t , X ) using Proposition 3.1 and [Wk This process defines an evolution of the.state @ ( t E) L1 of a multi-particle system having initial state e E Ll and generating functional R ( t ,X ) via the equation

where

[@k

@)IF0.

from which it follows that;

which is well defined as the form generated by the trace of products of the matrix c a n

(in the sense that the functionals coincide on A(")). Such a system has a clear interpretation as describing a multi-particle system whose evolution consists of independent branching and birth from the vacuum. We now consider all the Markovian multi-particle processes II ( t ): t 2 0 which are non-interacting. Due to the condition that the multi-particle states have n particle components in the symmetric algebra A(") it must be that any such process is described at time t 2. 0 by a multi-particle state ~ ( tof)the type

m=O k=O k l + . . .+k,=k

92

[&]Eo

where O k : M(") -+ M ( l ) ,e = E .C1 is some initial particle space and 'uk ( t )describes the state of the birthed particles. It follows that; m k=O k i + . . . + k , = k

and therefore P(") (t,X ) = T (t,X)@n' V (t,X ) . By taking n = 0 we see that V ( t ,X ) is positive definite and satisfies P (t,I) I for t > 0. It then follows that T ( t , I ) I (since otherwise P(") @,I) > I for n 2 N E N). The requirement that IIk) ( t )be a semi-group means that for n = 0 we have;

<

<

m

00

and hence

which gives us that:

+ +

V(t r,X) = V(t,T(r,X))V(r,X) T(t r,X)=T(t,T(r,X)) By considering the case n = 0 we see that

d -v dt

(t,X )

I

= F(O)( X )

t=O

and hence that: d -v dt

(t,X ) = F(O)(T (t,X ) ) v (t,X )

If we further assume that the generator

is conditionally completely positive and satisfies H ( X ) * = H ( X * ) then the multi-particle system is of the type described by the previous example.

93

6. Branching systems with interaction strength In this section we generalise the examples from the previous section to look at a specific class of interacting branching processes for which we can determine asymptotic solutions in the limit as the interaction strength tends to zero and correspondingly the number of particles tends to infinity. In the first example the Markovian multi-particle process was found to have a generator described by;

F(") ( X ) = (G ( X ) - C ) X@'" for a positive definite G (X) satisfying G (I)5 c. In the second example we saw that the generator was of the form n

F(") ( X ) =

C

X@j-l@

( H ( X ) - X B - B * X )Q9 X@('+j)

j=1

+

for some positive definite H ( X ) satisfying H ( I ) 5 B B*. In the third example the generator was the sum of the previous two, namely:

F(")(X) = G ( X ) - C X +

n

C X @ j - l @ ( H ( X ) - X B - B * X )Q9 X@("-j) j=1

In general we can always write the generator of a Markovian multiparticle process in the form

for some conditionally completely positive G("\") ( X ) , which we interpret as describing the rate of interactions involving k particles independently of the rest of the system, given that there are n. particles in total. The third example is a case in point, it describes the birth from the vacuum and single particle branching which is independent of the number of particles, and the state of the other particles. We will generalise the previous section in the following definition. Definition 6.1. The Markovian multi-particle process IIE( t ) is said to be a branching process with interaction strength E > 0 if its generator is described by a family of conditionally positive definite maps FJn) : B1 + which are analytic in the unit ball B1 and satisfy FJn) ( I ) 5 0 , Fin)( X ) * = F:") (X')of the form

94

where G(') ( X ) are analytic conditionally positive definite generating functions which can always be written in the form d(m)d(t)

w

($k)

A($)(i)* X@mA($) (i) - X@kK(k)- K(k)*X@k

( X )= m=O

i=l

for some A:) (i) :?dk) + ?dm)and w

E A('")satisfying:

am+'

m=O i=I

Our first example is the simplest branching process with interaction strength which is interacting. It consists of the independent pairwise branching interactions with a strength E > 0.

Example 6.1. Consider the generator described by the family F(n)( X ) = 0 for 0 5 n 5 1 and for n > 1

where

G ( X )=

w

4m)W

m=O

i=l

C

c

for some Am (i) : 3t(2)

A , (i)* XmmAm(i) - X @ 2 K- K * X B 2

X ( m ) and B

cc 03

E A(2)satisfying:

am+'

A , (i)* A , (i) IK

+ K*

m=O i=l

jFrom Theorem 4.1 such a family determines a Markovian multi-particle process. The equation of motion for the associated state generating functional RE( t ,X ) must satisfy, at least for positive X

where G ( X ) =

xiXiGi ( X ) @ ' . We write this in the form:

95

Let us assume that the initial state of the system is described by a state generating functional of the form

{: 1 {: 1

R, ( 0 , X ) = A ( X )exp - B ( X ) and look for solutions of the form

RE(t,X ) = A ( t ,X ) exp - B ( t ,X )

where B ( X ) is a conditionally positive definite dissipative analytic functional in B1 and A ( X ) is a positive definite contractive analytic functional in Bl. Under these assumptions the equation of motion for the state generating functional can be defined as the solution to;

equating powers of

E

we have:

Hence up to a term of order

E

we find that the state generating function is

96

determined by the solution to the equations

d -B d t ( t 7 X )= 2 ( G ( X ), (A8 (t,x))@ 2 ) with the initial conditions A ( 0 , X ) = A ( X ) , fi ( t , X ) = B ( X ) . These equations are recognised as the transport equation and Hamilton- Jacobi equation for a Hamiltonian system with Hamiltonian 1 2

H ( X ,P ) = -- (G ( X ),P@') where:

a

P ( t , X ) = -B dX

(t,X)

We now assume the existence of solutions Q t , Pt to the Hamiltonian system

satisfying the boundary conditions QO = Y and Po = &B ( Y ) for some Y E B1. Equivalently we assume Q,, P, are the solutions to the equations

97

due to the fact that, for example, for every

eE

We make the further assumption that not only is the Hamiltonian system solvable, but for each fixed t E I&+. and X E 23l there is a unique Y E 23l satisfying X = Qt. In finite dimensions this corresponds to the Jacobian

not vanishing. Due to this assumption we can write the Hamilton Jacobi equation in the form;

ddtB

(t,X )= --B (t,Q t ) dt .. /Qo=Y(t.X) =

((

&t,

a

a h (t,Q t ) Q.=Y(t,X)

the second equality being the explicit form of the total derivative &B (t,Qt). Then:

Likewise for

a (t,X) we have

d d - A (CX) = -A (t,Q t ) / dt dt Q,=Y(t,X)

98

Hence

and we have a solution:

Then subject to the stated assumptions being satisfied, there exists an approximate solution to the family of state generating functional R, (t,X) which describe the state of a multi-particle system undergoing independent pairwise interaction which can be written in the form:

ii, ( t , ~=)A ( t , x ) e x p We now investigate how closely RE(t,X ) approximates the exact solution R, (t,X ) . An exact solution of the form

R, (t,X ) = A (t,X ) exp must satisfy the equation

:

whereas our

=LA(t,X)

f& ( t ,X ) satisfies:

99

We use the Du Hammel principle to write

which can be written in the operator form

-@) 2

A ( X )2 ( t , X ) = ( A ( X ) - E where

@

A (t,X)

is the operation defined by:

Then, provided the right hand side converges, we can write;

(

A ( t , X ) = 1---@) iA;X)

-1

A(t,X)

the right hand side is given by

from which follows the upper bound

showing that the asymptotic solution does indeed converge to the true solution as E + 0. The above example illustrates the method used in the proof of the following theorem

Definition 6.2. We call the following Hamiltonian system a canonical pair

100

of quantum kinetic equations, Qt = X E B1,PO= Y E M ( l )

a ( K ( ~ ) * , ( P ~ Q T )p+ @~)) pr+ca(dPQT T

O0

m=O

Theorem 6.1. If the family

Fin): B1 + M(")describes the generator of

a Markovian multi-particle system with interaction strength E > 0 , and the initial state of the system is described by a generating function of the f o r m R C

c

1

( 0 , X ) = A ( X )e ~ p-B ( X )

where B ( X ) is a conditionally positive definite dissipative analytic functional in B1 and A ( X ) i s a positive definite contractive analytic functional in B', then there exists a solution RE( t , X ) of the state generating function at any t > 0 for which the canonical pair of quantum kinetic equations have solutions QT ( t , X ) : Qt = X , PT ( t , X ) : PO= &B (Y)ly=q,(t,x) f o r any X E B1.These solutions are given formally by the expressions R € ( t , X ) = e x p { f B ( t , X ) } (l--@)1 A (XI where

-1

A(t,X)

101

and 0 is the operation defined by:

Proof. The proof is a generalisation of the method illustrated by the previous example. We look for a solution to the equation of motion

-RE d ( t , X )= 1 dt

&

25(

( t , X ) ,G(") ( X ) )

&RE

m=O

satisfying the required initial condition, by looking for solutions of the form:

{: 1

R E ( t , X )= A ( t , X ) e x p - B ( t , X ) We find that under such an assumption

and throw away the higher coefficients of

and:

E,

looking for solutions to

102

We again associate the above equations with the Hamilton-Jacobi and transport equation for a Hamiltonian system with Hamiltonian

H ( X ,P ) = -

c m! * 1

( G ( m )( X ) ,Porn) m=O with Pt = &B (t,X ) satisfying the boundary conditions A (0, X ) = A ( X ) and B ( 0 , X ) = B ( X ) . The resulting Hamiltonian system can then be written in the form

from which it follows that

gdt B (t,X ) = dt

(t,Qt (t,X ) )

= (Qt(t,W,

a

( t , Q t ( t , X ) ) )- H (Qt ( t , X ),Pt ( t , X ) )

the second equality being the explicit form of the total derivative $ B (t,Qt). Then:

B (t,X ) = B (Qo (t,X I )

Likewise for

( t , X ) we have

103

where the right hand side takes the form:

In this case we know that Qt(t,X ) = and so we have

hence

and therefore:

To complete the proof we note that

c:=, 5 &(G(m)(Qt(t,X ) ) ,P R m )

104

whereas an exact solution for RE(t,X ) of the assumed type must satisfy

the result then follows from the Du Hammel principle as described in the previous example. 0

References 1. V. P. Belavkin, in: Mathematical Models of Statistical Physics [in Russian], Izdat. TGU, Tyumen', 1982: Dokl. Akad. Nauk SSSR 293, 18 (1987) 2. W. F. Stinespring, Proc. Am. Math. SOC. 6, 211 (1995) 3. G. Lindblad, Comm. Math. Phys. 48, 119 (1976) 4. J. T. Lewis and D. E. Evans, Dilations of Irreversible Evolutions in Algebraic Quantum Theories, Comm. Dublin Inst. for Adv. Studies, Ser. A, Vol. 24, 1997

A NOTE O N VACUUM-ADAPTED SEMIMARTINGALES AND MONOTONE INDEPENDENCE

ALEXANDER C. R. BELTON Lady Margaret Hall, Oxford OX2 6QA, United Kingdom E-mail: beltonamaths. ox.ac.uk A class of vacuum-adapted regular quantum semimartingales, with integrands which act as the conditional expectation on Fock space, are proved to possess increments which are monotone independent. The vacuum-adapted analogue of the Poisson process is shown to have increments which are distributed according to the monotonic law of small numbers.

1. Introduction

The vacuum-adapted analogue of Brownian motion (i.e., the solution to the quantum stochastic differential equation Z(0) = E(O), d Z = EdA + E d A f ) has increments which are distributed (in the vacuum state) according to the arcsine law [2, Section 4.41 (the result therein for Z ( t ) generalizes simply t o Z ( t ) - Z(s) for all t 3 s 2 0). The monotone Brownian motion of Muraki also has increments which are distributed according to the arcsine law [5, Section 4.21 and this observation suggests a connexion between monotone and vacuum-adapted processes. In Section 2 it is proved that if M is a regular vacuum-adapted quantum semimartingale, with integrands which act as the conditional expectation on Fock space, then M satisfies the identities

( M ( t )- M(s))’E(t) = ( M ( t )- M(s))’ = E ( t ) ( M ( t )- M(s))’ (an immediate consequence of vacuum-adaptedness) and

IE(s)(M(t)- M ( S ) ) ’ I E ( S ) = (a,( M ( t )- M ( s ) ) ’ W ( s )

<

for all p E N and all s, t E R+ such that s t. It follows that M is a monotone-independent processes in the sense of Muraki [7]. The relationship observed above between monotone and vacuumadapted Brownian motion gives rise to the thought that the 0-Poisson

105

106

process, i.e., the process M such that

M ( 0 ) = IE(0) and

dM =l E( dA

+ dA + dAt + dt),

should have increments distributed according to the monotone Poisson law, i.e., Muraki’s monotone law of small numbers [7, Section 41. This is proved in Section 3 and an explicit formula for the moments of that distribution is given. The presentation in this note follows the elegant approach to multidimensional quantum stochastic calculus due to Lindsay [3] which has evolved in articles by Lindsay and Wills. (Meyer [4, Section V.31 claims that this coordinate-free notation may be traced back to roots in work by Belavkin.) The treatment of [3] is simplified here, as the processes integrated in the sequel are bounded and act on Fock space as the conditional expectation; these integrals may be thought of as a form of vacuum-adapted Wiener integrals. It is straightforward to verify that the approach to multidimensional quantum stochastic integration given in [2, Section 5.41, based on the idea of bounded admissible collections of processes, is equivalent to the “coordinate-free” approach adopted in this note. 1.1. Conventions and Notation

All Hilbert spaces are complex, with inner products that are linear in the second argument and conjugate linear in the first. The Banach algebra of bounded linear operators on a Hilbert space H is denoted by B ( H ) . If E is an orthogonal projection then E’- := I - E is its orthogonal complement; I (or Ix) denotes the identity operator (on X).The restriction of a function f to a set X is denoted by f l x . The indicator function of a set A is denoted by X A , with the domain of X A being clear from the context. The set of natural numbers is denoted by N := { 1 , 2 , 3 , . . .}. Throughout, R is to be pronounced “vacuum”. 2. Preliminaries

Let k be a separable Hilbert space (the multzpplicity space) and let the elements of may be written as column vectors:

(3

E i(

and 2 : =

(:)

(A E C ,

2

E

:= C@k;

k).

R+ := [O,co) is any interval then L 2 ( I ;k) E L 2 ( I )@ k is the If I Hilbert space of k-valued, weakly measurable functions on I which are

107

square integrable. For f : I + k let

f : I + k be defined by

Let 31:= I?+ ( L 2 ( I ;k)) denote Boson Fock space over the Hilbert space L 2 ( I ;k) and define abbreviations

F

:= FR+,

Fs):= F[o,~), Fs,t := F[s,t) and Fp := Fp,..)

<

for all s,t E R+ with s t. Let ~ ( fdenote ) the exponential vector associated with f : recall that the linearly independent set { ~ ( f :)f E L2(Iw;k)} is total in F. There exist isometric isomorphisms k,,t : F F,)@F,,t@Fp which act on exponential vectors so that kS,t&(f)

= a s ) )

@ d f s , t ! @E(f[t)

V f E L2@+;k),

where fs) := f l p s ) , fs,t := fl[s,t) and f p := f([t,m) are the restrictions of the function f to the given interval. Let IE denote the conditional expectation on F,i.e., IE: IFP + B(F)is such that E ( s ) ~ ( f= ) E ( ~ X [ ~ , , ) ) V S E R+, f E L2(R+;k). Note that IE(s)lE(t) = lE(min{s,t}) for all s , t E R+. Throughout this note, a R-process is a weakly measurable function E: W + B(k €3 F)which is R-adapted, i.e.,

E ( s ) = (Ii, €3 IE(s))E(s)(Ii @ IE(s))

V S E R+.

If the function E : + B ( k ) is weakly measurable then E : s e E(s) @ IE(s) is a 0-process, the R-ampliation of E. Let p k and A Pk €3 13 denote the orthogonal projections on k and k €3 F,respectively, with ranges k and k €3 F. Define =’

E,X := A E A , E,X := A E A l , EO, := A l E A and E,O := A l E A l .

(1)

A R-process E is admissible if IIEIIt := IIE,”IIm,t

where

+ IIE,XIIz,t + IIEO,Iln,t+ llE,”lli,t< 00

V t E R+,

(2)

108

(The measurability condition on E is sufficient to ensure that these norms are well defined.) Theorem 2.1. If E is an admissible R-process then there exists a collection J E dA = E d A : t E W ) of bounded linear operators on 3 such that

(s,"

f o r all t E R+ and all f , g E L2(R+;k). firthemnore,

I/

fi

dAll

< IlEllt

V t E @,

where 11 . Ilt is as defined in Eq. (2). Proof. Let I = (1,. ..,dim k} (or I = N if dim k = MI) and let f := (0) U I. Suppose that ( e , : a E I) is an orthonormal basis of k and define operators P o : 3 + k @ z8 H d € 3 8 ,

Pa:F+kGz 8-ea€38

(UEI)

and Pa = Pi for all a E f; a quick calculation shows that Pau@8 = (e,, u)8 for all a E I, u E k and 8 E F and also that CaEn Pap" = A in the sense of strong operator convergence. It is easy t o verify that the column decomposition & = C @ k induces the block decomposition

and that

where

109

and

E; := paEzPb, E," := PaE,XPo and Ef := poE:Pb

V a , b E I.

(As k L @ 3 is naturally identified with 3,the two possible interpretations of E: will cause no confusion.) Since l2(I;7 )E 12(11)@F E k @ 3 , it follows immediately that {FF} is a bounded admissible collection of processes in the sense of [2, Section 5.41. To see that these processes are 0-adapted, note that PaIE = ( I i @ lE)Pa, and

SO

IEP'

= Pa(Ii €3 E),for all a E

%.Furthermore,

A(Ii €3 E) = Pk €3 IE = (Ii €3 lE)A

so A L ( I i €3 lE) = (Ii @ lE)AL;hence EEtIE = waEzPblE = Pa(Ii@ lE)AEA(Ii €3 IE)Pb = p a A ( I i€3 lE)E(Ii €3 E)APb = PaAEAPb = E t for all a, b 6 I. Similar working shows that EEFE = E; for all a,,B E i and the result follows by [2, Proposition 371. The following is the quantum It6 product formula for R-processes. Theorem 2.2. Let E and F be admissible R-processes and let M = S E dA and N = F dA. Then G is an admissible process, where

s

G = (P: €3 M)F + E(P:

@N)

+ EAF,

and M N = S G d A . Proof. This is a rewording of a result given in [2, Section 5.41; the process denoted above by G corresponds to the bounded admissible collection {G;} specified by [2, Equations 68-71].

Corollary 2.1. If M = S E dA then M" = S E(")dA for all n E N,where

E(") =

C

(P:

€3 M)"E(P: €3 M ) B

a+B=n-l

+

C

(P, €3 M ) ~ ( E A ) ~ ~ + ~€3 E M)? (P*

a+B+y=n-2

and the summation is over non-negative integers.

110

Proof. This follows by induction.

0

3. Result

The following are fixed throughout this section: let E: I&+ weakly measurable and such that IIPkEPkllcqt

+ B ( i ) be

+ I I P k E P k l l l 2 , t -k I I P k l E P k l l 2 , t + IIPklEPklII1,t<

00

for all t E W, let E be the R-ampliation of E, let’ M = J EdA and (for brevity) let M,,t := M ( t ) - M ( s ) for all s,t E rW+ with s t.

<

Lemma 3.1. If 6: X I+ (R, XR) is the vector state on B(F)corresponding to the vacuum vector R := e(0) then

IE(O)XIE(O) = +(X)IE(O)

vx E B(F).

Proof. This follows by a simple calculation. Proposition 3.1. If s,t E W are such that s

0

< t then

Proof.

Ssince

Corollary 3.1. If p E N and s , t E IW+ are such that s

M:,JE(t) = M$ = lE(t)M$

< t then

and lE(s)M:,,IE(s) = +(M:,,)lE(s).

111

Proof. The first two identities are immediate consequences of Proposition 3.1 and the fact that

W t ) = Ic,: ( k €3 )

€3 E(0)

1

ks,t.

The final claim follows from Lemma 3.1, Proposition 3.1 and the fact that

E(s) = q;( I F s ) €3 W )I F*,t€3 E(0)I F[*1k

0

t .

The following definition is due to Muraki [7]. Definition 3.1. A family of quantum random variables ( X i ) i E r (where I is a totally ordered set) is monotonically independent with respect to the state # if

for all i

< j, Ic < j

and p E N and

for all i, > ... > il > i, j n > > jl > i a n d p , p l , . . . ,p m,q l,.. . ,qn E N. (The possibilities m = 0 or n = 0 are allowed and are interpreted in the natural manner.) Theorem 3.1. The process M has monotonically independent increments: i f n E N and 0 s1 6 tl . . . 6 s, t , then (Msi,ti)y=l is monotonically

<

<

<

independent with respect to the vacuum state #: X e ( 0 , X n ) . Proof. Let i,j,Ic E ( 1 , . . . , n } be such that i < j and k ti t j - 1 s j , Corollary 3.1 implies that

<

<

Mai,ti

< j and l e t p E N.

Since

Mai,tiE(ti)= Msi,tiJE(ti)E(sj)= MS,,tiIE(sj)

1

and similarly MSk,tk= E ( s j ) M s , , t k .Hence, again by Corollary 3.1, MSi,ti

M:,,tj

MSk,th

= MSi,ti~sj)M,q.,,jIE(sj)MS,,tk

= Msi

,ti

#(MG

,tj

)IE('j l M s k , t b

Thus the first part of Definition 3.1 is satisfied.

= 4(M:j , t j ) M S i , t i

l'Sk

,tk.

112 and

For the second, let

let

Note that,

by Lemma 3.1, Colllary 3.1 and the fact that

et cetera, and so, by Lemma 3.1,

as required.

In principle, the mixed moments of the increments of M can be found by using monotone independence, which reduces the calculation to finding various q5(M:,) (see [6, Section 11 or [7, Section 21 for a nice explanation of how to do this) and these simple moments may be found using Corollary 2.1. In practice, the algebraic complexity may be formidable. 4. Example

s

Let k = CC and E(s) = ( i i ) for all s E E??, so that M := EdA is the R-adapted analogue of the Poisson process. A straightforward calculation, applying Corollary 2.1 to X[s,ml E dA, shows that m p ( t ):= $(M:,s+t) for s , t E W satisfies mo 3 1 , mp(0)= 0 and

s

k=O Hence m l ( t ) = t for all t E R+ and (mp+l - mp)’ = denotes the Laplace transform of mp, P+ 1 hP+l (x) = x-2

Jj=2

and

+ jx-1)

(1

Cp + l ) m p so, if eP

113

Proposition 4.1. If 1

Clp(t,x)

:=

p!

(E) ”--l

then m,(t) = p p ( t ,0) for all p E

(t - log(1 - 2))” (1 - x ) 2

N and t E W .

Proof. Note first that p1 (t,0) = t = ml ( t ) and P P ( t 7 2)

where Gq,&

2) := (d/dz)‘J(t- log(1 - 2))”. Hence,

if

then

But

d

G,+l,& 0) =

(a,)

p(t - log(1-

.))”-I

1-2

= qGq,p(t,O)+ P G q , p - l ( t , O )

(6)

and therefore Rp(t,O)= 0 for all t E R+. Furthermore, a simple inductive argument using Eq. (6) establishes the fact that G,,,(O, 0) = 0 for all p E N and all q E (0,. . . , p - 1). Hence pp(O,0) = 0 and the result follows.

114

Corollary 4.1. The R-Poisson process M is such that, in the vacuum state, Ms,t (0 3 t ) has probability distribution Y, the limit distribution of the monotonic law of small numbers [7, Theorem 4.11 (with parameter x = t - 5).

< <

Acknowledgments This note was inspired by a talk given by Dr Uwe Franz and discussion with Dr Anis Ben Ghorbal. These took place during the Conference on Quantum Probability and Infinite-Dimensional Analysis which was held at the Krupp-Kolleg, Greifswald, Germany, on 23rd-27th June, 2003; the hospitality of the organisers is gratefully acknowledged.

References 1. A. C. R. BELTON,Quantum R-semimartingales and stochastic evolutions, J . Funct. Anal. 187,94-109 (2001). An isomorphism of quantum semimartingale algebras, 2. A. C. R. BELTON, Quart. J . Math. 55, 135-165 (2004). 3. J. M. LINDSAY,Quantum stochastic calculus lecture notes, Preprint-Reihe Mathematik Nr. 8/2003, Ernst-Moritz-Arndt-Universitat Greifswald, 2003. 4. P.-A. MEYER,Quantum probability for probabilists, second edition, Lecture Notes in Mathematics 1538, Springer, Berlin, 1995. 5. N. MURAKI,Noncommutative Brownian motion in monotone Fock space, Commun. Math. Phys. 183,557-570 (1997). 6. N. MURAKI,Monotonic convolution and monotonic Lbvy-HinEin formula, preprint, 2000. 7. N. MURAKI,Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. TOP. 4, 39-58 (2001).

QUANTUM STOCHASTIC PROCESSES AND APPLICATIONS

MOHAMED BEN CHROUDA and MOHAMED EL OUED and HABIB OUERDIANE Dipartement de Mathematiques Faculti des sciences de !Funis 1060 !Funis, iPllnisia We use special chains of test functions and distributions spaces to give an analytic characterization theorem for continuous linear operators on these test spaces in terms of their symbols. Then, we define the notion of convolution of operators and we give explicit solutions of some quantum stochastic differential equations.

1. Introduction and preliminaries

Let X be a real nuclear F'rechet space. Assume that its topology is defined by an increasing family of Hilbertian norms {I. ,I p E N}.Then X can be represented as

where for p E N the space X, is the completion of X with respect to the norm .1, Denote by X-, the dual space of X,, then the dual space X' of X is represented as

X' = u

x-,,

PEN

and it is equipped with the inductive limit topology. Let N (resp. N,) be the complexification of X (resp. X,),i.e. N = X + i X and N, = X, iX,, p E Z. For any n E N we denote by Nhn the n-th symmetric tensor product of N equipped with the 7r-topology and by N$" the n-th symmetric Hilbertian tensor product of N,. We will preserve the notation 1., and I.l-, for the norms on N P and N?; respectively. Let 0 be a Young function on R+, i.e. 0 is continuous, convex, increasing function

+

115

116

and satisfies lim -= +a, see Ref. 7. We define the conjugate function

e* of e by

+w

x

For a such Young function 0 we denote by Gs(N)the space of holomorphic functions on N with exponential growth of order 6 and of arbitrary type, and by .F,(N’)the space of holomorphic functions on N’ with exponential growth of order 0 and of minimal type. For every p E Z and m > 0, we denote by E x p ( N P , 8 , m )the space of entire functions f on the complex Hilbert space N p such that

Then the spaces .Fe(N’) and Gs(N) are represented as

3e(N’) = p o j lim Exp(N-,, 8, m ) PEN m>O

Ge(N) = ind lim Exp(N,, 0, m). PEN m>O

The space .Fe(N’) and its dual F;(N’) equipped with the strong topology are called the test functions space and the distributions space respectively. We denote by << ., . >> the dual pairing between .Fh(N’) and .Fe(N’). It is easy to see that for every 5 E N , the exponential function eg : z I--) e(’*g) , z E N’ where ( , ) denote the dual pairing between N’ and N , belongs to the test space .Fe(N’) for any Young function 0.Then we define the Laplace transform of a distribution q5 E Fh(N’) by

L ($ )(J ):=< 4,eF

>> , J E N .

In Ref. 3, the authors prove the important duality theorem: the Laplace transform realizes a topological isomorphism of .Fh(N’) on (N). In this paper we establish the analytic characterization theorem of continuous linear operators from Fe(N’) into .Fi(N’) in terms of their symbols, and we give a criterion for the convergence of sequence of operators. Finally, as an application, we solve some quantum stochastic differential equations and characterize their solutions. We refer to Ref. 11 in the particular case where B(x) = x k .

117

2. Characterization theorems

2.1. The Operator symbol

In this section we characterize the space of continuous linear operators from .F,e(N’)into .FL(N’) using spaces of holomorphic functions with exponential growth. Let Ce = C(.Fe(N’),.F,L(N’))be the space of .F;(N’)-valued continuous linear operators on .Fe(N’). For p E lN and m > 0 we put

(((EII(e,p,m = SUP{(+C EYJ,$J

1 ; ( I q ( ( e , p , mI 1 and ll$lle,p,m I 1) , E

E

Leo

Then Ce,p,m = { E E Ce , IIIEllle,p,m< +m} becomes a Banach space with norm Ill.llle,p,m and we obtain

In the general t h e ~ r y , * J ~ ~ if ’we ~ ~take ’ ~ two nuclear F’rechet spaces X and V then the canonical correspondence E t)E K given by

( E u , v ) = ( E K , u @ v ) ,u

E

X,VE V ,

yields a topological isomorphism between the spaces C ( X , V ’ ) and (XBD)’. In particular if we take X = 2) = .Fe(N‘) which is a nuclear F’rechet space, then we get

C(.Fe(N’),.F,L(N’)) (.Fe(N’)8 .Fe(N’))‘.

(2)

We define the symbol of an operator E E Ce denoted by ,!? as follows

E(t,r))= << EK,ec Be,, B-,t,r)E N .

(3)

So, the symbol ,!? can be regarded as the Laplace transform of the kernel EK

Moreover, with the help of duality theorem, and equalities (2),(4) we obtain the following theorem Theorem 2.1. T h e symbol application :

Ce E

+ 6s. ( N )@ G e e ( N ) w,!?

realizes a topological isomorphism, i.e. let F be a function o n N x N with values an C, t h e n there exists a n operator E E Ce with F = ,!? if and only if

118

*

(i) f o r any t,t1,r ] , r]l E N , the function ( 2 , w) F ( z t + 51, wr]+ 111) is an entire function on C x C; (ii) there exist constants c > 0,m > 0 and p E JV such that

IF((, r ] ) l 5 c e"(ml~lp)+e'(ml~lp) ,

t,r] E N .

Remark This theorem is a generalization of the characterization theorem of operators given in Refs. 10, 11, 12 in the particular case 6(z) = xk and given in Ref. 1 using C.K.S spaces. 2.2. Convergence of operators

Recall that ( N ) @ 0s. ( N ) equipped with the r-topology is a nuclear F'rechet space, one of the properties of nuclear spaces is that the closed bounded sets are compacts15. Now we describe the convergence in 6s. ( N ) @ Be* ( N ) .

Theorem 2.2. Let ( f n ) be a sequence in G p ( N ) €3 G,g*(N). Then the following assertions are equivalents: (1) (f n ) converges in G e e ( N ) @ Go* ( N ) . ( N ) and is pointwise convergent. (2) (f n ) is bounded in 60. ( N ) @ (3) ( f n ) is bounded in G,g*(N)8 G p ( N ) and it has a unique cluster point in Be* ( N ) @ G,p ( N ) .

Proof ( N ) i.e. there exist p 2 0 Assume that (f n ) converges to f in ( N )@ and m > 0 such that limn++oo 11 f n - f II,g*,p,m = 0. In particular (f n ) is bounded in Ge*(N)@ Ge*(N)and is pointwise convergent. This proves 1) 2). Assume that ( f n ) has two cluster points f and g i e . , there exists two subsequences (f n l ) and (f n 2 ) of (f n ) which converge in Be* ( N ) 8 0s. ( N ) to f and g respectively. In particular the subsequence (f n l ) ( r e s p .f n 2 ) are pointwise convergent t o f (resp.g). Hence the sequence (f n ) does not pointwise convergent which is not true by assumption. This proves 2) 3). Finally, P ut S = {fn, n 2 0 } , S is a bounded set in ( N ) @ Go* ( N ) and ( N ) becomes a compact by consequence the closure 3 of S in ( N )@ set which contains the sequence ( f n ) . It is known15 that if a sequence has a unique cluster point in a compact set, then it converges. This completes the proof. w

*

+

As a consequence of theorem 2.2 we characterize the convergence of operators by the convergence of its symbols.

119

Corollary 2.1. Let (E,),>o be a sequence in Lo. Then ( E n ) converges in Lo if and only if ( 0 1 ) there exist p 2 0 , m > 0 and c 2 0 such that for every n E N Ign(J,q)I 5 c ~ z P ( ~ * ( ~+ oI *[ (Im~~ )q ~ p >E,V ), E N ; (02) f o r every

[, q

E N , limn++, &([, q) exists in C.

This result was proved in Ref- 11 with 8(x) = xk , x 2 0 , k ~ ] l , 2 ] . Remark A similar analytic characterization theorem for operator symbols and convergence of operators can be established if we replace the space Lo by the space Lo,,,, of continuous linear operators from Fe(N’) into FG(N’),where 8 and cp are two Young functions on R+. 3. The Convolution product of operators

If the Young function 8 satisfies limZ.++,

< +co,we get3

Fe(N’) L) L ~ ( x ’ ,L) ~ )F ~ ( N ’ ) ,

(5)

where y is the standard gaussian measure on X’. Consequently, to include the space L(Fe(N’),Fe(N’)) into Le we assume from now on that 8 satis< +co. In this case the function ( J , q ) t--) e-(tsq), J , q E fies lim+, N belongs to Bo*(N)8 Be*(N) N Go+,s*(Nx N ) see Ref. 14. Since Be*(N)8 Ge*(N)equipped with the pointwise multiplication is an algebra, theorem 2.1 shows that for every E1,E2 E Lo there exists E E Lo uniquely determined by g ( [ , q )= e - ~ ( t + q l € + q ) g l ( J , q ) ~ ~ ( sJ ,,qq )E, N .

(6)

The operator E defined in (6) is denoted by El * E2 and is called the convolution product of El and E2. We denote by E*, = E * E * ... * E , n times. Theorem 3.1. Let y be a Young function o n R+ which does not necessarily = +co and let f ( z ) = CnEN fnzn E E z p ( C , y , m ) f o r satisfy lim+, fnE*n E some m > 0. Then V E E Lo, the operator f * ( E ) := CnEN L(roee* )* .

Proof It is easy t o see that f ( k )E BToee*( N ) @ Groee* ( N ) . Then we conclude by theorem 2.1 that f * ( E )E C(,,,e*)*.

120

Corollary 3.1. Let E E Lo, t h e n the convolution exponential of the operator E defined by e*E := C $E*n as a n element of E L(,P)..

Remark 1) If an operator E E Lo is of degree k E N, k 2 2, i.e. e($,q) is a polynomial in t and q of degree k, then e*E E L, with cp(x) = x”-’. This result was proved in Ref. 11 with Wick product. 2) In infinite dimensional complex analysis2, a convolution operator on 3o(N’) is an operator which commutes with translation operators. Let x E N’, we define the translation operator T-, on To(”) by 7-xcp(~ =) cp(x

+Y) , 9 E N’ , cp E -7e(N’).

It was proved in Ref. 4 that T is a convolution operator on Te(N’) if and only if there exists 4 E FA(”) such that

T(cp) =

4 * cp >

vcp E To(”).

(7)

Proposition 3.1. Let E E Lo, we denote by E+ its adjoint operator, then the following assertions are equivalents: ( I ) T * E = T E , V T E L(3;,3;). (2) E+ * S = E+S , V S E L(T,9,3e). (3) E is a convolution operator. Proof The equivalent between 1) and 2) is obvious by duality. 1) j 3) For x E N’ we denote by T, the translation operator on 3,9(N’). Its symbol satisfies

?,(t,q) = a T,e<, e, >> = e(xic) ec,e, >>

- e(X>c)&( c + , , e + r ) ) . For every x E N’ we have

( E * T,)(t,q ) = e - ~ ( ~ + ~ ~ e + q ) ~ x ( t=, eq( )x i~e () E~( ,t ,qq ) ,

(8)

and

( E T x ) ( tq, ) =<<(ET,)ec, e,

>>= e(,%c)<< Eet, e , >>=e(xit),!?([,q ) .

(8) and (9) imply that

E * T, = ET,, V x

E N’.

(9)

121

On the other hand we know by hypothesis that

E*Tx=TxE,Q x E N ‘ . Then for every x E N‘ we get ET, = T,E, i.e. E is a convolution operator. 3) 2) Let E be a convolution operator, then by (7) there exists @ E FL(N’) such that E(cp) = @ * cp, V cp E .FO(N’), and we get

+

Z+(<,q)= << E+ec,e, >> = << ec, @ * e, >> = 3(q)<< ec,e, >>=3(q)ei(c+qzc+,).

Thus

-

(E+T)(<, q) = << E’TeC, e, >> = << Tee,Ee, >>=a Tee,@* e , >> = $(q) << Tec,e, >>=$(q)?(<,q) = e-1 (E+,L+q)?(t, q)$(q)ei(t+,>t+,) = e-+(c+,L+,)S(<, q ) f ( < 17) , =

w+* T)(<,q).

Let X = S ( R ) the space of rapidly decreasing functions. we denote by at and a t the annihilation and the creation operators r e ~ p e c t i v e l y . ~ ~ ~ ~ ~ ~ It is clear that for every t E R,at is a convolution operator. Then as a consequence of proposition 3.1 we get the following result

Corollary 3.2. For any T E L(.Fi,.?’i)and S E L(F’e,Fe) we obtain

T a t = T * a t = at * T , azS = a:

* S = S * at

+

.

Moreover, we have

a, * at = asat, a$

* at = a f a t ,

a, * a > = afa,, a: * a $ = a$.?.

3.1. Application to quantum stochastic diflerential equations

A one parameter quantum stochastic process with values in LO is a family of operators { Et, t E [0,TI} C Le such that the map t eEt is continuous. For such a quantum process Et we put

t n-l En= - x E * k=O

nEN*,

t

E

[O,T].

122

Since the process Et is continuous, {& ,s E [O,t]}becomes a compact set, ( N )@Be. ( N ) i.e. there exist p E IN , m > in particular it is bounded in 0 and Ct > 0 such that for every 5,q E N p we have

(t,q)I 5 Ct ee’

(mlclp)+e’

(mlqlp)

,v s E [o, t].

In particular I&(<,q)I <_ tCtee*(mlclp)+e’(m(qlp) V n E I N . In addition it is easy t o see that V C;,q E N , the sequence (&(&q))n converges to S,’.&(<,q)ds. Thus we conclude by corollary 1 that (En) converges in Ce. We denote its limit by

l E 8 d s := n-t+ca lim En in L O . Moreover, we have

/d’E8ds= l k d s , V t E [O,T]. Theorem 3.2. Let Lt and Mt be two continuous quantum stochastic processes in Lo. Then the ordinary differential equation

-dEt dt

E(O)=

+

Lt * Et M t , A E Lo,

has a unique solution E ( t ) E C(,P). given by E ( t ) = e* .f,’

Lads

*

(l

e* .f,’

- L d U

* Mads+ A ) .

(11)

Proof We apply the Laplace transform to equation (10) to get the following equation

Then

Finally we conclude by theorem 3.1 that E ( t )is given by (11)and it belongs to C(,S*)..

123 Example 1 If we take Lt = AG the Gross Laplacian and Mt = 0, then the solution of equation (10) is given by

E ( t ) = A * e*tAG, t > o . Since AG is a convolution operator we conclude by proposition 3.1 that E ( t ) can be also expressed by

E ( t ) = AetAG, t 2 0 . In what follows we consider X = S ( R ) . Example 2 Let L(i) be Lo-quantum stochastic processes for i E {1,2,3,4}, put Lt = a:Li at Li2)at a?Li3) L r ) and Mt = 0. Then equation (10) has a

+

4 +

+

unique solution in .L(,s*)*

given by

E ( t ) = e* So’ Lads * A.

(13)

Definition 3.1. A quantum stochastic process {Lt}tcR is adapted if

[ D,,Lt for any t E

R and y

E N’ with

I = [ q , L t 1 = 0, supp y E [t,+m[. In particular

Ltat = atLt and Ltat = atLt. Let {Lii’} be &-adapted quantum stochastic processes i = 1 , 2 , 3 , 4 and consider the quantum stochastic differential equation of It6 type dE E(0)

=(Lil)dAt +Ly)dAt =A,

t

+ Li3)dAt+ Li4’dt)E,

t

where At = So a f a s d s , At = So a,ds and A: = solution of equation (14) which will be adapted.

6afds.

(14)

We search a

Theorem 3.3. For i = 1 , 2 , 3 , 4 let {Lii’} C Lo be adapted stochastic quantum processes of convolution operators. Then the equation (14) has a unique solution in C(,e*). given by

E ( t ) = A * ezp*(

/ot

+

+

+

( a f L p ) a , L r ) a , aiLL3) Lp))ds)

124

Proof By means of symbols, we transform equation (14) to a usual differential equation

+ M L f 3 ) E t ) ( E , d$- (L14'Et)(Elrl).

(15)

On the other hand we apply the symbol map t o equation (10) discussed in example 2 dE

{ E/t,o dt

=(atLf')at = A E Lo,

+ Lf2)at + arLf3)+ ,514') * E ,

(16)

we obtain

-

then equations (15) and (17) coincide if

(LF'Et)(<,q)= e-(5'"'F)(E,~)~~(<,77)

,

i = 1,2,3,4

or equivalently if

Lfi) o Et = Lfi) * Et

,

i = 1,2,3,4.

Finally we conclude by proposition 3.1 that equation (14) is equivalent to w (16) and E ( t ) given by (13) is the unique solution of (14).

References 1. D.M. Chung,U.C. Ji and N. Obata : Higher powers of quantum white noises an termes of integral kernel operators. Infinite Dimensional Analysis, Quantum probability and Related Topics, V01.1, No.4 (1998) 533-559. 2. D. Dineen : Complex Analysis in locally convex spaces, Mathematical studies 57, North Holland, Amsterdam (1981). 3. R. Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui : U n the'orkme de dualite' entre espace de fonctions holomorphes d croissance exponentielle. J . f i n c . Anal, Vol. 171, No. 1 , pp. 1-14, (2000). 4. R. Gannoun, R. Hachaichi, P. Krde et H. Ouerdiane : Division de fonction holomorphe a croissance 8-exponentielle. Preprint, BiBos Nr: E 00-01-04, (2000). 5. T. Hida : , Brownian motion, Danslated from the Japanese by the author and T. P. Speed, Springer- Verlag, New York,1980.

125

6. T. Hida, H-H. Kuo, J. Potthof and L. Streit : White noise, an infinitedimentional calculus, Kluwer Academic Publishers Group, Dordrecht, 1993. 7. M-A. Krasnosel'ski and Ya-B. Ritickili : Convex functions and Orliez spaces. P .Noordhoff. Itd, Groningen, The Netherlands (1961). 8. P. Kr6e and R. Raczka : Kernels and symbols of operators in quantum field theory. Ann. I. H . P. Section A , Vol. 18, No. 1, (1978), p . 41-73. 9. H-H. Kuo : White noise distribution theory, CRC Press,Boca Raton, FL, 1996. 10. N. Obata : White noise calculus and Fock space. L.N.M 1577 (1994). 11. N. Obata : Wick product of white noise operators and quantum stochastic differential equations. J. Math. SOC.Japan Vol. 51, No. 3, (1999), p . 613-641. 12. H. Ouerdiane : Noyaux et symboles d'ope'rateurs sur des fonctionnelles analytiques gaussiennes. Japanese Journal of Math. Vol. 21, No.1, 223-234 (1995). 13. H. Ouerdiane : Alge'bres nucle'aires et e'quations aux dirive'es partielles stochastiques. Nagoya Math. journal, Vol. 151, 107-127, (1998). 14. H. Ouerdiane : Nuclear * algebras of entire functions and applications. BiBos Nr: 00-05-15 Univ. Bielfeld Germany (2000). 15. F. Trhves : Topological vector space, Distributions and kernels. Academic press New York. London (1967)

REGULAR QUANTUM STOCHASTIC COCYCLES HAVE EXPONENTIAL PRODUCT SYSTEMS *

B.V. RAJARAMA BHAT Statistics and Mathematics Unit,

Indian Statistical Institute (Bangalore Centre) R. V. College Post Bangalore 560059 India E-mail: bhatOisibang.ac.in

J. MARTIN LINDSAY School of Mathematical Sciences University of Nottingham University P a r k Nottingham NG7 2RD

U.K. E-mail: jmlOmaths.nott. ac.uk

Normal *-homomorphic quantum stochastic cocycles on a von Neumann algebra determine *-endomorphism semigroups and thereby product systems of Hilbert W*-bimodules. This product system is identified for a wide class of stochastic cocycles including all those whose Markov semigroup is norm continuous. When the stochastic cocycle is viewed as a dilation of its Markov semigroup the dilation is typically not minimal however in many cases it dominates an irreducible quantum stochastic dilation. Moreover when the semigroup is norm continuous it has irreducible stochastic dilations and, if the semigroup is also cosnervative, the product system of any such dilation is the same as that of the minimal dilation.

*Based on a talk given by JML in the Workshop Dilations, E-semigroups and Product Systems, Greifswald, Germany, June 2003. 1991 Mathematics Subject Classification. Primary 81825, 47A20 Keywords and phrases. Noncommutative probability, stochastic cocycle, Eo-semigroup, product systems, Hilbert modules, quantum stochastic

126

127

Introduction Quantum stochastic cocycles (with respect to a CCR flow) on a von Neumann algebra M , which are *-homomorphic, give rise to E-semigroups on the tensor product of M with the algebra of bounded operators on the symmetric Fock space of the flow ([L]). When M is a full algebra, the noise dimension space is one-dimensional and the cocycle is regular, the associated product system of Hilbert spaces (in the sense of Arveson, see [Arl]) has been shown to be isomorphic to an exponential product system in particular it is Type I ([Bz]). Here we identify the product system of Hilbert W*-modules (in the sense of Bhat and Skeide) for regular stochastic cocycles in the general case. Again we find that the system is isomorphic to an exponential product system - with Hilbert W*-bimodules replacing Hilbert spaces. In the case of full algebras the minimality of the E-semigroup, viewed as a dilation of the associated Markov semigroup, may be tested by an effective criterion - moreover every Markovian semigroup on the algebra (i.e. pointwise ultraweakly continuous semigroup of completely positive normal contractions) which is norm continuous has such a stochastic dilation, which is minimal ([Bl]). By contrast, minimal quantum stochastic dilation is typically not achievable for non-full algebras - at least not with the quantum stochastic calculus in its present form ([Par],[Mey],[L]). We have therefore introduced a notion of irreducibility for quantum stochastic dilations. Each quantum stochastic dilation (of a norm continuous Markovian semigroup on a von Neumann algebra) dominates an irreducibile one and, at least when the semigroup is unital, the product system of the irreducible dilation coincides with that of the minimal dilation. In this note we summarise these results; full proofs will appear elsewhere ([BLI,~]).

Notation. The symbol denotes algebraic tensor product, and 8 is reserved for tensor products of Hilbert spaces, von Neumann algebras and ultraweakly continuous completely bounded linear maps between von Neumann algebras. 1. Product systems of Hilbert W*-modules

Fix a von Neumann algebra ( M , b ) . Let (Et)t>o be a family of Hilbert W*-bimodules ouer M where EO= M ([Skl]). Thus each Et is a Hilbert W*-module over M and is also a bimodule over M which is balanced in the sense that the left and right actions of M commute. Concretely, Et is an ultraweakly closed subspace of B(b;Kt), for a Hilbert space Kt, and

128

there are sesquilinear maps

(t,rl) * (t,rl)

Et x Et + M, denoted satisfying

(5,rla)= (5,rl)a, namely (1,~) = <*Q. In view of the self-duality of Hilbert W*-modules there is a nice block matrix picture of the situation in which Et is realised as a corner of the von Neumann algebra

Here, and below, Ba denotes the von Neumann algebra of adjointable, and thus (completely) bounded, linear maps; IEt) consists of elements

15) : M + Et, a ++

Ea

(t E Et)

and (Etl consists of elements

(tl: E t + M,

17

*(t,~).

The Dirac notation here is in synch with the operator space structure present. For example each embedding Ba(FI;F!)L) CB(F1;Fz)is isometric ([Ble]). The family (Et)t>o - forms a product system of Hilbert W'-bimodules over M ([BhS]) if there is a two-parameter family of identifications (bimodule M-unitary maps):

Es O Et

+)

Es+t,

where O denotes ultraweak interior tensor product, which satisfy consistency conditions which ensure that the following diagram commutes:

J It is worth noting that the ultraweak interior tensor product coincides with the module weak-* Haagerup tensor product ([Ble]), so that the operator space structure is also respected by the above identifications.

129

An isomorphism of two such systems ( E t )and (Ei) over M is a family (At : Et + E;)t>o - of bimodule M-unitary maps satisfying the obvious compatibility conditions, namely that the following diagram commutes:

Es O Et AsaAtl

EL 0El

-

Es+t

E:+,

By M-isometric, (respectively M-unitary), for a linear map C : FI + F2 between Hilbert W*-modules over M we mean a module map (resp. isomorphism) satisfying

(equivalently, by polarisation, (CE,Cq)= (E, q)), similarly we shall say that an element is M-normalised if l l t j l ~= 1 ~ . A unit of a product system is a family (wt)t>o such that

<

wt E Et for each t

2 0 and

ws+t

= w s 0w t ;

it is M-nomnalised if each wt is M-normalised: (ut,Wt) =

1M-

A basic question for a given product system is whether it has ‘sufficiently many’, some or no units. In the case of product systems of Hilbert spaces this trichotomy is stated as follows: the system is either Type I or Type I1 or Type 111. The paradigm Type I product system of Hilbert spaces is the exponential system (Fk,[O,t[)t>O, where Fk,l denotes the symmetric Fock space (or exponential Hilbert space, in another parlance) over L 2 ( I ;k) for a Hilbert space k and subinterval I of !&, with Fk,[O,s+t[

F k , [ O , s [ @ Fk,[s,s+t[

Fk,[O,s[

@ Fk,[O,t[

(1)

giving the requisite identifications. For such systems multiples of certain exponential vectors:

provide suficiently many units in the sense that the following collection of vectors is total in F k := &$+ ([Arl]): & ( d l l [ t o , t l [ ) ~ . . . ~ ~ ( d_l,t,[) ,l[t,

nEN,di,...& E k,O=to 5 . . * i t , .

130

Analogously, an exponentid system of Hilbert W*-bimodules over M , is based on a Hilbert W*-bimodule F over M , and is the family (FF,[o,t[)t>o defined as follows ([BhS]). For each subinterval 1 of R+ and n E Z+define

rI := { a c I : #a < m>and rp) :=

cI:

=n>

so that for n 2 1 there is a natural bijection

rp)

+)

A?) := {s E P

:s1<

. - -< s n > .

Write I'for rl[p+Lebesgue . measure on A p ) thereby induces a measure on I??) and so, by stipulating that 0 is an atom of unit measure, a measure on Un20 I?) = rI. This is the Guichardet measure (called symmetric measure in [Gui]), integration with respect to which is conveniently denoted J . . . do. Now

f i , I:=

FQn@FJn), where FYI := L 2 ( r p ) ) ,

(3)

n20

with the convention Fo0 := M , may be viewed as a subspace of

and thereby inherits pre-Hilbert W*-bimodule structure from the interior tensor products:

.1cI . a2 := Ul$(

$9

:=

/- ($W>

$'(a))do. rr The product structure may be described in two ways, corresponding to the two views (3) and (4). Thus for cp E G,[o,a,[ and $ E PF,[,,,[, cp 0$ should be identified with the element of PF,[o,s+t[ defined by a1

0

*

)a2,

($3

++ cp( (a - t ) fLO, l4)@ $(a f l [O, tr),

where a - t := { s - t : s E a}. Or, in picture (3), for simple tensors x @ f E FonC&;![ and 71 €3 g E. FQm€3 F{$, the product element (x@ f ) 0( q @g) is identified with

(x@ 7)€a ( S t f @ 9 )

r[ti+,r

where St is the second quantised shift: (Stf)(a) = f(a- t ) for a E Notice that time is running to the left. (FF,[o,t[)t>ois a product system Letting FFJ be the completion of PF,I, of Hilbert W*-bimodules, all contained in the Hilbert W*-module FF := FF,w+.We call it the exponential system based o n F ; in [BhS] it is called

131

the time-ordered Fock module - to distinguish it from earlier experiments in module analogues of symmetric Fock space. The basic unit of this system the collection of all units of such a system which is ( d @ ( . ) lE~ Fp,[O,t[)t,O; are norm continuous (& maps JR++ FF)is indexed by M x F as follows. For each continuous unit w there is lc E M and q E F such that

where pt = etk ([LiS]), in clear generalisation of the situation with exponential systems of Hilbert spaces - see (2). By an E-semigroup on a von Neumann algebra ([Arz]) is meant a pointwise weak*-continuous semigroup of normal *-homomorphisms of the algebra (older terminology: eo-semigroup). If each map is also unital then it is called an Eo-semigroup. Following Skeide, who treated the unital (Eo) case ([Skz]), we next describe the product system of an E-semigroup 4 = ( $ t ) t > o on Ba(E) where E is a Hilbert W*-module having an M-normalised eLment:

which we fix. Set

Q = K ( ~ M I [)=( ) < I where

K

:M

: 77

e <(<,d,

+ Ba(E) is the embedding defined by

4.) := I<)a(tl = IEa)(
with a*

4t(Q)v. b := 4 t ( ~ ( a ) ) ~ b ,

defines Hilbert W*-bimodules (Et)t>O contained in E. These form a product system as follows. First, with < t h e fixed element ( 5 ) ,

x @ 4t(Qh * 4t(lX)(Jl)q

(6)

determines a one parameter family of bimodule M-isometric maps Bt : E 0Et + E . In view of the identity

4t ( I 4 s ( Q ) ~ ) ( < l ) q

= 4 s + t ( Q ) 4 t (I~)(
132

and the fact that Bt has range &(idE)E, each Bt restricts/corestricts to isomorphisms B,,t : E, 0 Et + Es+t and these identifications define a product system over E, with the associativity consistency conditions being covered by the identity

Bs+t 0 (idE OB,,t) = Bt 0 ( B , 0idEt). Moreover the E-semigroup is recovered via the identity

4 t ( X ) = B t ( x @ IEt)B;. We denote this product system by (Ef")t>o. Remark. To date there is no converse result to the effect that every product system of Hilbert W*-modules over a von Neumann algebra M is the product system of some E-semigroup 4 on the algebra of adjointable operators on some Hilbert W*-module with M-normalised element 6. Such a result would require the imposition of a measurability condition, as it does in the Hilbert space context ([Arl]). Under the assumption that d t ( I - Q ) 5 I - Q for t 2 0,

(7) the (pointwise weak*-continuous) family of completely positive contractions on M given by

Pt(a) = < 6 , 4 t ( w ) r , comprises a semigroup;

(tt := 4t (QK)t>Odefines a unit of the above product system; and the identity = (&,a *

tt)

holds. The E-semigroup 4 on B,(E) is thereby a dilation of the completely positive contraction semigroup P on M. The isomorphism class of the product system of Hilbert spaces associated with a n E-semigroup on B(H) is a complete invariant for such semigroups up t o cocycle conjugacy ([Arl]). For a Hilbert W*-module E over M, E-semigroups 4 and qS on Ba(E), are called cocycle conjugate if there - in B,(E) satisfying is a family (Wt)t>o

4 : ( q = Wt4t(T)W,'; WtW; = 4 i ( I ) and W:Wt = &(I); WS+t= WS4,(Wt);WO= I and t e Wt is SO-continuous.

(8)

133

Thus W is a strongly continous cocycle with respect to partial isometries in Ba(E) which intertwines 4 and 4’.

4 consisting of

Theorem 1.1. ([Skz]) Let E be a Hilbert W*-bimodule over a von Neum a n n algebra ( M ,I)) and suppose that E has a n M-nownalised element <.

Let

4 and 4‘ be E-semigroups o n Ba(E), with associated product systems

(E$’E)tLoand ( E f l E ) t ? orespectively.

(i) If 4 and 4’ are cocycle conjugate then their associated product systems are isomorphic. (ii) Conversely, if their product systems are isomorphic then the Esemigroups are cocycle conjugate modulo the continuity condition. For (i) (co-)restriction of each Wt gives the component isomorphisms. For (ii) set Wt = Bi(idE @At)&, in the notation (6), where At is the component isomorphism E?*€+ E f ’ E . 2. Quantum stochastic cocycles

By a quantum stochastic cocycie on a von Neumann algebra ( M ,I)) we shall mean a pointwise ultraweakly continuous family of completely bounded normal maps j t : M + M €3 B(&), for some Hilbert space k called the noise dimension space, satisfying .-

js+t

=3s

O

0s O jt

and jt(M)C M €3 B ( F k , [ O , t [ ) €3 I k , [ t , c o [ r

where ( a t ) t >-o is the CCR flow of index k, ampliated to M €3 B(Fk). Here, with Ran o, identified with M €3 B(Fk,[,,,[) and j , viewed as a map M + M €3 B(Fk,[,,W[), is the normal extension of j , to Rano,, so that

F,

RanTs

c M €3B ( F k , [ O , s [ )

€3 B(Fk,[s,w[) = M €3 B(Fk).

jF’rom the cocycle property it follows that Pt:=IEojt

and J t : = S o o t

(9)

form one-parameter semigroups of maps on M and M@B(Fk)respectively, where := 1~ €3 W k , Wk being the vector state given by the vector f l k := (1,0,0,. . .) in &. When j consists of *-homomorphisms J is an E-semigroup on M €3 El(&) = Ba(E) for E = M €3 I.&), which dilates P:

Pt = IE o Jt

o L,

134 L : M + M @ B(Fk) being ampliation. The cocycle is called regular when its associated Markov semigroup P is norm continuous. From the papers of Lindsay and Wills the regular *-homomorphic stochastic cocycles on M are completely characterised by their infinitesimal description.

Theorem 2.1. ([LWI-~])Let ( M ,fj) and k be respectively a won N e u m a n n algebra and separable Hilbert space.

+ B(z)@ M

satisfying

e(a*b) = e(a)*L(b) L(a)*e(b) e(a)*Ae(b),

(10)

If 8 is a linear ultraweakly continuous map M

+

+

where := C @ k, L i s ampliation M + B f i ) @ M and A is the orthogonal projection Pk @ I M , then the quantum stochastic differential equation djt = 50 8 d&;

j,(a) = u @ 1~~

(11)

where 5 = idB(Q €3 j t , has a unique weakly regular weak solution; the solution is strong and it forms a regular *-homomorphic quantum stochastic cocycle on M . Conversely, if j is a regular *-homomorphic quantum stochastic cocycle o n M , with noise dimension space k , then there is a unique linear completely bounded ultraweakly continuous map 8 : M + B(L)@ M such that j satisfies the quantum stochastic differential equation (11); moreower 8 satisfies (10). We refer t o j as the quantum stochastic cocycle generated by 8, and 8 as the stochastic generator of j.

Note. For this paper we are using an ordering convention for some of our tensor products which is different to the usual one - traditionally stochastic generators map M into M @ B f i ) . The present convention has the advantage of avoiding some of the tensor flips in the theory. Viewing B(L) @ M as a subspace of B(fj @ k @ fj), and writing 8 in corresponding block matrix form:

;[ ",I p

being ampliation M + B(k) @ M ) , T E B ( M ) is the generator of the Markov semigroup (Eo j t ) t > o , p : M + B(k)@ M is a *-homomorphism and S : M 3 Ik) @ M is a pderivation, with St defined by St(a) = S(a*)*. (Lk

135

In passing we mention that these stochastic dilations fit into the general picture described earlier, in the following sense. Lemma 2.2. The element [ = 1~ ‘8 I n k ) of the Hilbert W*-module E := M ‘8 I F k ) satisfies assumption (7) above, for the E-semigroup (Jt := 5 o fJt)t 2.0 : Jt(1-

Q) 5 I -Q

for Q = IE)(tI E Consider stochastic generators of the form

for a normal *-homomorphism p : M -+ B(k) €3 M . In case k = C it is well-known ([Hud]) that the stochastic cocycle generated by (13) has the simple form j t = pNt , in the sense that (jt(a)f)(fJ)= P#(‘”[o’tl)(4

(m),

for a E M and f E Ij @ F = L 2 ( r ;Ij). The cocycle continues to have a simple explicit form in the case of multidimensional k as the following result confirms. Proposition 2.3. Let j be the quantum stochastic cocycle o n M with generator (13). Then, under the identifications

b ‘8F

k

= @ (kBn €3

b)

‘8 F:;j[ ‘8 Fk,[t,m[,

n2.0

j is given by

.’?,(a)= @ pn(a) ‘8 I;;;[ €3 I k , [ t , o o [ n>O

where pn : M

+ B(kBn)€3 M

is defined by

with p ( i ) := idB(kB(i-1))@ p

2 1.

This proposition is key; its proof is a simple verification.

136 3. Product system of a stochastic cocycle

The strategy here is to view the stochastic cocycle as a perturbation (in the sense of [EvH] and [GLW]) of the stochastic cocycle generated by the number/exchange component of the cocycle generator, and thereby reduce the problem to pure number/exchange cocycles, by appealing to Theorem 1.l. In turn the explicit form of pure number/exchange cocycles given in the previous section leads us to the product system of such stochastic cocycles. Given a normal *-homomorphism p : M + B(k ) @ M , define a Hilbert W*-bimodule as follows:

Fk>j':= p ( l ) ( l k ) @ M) with a . p ( l ) q . b := p(a)qb, and standard M-valued inner product Now we need two Lemmas.

(t,q )

I+

(15)

("q.

Lemma 3.1. Let F = Fk,p, for a normal *-homomorphism p : M + B(k) @ M . In the notation (14) define Hilbert W*-bimodules FO := M and, for n 2 1,

Fn := p n ( 1 ) ( ( k @ n )@ M ) With u * pn(l)q . b := pn(a)qb. Then (a) The correspondence p n ( 1 ) (1x1 8.. . @ X n ) @ l M )

* p ( 1 ) ( 1 X 1 ) @ 1 M )0.. * @ p ( l )( I Z n ) @ l M )

extends to an M-unitary bimodule map Fn + Fan. (b) Under the resulting identifications, pn(1))(I.

x * Pn+m(l)(lX)@ pm(a)X)

Q a) 0

for x E kBn, a E M and

(16)

x E Fam = F,.

Lemma 3.2. Let k , p and (Fn)>o be as above. Then, with the identifications there, the product structure of the exponential product system for F = F k i pis determined by ( P n ( l ) ( l X ) @ a )@ f ) @ ( P m ( l ) ( I d@ b ) 8

9)

*

for x E kBn, y E kBm, a, b E M , f E F{$[ and g E F[$, where (St)t>o - is the semigroup of right shifts on F.

137

Using these we are able to identify the product systems of quantum stochastic E-semigroups. Proposition 3.3. Let j be a quantum stochastic cocycle on M with generator of the f o r m (13). Then the product system of its associated E-semigroup is (isomorphic t o ) the exponential system for the Halbert W*-bimodule F k @ .

Now consider a *-homomorphic stochastic cocycle j with general (bounded) generator (12). It follows from the Christensen-Evans Theorem ([ChE]) that there is d E Ik) @ M and h = h* E M such that .(a) = d * p ( ~ ) d h{d*d, U }

+ i[h,u ] , and

d(a) = da - p(a)d. Letting j o denote the quantum stochastic cocycle generated by [I p!Lk] i h - i d ' d d' and setting 1 = where p = p(l), the quantum stochastic -pd P~ differential equation

]

[

dWt = (Ic @ Wt)(idBci;,@ j f ) ( l ) d h t ; W O= I has a unique strong solution ([GLW]). Setting q5 = and q5' = J , where J o and J are the E-semigroups on B, ( M @ I&)) = M @ B(&) corresponding to j o and j respectively, it may be shown ([BLl]) that W satisfies (8). This establishes the following result. Proposition 3.4. Let J be the E-semigroup associated with a *homomorphic stochastic cocycle o n M with bounded ultraweakly continuous generator 9 = p-Lk . Then J is cocycle conjugate to the E-semigroup

[.

1

associated with the stochastic cocycle with generator

[I

In view of Theorem 1.1 these two propositions entail the following theorem. Theorem 3.5. The product system of a regular normal *-homomorphic quantum stochastic cocycle on a uon Neumann algebra is exponential. SpecificaEly, i f the cocycle generator is ;;p6_tLk] then the product system

[

is (-TF,[o,~[)~~o where F = Fk%p. 4. Irreducible quantum stochastic dilations

Let P = (Pt)t>o - be a Markov semigroup on M . A *-homomorphic quantum stochastic cocycle j on M , with noise dimension space k, is a dilation of P

138

if its Markov semigroup is P , that is lE& 0 j t = Pt, where & := idM @Uk. If P is norm continuous with generator T then j has bounded generator of the form [i In fact every norm continuous P has such a quantum stochastic dilation ([GoS]), but typically the dilation is not (and cannot be) the minimal dilation of P ([BL2]). This motivates the following definition. A quantum stochastic dilation j of a Markov semigroup P is irreducible when there is no other quantum stochastic dilation j' which is dominated by j in the sense that the corresponding E-semigroups satisfy (Jt - Ji) is

completely positive for all t 2 0.

Then the following results may be proved.

Theorem 4.1. Let P be a n o r m continuous Markov semigroup o n M . Each of its quantum stochastic dilations j dominates an irreducible quantum stochastic dilation. Specifically, i f [i is the stochastic generator of j then there is a projection q E B(k) 8 M such that the stochastic cocyop(p;-ck is t-homomorphic, is dominated by j, and is cle generated by irreducible.

LtLk 1

[

1

For the second result we assume unitality of the Markov semigroup.

Theorem 4.2. Let j be a n irreducible quantum stochastic dilation of a Markov semigroup P o n M . If P is norm continuous and unital then the product system of j coincides with that of the minimal dilation of P . The generic nonminimality of quantum stochastic dilations appears to us t o be a defect of quantum stochastic calculus in its current form. This might be rectified by refounding the calculus on more general Hilbert W*(and C*-)bimodules than the cut-down standard modules p(lM) (I k) @ M ). We view this as a fruitful direction for future research.

Acknowledgements We are grateful to Michael Skeide and to Adam Skalski for their comments on a preliminary draft. BVRB was supported by a fellowship under the Exchange Agreement between the Indian National Science Academy and the Royal Society of London, and by Visiting Fellowship GR/S57099/01 from the U.K. Engineering and Physical Sciences Research Council. The EU Research and Training Network Quantum Probability with Applications to Physics, Information Theory and Biology has faciliated useful discussions with fellow researchers.

139 References [Arl]. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC. 80 (1989)no.409.

[Arz]. W. Arveson, “Noncommutative Dynamics and E-semigroups,” Springer Monographs in Mathematics, Springer, New York 2003. [Bl]. B.V. Rajarama Bhat, Cocycles of CCR flows. Mem. Amer. Math. SOC.149 (ZOOl), no. 709. [Bz]. B.V. Rajarama Bhat, Product systems of one-dimensional Evans-Hudson flows, in “Quantum Probability Communications X” (eds. R.L. Hudson and J.M. Lindsay) World Scientific, Singapore 1998,pp. 187-194. [BLl]. B.V. Rajarama Bhat and J.M. Lindsay, The product system of a stochastic E-semigroup, in preparation. [BLz]. B.V. Rajarama Bhat and J.M. Lindsay, Irreducibility for quantum stochastic dilations, in preparation. [BhS]. B.V.Rajarama Bhat and M.Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top 3 (2000)no. 4,519-575. [Ble]. D.P.Blecher, A new approach to Hilbert C*-modules, Math.Ann. 97 (1997),pp. 253-290. [ChE]. E. Christensen and D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. SOC.20 (1979) no. 2,

358-368. [EvH]. M.P. Evans and R.L. Hudson, Perturbations of quantum diffusions, J. London Math. SOC.41 (1990)no. 2,373-384. [GLW]. D. Goswami, J.M. Lindsay and S.J. Wills, A stochastic Stinespring theorem, Math. Ann. 319 (2001)no. 4,647-673. [GoS]. D. Goswami and K.B. Sinha, Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra, Comm. Math. Phys. 205 (1999)no. 2,377-405. [Gui]. A. Guichardet, “Symmetric Hilbert Spaces and Related Topics,” Lecture Notes in Mathematics 267,Springer, Heidelberg 1970. [Hud]. R.L. Hudson, Quantum diffusions on the algebra of all bounded operators on a Hilbert space, in “Quantum Probability and Applications IV” , (eds. L. Accardi and W. von Waldenfels), Lecture Notes in Mathematics 1396, Springer, Heidelberg 1989,pp. 256-269. [Lan]. E.C. Lance, “Hilbert C*-modules, A toolkit for operator algebraists,” London Mathematical Society Lecture Note Series 210,Cambridge University Press, Cambridge 1995. [LiS]. V. Liebscher and M. Skeide, Units for the time-ordered Fock module, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001)no. 4,545-551. J.M. Lindsay, Quantum stochastic analysis, in “Lectures in the Spring [L]. School on Quantum Independent Increment Processes,” Lecture Notes in Mathematics, Springer, Heidelberg (to appear). [LWl]. J.M. Lindsay and S.J. Wills, Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory

140

Related Fields 116 (2000) no. 4, 505-543. [LWz]. J.M. Lindsay and S.J. Wills, Markovian cocycles on operator algebras, adapted to a Fock filtration, J. Funct. Anal. 178 (2000) no. 2, 269-305. (LWs]. J.M. Lindsay and S.J. Wills, Existence of Feller cocycles on a C*-algebra, Bull. London Math. SOC.33 (2001) no. 5, 613-621. [LW4]. J.M. Lindsay and S.J. Wills, Homomorphic Feller cocycles on a C*algebra, J. London Math. SOC.(2) 68 (2003) no. 1, 255-272. [Mey]. P.-A. Meyer, “Quantum Probability for Probabilists,” 2nd Edition, Lecture Notes in Mathematics 1538,Springer, Heidelberg 1993. [Par]. K.R. Parthasarathy, “An Introduction to Quantum Stochastic Calculus,” Monographs in Mathematics 85,Birkhauser Verlag, Basel 1992. [Ski]. M.S. Skeide, “Hilhert Modules and Applications in Quantum Probability,” Habilitation Thesis, Cotthus, Germany 2001. [Skz]. M.S. Skeide, Dilations, products systems and weak dilations, Math. Notes 71 (2002) 914-923.

EVOLUTION OF THE ATOM-FIELD SYSTEM IN INTERACTING FOCK SPACE

P.K.DAS Physics and Applied Mathematics Unit Indian Statistical Institute 203,B.T.Road, Kolkata- 700f08, Indaa e-mai1:daspkOisical. ac.in Here we discuss interaction of a single two-level atom with a single mode of interacting electromagnetic field in the Jaynes-Cummings model with the rotating wave approximation.

1. Introduction

Light is absorbed and radiated by atoms and one of the most fundamental problems in quantum optics is the interaction between the quantized electromagnetic field and an atom. But the real atoms being complicated systems it is often desirable to approximate the behaviour of a real atom by that of a much simpler quantum system. Sometimes only two atomic energy levels play a significant role in the interaction with the electromagnetic field and in many theoretical treatments it has become customary to represent the atom by a quantum system with only two energy eigenstates. Jaynes and Cummings considered a system consisting of a nonrelativistic two-level atom coupled to a single quantized mode of the electromagnetic radiation field under the dipole and rotating wave approximation. This simple model in quantum - optics is one of the few exactly solvable quantum mechanical models describing the interaction of matter with an electromagnetic field. The wave length of the field mode is assumed to be so long, compared with the atomic dimension that the dipole approximation can be made. Originally, Jaynes and Cummings wanted to study the QED predictions for an ammonia maser. But they failed to find an exact solution for this application with the help of simplifying assumptions described above. They had to make the additional approximation by removing transitions that correspond to processes which do not conserve energy. This approximation is called

141

142

the rotating wave approximation. In the rotating - wave approximation this model can be solved exactly. Exact solutions describing the dynamical behaviour of expectation values of variables such as the population inversion, the atomic dipole - correlation function and the mean photon number can be obtained in this case in the form of an infinite series. Starting with a cavity field in a coherent state and with the atom in its upper state it was found that repeated decays and revivals of the Rabi oscillations occur and that for certain times the field mode shows squeezing. The predicted collapses and revivals of the inversion oscillations are in agreement with the experiments done with Rydberg atoms in a microwave cavity. In recent years there have been several generalizations of the Jaynes - Cummings Hamiltonian in which the interaction between the atom and the radiation field is no longer linear in the field variables and are only particular cases of JCM in which the creation and annihilation operators of radiation field are replaced by deformed harmonic oscillator operators with the given commutation relations. Since the JCM is one of the basic models in quantum optics, its extensions in different directions are generally very interesting [6].In this paper, we study JCM in which field variables are taken as the actions on an interacting Hilbert space with the specified commutation relation. The work is organized as follows. In section 2, we have given definitions and preliminaries that will be used throughout the paper. In section 3, we discussed the interaction of a single two-level atom with a single mode of interacting field. In section 4, we described the evolution of the atominteracting field system by probability amplitude method. In section 5 , we gave a conclusion. 2. Definitions and Preliminaries

As a vector space one mode interacting Fock space I?(@) is defined by 00

r(@) = @@in >

(1)

n=O

for any n E N where @In> is called the n-particle subspace. The different n-particle subspaces are orthogonal, that is, the sum in (1) is orthogonal. Thenorm of the vector In > is given by

< nln >= An

(2)

{An} 2 0 and if for some n we have (A,} = 0, then A{}, = 0 for all m 2 n. The norm introduced in (2) makes r(@) a Hilbert space. where

143

An arbitrary vector f in r(C) is given by

f

= COlO > +Clll

> +c212 > +... +c,ln > +.. . llfll = (C,"==, < 00.

(3)

for any n with We now consider the following actions on r(C) :

A* is called the creation operator and its adjoint A is called the annihilation operator. To define the annihilation operator we have taken the convention o/o = 0. We observe that An < nln >=< A*(n- l ) ,n >=< (n - l ) ,An >= < n - 1,n - 1 >= . . . A,--1

(5)

and

By (2) we observe from (6) that A. = 1. The commutation relation takes the form AN+1

AN

AN

AN-1

[A,A*]= -- where N is the number operator defined by Nln >= n(n >. In a recent paper [l]we have proved that the set { n = 0 , 1 , 2 , 3 , . . .} forms a complete orthonormal set and the solution of the following eigenvalue equation

$$,

Afa = Qfa is given by (9)

T.

= Cr?o where $J(laI2) Now, we observe that

We call

fa

a coherent vector in r(C).

AN+1 AN AA* = , A*A = AN

We further observe that AA*.

(e

- &)

AN-1

commutes with both A*A and

144

3. Interaction of a single two-level atom with a single mode field

The interaction of a interacting single-mode quantized field of frequency Y with a single two level atom is described by the Hamiltonian in the dipole approximation

H = HF

+ H A + gHI

(10)

+

+

where H F = hvA*A and H A = ;twoz with HI = h(a+ a - ) ( A A * ) . Here A , A* are the interacting annihilation and creation operators for the photons at frequency v. The two - level atom is described by the usual spin - operators and the inversion operator az with w as atomic transition frequency. Also g is the coupling constant. The interaction energy HI consists of four terms. We drop the energy nonconserving terms corresponding to the rotating - wave approximation and obtain the simplified Hamiltonian as

H = hvA*A

+ -21h a , + hg(a+A + A*u-).

(11)

At this stage , for simplicity, we take h = 1 and consider the exact resonance case v = w . The resulting simplified Hamiltonian is 1 2

H = v(A*A+ -az) + g(a+A + A*o-).

(12)

H = Ho + H i

(13)

We write

where 1 2

(14)

+ A*o-)

(15)

Ho = Y ( A * A+ - a z ) and Hi = g(a+A

The Hamiltonian, given by equations (13), (14) and (15) describes the atominteracting field interaction in the dipole and rotating-wave approximation. It is convenient t o work in the interaction picture. The Hamiltonian, in the interaction picture, is given by

v = ,iHotHle-iHot

(16)

145

Then

v = ei[vA*A+1/2vuz]t~le-i[vA*A+1/2vur]t

- eivA'At~ei/2vuzt~le-i~A*At~e-i/2vuzt - eivA'At.ei/2vu,tg(a+A + ~ * ~ _ ) ~ - i v A ' A t . ~ - i / 2 v u = t = ge ivA'

At .ei/2vu,ta+Ae-ivA'

+geivA' At .ei/2vu,

- gei/2uuzta+e -i/2vuZt +geivA'AtA*e-ivA'At

Using

we get

At.e-i/2vu,t

tA*O-e-ivA'

At .e-i/2vuz t

eivA'AtAe-ivA'At .ei/2vaz ta-e-i/2vo,

t

(17)

146

ei/2vazto +e-i/2vazt = o+ + i / 2 v t [ o z , o + ] + ~ [ o z , [ o z , o + +... ll = Is+ i / 2 v t ( 2 a + ) -220+ .. . = o+ ivto+ @ g o + . ..

+

+ +

+

+ + = o+[l+ivt+ (zvt)2 2! +...I =o + p

(22)

From (17) we get

v = S[o+eivt e-iv( hx-N + ) t

N--l

A +A*eiv(e-*)to-e-ivt

I

(23)

4. Evolution of the atom-field system

In this section, we present probability amplitude method to solve the evolution of the atom-field system described by the Hamiltonian (23). The Pauli spin operators are:

We may define the following operators: 0-

= 1/2(o,

+ io,)

(::)

and

The o- operator takes an atom in the upper state into the lower state whereas o+ takes an atom in the lower state into the upper state. We begin with the following spin matrices:

o-=

(;;),

(;;),

ff+=

uz=

(

-1 0 l).

They satisfy the following commutation relations: [o+,Is-]=

oz, [o,,o+]= 2o+,

[Isz,o-]

= -%-.

Then we proceed to solve the equation of motion for 111, >, that is,

-i

at

’

= Vl11,>

(25)

147

At any time t , the state vector I+(t)> is a linear combination of the states la, > and Ib, >. Here la, > is the state in which the atom is in photons. A similar description the excited state la > and the field has is given for the state Ib, >. As we are using the interaction picture, , C~b , n . The state we use the slowly varying probability amplitudes c ~and vector is therefore

e

-&

-&

-&

k

Now,

and

Now we observe the following facts:

a+la > = la

>< bla > = 0

o+p > = la >< b[b> = la > a-la

a-lb We see the followings:

> = Ib >< ala > = Ib > > = Ib >< alb > = 0

148

From equations (29), (31), (32), (33) and (34) we observe that

Now comparing equations (26), (28) and (35) we get

149

From (36) we get

Or

Or

with

for

Now we assume a trial solution to get

or

S2 - A S - g2(-)An+l = 0 An whose solutions are

S= In (42) we take

fli

Af

4 A 2 + 4g2( h) An

(42)

2

A2 + 4g2(*)

i A+n t ( 2")

and write the solution as

i A-nn)t ( 2

ca,n(t) = A l e +A2e =A l e F . e W + A

~

~

F

.

~

~

= [ A l e w + Aze--i"]e% -in

t

+ i sin y}+ Az{cos y - i sin ?}ley = [(Ai+ A2) cos y + i(A1 - A2) sin

(43)

= [Al(cos

To find the constants

A1

and A2 we observe that Ca,n(O)

= A1

+ A2

(44)

150

Also we see that

Hence we see that

-ig/?Cb,n+l(O)

= A1 .i/2(A

+ 0,) + A2.i/2(A - Rn)

(46)

On solving (44) and (46) for A1 and A2 we find that

(47) and

(48) and

Subsstituting the values of

and

from above in (43) we find

finally

In a similar manner we find the value of ~ b , ~ + l (tot ) be

151

If initially the atom is in the excited state la > then c ~ , ~ ( O=) cn(0) and = 0. Here cn(0) is the probability amplitude for the field alone. We then obtain qn+1(O)

-

ca,n(t) = cn(o)[cos(?)

sin(F)]ey

(52)

or

Equations (52) and (54) give us all the relevent informations relating to the quantized interacting field and the atom. I ~ . , ~ ( tand ) l ~ I ~ b , ~ ( t are ) l ~ the probabilities that, at time t , the interacting field has n photons present and the atom is in levels la > and Ib > respectively. To obtain the probability p(n)that there are n photons in the interacting field at time t we take the trace over the atomic states: P(n) =IC~,~(+ ~ )I I C ~ ~ , ~ ( ~ ) I ~ 2 n t = I~n(0)l2[COS + (&I2 sin2(*)nI +Icn-1(0)l2

4g2(&)

a:_,

t

2 %-It

sin (7)

(55) The inversion W ( t )is given by

152

5. Conclusion

In conclusion, we have studied the atom-interacting field evolution in the dipole and rotating wave approximation by probability amplitude method and for simplicity we considered only resonance case. References

1. P. K. Das, Coherent states and squeezed states in interacting Fock space, International Journal of Theoretical Physics, vol 41, no. 06, (2002), MR. No. 2003e: 81091 (2003). 2. P. k. Das, Quasiprobability distribution and phase distribution in interacting Fock space, International Journal of Theoretical Physics, vol 41, no. 10, (2002), MR. NO. 2003j: 81098 (2003). 3. J. S. Peng and G. X. Li, Phase fluctuations in the Jaynes-Cummings model with and without the rotating wave approximation, Phys. Rev. A, vol. 45, no. 5, (1992), 3289 - 3293. 4. M. 0. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press,(1997). 5. Yariv, Amnon. Quantum Electronic, John Wiley and Sons,Inc. NY.(1967) 6. Crnugelj, J., Martinis, M., Mikuta-Martinis, V., Properties of a deformed Jaynes- Cummings model. Phys. Rev. A, vol. 50, no. 2, (1994), 1785-1791.

QUANTUM MECHANICS ON THE CIRCLE THROUGH HOPF Q-DEFORMATIONS OF THE KINEMATICAL ALGEBRA WITH POSSIBLE APPLICATIONS TO LEVY PROCESSES

V.K. DOBREV1t2, H.-D. DOEBNER3, R. TWAROCK4 School of Informatics, University of Northumbria, Newcastle-upon-Tyne N E l 8ST, UK, vladimir. dobrev0unn. ac.uk, Permanent address: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 7.2 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria, dobrev0inrne.bas.bg Department of Physics, Metallurgy and Material Science, Technical University Clausthal 38678 Clausthal-Zellerfeld, Germany [email protected] Centre f o r Mathematical Science, City University Northampton Square, London EC1 V OHB, UK r.twarockQcity.ac. uk A formulation of quantum mechanics on S' or its N-point discretisation Sh based on different types of q-deformations of subalgebras of the kinematical algebra of the system was discussed ( [l])in the framework of Bore1 quantisation. We review this method and introduce new Hopf q-deformations of the full kinematical algebra, i.e. q-deformations of both the subalgebras of position and of momentum observable. The implications of this deformed approach to the dynamics and the resulting evolution equations is assessed and compared with previous results including the non-deformed case. The presented algebraic method for q-deformations can be translated to LBvy processes on algebraic structures and the related evolutions; possible applications are outlined.

Key words: Hopf q-deformation, kinematical algebra, quantum dynamics,

153

154

Ldvy processes 1. Introduction

The quest for a discrete quantum mechanical theory is of interest both for conceptual and computational reasons, and different arguments in support of this have been discussed in [l]. There are various possibilities to construct discrete analogs to quantum mechanics, which rely on different guiding principles for the replacement of differential operators by suitable difference operators. The method adopted in [l]is of an algebraic nature, and is based on the kinematical algebra that represents position and momentum observables in the framework of Bore1 quantisation [2]. Via a q-deformation of this algebra, expressions of the quantum mechanical position and momentum operators are obtained in terms of q-difference rather than differential operators, i.e. expressions of the form

where e.g. f E C(R). Furthermore, the quantum dynamics based on these operators and a suitable generalisation of the first Ehrenfest theorem is given in terms of qdifference operators and corresponds to a discrete, nonlinear q-Schrodinger equation. The algebraic guiding principle based on q-deformations of the kinematical algebra leads to different discrete quantum mechanical theories for different choices of q-deformations. While a q-deformation of the subalgebra related to the momentum observable is necessary in order to obtain a quantum mechanical theory in terms of difference operators, a deformation of the subalgebra related to the position observable is not necessary for this purpose, which is minimal in the sense that it involves only a deformation of the subalgebra related to the momentum observable [3]. Therefore, initial attempts to construct a discrete quantum mechanical theory based on a q-deformation of the kinematical algebra have focused on a q-deformation of the kinematical algebra. However, from an algebraic point of view, this does not use the full potential of the algebraic guiding principle for the construction of a discrete quantum mechanical theory, and different options for this have been discussed in [l].Among these, an approach based on a Hopf q-deformation of the full kinematical algebra (see also [4])plays a distinguished role. It is the aim of this contribution to consider this case in further detail and to

155

compare the resulting quantum mechanical theory with the results in the minimal approach. We will show that the approach based on a Hopf q-deformation of the full kinematical algebra has interesting implications in the continuous limit. In particular, we discuss the quantum dynamics, which is given as a family of nonlinear q-Schrodinger equations, in greater detail and compare it with the family of Doebner-Goldin type nonlinear Schrodinger equations [5-81 that arise in the continous approach. The interesting feature of the ansatz based on a q-deformation of the full kinematical algebra is that both real part and imaginary part of the nonlinear Schrodinger equation are completely determined. This is in contrast to the situation in the undeformed case, where only the imaginary part of the nonlinearity follows from the Borel quantisation formalism, and the real part is inferred by some additional plausibility arguments related to the shape of the imaginary part of the nonlinearity [9].We show that in the continuous limit of our approach based on a q-deformation of the full kinematical algebra one obtains a nonlinear Schrodinger equation, in which the imaginary part of the nonlinearity coincides with the one in the Doebner-Goldin formalism and the real part falls in one of their proposed classes. In this way, this study may be understood also as a justification for the Doebner-Goldin models. Finally, we outline possible applications of these results to LBvy processes. Their investigation on noncommutative algebraic structures (Lie groups, quantum groups, bialgebras,. ..) is often intricate and involves representation theory, c.f., e.g., [lo-171. In this approach L6vy processes are characterized by their generators. Classifying such generators originating in our algebraic framework would be one of the main challenges. Another reason for the study of L6vy processes on algebraic structures is their relation t o evolution equations, c.f., e.g., [13-151. The paper is organised as follows. After a review of Borel quantisation in section 2, we discuss the Hopf q-deformation of the kinematical algebra and its implications on the quantum kinematics in section 3. In section 4 a corresponding quantum dynamics is derived and compared with the dynamics obtained via the minimal ansatz in [l].In section 5 we outline the possible applications to L6vy processes.

2. Short review of Borel quantisation with application to

S1

We consider (non relativistic, point-like) systems S moving and localized on a smooth manifold M (with measure p ) and specialize later to S1. The

156 exposition follows [l]. 2.1. The kinematics

To model possible localization regions of S on M and possible infinitesimal movements of the regions we choose for the regions Borel sets B from a Borel field B ( M ) and for the movements (smooth, complete) vector fields X E Vecto(M). These two geometrical objects are the building blocks of the kinematics K ( M ) of S:

-

I v

K ( M ) = (f?(M),Vecto(M)).

-

(2)

Borel sets are displaced through X by its flow @: as B' = {m'lm' = aT%),m E B , X E Vecto(M),7 E [0,1)}. For a quantisation of K ( M ) one has to construct a map Q = (Q, P) which maps the blocks in K ( M ) into the set SA(7-l)of self-adjoint operators on a Hilbert space 3t. It is reasonable to interpret the matrix elements of Q ( B ) ,i.e. ($,Q(B)$),$ E 3-1, as the probability t o find the system localized in B in a state $. The properties of B ( M ) and further physical requirements (e.g. no internal degrees of freedom) show that Q ( B )acts on 3-1 as the characteristic function x ( M ) of M , if 3t is realized via square integrable functions over M , i.e. as L 2 ( M ,d p ) . From the spectral theorem and from Q ( B ( M ) )we infer a quantisation map for C" ( M ,IR)

Q : C"(M,R)

-+ SA('W,

Q(fM = f$-

(3)

Hence we can use in the kinematics C"(M,IR) instead of B ( M ) , that is the kinematics can be viewed as an infinite dimensional Lie algebra, more precisely as a semidirect sum of the abelian algebra C" ( M ,IR) and a subalgebra of the Lie algebra of vector fields, and we denote it as K ( M ) (without tilde) in the following:

K ( M ) = C"" ( M ,R)EVecto ( M ).

(4)

To construct the quantisation map P for Vecto(M) we need further assumptions, which we will call P-assumptions following [l]: 0

0

The partial Lie structure (in connection with complete vector fields) of K ( M ) is conserved. The operator P ( X ) are - in analogy to the canonical quantisation in Rn - local differential operators.

157

With these P-assumptions we have the following result [2]: The P(X)are differential operators of order one with respect to a differential structure on the set M x C. We characterize this assumption (up to isomorphism) through Hermitian line bundles L over M with compatible flat connection V. Wave functions are sections ~ ( min )the bundle and L2( M ,p ) can be viewed as a space of square integrable sections. Unitary equivalent irreducible maps Q(..*”) - quantisations - are given by a bijective mapping onto the set

( a , D ) E 7 q M ) x R.

(5)

$ ( M ) denotes the dual of the first fundamental group of M , a topological quantity. D is connected with the algebraic structure of K ( M ) and characteristic for Bore1 quantisation. (a,D) are quantum numbers in the sense of Wigner. Q is labeled by these numbers, i.e. = (Q(a*D),P(criD)) and one has (m E M ) Q(a3D)

Q(”>”)(fb(m)= f(m)~(m)

+

P(*’”)(X)o(m)= (40% (-ii

(6)

+ D )div,g)

o(m),

(7)

which are self-adjoint operators on a common dense set. Here, V% denotes the connection in the line bundle L ( M ) over M and div, denotes divergence. Note that the quantum number D appears as a real factor in front of div,g and that the nontrivial topology yields the a dependence of ‘7%. 2.2. The dynamics

States of S are modeled via density matrices W , i.e. through trace class operators with Tr(W) = 1. We introduce a time dependence for W (in the Schrcdinger picture), which is based on Q(..*”), through a quantum anaof time dependent log to the classical relation between time derivatives functions f ( m ( t ) )and momenta, i.e. for M = R’

6

d

-f(z(t)) dt

-

PVf .

(8)

One can show [6] that (in the Schrijdinger picture) one has the following relation for expectation values (Exp,(A) = T r ( W A ) ) :

d -Tr(W(t)Q(*’”’(f)) = Tr(W(t)P(”’D) (Xgradf)) ,Vf E Cm(M,R) (9) dt

158

This is a restriction for the evolution of W ( t ) . For pure states it implies, under the condition that pure states evolve into pure states [6], the following generalized version of the first Ehrenfest relation

.

.

with a scalar product (., .) in L2( M ,dp). 2.3. The kinematical algebra K(S1) and a family of evolution equations

We consider now an application of Bore1 quantization to the case that the configuration space is S1. S1 is topologically nontrivial with n;(S1) = [0,27r), and we denote elements in 7r;(S1) as a. The flat line bundles over S1 are trivial, the vector fields are X = X(I$)-& E Vecto(S1) and the Hilbert space is L2(S1,dq). In these coordinates K(S1) is given by the generators

Q(UID)(f)$(I$)

= f(dM4)

P'"'D)(X)$(I$)= (-iX(I$)-&

(11)

+ (+

(y) +ax($)) $(I$X12)

+D)

To analyse the structure of K(S1) we use a Fourier transform F with z = &4: m

n=-w

fn

= f-n, X n = X - n . For the F-transformed quantum kinematics we find w

With the operators

Tn = zn

159

(14) can be expressed as

n=-w

c x, 00

P(*ID)(X)=

(LE

+ iDnT,)

(16)

.

,=-w

The generators T, are an Abelian Lie algebra which we denote as T,and for fixed cr E [0,27r) the L, LE fulfill the commutation relations

=

[Tm,T,I = 0

(17)

[L,, Tm] = mTm+n [Lm,Ln3 = ( n - m)Lm+n

and span an inhomogenisation of the Witt algebra W through T . This [18], where we use the index z to gives the algebraic structure of Kz(S1) indicate that it is given in terms of the variable z as opposed to the angle variable 4. We have from (17) (we have dropped the upper index ( a , D ) for convenience)

[Q(f),Q(g)1 = 0,

[P(X), P ( Y ) l = -.1’P([X,YI). Now we introduce the time dependence for pure states $($) E L2(S1,d4) [P(X),Q ( f ) l = -iQ(Xf),

and we evaluate the restriction (10) with

xgradf

= f’(4)”

d4

(I

G A): d4

This implies a generalized continuity equation of Fokker-Planck type for p = $$:

i

p = &j$” - $5))+ Dp” - crp’ = --(jt)’ + Dp”, d

(19)

where

corresponds for vanishing cr to the usual quantum mechanical current density on S1. This can be derived also by other methods based on Q(a*D)[5], [19],[9]. We use this information in (19) for a general ansatz for a Schrodinger equation of the type

i&$

= H$

+ G[$,$I$

160

in which H is a linear operator and G[$,$] can be written (formally) as a nonlinear complex function G[$,$] = GI[$,$] iG2[$,$] depending on $,$, their derivatives and explicitly on 4 and t . Hence G acts as a multiplication operator. This Ansatz leads to a family Fp of Schrbdinger d4) with G2 enforced by (19) equations [3], on L2(S1,

+

The real part G1 cannot be determined by Borel quantization. Hence a set of (natural) assumptions for G1 motivated by the form of the imaginary part Gz,has been introduced [5], [9]: (1) G1 is proportional to D , i. e. vanishing for D = 0. (2) G1 is a rational function with derivatives no higher than second order and occurring in the numerator only. (3) G1 is complex homogeneous of order zero, i. e. Gl[a!$,&$] = GI [$,$1 for all a! E C. These assumptions restrict G1 in the family Fp to the Doebner-Goldin family (DG-family) 3 D G [5] on S':

with free real parameters Dk, k = 1 , . . . ,5.

3. A discrete quantum kinematics based on a Hopf q-deformation of the kinematical algebra In [l]various q-deformations of the kinematical algebra in Borel quantisation have been discussed. As mentioned earlier, our focus in this contribution is on that version among these (see also [20,21]) which involves a q-deformation of the full kinematical algebra and hence exploits the maximal freedom. Since the corresponding operators are difference operators, it is possible to restrict the configuration space to an N-point discretisation of S1,i.e.

161

3.1. A Hopf q-deformation of the kinematical algebra

We discuss q-deformations of the subalgebras of the kinematical algebra related to the position and the momentum observable separately, and then consider the coupling between these two algebras in a next step. The q-deformation leads to a Hopf algebra, and is therefore called a Hopf qdeformation. 3.1.1. The position observables In the undeformed setting, the subalgebra of the kinematical algebra that corresponds to the position observables in the framework of Bore1 quantisation is given by the generators {T,} in (17). In the q-deformed setting, we augment this basis to {Tn,K,"},where the Abelian subalgebra K := {K,"} with s E Z corresponds to q-shift operators, i.e. operators that act by rescaling the arguments of a function f(x) according to

where x,q E S1.They have the property:

[K,",K;,] = 0,

(24)

with limq+l K," = 1. Moreover, the coupling to the generators {T,} is given by

K:T, = qsnTnK,Q

(25)

The q-analog to the generators T, is given via a restriction of the universal enveloping algebra of the algebra generated by the basis {T,, K,"}to monomials of the form T2,5:= T,K,". The generators in the q-deformed setting are hence related to the undeformed generators via the mapping MZ, which restricted to the subalgebra related to the position observables acts as follows:

The generators T& form a non-commutative algebra with the following commutation relations:

162

3.1.2. T h e m o m e n t u m observables The subalgebra of the kinematical algebra related t o the momentum observables is given by the Witt algebra with basis {L,} and commutation relations as in (17). We consider here a q-deformation, where the q-Witt algebra is generated by the basis {C&} with the following commutation relations:

As expected for consistency, these commutation relations reproduce the commutation relations of the Witt algebra in the continuous limit q + 1. The extra parameter j , which has been introduced during the deformation, vanishes in the limit q + 1. The generators in the q-deformed setting are hence related to the undeformed generators via an extension of the mapping M i to the subalgebra related to the momentum observables M i :

M:,i : Lz

Cz,i, (29) where n,j E Z and a E [0, 1). We hence have a mapping Mq = { M i ,MB,j} that acts on the full kinematical algebra. H

3.1.3. Coupling between the subalgebras: the Hopf q-deformation of the kinematical algebra As a first step, we consider the coupling between the q-Witt algebra in (24) and the algebra of shift operators in (28). One obtains a quadratic Lie algebra with basis {Cz,,, K & } ,with

K,QCZli = q""C"n93.Kf

(30)

It is a noncommutative and co-commutative Hopf algebra - Hopf-q-Witt algebra - 3tWq with coproduct A, counit c and antipode y given as follows:

A ( C g , j )= Cg,j 8 K&

+ K& 8 L$,j , A(KZ)= KZ 8 KZ

E ( C ; , ~ )= 0 , e(KZ) = 1 y(Ck,j) = -(K&)-'Ck,j(K&)-' , y(K;) = (KZ)-'

(31)

Further, we should couple the position observables {T,} . The latter have trivial co-algenra structure: A(T,) = T, 8 1 1 8 T, , e(Tn) = 0, y(T,) = -T, . Thus, one obtains the inhomogeneous Hopf-q-Witt algebra

+

163

with basis { L z , k K , L , Tz,,} and co-algebra relations of Tz,, : A(Tz,,)= T$,,@Kj+Kj@Tz,,, €(Ti,,)= 0, ~ ( 7 2 = , ~- ()K j ) - " T & ( K j ) - ' . The q-deformed kinematical algebra is then given in terms of the basis { L z , k , via a restriction of the monomials in the universal enveloping algebra of { K k ,T,} to {Tz,,},and the commutation relations are as follows:

x}

, is quadratic. Hence the deformed algebra with basis { L z , k Tz,,} We note that a restriction to {L&,Tm} would lead to the results discussed in [l],and it is the purpose of this contribution to discuss the changes in the quantum kinematics and quantum dynamics implied by the occurrence of the shift operators K L in the subalgebra related to the position observable. 3 . 2 . The quantum kinematics on S1

In this subsection we discuss possibilities to associate position and momentum operators to the q-deformed kinematical algebra in the previous subsection. Since on S1the position operators Q ( " > D ) (inf )(16) depend on the generators Tn and the momentum operators P("vD)(X)in (16) depend on the generators Tn and LE,j, a q-deformation of the quantum kinematics can be achieved based on the mapping M Q in (26) and (as),because it relates the basis of the kinematical algebra in the undeformed setting with the basis of the q-deformed kinematical algebra. In particular, M8,j induces the following mapping M b K from the undeformed quantum kinematics in (16) to the Hopf q-quantum kinematics induced by the q-deformation of the kinematical algebra.

M ~ : KQ ( f ) ++ 0;C.f)= C:='=_, fnM:(Tn) P ( X ) ++ P i ( X ) = Cz='=_, Xn (M:,j(LE) i w D M p ( T n ) ) .

+

(33)

{ M $ K ( Q ( f ) )M, b K ( P ( X ) ) }will be called the Hopf q-quantum Borel kinematics induced by Mq = { M : , M:,j}. Indeed, the mapping M $ K depends on two further parameters, j , s E Z, and expressions for the position and momentum operators in the Hopf q-quantum Borel kinematics are given by

164

where K,, respectively K,, is short-hand notation for the shift operators K,“ and K;. For each different choice of the deformation parameter q one hence obtains a different quantum kinematics. Of particular interest for applications are the cases where q is a root of unity, because in this case, the Hopf q-quantum Borel kinematics leaves the N-point discretisation Sh of the configuration manifold S1 invariant. In particular, in this discrete setting the parameter j in Lila) and s in K , have the following interpretation: contains - dependent on j - differences between different points of Sh,e.g. between next nearest neighbours for j = 2 or even further points, i.e. it measures how coarse-grained the discretization is. Similarly s is a measure for how coarsegrained the jumps initiated by the shift operators K , are. 4. The quantum dynamics with Hopf q-structure

In this section we derive a quantum dynamics that is compatible with the Hopf q-quantum Borel kinematics in (34). As an important building block in the derivation, an appropriately defined q-version of the first Ehrenfest theorem is necessary. In order to derive it, we introduce a symmetrization of T,K, and K,T, as follows: 1 Sm,s := -(TmK.g 2

+ KsTm).

(35)

It is given in terms of the shift operators K , and leads to the generators T , in the limit q + 1 as expected for consistency. Based on the symmetrised coordinates, we furthermore introduce the following symmetrised scalar product ( @ , S 4 ) s g r n :=

1 z{(@, S4) + ( S @ 4)) ,

7

(36)

where (,) denotes the usual scalar product as in section 2, S denotes a shift operator, S is its complex conjugate. This symmetrisation of the dependence on the shift operator S is sufficient to ensure that the quantum mechanical probability density p = is a real quantity in the deformed setting; without this symmetrisation, this would not be the case. With this definition of a symmetrised scalar product, the Hopf q-version of the first Ehrenfest relation reads:

165

where

Note that the coefficients r and s, that define the shift operators, are independent. 4.1. Derivation of a nonlinear q-Schrodinger equation

In order to derive the Hopf q-quantum dynamics, the left and right hand side of (37) have to be evaluated explicitly. 4.1.1. The right hand side of (37)

With N , as short-hand notation for zdZ,q-shift operators are given as qaNz and act on functions f(z) via

(40) Then the right hand side of (37) can be evaluated to yield an expression of q a N z f ( 4= f(q."z).

the form ( f , B )in the usual scalar product with

166

In order to obtain the last term in (41) we relate K, and K , such that K,Ks = 1. This ensures that the last term is a function of p = In the following we restrict ourselves to the case of a = 0 to keep the argument transparent. Furthermore, the a-dependence is not essential for the derivation of the nonlinear parts, so that it is not crucial for the main result, which is the derivation of possible real parts. In terms of $1 and $2, where $ = $1 i $ ~ , one finds for the part not depending on the parameter D in (38) the following expression:

4$.

+

4.1.2. The left hand side of (37) The left hand side of (37) can be computed with (13) to be of the form

at ($(t,z ) , Qi(f)$(t,z

)

) =~ at~($(t, ~ 21, f (z)Ks$(t,z))sym

~&($,f(~S+ d )((KS$Lf4)1 (fJt;(4FS$) + (KS4M)) =: (f,4 = =

(43)

where the real-valued functions f are time independent and hence are not affected by the operator at. The last line in (43) contains the usual scalar product.

167

In order to determine an expression for the quantity A in (43), one needs an ansatz for a nonlinear q-Schrodinger equation. We use an ansatz with H i linear in $ and Gi[$,$]nonlinear in $, 4:

i(&$)$ = (Hi$)(S$) + (G{[$,$l$)(R$)

(44)

S and R are shift operators like in (40),with parameters a in (40) given as a, and a,, respectively. Such shift operators typically occur in a q-deformed theory. (44) reduces to the corresponding expression for the evolution equation given in section 2 in the limit q + 1, if H! and Gi[$, $1 are such that they give H and G[G,$] in this limit. With (44) we get

4.2. The q-deformed evolution equation

Fulfilling the Hopf q-version of the first Ehrenfest relation (37) means equating the expressions for A and B in (42) and (45). We will consider linear and nonlinear terms independently.

The t e r n s connected with H; Assuming now that the linear operator H i , denoted by H , and the corresponding shift S are real, and using $ = i q ! ~ we ~, obtain that the terms on the right hand side of (45) involving the linear terms H are given by :

+

-(HK,$1 ) ( W 2 ) . (46) Equating this expression with B' in (42), we find uniquely ( H Q 2 )(SKS$JI)+(HK,$J2) (S$l)- (H$1)(SK,$2)

where E = f l . In particular, since K , = q s N z ,the parameter s is fixed in dependence on j . Note that these results differ from the nonHopf case considered in [3]. The formalism to derive a q-Schrodinger

168

equation corresponding to the ansatz (44) hence requires the selection s = Reinserting this into the Hopf q-quantum Bore1 kinematics in (34) means that the freedom in the position and momentum operator is not independent and linked precisely by this condition on s via the dynamics. The resulting part of the q-SchrBdinger equation is hence ( j E 2N) :

%.

(

f (qNz + q - N z ) 4.

(Hi$)(S$) = [ j N z ][ j % ]

(48)

The terms connected with Gi We focus now on the nonlinear terms Gi. In order to derive them, the terms in (45), which contain a dependence on G{[$, $1 := G1+iG2, are collected. The imaginary terms cancel as expected and one obtains with the R = R1 + iR2, $ the following real quantity:

GIhl(d1, $2,

+ G2h2($1, $2 ,R1,R2, Ks)

R1, R2, Ks)

(49)

where h1($1,$2,Rl,Rz,K,)

+ (Ks$2)(R2$2)

= (KS$l)(R2$1)

+(KS$2)(Rl$l)

- (KS$l)(R1$2)

+(41)(R2&$1) + ($2)

+($2 1(R1Ks$1)

(50)

(R2Ks$2)

- ($1 ) (R1Ks$2 )

and h2($1,42,Rl,R2,KS)

= (Ks$l)(Rl$l)

+ (KS$2)(&$2)

+(KS$l)(R2$2) +($l)(R~Ks$l)

+($1

(R2Ks$2

- (KS$2)(R2$1)

+ ($2)(R1Ks42) - ($2

(51)

(R2Ks$l).

It has to be equated with the terms proportional to D in (41). The corresponding identity implicitly contains GI, G2, as well as the shifts R1 and R2. Depending on the shifts R1 and R2, the components of the nonlinearity G1 and G2 can be obtained from the nonlinear terms:

FNL 3 (Gi[$,$]$)(R$)

= ((GI

+ iG2)$)(R1 + iR2)4.

(52)

In particular, the nonlinear q-Schrijdinger equation is then given as ( j E 2N, € 1 , €2 = f l ) :

i(&$)4= (q$)(s4) +FNL

(53)

169

with H{ and S as in (48). It is important to stress that the nonlinear term FNL depends on the choices for R1 and R2, and indeed for different choices of R1 and R2 different types of nonlinear terms are obtained. 4.3. The continuous limit

In this subsection we consider the continuous limit of our family of nonlinear q-Schrodinger equations. As before, linear and nonlinear case are treated separately.

The linear terms in the continuous limit A restriction to the linear terms in the q-evolution equation leads for all j in the limit q + 1 to the Hamiltonian obtained in section 2. It resembles the Hamiltonian obtained with the more restricted deformation in [l] up to a factor q e z j % , which now is compensated by the operator K,. The nonlinear terms an the continuous limit The limit q + 1 of the linear part of the evolution equation (48) has already been discussed above. In order to obtain the nonlinear part in the limit q + 1, it is necessary to expand the functions hl($l, $2 R1 ,R2, K,) and h2 ( $ 1 , $ 2 , R1 ,R2 K,) as well as the nonlinear terms in (42) in leading orders of h, where h is given implicitly by q = eh. For the nonlinear terms in (41) one finds the sum of the following two terms: B(1) = & $ / $ , / / I - $,/$///)h2 + O(h3) (54) B(2)= -Dp” + O(h)

-is

+

+

+

Moreover, using the ansatz R = R1 iR2 = (aqQNz bqBNz) i (cqTNZ dqdNZ)( a , b, c , d, a , ,f?, y, S real constants), one derives the expression

+

hi ($1

,$2, Ri, R2, Ks)= 2(c + d)($

+ $2) + 2h

((c + dk2; f CY +a) ($l$i + $ 2 4 ) +(aa f W ) ( $ 2 $ , ; - $I$;) + O ( h 2 ) . (55) A similar expression for h2 ( $ 1 , $ 2 , R1, Rz, K,) follows from (55) using hz ($19 $2, Ri ,Rz, Ks) = hi ($1, $ 2 , R2, Ri ,Ks). Based on these expansions one finds that in the limit q + 1 one always obtains the same imaginary part for the nonlinear functional.

170

It coincides with the expression that has been derived without qdeformation in the framework of Bore1 quantisation. In addition, one obtains results on the real part of the nonlinear functional. For nontrivial shift operators R, that is in the case R # 1, the following class of real parts occurs: Dp" 2P . Otherwise, if R = 1 is the trivial shift operator, the real part remains undetermined like in the undeformed setting as expected for consistency. 4.4. Comparison with previous results

The results derived for the real part of the nonlinear functional in the limit q + 1 is sensitive to the choice of shift operators in the q-deformed setting. In particular, q-deformations of the momentum subalgebra of the kinematical algebra as considered in [l]lead to results different from the ones that arise under a Hopf q-deformation of the full kinematical algebra and a related symmetrisation of the first Ehrenfest relation. The reason for this discrepancy lies in the fact that here one has obtained nonlinear terms in leading orders of h2 in (54), whereas the counterpart to this formula in [l]is given in leading orders of h, which have cancelled each other here due to the contributions from the symmetrisation. In this way, the formalism presented here contains more information and leads to more specific results. It is interesting to note that the real part derived here lies in the DG-class of real parts introduced in section 2. In contrast to this, there are two nontrivial classes of real parts for the non-Hopf q-deformation in [l],one of which coincides with the class derived here, and another one which does not fall into the DG-classes. The Hopf q-deformation of the full kinematical algebra together with a related symmetrisation of the first Ehrenfest theorem has, on the contrary, led to a specification of only one type of real part for the nonlinear functional which coincides with one of the classes in the DG-family of nonlinear Schriidinger equations. Moreover, it has led to the fact that the coordinates, that is the operators pn,are no longer commutative, and the Hopf q-deformation has hence resulted in a noncommutative theory. A method to derive the quantum kinematics on Sy through difference operators which is based on spectral triples is given in [22].

171

A further interesting feature of an approach related to a Hopf qdeformation of the full kinematical algebra is the fact that the q-SchrBdinger difference equations (53) adopt a more symmetric form due to the symmetrisation procedure. In particular, one obtains the following family of difference equations in dependence of j :

For comparison with the results related to a q-deformation of the momentum subalgebra of the kinematical algebra, see (96) in [l]. 5. Applications to LQvy processes Here we outline how to apply our formalism to Levy processes. Our starting point is the relation between the algebraic structures of the formalism developed here and results on Levy processes on compact and noncompact quantum groups obtained in [16,17]. First we note that in both formalisms there are Abelian subalgebras. In our case this is the infinitely generated Abelian Hopf algebra K . This could be related to the Cartan subalgebras K of quantum groups generated by group-like elements Ici, ki', (i = 1 , . . . ,n ) , in the formalism of Jimbo [23]. Since Cartan subalgebras are sub-Hopf-algebras of quantum groups G, the restriction of a Ldvy process on 6 to K is still a Ldvy process. We further restrict t o any of the group-like elements and relate the resulting Lkvy process with the subalgebra generated by any of our generators K j . For group-like elements one can use [12] to get explicit expressions without having t o solve any quantum stochastic differential equations, cf. [16]. Ldvy Processes on V,(G) have been considered for G = d ( 2 ) [16] and this may be applied to the subalgebra of (17) generated by L1,L-1,L0, which is isomorphic to the algebra sZ(2). Furthermore, there should be some relation between the real forms of sZ(2) and their quantum group deformations with real forms and deformations of subalgebras of the Witt algebra. Acknowledgments

V.K.D. and R.T. would like to thank the Alexander von Humboldt Foundation for financial support in the framework of the Clauthal-Leipzig-Sofia

172

Cooperation. V.K.D. was also supported in part by the Bulgarian National Council for Scientific Research grant F-1205/02 and R.T. by an EPSRC Advanced Research Fellowship.

References 1. V.K. Dobrev, H.-D. Doebner and R. Twarock, “Quantum Mechanics with Difference Operators” , Rep. Math. Phys. 50, 409-431 (2002). 2. B. Angermann, H.-D. Doebner and J. Tolar, , “Quantum Kinematics on

3.

4.

5. 6.

7.

8. 9.

10. 11.

12. 13. 14. 15. 16.

17.

Smooth Manifolds”, in: Lect. Not. Math. vol. 1037 (Springer, 1983) pp. 171208. V.K. Dobrev, H.-D. Doebner and R. Twarock, “A discrete, nonlinear qSchrodinger equation via Bore1 quantization and q-deformation of the Witt algebra”, J. Phys. A: Math. Gen. 38,1161-1182 (1997). R. Twarock, “A q-Schrodinger equation based on a q-Hopf deformation of the Witt algebra”, J . Phys. A: Math. Gen. 32,4971-4981 (1999). H.-D. Doebner and G.A. Goldin, “On a general nonlinear Schrodinger equation admitting diffusion currents” , Phys. Lett. A162, 397-401 (1992). H.-D. Doebner and J.D. Hennig, “A quantum mechanical evolution equation for mixed states from symmetry and kinematics”, in: Symmetries in Science VIII, eds B Gruber (Plenum Publ., New York, 1995) pp. 85-90. H.-D. Doebner and P. Nattermann, ,“Bore1 quantization: Kinematics and dynamics”, Acta Phys. Pol. B27, 2327-2339 (1996). P. Nattermann, W. Scherer and A.G. Ushveridze, “Exact solutions of the general Doebner-Goldin equation”, Phys. Lett. A184, 234-240 (1994). H.-D. Doebner and G.A. Goldin, “Properties of nonlinear Schrodinger equations associated with diffeomorphism group representations ” , J . Phys. A : Math. Gen. 27, 1771-1780 (1994). L. Accardi, M. Schiirmann and W.V. Waldenfels, Quantum independent increment processes on superalgebras, Math. 2. 198,451-477, 1988. M. Schiirmann, “A class of representations of involutive bialgebras”, Math. Proc. Camb. Philos. SOC.107,149-175 (1990). M. Schiirmann, White Noise on Bialgebras, Lecture Notes in Math. Vol. 1544 (Springer-Verlag, Berlin, 1993). U. F’ranz and R. Schott, “Diffusions on braided spaces”, J. Math. Phys. 39, 2 7 4 ~ ~ (1998). 6 2 U.Franz and R. Schott, “Evolution equations and Lkvy processes on quantum groups”, J . Phys. A: Math. and Gen. 31, 1395-1404 (1998). U. Franz and R. Schott, Stochastic Processes and Operator Calculus on Quantum Groups. (Kluwer Academic Publishers, Dordrecht , 1999). V.K. Dobrev, H.-D. Doebner, U. Franz and R. Schott, “Lkvy processes on U,(g) as infinitely divisible representations”, in: Probability on Algebraic Structures, eds. G. Budzban, Ph. Feinsilver and A. Mukherjea, Contemp. Math. vol. 261 (American Math. Society, 2000) pp. 181-192; [math.PR/9907016]. V.K. Dobrev, H.-D. Doebner, U. Franz and R. Schott, “Lkvy Processes on

173

U,(G)”, in: Proceedings of the International Workshop ”Lie Theory and Its 18.

19.

20. 21.

22.

23.

Applications i n Physics 111, (Clausthal, 1999); eds. H.-D. Doebner et all (World Scientific, Singapore, 2000; ISBN 981-02-4421-5), pp. 280-292. H.-D. Doebner and J. Tolar, “Infinitedimensional symmetries” , A n n , Phys. (Leipzig) 47, 116-122 (1990). H.-D. Doebner and G. Goldin, in: Proceedings of the First Geman-Polish Symposium on Particles and Fields, (World Scientific, Singapore, 1999) p.115. H. Hiro-Oka, 0. Matsui, T. Naito and S. Saito, “On the q-deformation of the Virasoro algebra”, TMUP-HEL-9004, (1990). S. Saito, “q-Virasoro and q-strings in: Quarks, Symmetries and Strings, eds. M. Kaku, A. Jevicki and K. Kikkawa (World Scientific, Singapore, 1991), pp. 231-240. H.-D. Doebner and R. Matthes, “Remarks on Spectral Triples Related to Difference Operators” in: Proceedings of the Fifth International Workshop Lie Theory and its Applications in Physics V, eds. H.-D. Doebner and V. K. Dobrev (World Scientific, Singapore, 2004), to appear. M. Jimbo, “A q-difference analogue of U ( g ) and the Yang-Baxter equation” Lett. Math. Phys. 10,63-69 (1985). ” )

ON ALGEBRAIC AND QUANTUM RANDOM WALKS *

DEMOSTHENES ELLINAS Technical University of Crete Department of Sciences, Division of Mathematics, GR-731 00 Chania Crete Greece E-mail: ellinasOscience.tuc.gr

Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are investigated and related. Based on minimal data provided by the underlying bialgebras of functions defined on e. g the real line R, the abelian finite group Z N , and the canonical Heisenberg-Weyl algebra hw, and by introducing appropriate functionals on those algebras, examples of ARWs are constructed. These walks involve short and long range transition probabilities as in the case of R walk, bistochastic matrices as for the case of Z N walk, or coherent state vectors as in the case of hw walk. The increase of classical entropy due to majorization order of those ARWs is shown, and further their corresponding evolution equations are obtained. Especially for the case of hw ARW, the diffusion limit of evolution equation leads to a quantum master equation for the density matrix of a boson system interacting with a bath of quantum oscillators prepared in squeezed vacuum state. A number of generalizations to other types of ARWs and some open problems are also stated. Next, QRWs are briefly presented together with some of their distinctive properties, such as their enhanced diffusion rates, and their behavior in respect to the relation of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is investigated in terms of the theorem of unitary extension of completely positive trace preserving (CPTP) evolution maps by means of auxiliary vector spaces. It is applied to extend the CPTP step evolution map of a ARW for a quantum walker system into a unitary step evolution map for an associated QRW of a walker+quantum coin system. Examples and extensions are provided.

1. Introduction

Random walks formulated in an algebraic f r a r n e ~ o r of k ~finite ~ ~ groups, ~~~~ bialgebras and operator algebras as well as in the framework of Quantum Mechanics,5~6~7i8~g~10i11~12~13 and references therein), are investigated. Min'Based on talk given in: Volterra-CIRM-Grefswald International Conference on Classical and Quantum Levy Processes: Theory and Applications, Trento Italy 27 Sept. - 3 Oct. 2003.

174

175

imal data for such constructions consist of a bialgebra14 and an integral (functional) defined on it, or alternatively of some Lie algebra, and two quantum systems modelling the walker and the coin system, together with a map modelling the coin tossing, that decides probabilistically the stepping of the walker. Examples of ARWs treated in the following subsections are walks on algebras of functions on R, on ZN, and on the canonical algebra HeisenbergWeyl hw15(section 2). For those walks we show how to define the entropy functional of their respective integral and/or Markov transition operator, and how to deduce that these are entropy increasing random walks by using arguments based on the interrelations relations between majorization bistochastic matrices and e n t r ~ p y . ' ~Moreover, ~ ' ~ ~ ~ ~ ~ ~ ~theoretic dea number composing of ARW on ZN is analysed, that is called prime decomposition,20 and refers to its factorization into products of similar smaller ZN-walks. Mathematically this decomposition is based on the Chinese Remainder Theorem and the co-associativity property. Also for the ARWs on R, in addition to the usual case of short range walk with nearest neighbor (NN) transitions (Polya walk),21 we discuss in our algebraic framework the cases of i) the NN centrally biased random walk (Gillis walk)22 and the case of symmetric random walk with exponentially distributed steps (LinderbergShu1er:LS walk).23 Finally, for the h w ARW, where its functional is constructed by means of the eigenstates of the annihilation operator of that algebra i.e the family of coherent state v e c t ~ r s ,the ~ ~continuous ,~~ time, or diffusion like limit, is obtained.15 This limit results into a trace preserving quantum master e q ~ a t i o n ~ for~ the 3 ~ density ~ matrix of a quantum boson system, which physically is identified with the evolution equation of an open quantum boson system interacting coherently with a classical electric field and incoherently (dissipative interaction) with a bath of quantum oscillators rigged initially into a squeezed vacuum state (squeezed white noise) .28,29,30 Section 3, gives a concise prescription of the concept of QRW, using the example of QRW on integers as paradigm.13 It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and coin systems, and the entropy increase without majorization effect of its probability distributions (pd). This section ends with a group theoretical scheme of classification of various known QRWs. In section 4, a relation connecting ARW and QRW is put forward. The

176

connection is grounded on the theorem due to Naimark that asserts the possibility of implementing in a unitary way a CPTP map operating on e.g the density operator of a quantum system.31 This unitary extension is realized in the original vector space of the density operator augmented by an auxiliary vector space, the ancilla space in the terminology of Quantum Information theory.32 Applied in the context of CPTP of a ARW,the ancilla space is identified with the quantum coin state space of a QRW.The Kraus generators determining the CPTP map of a ARW serve to built, albeit in non unique way, the unitary evolution operator of the associated QRW. This section concludes with the example of an explicit construction of a Q R W associated to the h w ARW,of section 2. Finally, section 4, summarized some of the results and gives some prospected applications of the ARW-QRW concepts and formalism.

2. Algebraic Random Walks

2.1. The case of R

=

Proposition 1. Let the bialgebra of real formal power series H Fun(R) generated by the coordinate function X , and let the positive definite functional 4 : H + R, defined as q5 = CiEZ pi&, with 0 5 pi 5 1, CiEZ p; = 1, where q5ad(f(x)) = f(cq), the functional that evaluates any function f E H , at the point cui = icu, defined for some step cu E R+, for i E 2. The n-step convoluted functional becomes

iEZ

where p(") = Dn-'p('), with the stochastic column vectors p ( k ) = (pi ( k )) i E ~ , = k 1 , 2 , 3,... and initially p ( l ) p = ( p i ) i E z . Also D = ( D i j ) = ( p ; - j ) , i,j E 2 a bistochastic infinite matrix (delta matrix), and 4 = ( 4 , ; ) i ~azcolumn vector. Majorization ordering among pd's is valid at each step i.e p("+l) 4 p("),and consequently the ARW is entropy increasing, namely, S(C$*"+') 2 S(q5*"), S(Tt+') 2 S(T;), where S($*") S(T;) S@")), with S(z) any Shur-convex function e.g the classical entropy i.e S(z) = 2i log 2;. Proof: Operating with convoluted functionals on some function f E

-

=

xi

177

H

yields

By induction we obtain the aimed relation $*" = $Tp(") = $*D"-lp('). Two important properties of the delta matrix are: i) bistochasticity, i.e the column and row sums is one, which is expressed by means of the column vector of units e = (. . . , 1 , 1 , 1 , . . .), that is left and right eigenvector of D, i.e De = e, eT = eTD, and ii) the shift property i.e D i j = pi-j = pi+l-(j+l) = Di+l,j+l. This property in the case of ARW in 2 governed by a pd with finite support, or in the case of a N C 2 finite dimensional walk (see remark below), amounts to a bistochastic matrix D. The functional of the walk at each time step c#P = CiEZpin)$ai, is characterized by the pd p(") = (pin))iEz, which in turn is determined by the bistochastic matrix D i.e p(") = Dp("-l) = D"-'p('). Let us assume that the pd's are of finite support (but see remark below), then by invocation of the theorem stating that two discrete pd's x = ( z ~ ) ~ , y = ( Y k ) k , that are connected by a bistochastic matrix D i.e x = Dy, are ordered by majorization i.e x 4 y, we conclude that the sequence of pd's { ~ ( ' ) , p ( ~ ) , .p. (.}~ ,) ,of site occupation probabilities resulting at each time step during the evolution of the walk is partially ordered by majorization i. e p(") 4 p(n-11, n = 1 , 2 , 3 , . . . . Let us adopt now the definition of the entropy of a functional to be the entropy of the pd that determines that functional i.e we set S($*") z S(T$) S(p(")),or more generally we do so for any convex function S : R + R of the type of the so called Shur-convex functions e.g the classical Shannon entropy S(p) = x i p i l o g p i , or the functions F ( p ) = x i p t , for any constant k 5 1, or F ( p ) = - nipi.17 By virtue of the theorem stating that x 4 y implies F(x) < F(y) , where F ( x) = x i f ( x i ) , for any convex function f : R + R, or otherwise said that the convex functions isotonic to maj~rization,~' we conclude that for the pd resulting from the random walk the majorization ordering is valid at each step i.e. p(") 4 p("-'), n = 1 , 2 , 3 , . . . , and this implies ordering for e.g their entropies i.e. ,903") > S(p"-l), and similarly for their functionals and transition operators. As majorization order implies entropy increase, it is considered as a measure of disorder, and this allow us to con16917318119

=

178

clude the ARW are getting more disordered in the course of time with respect to their site-visiting pd’s, which are getting more entropic, approaching, if left uninterrupted, to the uniform distribution of maximal entropy.

Remarks: 1) The assumption in the previous proof about the support of the pd’s been finite is not actually necessary. In fact in the proof given by Hardy et. a141 is stated that given p 4 q for finite sequence of q , p pd’s, use of Muirhead’s algorithm leads to a bistochastic matrix A , such that q = Ap, and then the proposition: H ( A p ) 2 H ( p ) for H information/entropy or more generally a Shur convex function, is applied. This proof is constructive and builds A in a number of steps not greater than the lengths of p , q , therefore for infinite pd’s a theorem not using bistochastic operators for the characterization of majorization is needed. Such a theorem is provided in Ref. 42. 2 ) For the above proposition the corresponding Markov transition operator defined as T+= (4 @id) o A, is equal to T4 = C i E Z p i e a i A For . the simplest case of p* = ( p , 1 - p ) , a&= f a , and all other p’s and a’ s been zero, the continues limit limn--too = (T4)n T r has been obtained that leads to a diffusion equation.33s334s15 3) The above general algebraic setting implies that the stepping probability matrix D = ( D i j ) = (pi-j), i , j E Z, would be expanded in the enveloping algebra U ( e ( 2 ) ) of the Euclidean Lie algebra e ( 2 ) M is0(2), spanned by monomials of its generators { E+,E- ,L } , that satisfy the defining commutation relations43

An irreducible matrix representation of those generators in the Hilbert space 31 Zz(Z) spanned by the eigenvectors of the ”distance operator’’ L, is useful in expressing the D matrix for various random walks, and look for solutions by means of e.g the Fourier method. This irrep in the canonical basis of 31, and using the same symbol for abstract generators and their matrices reads:

=

mEZ

mEZ

Next we present three different random walks in R, that can be used as show cases of the scheme of ARW presented here. In these concrete examples the defining the walk transition probability matrix D ,is written as an element of the U ( e ( 2 ) ) algebra. No attempt will be made t o give an

179

algebraic solution for the problem of finding the n th-step site occupancy probability distribution, as this can be solved by other means. The examples include: i) the simplest case of symmetric nearestneighbor (NN) random walk (Polya-walk2’); one of its deformations, ii) the NN centrally (site n = 0) biased random walk (Gillis-walk22), which refers t o a solvable case of a walk with no translational invariance, and stepping probabilities with a bias which has power law decay, or more specifically which decays in proportion from the origin of coordinates. The &-deformation parameter is chosen so that when E > 0, the walk is biased to enhance returns to the origin, while if E < 0, escape from the origin is enhanced, and iii) a symmetric random walk with non- nearest-neighbor transitions with transition step length decaying according to an exponential law ( L S - ~ a l k ~Explicitly ~). we have: 1 ) Polya-walk: symmetric nearest-neighbor ( N N ) random walk with Markov transition operator 1

D p = -(E2

+ E+)

(5)

with matrix elements the inter-site transition probabilities 1 1 DP(l,l ) = p l J I - 1

1

+ 24,,I+l.

(6)

2 ) Gillis-walk: nearest-neighbor centrally (site n = 0) biased random walk with Markov transition operator 1 D G = D G ( E )= -2( E - + E + ) + ( E - - E + ) z N E p , ,l -1<&<1(7)

where PO = eoei, P,,-’- = 1 - PO,the projection operators in the vector eo, and its orthogonal subspace respectively in 12 (Z), and with matrix elements the inter-site transition probabilities

3 ) LS-walk: symmetric random walk with exponentially distributed steps with Markov transition operator

180

with matrix elements the inter-site transition probabilities

= 0,

1=

i.

Use of the eigenvector equations E*e = e, et = etE*, leads to the conclusion that the transition matrices of the Polya, and LS random walks i.e the matrices D p , DR, and DLS respectively are bistochastic, while that of the Gillis walk DG, is column stochastic. Also from the definitions the following two limits are deduced: lim,,o D G ( E )= limE-.,m D L S ( E )= D P . Generalizations: The 2D generalized Gillis random walk with entanglement. This model describes a 2D NN random walk with a biased towards a point placed at (ml,m2) coordinates on the plane. The bias is an attractive or repelling one depending on the sign of two parameters (€1, ~ 2 )in reference t o the motion along z,y axes respectively, and its strength decays following an inverse power law with characteristic exponents ( a l ,a2) correspondingly. This is summarized by writing explicitly all parameters in the Markov transition operator DgD(&l, ml, U I ; E Z , m2,u2), and their , j , u j } E { (-1, l ) , Z, Z+}, j = 1,2. The transition domain of values { ~ j m matrix along each axis is

1 D ~ ( ~ j , m j ,=~ -(Ej) 2

+ E+) + (E- - E+) 2 ( N - j E

mjl)aj

P;,

j=1,2

(12) where Pj = eje;, Pk = 1 - Pj, the projection operators in the vector ej, and its orthogonal subspace respectively in /2(Z). The 2D transition operator consists of an entangled32 convex combination of two factorizable 1D transition operators with the parameters of position of bias site, characteristic decay exponent, and decay strength c.f { m j , a j , ~ j , }j, = 1 , 2 interchanged, it reads (0 5 Q 5 1)

with matrix elements the inter-site transition probabilities

181

(14)

(Zi,Zi)

(Zi,Zi)# (mz,ml),for

with # (ml,m2), for the q component and the (1 - q) component of the convex combination. Also

1 1 D&D(Z1,2; = m1,2;21,1; = m2,i) = q(5&l,ml-i + 2&1,ml+i)

the matrix elements with (Z;,li) = (ml,mz), for the q component and (Z1,Z;) = (mz,ml), for the (1- q ) component of the convex combination. The role of bias and that of the entanglement of the two 1D walks, can be investigated in a n algebraic manner in terms of tensor product representations of the iso(2) algebra, and it will be given elsewhere.44 2.2. The case of ZN

In this section we give a brief study of algebraic random walks on abelian groups ZN,3 using their underlying bialgebra structure, and further investigate possible forms of their decomposition into simpler and dimensionally lower ARWs, based on number theoretic properties of N . Proposition 2. Let the multiplicative abelian group ZN = {e,g,g2,. . . ,gN-'} and the bialgebra H C(Z,) F u ~ ( Z N )= C N-1 span({gi}i=o ), with dual algebra H* C Z N Fun(Z&), with pairing given by evaluation. Let the positive definite functional (state) of a ZN random walk I$ = &o,N-llpigZ, identified as weighted sum of elements of H , with 0 I pi I 1, &O,N--ll pi = 1. By means of the column vectors p = ( p i ) i ~ [ o , ~ - l ] ,= G (gi)iEIO,N-l],we write it as I$ = GTP,where T denotes transpose. Then the n-th step convolution becomes I$*n = GTDn-l p, where D = (Dij) = ( ' p i - j l m o d ~ ) , i, j E [0, N - 11 a circulant bistochastic matrix, to be called delta matrix.

=

Proof: Straightforward (c.f. Ref. 3).

=

=

182

Remarks: 0) The delta matrix is more precisely a circulant bistochastic matrix,45 that can be treated as an element of finite Heisenberg groupHN, 2o and this leads to an explicit solution for the dynamics of ZN walk. 1) Recall the following version of the Chinese Remainder Theorem (CRT)46: let N = N1N2, the decomposition of positive integer N , into product of coprimes N1, N2, then working with the abelian additive groups of numbers Z N ~ ,mod ~ , N1,2 respectively, we can introduce the unique bijection

6 : ZN

+ ZN, x

ZN,,

6 ( ~=) (x - p N 1 , ~ aNz),

(16)

p , a E 2, that maps the numbers of Z N , into the ordered pair of its

remainders after division by N1, N2. Its dual map is the inverse p 6-’ = ZN, x ZN, + ZN,which constructs the unique number x from its remainders a,b), with respect to the divisors N1,N2, as p ( a , b ) = aNT(N1)+ bNT N z ) = x , where cp(c) = # Ic 2 m E Z+; GCD(m,c ) = 11, the Euler function of given integer c, that equals the number of co-primes

r

less or equal t o c. 2) The same factorization is valid for abelian multiplicative groups i.e ZN ”= Z N ~@ Z N ~ if, N = NlN2, and N1, N2, are relative primes. 3) Let for later use introduce now the map 6 + Va, that uniquely determines from the CRT bijection 6, with 6 ( i ) = ( i l , i 2 ) , the isometric matrix & : C N + C N @ ~ C N ~written , in the canonical basis as20

and its inverse

for which we have

and

Notation: in the sequel and in order to distinguish the space dimensionality N , referring to certain e.g functional, operator, probability vector, co-multiplication etc, we will denote it by $“I, T“],and P [ N ] respectively. We can now state two necessary and sufficient conditions, in order to obtain an isomorphic prime decomposition of a ZN ARW governed by a pd +I, into a product of two others ARWs Z N ~ Z, N ~ with , respective pd’s q N ] = qNl] 8 q N z ] . The first condition is number theoretic and is about

183

the compositeness of the dimension number N of the probability distribution P [ N ] that generates the ARW on Z N , while the second condition is about its factorization into a tensor product of two pd's of appropriate dimensions. Proposition 3. a) Let N = N1 N2 be the prime factorization of a positive integer N , then if in addition to the isomorphism of abelian groups of ZN M Z N , @ ZN,, we consider a probability distribution (pd) q N ] factorizable into a product Of two others pd's q ~ ,P][ N , ~namely ], such that V ~ ~= N 4 I~ ~ q N 2 ] , or p!"] = p ~ ~ ] then p ~ the ~ ]functional , of a ZN algebraic random walk $"I = &zNp\N1gjNl, with 0 5 ptN1 5 1,EiEZN pi"' = 1, factorizes for every step n, namely $fG1 M $iGll @ $;G2],and similar factorization is valid for the transition operator i.e T b ! M ThI1 @ ii) Let Nl, N2 and N3, be the co-prime factors of some positive integer N , then the decomposition ZN M ZN, @ Z N ~ @ ZN, x Z N , N ~@ Z N ~M Z N ~@ Z N , N ~ is , co-assosiativel* as indicated in the last, two equations, and if a pd q N ] is considered for which the finest decomposition V 6 P [ N ]= P",] @ q N 2 1 @ P [ N ~ in]terms , of three others pd's is true, then the associated Z N ARW is also decomposed at any step n, namely for its functional and transition operator respectively, the following co-associative factorizations are valid

Th2].

4[14"11

proof:

2)

@

@ 4[N3]

$[NlNz] @ 4[N3]

$[N1]@ $[NzN3],

(19)

The N = N1 N2 case: By means of the factorization property

of the pd viz.

we deduce that

i,jEZN

and therefore &DblV'

%

D b l l@

for the delta matrix. This yields

18

the factorization of the functional $[N] =

G ~ ] P [ N= ] G~]V~;:VP[N]

and similarly for the transition operator (with T(X@ y ) = y TIN]= (id @ 7 @ id) 0 ( 4 1 ~@ 1 id[^])

" (41~x1@

0

@ x)

A[N]

"

@ $[Nz] @ id[N~]) O ( A [ ~ l@ ] A [ N ~ ] ) T [ N ~@] T

[~~]. (23) ii) The N = NlN2N3 case: Applying the CRT to N, in two different ways yields for the abelian group ZN two factorizations i.e ZN FZ ZNl @ ZNz @ ZN3 .ZN~N @ ~Z N ~ M ZN1 @ ZNZN3. This can equivalently be expressed by means of the bijection 6, as id[~l]

( b [ ~ l , ~@zid) ] 6 [ ~N1Z , N ~= ] (id @ d[N2,~3]) O ~ [ N I , N z N ~ ] .(24) Performing the factorization of the pd PIN], as before twice we have (v~[N1 ,N,, which implies that

@

1 ) V J [ ~ 1 ~'IN] Z , ~=3'[~ NI]

@ I'N

21

@ '[Ns]

(25)

(&[Nl,Nz]@ 1)~[NlN2,N3] Db]((fi[N1,NZJ @ 1)h[N1N2,N~] It

"D k l ] @ D b Z l

(26) D$h]. This factorization of the delta matrix leads to an equivalent factorizations of the functional at each time step of the ZN walk, i.e @

$:El " 4iElNzl @ 4:E3] " $;Ell @

" 4igll@ $iE2]@ 4iE3]

(27) Similar decompositions can be obtained for Markov transition operators. #!i:2N3]

Remark: Interpretation of this result implies that the original random walk ZN, is isomorphically decomposed into the product of : i) three similar walks ZN E ZNl @ ZNz @ ZN3, ii) or into the product of two walks of dimensions N1N2 and N3 i.e ZNINz@ ZN3 and iii) or into the product of two walks of dimensions Nl and NzN3 i.e ZN, @ ZNZN3.AS an example we take the Z6 random walk on the vertices of a canonical hexagon the statistics of which is determined by a 6-dimension pd vector Pp1. If this pd has been chosen so that there are two other pd's one that generates a Z3 ARW on the vertices of a canoni3-dimensional P13] cal triangle, and one 2-dimensional PI2] that generates a Z2 walk on two @ PlzI,then we can decompose the ARW points, such that PI6]= PI3]

185

on the canonical hexagon as a product of two walks one on the triangle times one on the two-point set. Similar interpretations can be given to the prime decomposition of an A R W on a canonical polygon. E. O 2 3 @ Z4 @ 2 5 E Z12 @ 2 5 2 3 @ 220~'. g the decomposition Z ~ 2 Problem: Let N = N1NZ be the prime factorization of a positive integer N, for which the isomorphism ZN M ZN, @ Z N ~is, valid, let US consider an ZN A R W generated as previously by a pd q N ] . It this pd is not factorizable into a product of two others pd's as in the remark above, but instead there are two pairs of pd's qN1], PArl,and q N z ] , PAzl,such that the original pd is a convex combination of them i. e (0 5 4 5 l),

In the terminology of quantum i n f ~ r m a t i o n , we ~ ~ have here two

ARWs Z N ~Z, N ~ that , are classically correlated (cc) and form a probabilistic decomposition of the A R W in ZN, and write symbolically ZN qzN, @ + (1- q)zhl@ zh2.A number of interesting problems arise in this context: the total dynamics and reduced dynamics of the components of the ZN walk; the problem of construction of measures of correlations among the components of ZN walk; the problem of information(majorizati0n) dynamics and information exchange among the pd's components of the ZN walk; the problem of asymptotics of the ZN walk. 2.3. The case of hw-Algebra

A bialgebra14 A

= d ( p ,q,A, E ) over a field k is a vector space equipped with an algebra structure with homomorphic associative product map p : AxA A, and a homomorphic unit map q : k + A, that are related by p o (q @ id) = id = p o (id @ q), together with a coalgebra structure

+

A I8 A and a with a homomorphic coassociative coproduct map A : A homomorphic counit map e : A + k, that are related between them by (E I8 id) o A = id = (id I8 E) o A. Both products satisfy the compatibility condition of bialgebra i.e ( p @ p ) o (id I8 T @ id) o (A A) = A o p, where T ( Z I8 y) = y @ z stands for the twist map. If q or E is not defined in A we speak about non unital or non counital Hopf algebra. Suppose we have a functional 4 : A + C , defined on A, let us define the operator T4 : A + A as T+= (4 @ id) o A, then E o T4 = 4, namely the counit aids to pass from the operator to its associated functional. From this relation we can define the convolution product .1c, * 4, between functionals

186

as follows 33: ($J@id) o A o ( 4 @id) o A = ( 4 @ $J) 0 (id@ id@ E ) 0 (id@ A) 0 A

E 0 T$T$ = E 0

= ( d @ $ J ) o A= 4

*$,

(29)

and in general E o T; = E o T p = 4*". These last relations imply that the transition operators form a discrete semigroup with respect t o their composition with identity element T, = i d (due to the axioms of bialgebra) and generator T$,while the functionals form a dual semigroup with respect to the convolution with identity element e and generator 4, and that these two semigroups are homomorphic to each other. We recall now the Heisenberg- Weyl algebra hw and its structural maps: this is the algebra of the quantum mechanical oscillator and is generated by the creation, annihilation and the unit operator {at, a, 1) respectively which satisfy the commutation relation (Lie bracket) [a,at] = 1, while 1 commutes with the other elements. This algebra possesses a non counital bialgebra structure, cf. Ref. 3, chapt. 3, with comultiplication defined as

~ ( n - l ) a= n - + ( u @ ... @ 1 + 1 @ a8 . - .@ 1 + 1 @ ... B U ) , ~ ( n - l ) a t= n - i ( u t 8 . . . @ 1 + 1 8 . .. @ 1 + 1 @ . . . @ a t ), A1 = 1 @ 1 ,

(30)

where as indicated above the maps the creation/annihilation operators into the n t h fold tensor product of the algebra and adds appropriate factors (also c.f. Ref. 47). Let us also define the number operator N = at u with the following commutation relations with the generators of hw: [ N ,at] = at , [ N ,a] = -a. The module which carries the unique irreducible and infinite dimensional representation of the oscillator algebra is the Hilbert-Fock space 3 1 which ~ is generated by a lowest (or "vacuum" ) state vector 10) E 3 1 and ~ is given as 3 1 =~ {In)= 10) ,nE Z+}. The functionals we intend t o use will be defined by means of the canonical coherent state vectors of the hw algebra so in the sequel we give a brief introduction to the concept of coherent state vectors (CSV) on Lie groups: consider a Lie group (7, with a unitary irreducible representation T ( g ) ,g E 8, in a Hilbert space 31. We select a reference vector I+o) E 31, t o be called the "vacuum" state vector, and let 80 C 8 be its isotropy subgroup, i.e for h E Go, T ( h ) = eip(h)IQo). The map from the factor group M = 8/80t o the Hilbert space 31, introduced in the form of an orbit of the vacuum state under a factor group element, defines a CSV

I+o)

187

) .1

= T(Q/Qo) IQo) labelled by points z E M of the coherent state manifold. Coherent states form an (over)complete set of states, since by means of the Haar invariant measure of the group Q viz. d p ( z ) , z E M , they provide a resolution of unity, 1 = d p ( z ) I.)( 21. As a consequence, any vector JQ) E % is analyzed in the CS basis, I*) = dp(z)!I!(z)Iz), with coefficients !I!(x)= (z1Q). We should note here that the square integrability of the vectors !I! will impose some limits on the growth parameters of the functions @(z)at the boundary of manifold M (cf. Refs. 24, 25 and references therein). The hw-CS is defined by the relation

sM

sM

It is an (over)complete set of normalized states with respect to the measure &(a) = ie-la12&a for the non-normalized CS, and a E M = HW/U(l) M C is the CS manifold. Since a)1.1 = a la), M is the flat canonical phase plane with the standard line element ds2 = dad&. Also the symplectic 2-form w = ida A d& is associated to the canonical Poisson bracket { f,g} =

i(a,pg - &a:).

The density operator (state) p which would be used to determine functionals of some operator bialgebras A, is defined generally as follows : Let a Hilbert vector space 31 that carries a unitary irreducible representation of A of finite or infinite dimension. The set of density operators

S = { p E End(%) : p 2 O,pt

= p , t r p = l}

,

(32)

namely the set of non-negative, Hermitian, trace-one operators acting on

3t form a convex subspace of End(%), which is the convex hull of the set Sp = { p E S,p2 = p }

%/U(l) ,

(33)

namely of the set of pure density operators (states), that are in one-to-one correspondence with the state vectors of %. The density operator to be used in the case of h w walk uses the pure density operators Ifa)( k a1 E Sp and is a convex combination belonging to the convex hull of Sp i.e 0 <_ p 5 1,

P = P b ) ( a l + (1 -PI

I-a)( -

4

f

(34)

Let 4(.) = T r p ( . ) G < p , . >, a functional defined on the enveloping Heisenberg-Weyl algebra U ( h w ) ,where p = p la)(a1 (1 - p ) l-a)( - 011, i.e the p density operator is given as a convex sum of pure state density

+

188

operators. The action of the transition operator T4 = (4 c3 id) o A on the generating monomials of U ( h w ) (where we ignore the numerical factors in the comultiplication of eq.(30)) reads,

T4((at)"a") = (4 8 id) 0 A((U~)"U") =

it:9 (y ) (7)

[pa*%$+ (1 - p)(-CY)i(-CY)j](at)"-ian-j

i=o j=o

= p(at

+ CY*)"(a + a)" + (1- p ) @

- a*)"(a - a)"

.

(35)

For a general element f ( a , a t ) E U ( h w ) that is normally ordered, namely the annihilation operator a is placed to the right of the creation operator at, denoted by f ( a , a t ) = Cm,n20cmn(at)"an,the action of the linear operator T4 becomes

T4(f(a, at)) = p f ( a + CY, at + a * )+ (1 - p)f(a - CY, at - a*)

(36)

By means of the CS eigenvector property and the normal ordering of the f element we also compute the value of functional viz.

4(f(a,at)) = Pf(Q1,a*)+ (1 - P)f(--(y, -a*>.

(37)

Let us consider the displacement operator D , = east-,*' = - eA, with A = crat - Za which acts with the group adjoint action on any element f of the U ( h w ) algebra v ~ z . ~ ~ = ead(,at-,*a) (f)= eadA= D , f D L ,

~ d ~ , (=j, , ,)a t - a * a ( f )

(38)

where ad(X)f= [X, f] and ad(X)ad(X)f = [X, [X, f]]and similarly for higher powers, stands for the Lie algebra adjoint action that is defined in terms of the Lie commutator. Similarly the group adjoint action in terms of the displacement operator on the generators of U ( h w ) reads AdD*,(a) = a CY and AdD*, ( a t ) = at F a*.By means of these expressions we rewrite the action of the transition operator as

+ (1 - p)AdD,If(a,at) + (1 - p ) e a d A ] f ( aat). ,

T4(f(a, at)) = b 4 d D - a =

(39)

Next we compute the limiting transition operator

Tt

T4, n+m

lim T$ =

n+cc

t t + -y(adat)' + -y(~da)~n n t

- IyI (ada a h t

n

+ a h t adat)]".

(40)

189

In the last expression we have introduced the parameters of continuous time t E R and the drift and diffusion terms respectively c,y E C by means of the relations

and have performed the limits a + 0, n + 00, with t, c, y been fixed. We have also been used the limit limn+m(l f ) = e z , to obtain the continuous time Markov transition operator Tt = etL, with its L generators as it is obviously identified in the equation below

+

Tt = etL E expt [ 1 + a d h + y(adat)’

+

- IyJ(adu a h t

+ adut adut)]

(42) By construction Tt is the time evolution operator for any element f of U ( h w )i.e f t = Tt(f) and forms a continuous semigroup TtTtl = Tt+t,under composition. This yields the diffusion equation obeyed by ft, which will be taken to be normally ordered hereafter. By time derivation of the equation + t ( f ) =< p , f t

>=< p , etadcf >=< e-tadct P , f >=< p t , f >

7

(43)

we obtain the diffusion equation -$f t = L f t , as well as the dual one satisfied by the p density operator viz. -$pt = Ltpt. Explicitly the quantum master evolution equation for the density matrix reads d -p(t) = [cat - Za, p ] y(at2p pat - Patpat) y(a2p pa2 - 2apa) dt - JyI( ( 2 N + 1) p p(2N 1) - 2atpa - 2 4 ) . (44)

+

+

+ +

+

+

The obtained equation is similar to the quantum master equation that describes the trace preserving dynamics of the reduced density matrix operator of single mode of the electromagnetic field interacting coherently with classical electric filed while it is immersed in a bath of quantum oscillators.28 The decay of the field mode is influenced by the kind of initial condition the reservoir oscillators are put in. To analyze the physical content of that equation we rewrite it below by separating its right hand side into three lines i.e d -p(t) = [cat - a, p] dt - IyI (atup pat, - 2apat aatp paat - 2atpa)

+

+ + + y ( d 2 p+ pat - 2atpat) + Y(a2p + pa2 - 2apa).

(45)

190

The first line gives the coherent interaction of the mode with the classical electric filed of intensity c, as described by the commutator of density operator with the Hamiltonian term. It is neglected for balanced walk. The second line is a typical part of a master equation describing mode decaying for reservoir oscillators in thermal e q ~ i l i b r i u m .The ~ ~ last line is related to the case where the reservoir is prepared in a squeezed vacuum state.28 Closing we should notice that the above quantum master equation can be transformed into a Fokker-Planck partial differential equation for some quasi-probability function e.g P, Q, or Wigner function associated with the density operator e.g 28.

3. Quantum Random Walks 3.1. The case of 2

In a quantum random walk there are two dynamically coupled systems: the walker systems described by a Hilbert space H,, and the coin system also described by a 2D Hilbert space H , M C2 =span( I >, I - >). Let the uni-

+

7

tary matrix U operating in H,, e.g UH = 1 Jz ( l1 - 1 , or Uq = QUH, the Hadamard (Fourier) transform and the - rotation matrix respectively. If we denote by Pk = >< -f, the projection of an orthogonal partition of H,, and by S,, two step operators in the walker’s space H , (explicit examples determined below), we introduce the unitary one-step evolution operator acting in the space H , @ Hw in the combined coin+walker system:

4

In the equation above the upper/lower signs correspond to the choices

U H ,U s , respectively. If initially the two systems are decoupled, their density matrices are factorized i.e pc @. @, Then if we assume that pc = I f >< @I, is a projective density matrix with If >= a ] + > +bl- >, a normalized coin state vector, then the 1-time step of the QRW is considered as a completely positive trace (CPTP) map E,,, operating on the walkers’ density operator, obtained by partially tracing out (“forgetting”) the coin system i.e

191

Unitarity of V, implies that the two probabilities p+ = TrP*Up,UtP* = f 2Reub 2 0, p+ p- = 1, are determined by the coin system state vector variables and in turn they determine the Kraus generator,31 ( G S + ,,/jjZS-), of CPTP evolution map. Four sources of choices are implicit in the above prescription of QRW: the choice of the initial coin+walker state vectors, the choice of unitary U, the choice of definition of time step in terms of the tracing of the coin system (to be investigated in detail below), and finally the choice of step operators Sr in the walker's space, that determines the king of the QRW under investigation. Next we choose to turn to a Hadamard random walk on integers with dynamical algebra the Euclidean algebra e(2) = is0(2), with step operators (fiS+, 5%) = (&jE+, d m E - ) , and third element the "distance" operator L. The latter is the interesting quantum observable the quantum moments of which are used to compare classical and quantum walks. We examine three possible tracing schemes: i) the classical scheme that promptly traces the coin system after each V action, leading to the CRW

+

E V ( P $ ) ) = Trc(Vpc 8 p$-"Vt)

= pE+@,,, (n-l)E+ +

+ (1-p)E-@$-l)Et, (48)

which produces the diagonal sequence of density matrices

with diagonal elements the probabilities of site occupancy, given by the ( p , 1 - p ) classical Pascal triangle, ii) the scheme that traces the coin system by increasing delays i.e after an increasing number of actions of V operator, leading t o a walk designated by QRWl EVN

(&I)= Trc(VNpc8 p$"VtN) =

C Sm e, (O)S(N)t, (N)

(50)

m=f

which produces the sequence of density matrices (51) iii) and the scheme of delaying the trace of coin system by exactly one action of V operator, leading to a walk designated by QRW2

192

which produces the sequence of density matrices

{P P ,

EV2 (PL?), 4

2

(Pi?) ,4

2

( P P ) ,4 7 2 ( P 9 ,...} .

(53)

Next proposition deals with the time evolution of the pd (Pi"),

=

(ml Im) , made of the diagonal elements of the density matrices in the course of the walk QRWl and QRW2.13 Remarks: 1)From the above treatment can been shown that the QRWl reproduces along the diagonal elements of the sequence of evolving density matrices of the walker system the pd of the 1D Hadamard random walk and its diffusion rate13 2) by expressing the evolution unitary operator V , as the exponential of a hermitian operator H i.e V = eiH, which describes an interaction between coin and walker quantum systems, the various schemes of partial tracing can be physically implemented by choosing the length of interaction time e.g the QRW2 scheme requires an interaction time t = 2 for the evolution operator vt = exp (itH).44 Proposition 4. There exists bistochastic matrices A, = SO030 S 1 o s 1 , and A, = BOo&+B1 OBI,which determine the pd drawn from the diagonal elements of the evolving density matrices of the walks QRWl, and QRW2 respectively, by means of the respective equations

+

P:n+l) = A,P:")

+ (E+- E-)M(")P,(O),

p:("+l)= A 9 p(n), 9

(54) (55)

where

and

Above the element by element or Hadamard product A = A o B , defined between matrices A , B of the same size by ( A o B)ij = AijBij, has been used." The study of the pd obtained from the previous proposition reveals two novel aspects of the models QRWl and QRW2. First the aspect of breaking the condition majorization-implies- entropy increase, and that of enhanced diffusion rates. Before closing this section we give a brief demonstration of the latter one (detail investigation together with relevant references can be found in Ref. 13).

193

Enhanced Diffusion Rates:Let us consider the mth order statistical moment of the distance operator < Lm > n = Tr(p?)Lm) , at the nth step. Assume we have a symmetric walk with < L > n = 0, so that the standard deviation at nth step is on = d m . For the CRW we have that 0,"= &, and from the pd's taken by the previous proposition we obtain the respective standard deviation for the walks QRWI and QRW2, expressed in terms of their classical counterparts, for the first five steps:{ayRW' = QRWi - C QRWI = 0F,04QRWi = &j/20,C,0,&RWi = - 0 2 703

OF, 0 2

1,

QRW2

and {q

=

c

QRWz

01 9 0 2

= r

o

512

c 0QRW3

=

&g,02 A%?!

=2@}. From that we deduce that there is a quadratic speed up of the spreading rate in the case of QRWl with respect to CRW,and that rate is even bigger for the case of QRW2.Finally the asymptotic (n >> l ) , growth values are m a 4c >ff5 QRW2

turn out to be

-d

w

0,".

4. Relation of Algebraic and Quantum Random Walks

This section will put forward a relation between the two types of random walks under investigation so far i.e the ARQ and QRW.The relevant theory here is Naimark's extension theorem that allows to express in a non unique manner a positive trace preserving map, operating by means of its Kraus generators on a density matrix describing the state of some quantum system in a certain Hilbert space, by a unitary operator acting on a extension of the original space. Stated in the language of random walks the extension theorem assumes a ARW described by a CPTP map E(e,,,) = Cm=fPmSme,,,SA, operating on the density matrix of walker system with its Kraus generators (&S+, S S - ) , defined to act on Hilbert space H,. It is further assumed as usually for ARWs,that the step generators S* are related to and algebra of operators that needs to be specified. Then a unitary operator V is considered acting on H , @ H,,, i.e an extension of the original space by an extra or ancilla space H , M C2 =span(l+ >, 1- >), which in the context of QRW stands for the coin system. Let a pure density matrix in the coin system pc = I@ >< @I. Then the extension theorem provides a unitary representation of the CPTP i.e Ev(ew) =

C PmSmezus;

= Trc(V@c8 e w v + ) .

(58)

m=f

The unitary operator provides the Kraus generators as G S , = (ml V I@), up to a local unitary operator W, i.e the transformation V -+

194

W €3 1, V,provides the same generators. For the case of ARQs the unitary operator V is specifically expressed by means of the coin states projections

P&,and the unitary matrix U(p*) =

Em=*

6 6 of the coin space, as (6-6)

V = PmU €3 Sm.In particular the step operators are unitary and inverse to each other i.e S+ = (S-)t.This unitary representation can be extended t o products of CPTM maps by first defining the unitaries

C

v@'

PmU

QD PnU QD SmSn.

(59)

m,n=&

Then we obtain

&(ew)

=

C

PmPn

SmSnew(SmSn)t= TTcQDTTc(V~Vec~ecQDewVt€3Vt),

m,n=f

(60) This unitary extension of E', requires a double ancilla space or two coin quantum systems coupled, so the total space is Hc 63Hc @ H w .In the general case the kth power of the CPTP map of an ARW can be implemented unitarily by extending the original walker space by k anchillary coin systems, so the total space becomes HFk QD H,, and the total unitary operator is

v@L"

C

PmlU

€3

... B PmkU

QD Sml...Smk

...,mk=f

ml,

(61)

with unitarity condition V@'"VBkt = lyk€3 1,. Then we obtain for the kth step of the QRW as described by k successive actions of its CPTP map, a unitary realization which involves tensoring of quantum walker t o k quantum coin systems, followed by a coupling of them by a unitary evolution operator on the space of coins+walker composite system, and finally a decoupling of coins from the walker system, taken by partially tracing with respect t o the coin Hilbert spaces. The partial tracing corresponds to coin tossing in an ordinary random walk, and results into a density matrix for the quantum system of the walker, which further may provide statistics of various quantum observables of the walk. Explicitly the unitarization of ARW reads Et(ew)

=

C

pml...pmkSm,...Smkew(Sm,...Smk) t

...,mk=f

ml,

= TrFk((V@"~Fk @ e,V@'"'),

and can be identified with a QRW.

(62)

195

Let us remark at this point that an equivalent decomposition of the unitary Vmkwould be

n k

VB'" =

~

i

where ,

~i

= ( P + u )@ ~(s+)k+l+ ( P - u ) ~ @(s-)k+1,

i=l

and the subindex denotes the position of the embedding of the respective operator into the k-fold tensor product. In fact each of these operators Wi E End(Hpk @ H w ) , provide a new decomposition of the C P T P map i.e ~k = E W ~ W ~ . . . ,Wwhich ~ is equivalent to a nonstationary QRW with k different unitary evolution operators empoyed in order to construct the k - t h step. As a matter of fact this new decomposition helps to account for the type of quantum entanglement involved between coin and walker systems. Let us take the simplest k = 2 case, where Vm2= WlW2, with

Wl = (P+ @ 1, @ s++ P-

@ 1, @ s-) u @ 1, @ 1, ,

W2= ( ~ c @ ~ + @ s + + l c @ P - @ s - ) l c @ u @ l w . The action of these operators on the product density matrices pc @ pc @ pw is akin t o theaction of some control-control-S* type of non-local operator, which uses the two coin states as control spaces and the walker state as the target space, preceded by the local unitary operator U which acts on the control spaces and creates appropriate superposition of coin states. These actions generate quantum entanglement and can be described by the quantum circuit of Fig.1.c below. For purpose of comparison in Fig.l.a, we have included the corresponding circuit that generates the four entangled bipartite Bell states upon action of the composite operator UCNH @ 1 on the four orthogonal product qubit states, and on Fig. l b the circuit that corresponds to the unitary V = (P+@ S+ P- @ S-)U @ 1, of the EV map.

+

Though the topic of the entanglement in the QRWs will not be investigated further here, it should be obvious from the above analysis that the CPTP map that implements some discrete time-step of a QRW,also generates quantum entanglement that can be studied by appropriate circuits and evaluated by effective measures, as usually is done in other cases of coupled quantum systems. Although quantum correlations have been generally accepted to be the common cause of all novel effects in the QRW performance, the exact evaluation of the entanglement resources needed in the course of a QRW is still an open problem.

196

Fig.1 .a

Fig. I .b

Fig. 1.c Figure 1. Fig.1 (a) Circuit generating entangled Bell states from product states using local Hadamard and non-local control-not gate; (b) Circuit generating the unitary evolution of the 1-step map E V ; (c) Circuit generating entangled coin-walker states from corresponding factorized ones using local unitary (e.g Hadamard gate in the case of the homonymous QRW), and non-local control-control-& gates, that form the map of the 2-step QRW E $ = E W ~wZ.

To establish further the connection among ARQs and QRWs we give four particular examples: a) the Euclidean QRW13 [iso(2)-QRW]: with the distance operator L Im) = m Im) , with its eigenspace HE =span{ Im) ,m E Z}, and its dual phase operator @ Icp) = cp 19) , with its eigenspace H,” =span{Icp) ,cp E [0,27r); related by a Fourier transform with H i . Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the distance random walk on Z, with C P T P map constructed with Kraus generators been the step operators in the distance operator eigenstates i.e S& Im) E& Im) = e f i 6 Im) = Im f 1), and the phase random walk on the circle S, with CPTP map constructed with Kraus generator been the step operators in the phase operator eigenstates

g},

=

197

i.e S* 19) e f i L 19 f I) ; ii) the Canonical Algebra QRW [ hw-QRW]: with the position operator Q Iq) = q Iq) , with its eigenspace H,& =span{ Iq) ,q E R; dq}, and its dual m o m e n t u m operator P Ip) = p Ip) , with its eigenspace H,' =span{ Ip) , p E R;d p } , related by a Fourier transform with H,&. Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the position random walk on R, with CPTP map constructed with Kraus generators been the step operators in the position operator eigenstates i.e Sk Iq) e f i P lq) = Iq f 1), and the m o m e n t u m random walk on R, with CPTP map constructed with Kraus generator been the step operators in the momentum operator eigenstates i.e Sk lp) ehiQ lP) = IP f 1); iii) the M - dimensional Discrete Heisenberg Group QRW [~M-QRW]: with the action operator N In) = n In) , with its eigenspace H," =span{ In > , n E ZM}, and its dual angle operator 0 IS,) = 29, IS,) , with its eigenspace H," =span{lS,), 19, E &ZM}, related by a finite Fourier transform with H,". Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the action random walk on ZM, with C P T P map constructed with Kraus generators been the step operators in the action operator eigenstates i.e S* In) K e *ie In) = h*l In) = In f 1), and the angle random walk on & Z M , with CPTP map constructed with Kraus generator been the step operators in the momentum operator eigenstates i.e S* 18), gfl = e * w IS,) = IS,*,) ; iv) the Coherent State QRW15[ CS-QRW]: with the annihilation operator a la) = a la) , with its eigenspace H: =span{la) ,a E C; %}.Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the annihilation random walk on C, with CPTP map constructed with Kraus generators been the step operators in the annihilation operator eigenstates i.e Sh la) = eJ(*') la) = D A la) ~ = la f 1).The step operators here identified as special case of the canonical coherent state displacement operator D*g = e*DatFBa = eJ(*g), have based the indicated step property on the following operator identity D,Dg = e-iaxfiDa+g, applied for the case of co-linear a,,B vector on complex plane. To give an explicit identification of the ARW based on the hw algebra constructed in Ref. 15, as a quantum random walk, and in particular as a CS-QRW, we make the following choices: the transition probabilities are p+ = p , p - = 1 - p , the coin state is pc = 10 >< 01, the U operator is U ( p , l - p ) , and the step operators are CS displacement operators with

=

IS,)

198

steps k/3 E C. Then the total 1-step evolution operator in the coin+walker system is V = PU , 18 S, = fiD+P c p D + f i ) , a n d t h e G D - P -@-s reduced walker evolves in 1-step by the CPTP &v(e,,,) = p D + g ~ ~ D i ~

Em=*

(

+

(1 - P)D-aeWD!P = T T c ( V Q c 63 e w v ' ) . For n steps the evolution of the walker has been chosen in Ref. 15 to be +((e,,,). It is important to notice that this is a choice based on the ARW construction methodology, and that our present treatment of the same walk as a QRW, sees the $(e,,,) type of evolution to result from a partial tracing of the coin system at every step. Our previous discussion of other types of tracing schemes motivates the study of CS-QRWs with delayed tracing, in order t o investigate phenomena such as enhanced or anomalous diffusion in ARWs. This problem will be taken up elsewhere. 5 . Discussion

We have outlined a mathematical framework where the conception of random walk and its associated statistical notions, and equations of motion, can both be studied in an algebraic and quantum mechanical manner. ARWs and QRWs appear to be two aspects of the same mathematical device, so their interconnection serves to conceptually clarify the common ground between them and to enrich the heuristics of formulating new problems and methodically searching for their solutions. Quantum random walks are important both as quantum algorithms to experimentally be realized and as modules in a general quantum computing algorithm-devise that could outperform some classical rival. The connection ARW-QRW could serve to generalize, unify and compare such algorithms. Also Quantum Information Processing concepts and tools, could be developed for ARW-QRWs. The step taken here is only a preliminary one towards developing such a theory. Finally, ARW-QRWs come with lots of free choices for its constituting parameters. To mention only one expected application in the field of Open Quantum Systems, we should emphasize the importance of choosing the functional in e.g the hw ARW. Various choices of functionals in terms of types of coherent state vectors, combined together with various choices of ordering the operator basis in the enveloping algebra U(hw), i.e normal, antinormal, symmetric etc, could serve as a guiding rule for constructing quantum master equations for open boson systems interacting with various types of quantum mechanical baths.

199

6. Acknowledgments

I wish t o thank the organizers of the Volterra-CIRM-Grefswald Conference for the opportunity to give a talk. Discussions with L. Accardi, U. F’ranz, R. Hudson, M. Schurmann, and with my collaborators A. Bracken and I. Tsohantjis, are gratefully acknowledged. I am also grateful to the anonymous referee for suggesting eq.(63). References 1. P. A. Meyer, Quantum Probability for Probabilists (Lect. Notes Math. 1538), (Springer, Berlin 1993). 2. M. Schiirmann, White Noise on Bialgebras (Lect. Notes Math. 1544), (Springer, Berlin 1993). 3. S. Majid, Foundations of Quantum Groups Theory (Cambridge Univ. Press, 1955), ff. chapter 5. 4. U. Franz andR. Schott, Stochastic Processes and Operator Calculus on Quantum Groups, (Kluwer Academic Publishers, Dodrecht 1999). 5. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p.37. 6. D. Aharonov, A. Ambainis, J. Kempe and U. Vasirani, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p.50. 7. A. Nayak and A. Vishwanath, arXive eprint quant-ph/0010117. 8. J. Kempe, Proc. 7th Int. Workshop, RANDOM’OJ, p.354 (2003). 9. A. M. Childs, E. Farhi and S. Gutmann, Quantum Information Processing 1,35 (2002). 10. B. C. Travaglione and G. J. Milburn, Phys. Rev. A 65, 032310 (2002). 11. B. C. Sanders, S. D. Bartlett, B. Tregenna and P. L. Knight, Phys. Rev. A 67,042305(2003). 12. J. Kempe, Contemp. Phys. 44, 307 (2003). 13. A. J. Bracken, D. Ellinas and I. Tsohantjis, J. Phys. A : Math. Gen. 37, L91(2004). 14. E. Abe, Hopf Algebras (CUP Cambridge 1997). 15. D. Ellinas, J. Comp. Appl. Math.133, 341 (2001). 16. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic Press, New York, 1979). 17. R. Bhatia, Matrix Analysis (Spinger-Verlag, New York , 1997). 18. P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order: Double Stochastic Maps and Unitary Mixing (Dordecht, Boston, 1982). 19. M. A. Nielsen, An Introduction to Majorization and its Applications to Quantum Mechanics (unpublished notes). 20. D. Ellinas and E. Floratos, J . Phys. A : Math. Gen. 32,L63 (1999). 21. B. D. Hughes, Random Walks and Random Environments Vol. I, (Clarendon Press, Oxford 1995) 22. J. Gillis, Quarterly J. Math., (Oxford, 2nd series)7, 144 (1956). 23. K. Lakatos-Lindenberg and K. E. Shuler, J. Math. Phys. 12,633 (1971).

200 24. J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific, Singapore (1986) 25. A. Perelomov, Generalized Coherent States and their Applications, (Springer - Verlag, Berlin 1986). 26. E. D. Davies, Quantum Theory of Open System, (Academic, New York, 1973). 27. G. Lindblad, Non-Equilibrium Entropy and Irreuersibility, (Reidel, Dordrecht 1983). 28. M. 0. Scully and M. S. Zubairy, Quantum Optics, (Cambridge Univ. Press, Cambridge 1997), p. 255, 448, 453. 29. S. Stenholm, Phys. Scripta T12(1986). 30. J. Gea-Bauacloche, Phys. Rev. Lett. 59,543 (1987). 31. K. Kraus, States, effects and operations (Springer-Verlag, Berlin, 1983). 32. M. A. Nielsen and 1. L. Chuamg, Quantum Computation and Quantum Information, (Cambridge Univ. Press, Cambridge 2000). 33. S. Majid, Int. J . Mod. Phys. 8 , 4521-4545 (1993). 34. S. Majid, M. J. Rodriguez-Plaza, J . Math. Phys. 33,3753-3760 (1994). 35. P. Feinsilver and R. Schott, J. Theor. Prob. 5,251 (1992). 36. P. Feinsilver, U. F’ranz and R. Schott, J . Theor. Prob. 10, 797 (1997). 37. U . Franz and R. Schott, J . Phys. A: Gen . Math. 31 , 1395 (1998); 38. U. Franz and R. Schott, J. Math. Phys. 39, 2748 (1998). 39. D. Ellinas and I. Tsohantjis, J . Non-Lin. Math. Phys.8Suppl. 93(2001). 40. D. Ellinas and I. Tsohantjis,Inf. Dim. Anal.- Quant. Prob. 611 (2003). 41. G. H. Hardy, J. E. Littlewood and G. Polya, Messenger Math. 58, 145 (1929). 42. H. Sikic and M. V. Wickerhauser, Appl. Comp. Harm. Anal. 11, 147 (2001). 43. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, (Wiley, New York 1974). 44. D. Ellinas, et. al, work in progress. 45. P. J . Davis, Czrculant Matrices, (Wiley, New York 1979). 46. M. R. Schroeder, Number Theory in Science and Communication, (Springer, Berlin 1997). 47. C. D. Cushen and R. L. Hudson, J . Appl. Prob. 8, 454 (1972). 48. This second line of the equation has the form a Lindblad type master equation taken in the large temperature limit, where for the average number of thermal photons we have A NN A + 1 = (yI,c.f. Ref. 28. 49. H. Risken, The Fokker-Planck Equation, (Springer, Berlin 1996). 83 (1990).

DUAL REPRESENTATIONS FOR THE SCHRODINGER ALGEBRA

PHILIP FEINSILVER Department of Mathematics Southern Illinois University Carbondale, IL. 62901, U . S . A . RENE SCHOTT IECN and LORIA UniversitC Henri PoincarC-Nancy 1, BP 239, 54506 Vandoeuvre-lks-Nancy, fiance. Starting from the multiplication table for a basis of a Lie algebra, we show how to find a realization as vector fields in the case there is a Lie flag. Then we show how this is connected to finding coordinates of the second kind. We find the explicit form for coordinates of the second kind for the Schrodinger algebra in a particularly useful basis. These coordinates are related to dual vector fields that form an abelian Lie algebra generating an associated family of polynomials. Families of polynomials in quantum variables arise by specialization of the parameters of the first kind corresponding to quantum observables.

201

202

1. Introduction

The Schrodinger Lie algebra plays an important role in mathematical physics and its applications. It has been introduced and investigated as the algebra of symmetries of the free Schrodinger e q ~ a t i o n . The ~~~ re-J sulting structure of the semidirect product of the Heisenberg algebra and sl(2) was investigated in Ref. 11. Recently this algebra has appeared in the context of the square of white noise.’ And G. Pap is using it to compute the Fourier transform of Brownian motion on Heisenberg groups.15 This article is composed of two parts. In Part I, we present the calculation of dual representations of a Lie algebra using Lie flags and illustrate its application to the Schrodinger algebra. In Part 11, we recall the Splitting Lemma for coordinates of the second kind. For the Schrodinger algebra, these coordinates are found in two steps. First using a basis that reflects the semidirect product structure of the algebra the appropriate equations are solved. Since the “standard basis” is a permutation of that first basis, the coordinates of the second kind for the standard basis can be found using group theory. These coordinates are an essential part of a generating function for a family of polynomials giving a representation of a n interesting associated abelian Lie algebra. 2. Dual representations

First, dual representations, particular realizations of a Lie algebra in terms of vector fields, are defined along with the corresponding .rr-matrices. Then we recall the definition of Lie Aag and how it is used to find a dual representation. This approach is carried out for the Schrodinger algebra. The main advantage is that the coefficient matrices of the vector fields can be computed directly in terms of “partial adjoint matrices” which allow for fast symbolic calculations. 2.1. Dual representations and r-matrices

For a given Lie algebra 8, with basis {[I,. . . ,&}, we have the corresponding PBW-basis for the universal enveloping algebra, namely ordered monomials <,”l . . .tyd, with ni 1 0, 1 5 i 5 d. The generating function of this basis

g(A;<) = exp(A1J1) exp(A2J2) . . * exP(AdSd)

203

are local group elements, i.e., they give a representation of the Lie group near the identity, equivalently, for ( A l , . . . ,Ad) near the origin in Cn.The local coordinates (Al, . . . ,Ad) are coordinates of the second kind. Now, left multiplication by a basis element, dualizes to an action on g as some type of generalized differential operator. However, by Lie’s theorems, we know that in fact it acts as a vector field, a first-order partial differential Thus, there is a matrix of coefficients r t ( A ) such that operator,

[it.

Similarly, multiplying from the right we have vector fields C and r-matrix

r * ( A )such that

We call these the left and right dual representations, respectively, of Q. We will see how to compute these quickly if the Lie algebra has a special structure. 2 . 2 . Commutation relations: Kirillov matriz, adjoint

representation Once a basis has been chosen, the Lie algebra is defined by commutation relations. In other words, in terms of structure constants, c $ , determined by k

It is convenient t o summarize the commutation relations in the form of a matrix, K, the Kirillov matrix. The commutation relations, Eq. (l),yield matrix entries k

linear forms in the variables {zk}. (Warning: these are purely formal and have nothing to do with the zvariables used below for representations on functions.)

204

It is usual t o interpret Eq. (1) as giving the action of a linear map ad(&) on &, i.e., k

The matrices, ti, of the linear maps ad(&), 1 5 i . representation of 6. Thus, (&)jk =

dk

2.3.

5 d , are the adjoint

Lie flags

Definition 2.1. A Lie algebra 6 has the flag property if there is an increasing chain of subalgebras .Ci

{o}

c LCt c ,& c ' . c L d

=6

(2) each of codimension one in the next. Such a flag is an increasing Lie flag. '

Note that this is a flag in the usual sense of D as a vector space, but is rather stringent as each Li must be closed under Lie brackets. Suppose that {&, . . . ,&} is a corresponding adapted basis for (2), i.e., for 1 5 i 5 d , {&, . . . ,ti} is a basis for Ci. Then, reversing the order of the basis gives a decreasing Lie flag. Every solvable, in particular every nilpotent, algebra has an increasing Lie flag. The Lie-Engel Theorem guarantees even more, namely, the existence of a flag of ideals. (See Humphreys l 3 for background and proofs.) There are Lie algebras that are not solvable yet which have the flag property. For example, s1(2), with basis { E - , E+, H } and commutation relations [E-,E+] = H , [H,E*] = fE* admits the Lie flag with adapted basis { E + , H ,E-}. Direct sums of sl(2) thus have the flag property. The main example of this paper, the Schrodinger algebra, is another example of this phenomenon. 2.4. Dual representations and flags

iz

Given an increasing flag, with adapted basis {[I,. . . ,&}, denote by the transpose of the matrix of & in the adjoint representation restricted to the subalgebra Li. I.e., columns i 1 through d of are zero'd out and then the matrix is transposed. In terms of the structure constants the entries of are

+

ti

205 with the condition that j , k 5 i, otherwise null. Dually, for a decreasing flag, we denote by the transposed matrix of the restriction of the adjoint action of & to the subalgebra Cf = span{&, . . . ,Jd}, i.e., the first i columns are ctj as are zero’d out, then the matrix transposed. So the entries of in equation (3) except with the condition j , k 2 i otherwise null.

(it

(!

We recall the main theorem from Ref. 9, p. 33 :

Theorem 2.1. For the dual representations we have: (1) Given a n increasing flag, the pi-matrix f o r the right dual is given by

(2) Given a decreasing Jag, the pi-matrix f o r the left dual is given by r t ( A ) = exp(-Al[f) exp(-A2&) ... exp(-Ad(fi)

Remark 2.1. This approach, along with a MAPLE implementation, is presented in Ref. 10. 2.5. Dual representations for the Schrodinger algebra

Order the basis as follows: cl=M,

&=K,

&=G,

&=D,

[.5=Pz,

&=Pt

We call this the standard basis for the Schrodinger algebra. With rows and columns labelled by the corresponding operators, we have the following multiplication table

Notice that this is a variant of the kirillov matrix using the original elements rather than x-variables.

206

Observe that the basis is adapted to an increasing Lie flag. To use Theorem 2.1 we need the transposed partial adjoint matrices. For example, for (4 the adjoint matrix and the transposed partial adjoint matrix are

i4

=

0000 0 0 0200 0 0 0010 0 0 0000 0 0 0000-1 0 0 0 0 0 0-2

000000 020000 .$ = 0 0 1 0 0 0 000000 000000 000000

and

It is not hard t o see that ll = $3 and & = ( 5 while (,*= (; = 0. Of course $1 = 0 as is central. Exponentiating and multiplying according t o Theorem 2.1 yields the matrix T*,coefficients of the vector fields of the right dual representation: . 1

0

0

0

0

0

1 2 As2 e(2A4)A5 eA4 A6 As A5 As2

A5 0 0 0

0 0 0 0

eA4 0 0 1 0 0 0 0

A6 A5 1 0

0 2A6 0 1

In other words, we have the realization as vector fields given by

M * = al K* =

+ e2A4a2+ A5eA4d3+ -4684 + &A585 + eA4d3+ A685 D* = 8 4 + A5d5 + 2A6b6

+A:&

G* = A5dl

P; = a, P; = a, with

a,

denoting

a -.aAi

Of particular interest here is the fact that the

“physical realization” of the Lie algebra, in space-time variables, comes naturally from the right dual representation as follows. Make the correspondences A5 + 2 and A6 + t. Then acting on functions of the form enAle-dA4f(A5, A6) we recover the form given in Ref. 6, for example. Note that the variables A2 and AJ are ignored. We do not know if they carry any physical meaning. 6112114

207 2.6. Double dual

Another way to get a representation of the Lie algebra is to use the double dual. Observe that the action on the enveloping algebra from the left is a Lie anti-homomorphism. So we dualize the vector fields by exchanging Ai H a;, in other words, by taking the algebraic version of Fourier transform.

If r* is known, then rt can be found via the exponentiated adjoint representation % ( A )= g(A;t ) ,the group element formed using the adjoint matrices. In fact, denoting transpose by T, we have the relation (%)T = r* (rt)-l (Ref. 9, p. 37). Thus we find for the Schrodinger algebra in the standard basis the left dual pi-matrix 1 0 0 0 1 0 0 0 1 0 2A2 A3 A3 0 A2

f A:

0 0 0 0 0 0 1 0 0 eA4

0 0 0 0 0

A: A2A3 A2 A3eA4e2A4

For clarity in the double dual, we can use x’s as Fourier-transformed variables taking A;

a + a;, & + xi. In the resulting expressions, ai denotes -.dX;

Then we will have a representation in terms of dual vector fields in the variables {xi}. So, in rt, we replace each A; by &, then formally multiply the resulting matrix with a column vector of the variables xi, and finally, put all partials to the right, “Wick ordering”, to arrive at:

This realization has special features:

208 (1) There is an associated family of polynomials that provide a representation of the boson commutation relations, BCR. By this approach, to a given Lie algebra there is associated an abelian Lie algebra in a natural way, so that the corresponding group elements act the same on the constant function 1. This is explained in Sec. 3.3. (2) The double dual generally is suitable for constructing induced representations of the Lie algebra. See Ref. 9, pp. 40-41.

3. Splitting formula for the SchrSdinger group Now we will find the coordinates of the second kind for the standard basis. The idea is that in semidirect product form the equations for the coordinates of the second kind can be readily solved. Then those for the standard basis can be found using group theory.

3.1. Splitting lemma As a vector space with basis ((1, . . . ,&}, a typical element of G has the form X = ai&. Thus exp(X) is a group element in a neighborhood of the identity. The ai are coordinates of the first kind.

xi

If we write the group element in terms of coordinates of the second kind, we have effectively factorized or split the exponential into a product of oneparameter subgroups. Thus the lemma relating the two types of coordinates is called the splitting lemma. This means finding the change-of-coordinates mapping a + A ( a ) , with a = ( ~ 1 , .. . ,ad). For X = a&, A ( a ) is the map of coordinates determined by

xi

exp(X) = g(A;I ) = eAl(a)tl . . . eAd(a)td

Remark 3.1. This factorization is the principal object of the Wei-Norman theory. l6l1’ Let 0 denote the group law:

The “flow of the group law” z ( t ) = A ( t a ) 0 A , satisfies the equations x j = Z k( Y k x l j ( z ) ,with initial condition z(0) = A . Thus we formulate the

209 Lemma 3.1. Splitting Lemma For X = ai&, consider the factorization

Ci

exp(X) = g(A; <) = eAl(a)
Let rt denote the coeficient matrix (pi-matrix) of the left dual representation. To find the coordinate map a! -+ (Al(a!), . . . ,Ad(a!)), solve the diferential equations

with the initial conditions x l ( 0 ) = ... = ~ ( 0 =) 0 . Then Ai(a) = zi(l), for 1 5 i 5 d. As well, we can use the right dual, 7 r * ( x ) ,in the above system of differential equations. This gives the flow of the group law composing on the right, which with zero initial conditions again gives the coordinate mapping A(a) at t = 1. 3.2. Coodinates of the second kind found

For the basis { & } i = ~. .,. , 6 we have the differential equations for the coordinates of the second kind Ai

Note that even though the equations for A2, A4, A6 may be solved independently, there is still coupling with the other three variables. If instead we use the basis {qi}i=1,,.,,6with

which corresponds to the semidirect product structure, then we have the

210

system of equations for the coordinates of the second kind Bi 1 1 = a1 a3 B2 - - 0 4 B32 + - B22 2 2

+

&=&z-a4B3+&jB2

B 4 =a4+2(115B4f(Y6B42

B5

= a5

+ a6B4

& = a6 e2B5 so now the two groups of variables are completely decoupled. (We thank G. Pap for this observation.) The equations for B4, B5, B6 are for coordinates of the second kind for sl(2) which, e.g., have been computed previously (cf. Ref. 8, p. 32). And the equations for B1, B2, B3 cam be resolved without too much difficulty, noting that those for B2 and B3 are linear. Let us set d2 = Pt - P 4 P 6 , r = (tanhS)/b, and B1

= P1+

sinh 6 - 6 0 P2 263

P2 P3 P 4 P5

CT

= (1 - sech6)/J2. Then

1

+ 2 B2B3

P3 P5 P 6

Now to fin the splitting formula for the

we equae

21 1

In other words,

For the group element, we have the ordered products

Basically, we want to shift P, to the right, from position 3 to position 5. From the adjoint action espm Ke-8Px = K sG (s2/2)M we have

+ +

,sP, et K = et K estG

es2tM/2,spa

D] = Pz,we have the dilation effect And from [Pz, esPaetD = etD exp (set P,) The following relations result:

So substituting and some rearrangements yields Theorem 3.1. Let 62 = as - ~ 2 ~ 1 6T , = (tanh6)/6, and LT = (1 - sech 6)/62. For the basis (1 = M , (2 = K , (3 = G , t 4 = D , (5 =

212

P, , & = Pi, the coordinates of the second kind have the form

3.3. Basic polynomials and quantum observables Taking the results of the above theorem, we can form the generating function

with multi-indices n = (nl,.. . ,n6) in the expansion. The quantities y,(x) are “basic polynomials” or “canonical basis polynomials” since they yield a representation of the Heisenberg algebra, or BCR, boson commutation relations (Ref. 9, pp. 19-20. See also Ref. 5 for various points of view). The formalism goes like this. We have variables {xi} and their correspondd Acting on smooth functions, ing partial derivative operators Di = -. axi they satisfy the BCR: [ D j , X i ]= 6ijI [Dj, Di] = 0, [ X j , X i ] = 0, where Xi denotes the operator of multiplication by xir and I is the identity operator. Write z = (21,.. . zd) and D = (01,. .. ,Dd). Now, for any function V = (Vl(z),.. . ,Vd(z))holomorphic in a neighborhood of the origin in C d , we have a representation of the BCR by setting Vj = V , ( D ) and U, = C k x k W k i ( D ) where W is the inverse Jacobian, satisfying

213

C

av, wkj = 6ij.

They satisfy the BCR

k

[vj,K] = SijI,

[vj,K] = 0,

[ y j , y z ]= 0

Of importance is that the variables Y , commute - they form a n abelian Lie algebra - and satisfy

i

k

where U is the inverse of V in the sense of functional composition: V ( V ( z )= ) z in a neighborhood of the origin. Now we use Theorem 3.4 of Ref. 7 to the effect that the exponential of the double dual acting on the constant function 1 is e x p (C i ziAi). That is,

i

k

If we interpret A(a) as V(w), we see that the yn are the basis for a representation of the BCR with raising operators Y , and lowering operators vj according t o the correspondences z with A and V with a. Note that the yi can be found explicitly from the inverse Jacobian of the map A + a, equivalently from the Jacobian of the map a + A expressed in terms of the variables A.

+ +

+

As observed in Ref. 11, the elements XI = P t D K and X2 = G P, commute and, in fact, a Hilbert space can be constructed on which they act as selfadjoint operators. In other words, they provide quantum observables. Thus, let us specialize the coefficients accordingly: a1 = 0,122 = a4 = a6 = z1 and a3 = a5 = 2 2 . For the double dual representation, we thus have

To use Theorem 3.1, note that 6 = 0. Thus, T reduces to 1 and We get

0

to 1/2.

214

These yield

(Compare this with Ref. 11, Appendix, Eq. A.2 .) The point is that now we have the whole context available. We remark two features: (1) The symmetry properties of the special polynomials, mixed Hermite and Laguerre polynomials in this case, are apparent from the contextual Lie apparatus. (2) The basis states for the observables, the special polynomials noted in #1, are seen indeed as specializations of general basis states, “basic polynomials”, of all six variables corresponding to the basis of the Lie algebra.

The above properties hold in general for any Lie algebra and, particularly, for observables built by specialization from basis elements of a Lie algebra (e.g., Ref. 4). This turns out to explain the results in Ref. 8 where orthogonal polynomials for the classical “Bernoulli-type” probability distributions are constructed as basis states for Lie algebras s1(2), Heisenberg algebra, and the oscillator algebra. The present theory thus gives a setting for general Lie algebras.

215

References 1. L. Accardi, U. Franz and M. Skeide, Renormalized squares of white noise and non-gaussian noises as LBvy processes on real Lie algebras, Comm. Math. Phys. 228, no. 1, 123-150 (2002). 2. A.O. Barut and R. Qczka, Theory of group representations and applications, 2nd ed., PWN, Warszawa, 1980. 3. A.O. Barut A 0 and B.-W. Xu, Conformal covariance and the probability interpretation of wave equations, Phys. Lett. 82A,5, 218-220 (1981). 4. S. Berceanu and A. Gheorghe, On equations of motion on compact Hermitian symmetric spaces J. Math Phys. 33,3, 998-1007 (1992). 5. A. Di Bucchianico, Probabilistic and analytical aspects of the umbra1 calculus, CWI tract, 119, Amsterdam, 1997. 6. V.K. Dobrev, H.D. Doebner and Ch. Mrugalla, Lowest weight representations

7.

8. 9.

10.

of the Schrodinger algebra and generalized heat/Schrodinger equations, Rep. Math. Phys. 39,2, 201-218 (1997). Ph. Feinsilver and R. Schott, Vector fields and their duals, Adv. in Math. 149,2, 182-192 (2000). Ph. Feinsilver and R. Schott, Algebraic structures and operator calculus, v. 1, Representations and probability theory, Kluwer Academic Press, 1993. Ph. Feinsilver and R. Schott, Algebraic structures and operator calculus, v. 3, Representations of Lie groups, Kluwer Academic Press, 1996. Ph. Feinsilver and R. Schott, Symbolic computation of Appell systems on the Schrodinger algebra, Proceedings of RIMS Symposium on Algebraic Systems, Languages and Computation, Kyoto 2000, Ed. M. It8, RIMS 1166, 59-66

(2000). 11. Ph. Feinsilver, J. Kocik and R. Schott, Berezin quantization of the Schrodinger algebra, IDAQP 6,1, 57-71 (2003). 12. C.R. Hagen, Scale and conformal transformations in Galilean-covariant field theory, Phys. Rev. D5,2, 377-388 (1972). 13. J. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1980. 14. U. Niederer, The maximal kinematical invariance group of the free Schrodinger equation, Helv. Phys. Acta 45,802-810 (1972/73). 15. G. Pap, Fourier transform of symmetric Gauss measures on the Heisenberg group. Semigroup Fomm 64,130-158 (2002). 16. J. Wei and E. Norman, On global representation of the solutions of linear differential equations as a product of exponentials, Proc. A.M.S. 15, 327334 (1964). 17. J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Math. Phys. 4, 575-581 (1963).

HARMONIC ANALYSIS ON NON-AMENABLE COXETER GROUPS

GEROFENDLER Gero Fendler Finstertal 16 0-69514 Laudenbach Germany

1. Introduction In this note we present certain geometric aspects of Coxeter groups, which lead to results on the regular representation of Coxeter groups. We are merely interested in non-amenable Coxeter groups and shall briefly discuss: (1) weak amenability of the Coxeter group (2) a Haagerup inequality, (3) simplicity of the regular C*-algebra.

This note is a short report on results obtained in Refs. 7 and 8 and for complete proofs we refer to these papers. 2. Coxeter groups

A Coxeter group (G,S) is a group G with a set of generators S and a presentation: s2=e,

(st)m(8gt) =e,

SES s , t E S, s

#t

where m(s,t) E { 2,3 ,4 ,. . .,oo} is symmetric: m ( s , t ) = m ( t ,s),

Examples

216

s,t E S.

217

(1) Reflection Groups Assume that V is a finite dimensional real vector space endowed with a scalar product (., .) : V x V R. For a E V \ (0) denote s, : V 3 V the reflection defined by:

+

It stabilises pointwise the hyperplane H , = {r) E V : (a,?) = 0) and inverts a: s,(a) = -a. A finite group generated by reflections is known to be a Coxeter group. (2) Dihedral Groups Now let V = R2 be the usual euclidian plane, 0 = 2,where m E {2,3,4,. . .}.

For m as above the reflections s, and sp generate a finite group of order 2m. If 0 is not a rational multiple of T , then s, and sp generate the infinite dihedral group.

Ha

(3) Symmetric Groups On V = R" the symmetric group S, acts by permuting the coordinates. It is generated by the transpositions Ti=(iri+1)

,

l
Denoting €1,. . . ,en the standard basis Ti(Ei

- E i + l ) = E i + l - ei

defines a reflection which stabilises Hi = {C; : [ I ( e i - ei+l)} pointwise. 3. Geometric Representation

Let (G,S) be a Coxeter group. We assume #S < 00 and on the finite dimensional real vector space V = @,gsRaswith basis {a,: s E S } we

218

define a bilinear form B : V x V

B (as,at)=

{

+ R by 1 if s = t

if s # t , m(s,t ) < 00 -1 if m(s,t)= 00.

- cos

(1)

For each s E S we define a reflection on V by

a,< = t - 2 B ( a s , J ) a s

,

< E v.

(2)

Then there hold true: 0

0

0

V = Ra, @ H,, where H , = { q E V : B(a,,q) = 0) is stabilised pointwise by us and usas= -as. s I+ (T, extends multiplicatively to a representation u : G + Gl(V) of the Coxeter group. This representation is faithful and its image is a discrete subgroup of Gl(V).

Remarks: B is strictly positive definite if and only if G is finite. B is positive semidefinite, with non-trivial kernel, if and only if G is a semi-direct product of an additive group of translations and a finite Coxeter group acting on the former.The group is called an affine Coxeter group in this case. B is strictly indefinite if and only if G is of exponential growth (equivalently: non-amenable) . 4. Adjoint Representation We dualise the representation

and shall merely work with Let for each s E S:

(T

:G

+ Gl(V) as usual by

IT* instead

2, = {f E

of

(T,

v* : f ( a , ) = O),

A, = {f E V* : f(cr,)

> 0).

Denote

C = ngEG gA, - the fundamental chamber D=C\{O}and U =U

g E g ~D -

the Tit’s cone :

219 o

C is a simplicia1 cone, its faces are the sets 2, n D.

o

U is a convex cone, D is a fundamental domain for the action of G on it.

o

We define a family of hyperplanes in V * :

31 = UgEG gZ8. Then, o a closed line segment [u,c]

cU

between points u , c E U meets only

finitely many members of 3c. o Moreover, for any c E C: where l(g) = inf{k : g = s1 . . . sk, si E S} denotes the usual length with respect t o the generating set S. For the readers convenience we state and give a proof of the following theorem due to Bozejko6.

Theorem 4.1. For each 0 5 r 5 1 the f i n c t i o n g I+ nite.

is positive defi-

Proof: One can write

ZEN

where c E C is arbitrary and

Xh

is the characteristic function of

N~ = { Z E 31 : [ h c , ~n]z # S}. Hence 1(.) is a negative definite and, by a theorem of Schoenberg, rl(.)a positive definite function (for 0 5 r 5 1).

5. A Product of Trees Now, by o*,the Coxeter group G may be considered as a discrete subgroup of Gl(V), where dim(V) = #S < 00. Hence there exists, e.g. by Selberg's Lemma, a torsion free normal subgroup J? of finite index in G. Next we apply a lemma due to Jaffe and Millson13.

Lemma 5.1. For g,g' E

r

and Z E 3c either g Z U g'Z = 0 or g Z = g'Z.

220

We may decompose %! into different r-orbits

where Xi = {yZi : y E Zi} for some Zi E 74. For each i E { 1, . .,A} we define a graph 71 with vertices the connected components of U \ UzExiZ and edges the hyperplanes Z E Xi. An edge -Z E E(7;) connects the vertices CO, C1 E V ( 5 )if, as subsets of U ,ConC1 C 2. The Lemma of Jaffe and Millson shows that each X, i E (1,. . ., A } is a tree. Moreover I' acts on each tree by simplicia1 automorphisms and freely on the product

.

6 = 71 x . . . x rn. A tree may be seen as a chamber system Edges

t)Chambers

Vertices c--)Facets. The above product of chamber systems is a building of type A1 x . . . x 21. Its chambers are A-tuples of edges: ( e l , . . , e n ) , ei E E(X) the facets of the above chamber are ( e l , . .. , e j - I , u j , e j + l , . . . ,e n ) , wj E V ( X ) ,j = 1 , . .., A . We obtain an embedding of the Coxeter group into the building:

.

Vl

i ' L g'---,gc

:

VEV(6).

L/' VA

This is even a metric embedding of the Cayley-graph into the 1-skeleton of the building, where the metric on 6 is obtained from the natural metrics on the trees:

+...+ dn(un,~n),

d(u,v) = & ( w , v I )

where v = (211,. . . ,vA), u = ( ~ 1 ., .., U A )E V ( 6 ) . Remark: i) r acts freely on the vertices of 9. ii) No non-trivial subgroup of r has a bounded orbit.

22 1

6. Weak Amenability

For functions f , h : G

+ C their convolution is defined by: f *MY) =

c

f(z)h(z-ly).

ZEG

For summable f : G -b C we denote X ( f ) : Z2(G) + Z2(G) the associated convolution operator X ( f ) h = f * h. The regular (or reduced) C*-algebra C,*(G) is the just the operator norm closure of { X ( f ) : f E Z1(G)}. Its second commutant, the regular von Neumann algebra VN,(G), has a predual identified with the Fourier algebra A(G) = {cp = k * h : k , h E Z2(G)}. The duality is given by

< X(f), cp >=< X ( f ) h ,k >=

c

f(z)cp(z).

ZEG

With respect to pointwise multiplication of functions A(G) is a Banach algebra. A bounded function m E Z"(G), which with respect to pointwise multiplication multiplies A(G) into itself, is called a multiplier of the Fourier algebra. If the operator, dual to this multiplication, acts completely bounded on the regular von Neumann algebra, then the function m is called a completely bounded multiplier of A(G). By 11 m llcb we denote its completely bounded norm.

Definition 6.1. The group G is called weakly amenable if there exist, a constant c > 0 and a net ( m i ) i G lc A(G) such that micp

+ cp

Vcp E A(G) and IImi llcb I c V i E I.

Remarks: i) Any amenable group is weakly amenable, since for an amenable group A(G) has a bounded approximate identity. ii) The free group on two generators is weakly amenable. iii) In C*-algebraic terms there are the following equivalences respectively implications:

A discrete group G is amenable if and only if C,*(G)is nuclear. The group G is weakly amenable if and only if C,*(G) has the completely bounded approximation property.

If G is weakly amenable then C,*(G) is exact. Theorem 6.1. A finitely generated Coxeter group is weakly amenable.

222

We shall only sketch the proof of the theorem and refer the reader to7 for details. is positive definite. Hence there For 0 < T < 1 the function cpr(g) = exists a unitary representation r,.of G on a Hilbert space H , such that: V r ( g ) =< r r ( g ) t , t

>

for Some

E Hr.

Hence the functions cp, are completely bounded multipliers of A(G) with IIcp, llca = cpr(e) = 1. For T -+ 1 we have that p,t+!J-+ t+!J for all $J E A(G) but for T close to 1 the functions do not belong to A(G). The point of the proof is to cut them off in a proper way to control the complete bounded norm of the cut offs. From these one can then select a net in A(G) which fulfils the conditions of the definition. For z E C with ( z ( < 1 define pz(g) = ~ ' ( 9 ) . As a consequence of the tree structure of the graphs 7; the representations r, can be modified to yield non-unitary but uniformly bounded representations rz on H,, where T = Rez is the real part of z , with

For N E N let FN(eit) = C l k l l N ( l- &)eikt define t+!JN,r=

1

2a

denote the Fejer kernel and

FN(eit)cpreit dt

E A(G).

It is not hard to see that from these functions the net required in the definition of weak amenability can be chosen. 7. Haagerup Inequality In 1979 Haagerupg proved for the free group on two generators remarkable inequality:

II f * h 112 I (n+ 1111f llzll h 112,

IF2

the

h E Z2(F2),

whenever suppf c {u : l(u) = n}. Since then, inequalities of this kind have been proved for Gromov hyperbolic groups by Jolissaint12and de la Harpell. Their stability under free products and group extensions have been studied by Jolissaint. Let (G, S) be a group with a finite set of generators and associated lengthfunction 10.

223

Definition 7.1. We say that (G, S) satisfies a a Haagerup inequality whenever there exist C > 0, IC > 0 such that for all f with supp f c {u : 1(u) = n}:

I1 f * h 112 < C ( n+ 1)“11f llzll h 112,

h E Z2(G).

(3)

There are equivalent: o G satisfies a Haagerup inequality o There exist CI> O , I C ~> 0 such that for all finitely supported

f :G +

C:

>0

, >~ 0 such that for all finitely supported nonnegative symmetric f : G + R+:

o There exist C2

Here f (2k) denotes the 2k-th convolution power of f ,further one may suppose in the last statement that 11 f 111 1. That is, f is a symmetric probability measure on G and f(2k)(e)is the probability that the random walk with transition probabilities p(g, h) = f(g-lh), starting at e, will return to the identity in 2k steps.

<

Theorem 7.1. A Coxeter group satisfies a Haagerup inequality. About the proof: Generalising a theorem of Rammage, Robertson and Stegerl6 , a finitely generated group acting freely on an A, x . . . x A,building satisfies a Haagerup inequality. Since G / r in the extension 1 + l? + G + G/I’ + 1 is finite, a result of Jolissaint12 implies the assertion. Remark: Taking #S = 4 and m(s,s’) = 3 if s # s’, we obtain an example, which, by computations of Bestvina4 has virtual cohomological dimension two. Hence, here I? can not act freely on a single tree. This group is not Gromov hyperbolic either. Conjecture: The best possible value of the constant IC in (3) is qvcd(G). 8. Simplicity of the regular C*-algebra

If G is a discrete group then C:(G), the operator norm closure of {X(f) : f E Z1(G)}, is a C*- algebra with a canonical trace

224 TG : c,*(G)

+ c,

given by T G ( ~ = ) 0 if y # e and TG(e) = 1. If G is amenable, or more generally contains an amenable Go a G, Go # {e}, then trivG 4 XG, resp. Ind(trivG,) 4 XG, and the kernel of those representations is a non-trivial ideal. In contrast to this, a C*-algebra is called simple if it contains no non-trivial two-sided ideal. Examples of groups with simple reduced C*-algebra are necessarily non-amenable: (1) free group on TI 2 2 generators - Powers15, (2) appropriate free products - Paschke, Salinas14, (3) fichsian groups - Akemad , (4) torsion-free Gromov hyperbolic - Gromov, de la Harpel', ( 5 ) Zariski-dense subgroups of connected real semisimple Lie groups without compact factors - Bekka, Cowling, de la Harpe2.

Now consider conjugacy classes of elements:

C(Y) = {gw-l

:9

E GI.

Then, Gf = {y : #C(y) < 0 0 ) a G is amenable, hence for the simplicity of C,*(G) it is necessary that G is icc, i.e. #C(y) = 00 if y # e . Clearly, if G is a finite or S n e Coxeter group then G is amenable, hence C;(G) not simple. Hence we shall assume that the canonical bilinear form B defined in (1) is indefinite, furthermore we assume that it is non-degenerate. To deal with the non reducible case we suppose that the Coxeter group (G,S) is indecomposable. i.e. for all S1,S2 C S there are s1 E S1 and s2 E S 2 such that s1, s2 do not commute. De la HarpelO has shown that under these conditions the restriction of the canonical representation olr : r + Gl(V) remains irreducible whenever I'c G is a subgroup of finite index. It can be shown then, that this is true also for the complexification o @ Id : G + Gl(V @ C). Theorem 8.1. With the above conditions G is an icc-group.

VN,(G) a factor.

Proof: Let e # w E G and 2, be its centraliser. Then

#C(w) < co

* (G : 2,) < * o(w)= Id by irreducibility of olz, * w = e by faithfulness of o. 00

Hence

225

Now G is icc, then a theorem of Bekka and de la Harpe3 shows that CJ(G) is simple, with unique trace if and only if this holds true for C,*(I'),where I' is a torsion free, normal, finite index subgroup as in section 5. We shall concentrate on I?. Taking the trees . . , 7 as ~in that section we consider the action of I' on each of the trees. Bass-Serre theory shows:

x,.

Lemma 8.1. I' contains solvable subgroups if one of the trees has only vertices of degree at most two. On the other hand the theorem of Lie-Kolcbin allows to prove:

Lemma 8.2. rcontains n o non-trivial solvable subgroup. Hence all trees have vertices of degree at least three. A discussion of the action of r on the boundary of the trees yields:

Lemma 8.3. Assume h E I' and let F c C(h) be finite then there is v E I? such that < k,v >= Z * Z for all k E F . Then it is known, see Ref. 2, that there exists a C k E F: j

> 0 such that

for all

j

The above arguments show that I? is an ultraweak Powers group as considered by Boca and Nitica5. As a consequence C* (I?) is a simple C*-algebra, and as already mentioned we infer:

Theorem 8.2. Let (G,S) be a indecomposable Coxeter group, such that the canonical bilinear form is indefinite but non-degenerate, then C,*(G)is simple with unique trace. References 1. C. Akemann. Operator algebras associated to Fuchsian groups. Houston J. Math., 7:295-301, 1981. 2. M. Bekka, M. Cowling, and P. de la Harpe. Some groups whose reduced C*-algebra is simple. Publ. Math., Inst. Hautes Etud. Sci.,80:117-134, 1995. 3. M. B. Bekka and P. de la Harpe. Groups with simple reduced C*-algebras. EXP.Math., 18:215-230, 2000. 4. M. Bestvina. The virtual cohonolocical dimension of Coxeter groups. In Geometric Group Theory I, Cambridge, 1993. Cambridge Univ. Press. 5. F. Boca and V. Nitica. Combinatorial properties of groups and simple C*algebras with a unique trace. J. Operator theory, 20:183-196, 1988.

226 6. M. Bozejko. Positive and negative definite kernels on groups. Lectures at the University of Heidelberg, 1987. 7. G. Fender. Weak amenability of Coxeter groups. arXiv:math.GR/0203052, 2000. 8. G. Fender. Simplicity of the reduced C*-algebra of certain Coxeter groups. Ill. J. Math., 47(3), 2003. 9. U. Haagerup. An example of a non-nuclear C*-algebra which has the metric approximation property. Invent. Math., 50279-293, 1979. 10. P. d. 1. Harpe. Groupes de Coxeter infinis non atFines. Exp. Math., 5:91-96, 1987. 11. P. d. 1. Harpe. Sur les alghbres d’un groupe hyperbolique. C. R. Acad. Sci. (Paris) Serie I, 307:771- -774, 1988. 12. P. Jolissaint. Rapidly decreasing functions in reduced C*-algebras. !?+am. Amer. Math. SOC.,317:145-157, 1990. 13. J . J. Millson. On the first Betti number of a constant negatively curved manifold. Ann. of Math., 104235-247, 1976. 14. W. Paschke and N. Salinas. C*-algebras associated with free products of groups. Pacific J . Math., 82:211-221, 1979. 15. Powers, R. T. Simplicity of the C*-algebra associated with the free group on two generators. Duke Math. J., 42:151-156, 1975. 16. J. Ramagge, G. Robertson, and T. Steger. A Haagerup inequality for A1 xA1 and A2 buldings. Geometric And finctional Analysis, 8:702-731, 1998.

A LIMIT THEOREM FOR CONDITIONALLY INDEPENDENT BEAM SPLITTINGS

K.H. FICHTNERYOLKMAR LIEBSCHER~MASANORI OHYA

1. Introduction

Consider a Boson Fock space r ( L 2 ( G p, ) ) . Then the coherent state $f to a wave function f describes a system of particles in the same one-particle state f/ llfll in the space G. For testing whether two coherent states 4f and $f‘ with l l f l l = Ilf’ll (i.e., the mean number of particles is equal) belong to the same one-particle state or f If’, one could use the following procedure. One applies a conventional beam splitter which transforms the state $f @ $f‘ into $(f-f’)/JZ @ $(f+f‘)/J;i. If f = f ‘ the state 4(f-f’)/JZ is the vacuum $O. Thus measurement using the projection onto the vacuum function in Fock space yields a distribution concentrated on 61. If f I f’,the same measurement has under the state $(f-f’)lJZ the distribution e-llf1126, (1 - e - ~ ~ f ~ ~ 2For ) 6 high ~ . numbers of particles, i.e. llf1I2 = llf’1I2 >> 1, we can therefore distinguish between the two situations with high confidence. The basic idea of this procedure was applied to quantum teleportation [11,12] as well as in the modelling of brain recognition processes, see [lo]. But, especially in the latter case, the condition l l f l l = Ilfll‘ is not realistic because the density of the state representing the input-information differs from the individual density of excited neurons in the brain. Therefore there is need for a preprocessing step which makes the mean number of particle of both states equal to each other. Standard quasifree transition operators

+

* Friedrich-Schiller-Universitat, fakultat fur mathematik und informatik, institut fur angewandte mathematik, D-07743 Jena (Germany) t GSF -National research centre for environment and health, institute for biomathematics and biometry, ingolstadter landstr. 1, D-85758 Neuherberg, Germany, emai1:liebscherQgsf.de *department of information sciences, science university of Tokyo, Noda city, Chiba 2788510, Japan. e-mail: ohyaQis.noda.sut .ac. j p

227

228

are not sufficient for this purpose since then Q(q5f) = q5Tf+h for a linear operator T and some h. In the present paper, we present a limiting procedure based on conditionally independent beam splittings [8],which is a weak generalisation of conventinal beam splittings which achieves the goal of transforming all high mean intensity f to the same lower intensity f' for wave functions f which are constant. The consideration of locally normal states ensures that the mean particle number may be 00 as a prerequisite for limit theorems. A generalisation of the limit theorem to non-constant f is straight forward but tedious and left for the future. 2. Basics

First we introduce Boson Fock spaces in the language of counting measures. This seems to be a convenient description of these spaces as far as particle exchange mechanisms, like introduced in [8] and used for the main result of this paper, are concerned. For details on this representation of Fock spaces we refer t o [6,7,13],see also [18,19,20,21,22]. Let G be an arbitrary complete separable metric space and 6 its aalgebra of Bore1 sets. Further, let p be a locally finite diffuse measure on [G,631,i.e. p ( K ) < 00 for bounded K E 6 and ~((2))= 0 for all singletons x E G. Especially we can apply our results in the standard setting G = Rd and p = Cd is Lebesgue measure. By M we denote the set of all finite integer-valued measures on [G,651, i.e. M is the set of finite counting measures. Since each cp E M can be . - . 6,, for some n E N and xj E G written in the form cp = 6,, (where 6, is the Dirac measure in x) the elements of M code the finite point configurations in G. We equip M with its canonical a-field !TI - the smallest a-field containing all sets of the form {'p E M : cp(K) = n},K E 6 , n E N. On [ M ,!TI] we introduce the exponential measure F" [2] by setting

+

+

Hereby, o is the empty configuration in M , i.e. o(G) = 0. We call M := L 2 ( M ,9X,FP) symmetric Fock space over G, a similar definition of the Fock space one can find e.g. in [Zl]. For a function g : G C we introduce the exponential vector qg : M C through

-

-

229

If G is not bounded one can introduce a quasilocal algebra which allows to model infinite Boson systems. For any measurable K E 6 there is the Hilbert space M K = L 2 ( M ~ , M n M K , F ~ ~ M where K ) MK = {'p E M : p(Kc) = O}. It is well-known that M 2 M K 8 MKc such that $g 2 $gx, 8 qgXKc. This factorisation allows t o define local algebras dK C ( M K )8 1 [ ~ ,for~ bounded K as well as the quasilocal C*-algebra A as the uniform closure of the union of all local algebras. In this work, we are only interested in special states on A, the coherent ones. The space Lt,(G, p ) of locally square integrable functions consists of all p-equivalence classes of measurable functions g such that lK /gI2dp < 00 for bounded K. For g E L t c ( G , p ) the coherent state $9 is the locally normal state given by

$ g ( A )= e-11g112 ($g,A$g),

( A E dK,K E 6, bounded ).

(3) The coherent state $ = 4° is the vacuum state. Note that the so-called position distribution Q,pwill be a Poisson point process, see [5].

-

We follow [8] for the definition of exchange operators. For a map C we define an operator Uv by

Y

:

M4

uv$('pl, V2) =

v(7l

A 'p1771 A q 2 7 7 2 A ( P I 7 7 2 A (P2)$(71 9 7 2 ) (4)

7 1 +72=991+(P2

using the natural lattice structure of M related to the ordering 'p1 5 'p2 ++ 'pl(B) 5 p2(B)VB E f3. Especially, we are interested in conditionally independent beam splitting derived as & from kernels b

-

b ( p l , p27 9 3 , (P4) = ~bl(.,(~)((Pl)~b,(.,~)((P2)~b~(.,~)((P3)$b~(~,(~)((P4)

+ +

(5)

where we use the convention cp = cp1+ p2 cp3 cp4 and bi : G x M C, i = 1 , 2 , 3 , 4 . Conventional beam splittings correspond to bi depending on the first argument only or, typically, even constant bi [S]. We want to apply such operators for various regions A. So suppose we have functions Pi : [0,00) HC such that

From [8] we derive easily

Lemma 2.1.

is unitary if

fulfuil

230 Therefore, we can use U ~ to A transform states (on d A @ d A ) if (7) is fulfilled. Convergence of states we understand as convergence of the local states in norm, i.e. w, t-+ w iff n+w

II(wn)K

- WKII

( K E 6,bounded ).

,two,

This topology is metrizable if we restrict to so-called locally normal states like the coherent ones [l]. A + G denotes the limit over the net of bounded Bore1 sets, i.e. convergence for each monotously increasing sequence (An)nENsuch that for all bounded K E 6 there is n E N with A, 2 K . 3. The Results

We define for all bounded A E 6 a channel (conditionally independent exchange) Q ~ , Aon C(M @ M ) by ) Q~,A(AA@AAc) = (U~AAAU*A)@AAC,( AA E C ( M A @ M A ) , A AECC(MAC@MAC)

where bA is given through ( 6 ) and Pi fulfils (7). For z E C , denote the coherent state @Xc briefly by qY.

Theorem 3.1. Suppose Pi, i = 1 , 2 , 3 , 4 fulfil(7) and z E C is such that all Pi are continuously differentiable in a neighbourhood of 1 ~ 1 ' . Then

with

d

m

Example 3.1. If P1(r) = for some a > 0, we find that the first component of all states limb+G Qi,A(qY @ $'), 1z1' > a, is a coherent state with fixed mean intensity Ic11' = a. If further PZ = this holds for the second component as well. This is the result of interest mentioned in the introduction. Clearly, we can apply the above theorem with this P unless 1z1' = a. The next results study the limiting behaviour of the left hand side in (8) for this and similar situations. First, we weaken the continuity condition and get a mixing effect.

d

m

,

Theorem 3.2. Suppose z E R and 6 > 0 are such that all Pi have left and right limits as r --+ 1z1' and

231 0

in the internal (1zI2- 6, 1zI2],the functions Pt,

0

are continuously differentiable. in the interval [ ! . I 2 , 1zI2 6 ) , the functions P;,

+

are continuously differentiable. Then

with


czf. = P2(1.I2 f0 ) z This is still not sufficient for Pl(r) = d m , as /32 is not differentiable at points r where IP1 ( r )I = 1 and P1 is differentiable with ( r f 0) In these points I/32(r = O(&). Thus we need also the following

+I)&

# 0.

Theorem 3.3. Let z E R and /3 be such that

IPl(IZl",l

O

=1

/3 is continuous in a neighbourhood of (tl2 is differentiable in a left and a right neighbourhood (with limits of the derivative in lz12) and 8 z ( 1 4 2 + E ) has left and right limits in 0. E H 6

Then, with <1,2 given by (9), lim Q ; , ~ ( @8 4') =

A+CC

@ q+.

4. Proofs

Proof. (of Lemma 2.1) In [8] it was proven that if b is given by (5) and all bi are bounded then Ub is unitary if and only if for F-a.a. 'p E M for all x E SUPPCP

232 The

2.f

part of this result translates directly into (7) if (6) is applied.

Proof. (of Theorem 3.1) Let K be fixed, then A -, G shows that A 3 K eventually and C(A\K) -+ 00. We will set to = C(K)and t = C(A\K) 4 00. We have t o compute the kernel of Q Z , , ( V @ + ' ) K U K . First we determine the kernel p of Q>,,(@ @ ~ O ) A , A . By definition, for any f E L2(G, v),

W$f c2 $O)((Pl,

c

-

(P2)

$01 ( 7 1 A ( P l ) $ b (71 A (P2)$03 ( 7 2 A (Pl)$fk(72 A (P2)$f(71)$0(72)

71+72=v1+v2

- $01 (v1+v2) ((P1)%2(v1+p2) ((P2)$f((P1 - ~01(v1+vz)f((P1)~02(vpl+vz)f((P2).

+ Cp2)

This yields p ( ' ~ 1 (, ~ 2( , ~ 3( , ~ 4 = ) e-llzxAAlla$p(

vi+v2)zxA((~1)$ba(vi+vz)zxA((~2)

$b: (v3++'4)zXA ((P3)$bt(p3+v4)zX~ (p4)

where XA denotes the indicator function of the set A. We find that the state G&(V @ + O ) K has again property ( 7 1 ) [17,15],i.e. its kernel po depends only on particle numbers: pO((p1, ( ~ 2( ,p 3 , p 4 )

c

=

e-lz12totk+k'z'+l' ck,kt,Z,l' t

k,k',l ,I'EN

(l(Pll)X{k'})( ~ ' P 2 ~ ) x { Z } ) ( I ~ 3 ~ (1941) )x{l~}) where [(PI = ( P ( K )We . find from (for a precise definition of this formula in the general case see 114,161) x{k})

po((Pli(P2,(P3,(P4)

= 1~~\,(d[@l,d2])p((P1+dl,lpz+d2,(P3+@1,(P~+@2)

that (denoting @(k) = & ( k / C ( A ) ) ) t

Ck,kr,l,l'

=

-1zIZt

Fi\K(d[+l,@21) k+lGlI

+ 1+11+

1@21)

P,^@ + k'

k'+ Ik2 I

+ 1@11+ 1@21) + k' + l@l)+ J @ 2 J ) Z + v 3 ; ( 1 + 1' + 1@11 + 1@21)"+Id1 1 I421@1+82I

P:'@ + k' P:'(k

1

Now is it enough t o prove convergence of ck,k!,l,l, as t + 00 to a suitable limit. The reason is the following lemma which can be found in [3]. Lemma 4.1. If w,, w are normal states on B(3-1) such that for a total set

T23-1 wn(P+,+)n~mw(P+,+~)i ( 1cI,1cI' E T )

(10)

233

then ( w , ) , ~ ~ converges an n o r m t o w . We apply this lemma with the total set consisting of vector being in the span of {&xK : z E C} resp. orthogonal to it. For the orthogonal ones, on both sides (10) there is zero, for the others we use convergence of c : , k , , l , / , We find from definition of F and Pi for k,k‘, 1,l‘

We see for random variables X t , X t ck,k’,l,ll

~

1

~

that 1 2

~

~

=

(P(

+ty;xt

) , P ( l +l’+X,))X’

t

+ to

Additionally, we know X t / t +%(zI2. Upto here only the distribution t-w matters, but we can assume for convenience (due to the Skorohod r e p resentation Theorem, cf. e.g. [4]), that the convergence is actually almost surely on some artificial probability space. Due to (7) we find pz(r)pz(r’)( 5 1, such that we can apply the dominated convergence theorem. We want to find the limit by Taylor expansion. First, let p be one of Pi and apply the expansion

I~/?I(T’) +

P(2

+

E)

= P(2)

+

EP‘(Z)

+

O(E)

(11)

234 in an interval around \z(’ leading to

-

!l?Zpi(r)p:(r)= 0. This leads

Differentiating the first line in (7) yields to

k + k’ + X ” , $ ( P ( t+to

=1

+ 1’ + X t ) ) t

+to

+ o(t-1)

and thus

k+k’+Xt l+l’+Xt )) (@( t to ),p( t +to

+

xt =1

+ o ( X t / t ) = 1 + o(1).

This Shows

Since the left hand side gives the correct matrix element for (q5f11(lz12)z q5flz(lz12)z) K this completes the proof.

@

Proof. (of Theorem 3.2) We apply the same method, but derive the limit of c ; , ~ , , ~differently. ,~, It holds by the central limit theorem even

for some random variable Y and again we assume almost sure convergence on some artificial probability space. Rewriting

xt = ( Z l 2 t + (21JiY + o ( J i ) ,

(t

--+

00

)

we derive that

P ( X t / t > 1z12 eventually ) = 1/2 = lim P ( X t / t < 1z12 eventually t+m

1.

On both sets we may apply the above scheme to derive the result. Proof. (of Theorem 3.3) Here it is crucial to estimate for some 0 E C P2(12I2

+ E ) = m +4&)

(12)

235 what leads to (separately on {Y

since consideration of leads to

> 0)

and {Y

< 0))

1 for small

References 1. 0. Bratteli and D. Robinson Operator Algebras and Quantum Statistical Mechanics I 2nd edition Springer Berlin Heidelberg New York 1987 2. D. Carter and P. Prenter Exponential spaces and counting processes Z. Wahrscheinlichkeitstheor. Verw. Geb. 21:l-19 1972 3. E. Davies Quantum Theory of Open Systems Academic Press New York 1976 4. S. N. Ethier and T. G. Kurtz Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics John Wiley & Sons New York etc. 1986 5. K.-H. Fichtner and W. Freudenberg Point processes and the position distribution of infinite boson systems J . Stat. Phys. 47:959-978 1987 6. K.-H. Fichtner and W. Freudenberg Characterization of states of infinite boson systems I. On the construction of states of boson systems Commun. Math. Phys. 137:315 - 357 1991 7. K.-H. Fichtner and W. Freudenberg Remarks on stochastic calculus on the Fock space In L. Accardi, editor, Quantum Probability and Related Topics VII World Scientific Publishing Co. Singapore, New Jersey, London, Hong Kong 1991 pages 305 - 323 8. K.-H. Fichtner, W. Freudenberg, and V. Liebscher On Exchange Mechanisms for Bosons submitted 1998 9. K.-H. Fichtner, W. Freudenberg, and V. Liebscher Time Evolution and Invariance of Boson Systems Given by Beam Splittings Infinite Dimensional Analysis, Quantum Probability and Related Topics 1(4):511-531 1998 10. K.-H. Fichtner, W. Freudenberg, and M. Ohya Recognition and teleportation pages 85-107 11. K. Fichtner and M. Ohya Quantum teleportation with entangled states given by beam splittings Commun. Math. Phys. 222(2):229-247 2001 12. K. Fichtner and M. Ohya Quantum teleportation and beam splitting Commun. Math. Phys. 225(1):67-89 2002 13. W. Freudenberg On a Class of Quantum Markov Chains on the Fock

14.

15. 16.

17. 18. 19.

20. 21.

22.

Space In L. Accardi, editor, Quantum Probability & Related Topics IX World Scientific Publishing Co. Singapore 1994 pages 215 - 237 V. Liebscher Diagonal Integration and Diagonalized Versions Forschungsergebnisse der Fakultat fiir Mathematik und Informatik Jena Math/Inf/96/38 1996 V. Liebscher Beam Splittings, Coherent States and Quantum Stochastic Differential Equations Habilschrift FSU Jena 1998 V. Liebscher Diagonal Versions and Quantum Stochastic Integrals on the Symmetric Fock Space with Nonadapted Integrands Prob.Th.Rel.Fields 112:255-295 1998 V. Liebscher Using weights for the description of states of boson systems in preparation 2001 J. Lindsay Quantum and Noncausal Stochastic Calculus Prob. Th. Re1. Fields 97:65-80 1993 J. Lindsay and H. Maassen An integral kernel approach to noise In L. Accardi and W. uon Waldenfels, editors, Quantum Probability and Applications I11 volume 1303 of Lecture Notes in Mathematics Springer-Verlag Heidelberg 1988 pages 192 -208 J. Lindsay and H. Maassen The stochastic calculus of Bose noise Preprint 1988 H. Maassen Quantum Markov processes on Fock space described by integral kernels In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications I1 volume 1136 of Lecture Notes in Mathematics Springer Berlin, Heidelberg, New York 1985 pages 361 - 374 P. Meyer Quantum Probability for Probabilists volume 1538 of Lecture Notes in Mathematics Springer Berlin Heidelberg New York 1993

ON FACTORS ASSOCIATED WITH QUANTUM MARKOV STATES CORRESPONDING TO NEAREST NEIGHBOR MODELS ON A CAYLEY TREE

FRANCESCO FIDALEO Dipartimento di Matematica Universitd di Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy, E-mail: fidaleoOOmat.uniroma2.it FARRUH MUKHAMEDOV Department of Mechanics and Mathematics National University of Uzbekistan Vuzgorodok, 700095, Tashkent, Uzbekistan, E-mail: far75mOOyandex.r~ In this paper we consider nearest neighbour models where the spin takes values in the set @ = {q1,v2,.,.,qq} and is assigned to the vertices of the Cayley tree .?I The Hamiltonian is defined by some given A-function. We find a condition for the function A to determine the type of the von Neumann algebra generated by the GNS - construction associated with the quantum Markov state corresponding t o the unordered phase of the A-model. Also we give some physical applications of the obtained result. 2000 Mathematical Subject Classification: 47A67, 47L90, 47N55, 82B20. Keywords: von Neumann algebra, quantum Markov state, A- model, Cayley tree, unordered phase, Gibbs measure, GNS - construction.

1. Introduction It is known that in the quantum statistical mechanics concrete systems are identified with states on corresponding algebras. In many cases the algebra can be chosen to be a quasi-local algebra of observables. The states on this algebra satisfying the KMS condition, describe equilibrium states of the quantum system. Basically, limiting Gibbs measures of classical systems with the finite radius of interactions are Markov random fields (see e.g. Ref. 8, 22). In connection with this, there arises a problem to construct analogues of non-commutative Markov chains. In Ref. 1 Accardi explored

237

238 this problem, he introduced and studied noncommutative Markov states on the algebra of quasi-local observables which agreed with the classical Markov chains. In Ref. 3, 15,2, modular properties of the non-commutative Markov states were studied. In Ref. 10 Fannes, Nachtergale and Werner showed that ground states of the valence- bond- solid modles on a Cayley tree were quantum Markov chains on the quasi-local algebra. In the present paper we will consider Markov states associated with nearest neighbour models on a Cayley tree. Note the investigation of the type of quasi-free factors (i.e. factors generated by quasi-free representations) has been an interesting problem since the appearance of the pioneering work of Araki and Wyss5. In Ref. 21 a family of representations of uniformly hyperfinite algebras was constructed, which can be treated as a free quantum lattice system. In that case factors corresponding to those representations are of type I&, X E (0,l). More general constructions of product states were considered in Ref. 4. Observe that the product states can be viewed as the Gibbs states of Hamiltonian system in which interactions between particles of the system are absent, i.e. the system is a free lattice quantum spin system. So it is interesting to consider quantum lattice systems with non-trivial interactions, which leads us, as mentioned above, to consider the Markov states. Simple examples of such systems are the Ising and Potts models, which have been studied in many papers (see, for example, Ref. 24). We note that all Gibbs states corresponding to these models are Markov random fields. The full analysis of the type of von Neumann algebras associated with the quantum Markov states is still an open problem. Some particular cases of the Markov states were considered in Ref. 11,14,17,18,19. The present paper is devoted t o the type analysis of some class of diagonal quantum Markov states, which correspond to a X-model on the Cayley tree, in which spin variables take their values in a set @ = {ql, ..., q q } ,where r]k E k = 1,.. . ,q. Observe that the considered model generalizes a notion of X-model introduced in Ref. 23, where the spin variables take their values f l .

2. Definitions and preliminary results

The Cayley tree rk of order k 2 1 is an infinite tree, i.e., a graph without cycles, such that each vertex of which lies on k 1 edges. Let I? = (V,A), where V is the set of vertices of rk,A is the set of edges of I?. The vertices x and y are called nearest neighbor, which is denoted by I =< x , y > if

+

239 there exists an edge connecting them. A collection of the pairs < z, z1>, . . . , is called path from the point z to the point y. The distance d(z,y), z, y E V, on the Cayley tree, is the length of the shortest path from z to y. We set Wn = {Z E v J ~ ( x2'), = TI},

Ln = ( 1

=< Z , Y >E L ~ z , YE Vn},

for an arbitrary point xo E V. Denote 1x1 = d(x,xo), x E V. Denote S(Z) = {y E Wn+l :d ( ~ , y = ) I}, x E Wn. This set is made of the direct successors of x. Observe that any vertex x # xo has k direct successors and xo has k + 1. Theorem 2.1. There exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k 2 1 and the group Gk+l of the free products of k 1 cyclic groups of the second order with generators

+

al,a2,.**,ak+l.

Consider a left (resp. right) transformation shift on follows. For go E Gk+l we put

T g o h= goh (resp. T g o h= hgo,), h E

Gk+l

defined as

Gk+l.

It is easy to see that the set of d l left (resp. right) shifts on Gk+l is isomorphic t o the group Gk+l. Let @ = { q l , q 2 ,...,qQ}, where ql,q2,..., qq are vectors in RqP1, such that

We consider models where the spin takes values in the set @ = { q l , q 2 , ...,q q } and is assigned to the vertices of the tree. A configuration o on V is then defined as a function x E V + o(x) E @; the set of all configurations coincides with R = &. The Hamiltonian is of an A-model form : HA(0)

=

c

<X,Y>

A(+),

4Y); J ) ,

(2)

240

where J E Iwn is a coupling constant and the sum is taken over all pairs of neighboring vertices < z,y >, a E R. Here and below A : @ x @ x Rn 3 R is some given function. We note that A-model of this type can be considered as a generalization of the Ising model. The Ising model corresponds to the case q = 2 and X(Z, y; J) = -J X Y . We consider a standard a-algebra 3 of subsets of R generated by cylinder subsets, all probability measures are considered on (0,3).A probability measure p is called a Gibbs measure (with Hamiltonian H A )if it satisfies the DLR equation: for n = 1,2, ... and a, E GVn:

where

Y $ ~ ~ + , is

the conditional probability

(0,) YWIW,+l vn

= 2-l (w IWn+l) e ~ ( - P ~ ( IaIwsIwn+1)1.

where /3 > 0. Here snlvn and WI W , + ~ denote the restriction of a,w E R to V, and Wn+l respectively. Next, a, : 2 E V, 3 a,(z) is a configuration in V, and H(a,,llwl~,,+,)is defined as the sum H(o,) +U(~,,WIW,,+~) where

c

H(an) =

<X,Y>€L,

U(an,WIwn+l) =

A ( ~ n ( ~ ) , % b J), );

c

A(aTa(z),w(y); 4.

<x,y>:X€Vn,yEWn+l

Finally, Z ( w l ~ , + ~stands ) for the partition function in V, with the boundary condition w I : Z(wlwn+l) =

C

e~(-PH(~~IIwIwn+A. cnEOVn Since we consider nearest neighbour interactions, the Gibbs measures of the A-model possess a Markov property: given a configuration w, on W,, random configurations in V,-l and in V \ Vn+l are conditionally independent. It is known (see Ref. 26) that for any sequence dn) E R, any is a Gibbs measure. Here limiting point of measures

52,,)

IWn+l

is a measure on fl such that Vn’

0,

otherwise.

> n:

52 ,wn+l n,

24 1

We now recall some basic facts from the theory of von Neumann algebras. Let B ( H ) be the algebra of all bounded linear operators on the Hilbert space H ( over the field of complex numbers C). A weak (operator) closed *-subalgebra N in B ( H ) is called von Neumann algebra if it contains the identity operator 1. By Proj(N) we denote the set of all projections in N . A von Neumann algebra is a factor if its center

Z(N := {x E N : xy = yx,

Vy

EN}

is trivial, i.e., Z ( N )= { A 1 : X E C}. The von Neumann algebras are direct sum of the classes I (In,n < 00, Im), I1 (IIl,II,) and 111. Further, a factor is of only one type among these listed above, see e.g. Ref. 30. An element x E N is called positive if there is an element y E N such that x = y*y. A linear functional w on N is called a state if w(x*x) 2 0 for all x E N and w(1) = 1. A state w is said to be normal if ~ ( s u p x , ) = supw(x,) for any a

a

bounded increasing net {xa} of positive elements of N . A state w is called trace (resp. faithful) if the condition w(xy) = w(yx) holds for all x,y E N (resp. if the equality w(x*x) = 0 implies x = 0). Let N be a factor, w be a faithful normal state on N and aW , be the modular group associated with w (see Definition 2.5.15 in Ref. 6). We let I?(&) denote the Connes spectrum of the modular group a; (see Definition 2.2.1 in Ref. 6). For the type I11 factors, there is a finer classification. Definition (Ref. 9). The type I11 factor N is of type (i) 1111,if r(aw) = (ii) IIIx, if r(aw) = {nlogX,n E Z}, X E (0,l); (iii) I&, if r(aw) = (0); see, e.g. Ref. 6, 28 for details of von Neumann algebras and the modular theory of operator algebras.) 3. Construction of Gibbs states for the A-model

In this section we give a construction of a special class of limiting Gibbs measures for the A-model on the Cayley tree. Let h : x + h, = (hl,,, h2,z, ...,hq-l,z) E be a real vector-valued function of x E V . Given n = 1 , 2 , ... consider the probability measure ~ ( on aVndefined by /Jn)(an)= Z,-l exp{-PH(an)

+

c

XEW,

hzo(x)}

7

(3)

~

1

242

Here, as before, a, : x E V, partition function:

+ on(x) and

2, is the corresponding

X€W,

e n EQv,

The consistency conditions for p(,)(o,), n 2 1 are ~ p ( ~ ) ( a , - l , a ( n=) p(n-l)(a,-l), )

(4)

a(")

where a(n) = { a ( x ) , xE Wn}. Let V1 c V2 c ... Ur?, V, = V and p 1 , p 2 , ...be a sequence of probability measures on aV1, Ovz, ... satisfying the consistency condition (4). Then, according to the Kolmogorov theorem (see, e.g. Ref. 25), there exists a unique limit Gibbs measure jJh on R such that for every n = 1,2, ... and on E aVnthe equality holds

0

P {alv, = an}

= P(%,).

(5)

Further we set the basis in RQ-l to be ql, 72,..., qq-l. The following statement describes conditions on h, guaranteeing the consistency condition of measures p(,) (a,). Theorem 3.1. The measures ,u(,)(a,), n = 1,2, ... s a t i s b the consistency condition (4) if and only if f o r any x E V the following equation holds:

h', =

c

F(h',,X) >

(6)

l/ES(X)

Here, and below ha stands f o r the vector *hx and F : RQ-' function is F(h;X)= (Fl(h;X),..., Fq-l(h;X)),with

+ RQ-' (7)

Fi(h,,h2,...,hq-,;X)

i = 1,2, ...,Q - 1, h = ( h l ,...,hq-1). The proof uses the same argument as in Ref. 18,19. Denote

V = { h = (h, E EXq-'

:x E

F(h,,X), V X E V } .

V ) : h, = l/ES(X)

According to Theorem 3.1 for any h = ( h x , x E V ) E V there exists a which satisfies the equality (5). unique Gibbs measure

243

If the vector-valued function ho = (h, = (0, ...,0 ) , x 6 V)is a solution, i.e. ho E 2) then the corresponding Gibbs measure p p ) is called the unordered phase of the X-model. Since we deal with this unordered phase, we have to make an assumption which guarantees us the existence of the unordered phase. Assumption A. For the considered model the vector-valued function ho = (h, = (O,O, ...,0 ) , x E V) belongs to 2). This means that the equation (6) has a solution h, = ho = 0,x E V. According to Theorem 2.1 any transformation S of the group Gk+l induces an automorphism ,$ on V. By &+I we denote the left group of shifts of Gk+l. Any T E G k + l induces a shift automorphism 5! : + by (Fa)(h)= a ( T h ) , h E Gk+i, 0 E R . It is easy to see that p e l o = p o('1 for every 5! E G k + l . AS mentioned above, the measure p c ) has a Markov property (see, Ref. 27). Assumption B. We suppose that the measure p p ) enables a mixing property, i.e. for any A, B E 3 the following holds

Note that the last condition is satisfied, for example, if the phase transition does not occur for the model under consideration.

4. Diagonal states and corresponding von Neumann algebras

Consider C*-algebra A = @lykMq(c),where Mq(C!)is the algebra of q x q matrices over the field C! of complex numbers. By e i j , i , j E {1,2, ...,q} we denote the basis matrices of the algebra M q ( C ) . We let CM,(@)denote the commutative subalgebra of Adq(@)generated by the elements eii i = {1,2, ...,q } . We set CA = @ykCMq(C). Elements of commutative algebra CA are functions on the space R = { e l l , ..., eqq}". Fix a measure p on the measurable space (0,B ) , where B is the 0-algebra generated by cylindrical subsets of R. We construct a state w p on A as follows. Let P : A + CA be the conditional expectation, then the state w p is be defined by wp(x) = p ( P ( z ) ) ,x E A, here p(P(x)) means an integral of a function P ( z ) under measure p, i.e. p(P(x)) = J, P(x)(s)dp(s) (see Ref. 29). The state is called diagonal.

244

By w c ) we denote the diagonal state generated by the unordered phase pt;"). The Markov property implies that the state up)is a quantum Markov state (see 3). On a finite dimensional C*-subalgebra Avn = @vnMq(C) c A

we rewrite the state

up) as follows

where tr is the canonical trace on Av,. The term X(a(x)o(y);J) in (4) is given by a diagonal element of Mq(C) @ Mq(C) in the standard basis as follows B(1) 0 . . .

P X o ( x ) ,4 4 ) ;

4=

(10)

. .. Here, B ( k )= ( b i j , k ) : , j = l , k = l,dots,q are q x q matrices, and

Consequently, using (9) and from (10),(11) (cp. Ref. 26, Ch.1, $1)the form of Hamiltonian fi(Vn) in the standard basis of Av, (i.e. under the basis matrices) is regarded as

0 Denote M = 7r (X)(A)",where T WO

(A)

WO

. ... . . . .B('3)

0

is the GNS - representation asso-

ciated with the state w c ) (see Definition 2.3.18 in Ref. 6). Our goal in this section is to determine a type of M . Remark. In Ref. 29 general properties of a representation associated with diagonal state were studied, but there concrete constructions of states were not considered. In Ref. 20 a deep classification of types of factors generated by quasi-free states has been obtained. For translation-invariant Markov states the corresponding type analysis has been made in Ref. 13.

245

The investigation of the type of factor arising from translation invariant or periodic quantum Markov states on the one dimensional chains is contained in Ref. 11. Now we define translations of the C*-algebra A . Every T E !&+I induces a translation automorphism rT : A + A defined by

Z€V"

XEV,

Since measure 111"' satisfies a mixing property (see (8)), then we can easily obtain that w?) also satisfies the mixing property under the translations { T T } T ~ B ~,+ i.e. , for all a, b E A the equality holds

According to Theorem 2.6.10 in Ref. 6, the algebra M is a factor. We note that the modular group of M associated with is defined by

up)

(A) CTYO

(2)=

where R(A) =

lim exp{itfi(A)}zexp{-itl?(A)},

Atrb

C

x

EM

,

(12)

It is well know that the last limit exists

<x,y>EA

if a suitable norm of the potential l? is finite (see Theorem 6.2.4. in Ref. 7). First of all, we recall the definition of the norm of a potential Q = CxCrb Q(W

where d > 0. Here @(X)E Ax = @ x M q ( C ) . NOW we compute 1lRlld:

m~ w,k 1 lOgpij,kI < co .

Hence the norm of l? is finite, therefore the limit in (12) exists. By M u one denotes the centralizer of w:'), which is defined as (A)

M" = {z E M

:

op

(z) = 2,

t E R} .

Since up)is Gibbs state, according to Proposition 5.3.28 in Ref. 7, the centralizer M u coincides with the set M ( A ) = {z E M : w F ) ( z y ) = ~ F ) ( y z ) y, E M } , (13) wo

246

where we denote by w r ) also the normal extension of the state under consideration t o all of M . By II[n]we denote the group of all permutations y of the set V, such that $22)

= 2, 2 E

w,

Every y E D[n] defines an automorphism ar : M

.rcn,”,,

a,) =

+M

by

n,,, 8

Mp(@) = id,

where id is the identity mapping. Denote

so = UIarIY E II[nl}. Simply repeating the proof of a proposition in Ref. 16, we can prove the following Lemma 4.1. T h e group

Go = {a E SOI wr’(a(x))= w p ) ( x ) ,

acts ergodically o n M , i.e. the equality a(.)

2

EM},

= x, a E Go implies x = 8 i ,

e E c. Lemma 4.2. T h e centralizer M u i s a factor of type IIl.

Proof. From the definition of the automorphism a7 (see (14)), it is easy to see that every automorphism a E Go is inner, i.e. there exists a unitary u, E M such that a(.) = u,xuL,x E M . From the condition up)o a = we find

up)

It follows from (13) that u, E M ( A ) . According to Lemma 4.1 the group *O Go acts ergodically, this means that the equality u,x = xu, for every a E Go implies x = 81,B E C. Hence, we obtain {u,la E Go}’ = Cl. Since Mu’ c {u,}’ we then get

M‘‘nM=ci. In particular Mu’ n M u = @I.This means that M“ is a factor.

I7

247

NOW we are able to prove main result of the paper (compare with the analogous result in Ref. 11). Theorem 4.1.

(i) If the fraction

is rational f o r every i, j , m, 2, k , p , u , v E 1,2, ...,q, whenever the denominator is different f r o m 0, then the von Neumann algebra M associated with the quantum Markov state corresponding t o the unordered phase of A-model (2) o n a Cayley tree is a factor of type IIIe . (ii) If M is a type of 1111factor, then all the fraction cannot be rational. Proof. It is known (see Proposition 2.2.2 in Ref. 9) that Connes' spectrum r(a)of group of automorphisms Q: = { Q : ~ } ~of~von G Neumann algebra M has the following form

r(Q)= n { S p ( a e ) l e E P r o j ( Z ( M " ) ) , e # O},

(15)

where a e ( z )= Q:(eze),z E e M e and Z ( M " ) is the center of subalgebra

M" = {z E M : a g ( z )= z,

9 E G}.

Here, Sp(a) be the Arveson's spectrum of group of automorphisms a: (see for more details Ref. 9,28). By virtue of Lemma 4.2 we have Z ( M " ) = Cl. The equality (15) (a) implies r(&))= sp(ow0 1. We now consider the operator H(V,) = C CP,,. We let ELn

Sp(fi(Vn)) denote the spectrum of the operator R(Vn).Setting (A)

oyo 'n(z) = exp{itfi(V,)}a:exp{-itfi(V,)},

a: E M

,

we obtain S p ( a W ~ ' l= n )Sp(fi(V,))-Sp(fi(Vn)) = {A-p : A,p E Sp(fi(V,))}. (16)

It is clear that ,BA(qi, q k ; J ) E Sp(&(Vn)),V i , k E 1 , . . .,q. Formula (16) then implies that Sp(oWr)9")is generated by elements of the form A ( r l ~ , ~ ~ ; J ) - A ( r l r C , r l ~ ; iJ,)j , k , Z E l , . . . q *

248

Since XX ((V9 ~b~,Tp;J)-X('lu 9j'J)-~(Vm~91'J) is a rational number, then there is a number t9u ; J ) y E (0,l) and integers mi,j,k,l E 2, (i,j,k,1 E {1,2, ...,a}) such that X(77i,qj;J) - X(77k,77l3 J ) = mi,j,k,llOgy*

(17)

Hence we find that an increasing sequence { E ( n ) }of subsets Z such that E(-n) = -E(n) and Sp(H(V,)) = {mlogy}mEE(n)is valid. It follows that Sp(o"b*)) c (nlog7)nEz. Hence there exists a positive integer d E Z such that we have (A)

qawo

) = { n log y d } n G z .

This means that M is a factor of type 1110, 8 = rd.

0

5. Applications and examples 5.1. Potts model

We consider the Potts model on the Cayley tree regarded as

rk whose Hamiltonian is



where J E R is a coupling constant, as usual < 2, y > stands for the nearest neighbor vertices and as before a ( . ) E @ = (71,q 2 , ...,qP}. Here 6 is the Kronecker symbol. Equality (1) implies that

for all z,y E V. The Hamiltonian H ( a ) is therefore

H ( a )= -

c

J'a(z)a(?/) ,

(18)

EL

where J' = q-lJ. 9 Hence the A-model is a generalization of the Potts model, that is in this case the function X : @ x @ x R + R is defined by A(z, y; J ' ) = -J'(z,y). Here, z, y E Rq-' and (2, y) stands for the scalar product in Rq-'. From (1)it is easy to see that

249

From (19) and (6) we can check that the assumption A (see section 3) is valid for the Potts model. So there exists the unordered phase. From (19) we can find that the fraction A(qi ,qj ;J’)- Vqrn,qi; J’) takes values A ( q k , q p ; J ’ ) - A(qzt,%J; J’Y k l and 0. So by Theorem 4.1 a von Neumann algebra M is a I&-factor.

{(if?)}.

From (17) we may obtain that I#J = exp -

Hence we obtain the

following

Theorem 5.1. The von Neumann algebra M corresponding t o the unordered phase of the Potts model (18) o n a Cayley tree is a factor of type III@h,f o r some k E Z, k

> 0, where

# IJ

= exp

Remark. If 4 = 2 the considered Potts model reduces to the Ising model, for this model analogous results were obtained in Ref. 17. For a class of inhomogeneous Potts model similar result has been also obtained in Ref. 18. 5.2. Markov random fields

In this subsection we consider a case when A ( x , y) function is not symmetric and the corresponding Gibbs measure is a Markov random field (see Ref. 27). Let P = ( ~ i j ) & =be ~ a stochastic matrix such that p i j > 0 for all i,j E 1,d. Define a function A ( x , y ) as follows: m i , q j ) = - 10gpij 7

(20)

for all i, j E { 1,.. . ,d } . From now on, we will consider the case /3 = 1 and 4 = d. It is easy to verify that Assumptions A and B, for the defined function A, are satisfied. By p we denote the corresponding unordered phase of the A-model. Observe that if the order of the Cayley tree is k = 1 then the measure p is a Markov measure, associated with the stochastic matrix P (see Ref. 27). By w,, one denotes the diagonal state corresponding to the measure p on C*-algebra A = @ , r b n / i d ( c ) .

Theorem 5.2. Let P = (pij)&=l be a stochastic matrix such that pij > 0 for all i, j = 1,. . .d and at least one element of this matrix is different from 112, and w,, be the corresponding Markov state. If there exist integers rnij i, j E (1,. . . ,d } , and some number 01 E (0,l) such that

250 then T,,,,(A)” , is a factor of tppe 1110 for some 6 E (0’1).

In order t o prove this theorem it should be used the rationality condition of Theorem 4.1. Namely, if the condition (21) is satisfied then using (20) one can see that the rationality condition holds, so we get the assertion. Remark. If all elements of the stochastic matrix P equal to 1/2 then the corresponding Markov state up is a trace and consequently T+ (A)” is the unique hyperfinite factor of type 111. Remark. There is a conjecture (see for example, Ref. 31) that every factor associated with GNS representation of a Gibbs state of a Hamiltonian system having a non-trivial interaction is of type 1111. Theorems 4.1 and 5.2 show that the conjecture is not true even if the Hamiltonian has nearest neighbour interactions. Acknowledgements The final part of this work was done within the scheme of NATO-CNR Fellowship at the Universita di Roma ’Tor Vergata’. We are grateful to L. Accardi and E. Presutti for useful comments and observations.

References 1. L.Accardi, On noncommutative Markov property, f i n k t . anal. i ego proloj. 8(1975), 1-8. 2. L.Accardi,F. Fidaleo, Non homogeneous quantum Markov states and quantum Markov fields, J . finct. Anal. 200 (2003), 324-347. 3. L.Accardi, G.Frigerio, Markovian cocycles, Proc. R.Ir. Acad. 83A(1983), 245273. 4. H. Araki, E.J. Woods, A classification of factors Publ. R.I.M.S. Kyoto Univ. 3(1968), 51-130. 5. H. Araki, W. Wyss, Representations of canonical anticommutations relations. Helv. Phys. Acta 37(1964), 136-159. 6 . 0. Bratteli, D. Robinson, Operator algebras and Quantum Statistical Mechanics I., Berlin: Springer-Verlag, 1979. 7. 0. Bratteli, D. Robinson, Operator algebras and Quantum Statistical Mechanics II. Berlin: Springer-Verlag, 1981. 8. R.L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity, Theor. Probab. Appl. 1 3 (1968), 197-224. 9. A. Comes, Une classifications das facteurs de type 111, Ann. Ec. Norm. Sup. 6(1973), 133-252. 10. M.Fannes,B. Nachtergaele, R.Werner, Ground states of VBS models on Cayley trees, J.Stat. Phys. 66(1992), 939-973.

251 11. F. Fidaleo F., F.Mukhamedov, Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras, preprint, 2004. 12. N. Ganikhodjaev, A group representations and automorphisms of Cayley tree, Dok1.Akad.nauk Rep. Uzb. 1990, 3-6. 13. N.N.Ganikhodjaev, F.M.Mukhamedov, Markov states on quantum lattice systems and its applications, Methods of f i n c t . Anal. and Topology, 4(1998), n.3, c. 33-38. 14. V.Ya.Golodets, S.V.Neshveyev, Non-Bernoullian K-systems, Commun. Math. Phys. 195( 1998), 213-232. G.N.Zholtkevich, On KMS-Markovian states, 15. V.Ya.Golodets, Theor.Math.Phys. 56(1983), 80-86. 16. W. Kriger, On the finitary isomorphism of Markov shifts that have finite coding time, 2.Wahr.uerw. Geb. 65 (1983) 323-328. 17. F.M. Mukhamedov, Von Neumann algebras corresponding translation - invariant Gibbs states of Ising model on the Bethe lattice, Theor. Math. Phys. 123(2000), n.1, 489-493. 18. F.M.Mukhamedov, U.A.Rozikov, Von Neumann algebra corresponding t o one phase of inhomogeneous Potts model on a Cayley tree, Theor.Math.Phys. 126(2001), n.2, c. 206-213. 19. F.M. Mukhamedov, U.A. Fbzikov, On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras. J. Stat. Phys. 114(2004), 825-848 20. T.Murakami, S.Yamagami, On types of quasi-free representations of Clifford algebras, Publ. R.I.M.S. Kyoto, Uniu. 31(1995), 33-44. 21. R. Powers, Representation of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. 81(1967), 138-171. 22. C. Preston, Gibbs states on countable sets (Cambridge University Press, London 1974). 23. U.A. Rozikov: Description of limiting Gibbs measures for A-models on the Bethe lattice, Siberian Math. Jour. 39(1998), 427-435. 24. C.M. Series, Ya.G. Sinai: Ising models on the Lobachavsky plane, Commun. Math. Phys. 128(1990), 63-76. 25. A .N.Shiryaev, Probability, M.Nauka, 1980. 26. Ya.G. Sinai, Theory of phase transitions: Rigorous results, Pergamon, Oxford, 1982. 27. F.Spizer, Markov randon fields on trees, Ann. Prob. 3(1975), 387-398. 28. S. Stratila, Modular theory in operator algebras, Bucuresti, Abacus Press, 1981. 29. S.Stratila, D.Voiculescu: Representations of AF-algebras and of the group U(m), Lec. Notes Math. 486(1975), Springer-Verlag. 30. StrtitilB S., Zsid6, L. Lectures on uon Neumann algebras, Abacus press, Tunbridge Wells, Kent, (1979). 31. M. Takesaki, Automorphisms and von Neumann algebras of type 111. Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 111-135,Proc. Sympos. Pure Math., 38, Amer. Math. SOC.,Providence, R.I., 1982.

ON QUANTUM LOGICAL GATES ON A GENERAL FOCK SPACE

WOLFGANG FREUDENBERG Brandenburgische Technische Universitat Cottbus, Institut f i r Mathematit, Postfach 101344, 03013 Cottbus, Germany, E- Mail: [email protected] MASANORT OHYA Department of Information Science, Tokyo University of Science, Noda City, Chiba 278, Japan, E-Mail: [email protected] NOBURO WATANABE Department of Information Science, Tokyo University of Science, Noda City, Chiba 278, Japan, E-Mail: watanabe0is. noda.tus. ac.jp In this paper we investigate quantum logical gates of Fkedkin type. The information 0 and 1 will be encoded by coherent states on a general Fock space, the interaction between the inputs will be a general beam splitting procedure. The interaction in the control gate is given by a unitary operator. The aim of the paper is to give examples for quantum channels transmitting some information such that the gate will fulfill the truth table. For the output we get simple explicit expressions. Our aim is to construct a gate using the well-known splitting procedures and coherent states.

1. Introduction In [15]Fredkin and Toffoli proposed a logical conservative gate on the base of which Milburn [25] constructed a quantum logical gate using a Kerr medium. The gate is composed of two input gates 11, 12, and one control gate C. Two inputs come t o a first beam splitter and produce two outputs. One of them passes via a control gate to the second beam splitting

252

253 apparatus, the second output (of the first beam splitting procedure) passes directly to the second beam splitter. After the second beam splitting we will have two final outputs 01. 0 2 .

Figure: Model of a Redkin-Tofloli-Milburn gate The aim of the paper is to give an example for quantum channels transmitting some information such that the gate will fulfill the truth table given below. Hereby, the signals 0 and 1 will be encoded by two different coherent states. One of these states also could be the vacuum state. The control gate if switched on will change the information ( 0 , l ) into (1,O) but leave the informations (0,O) resp. ( 1 , l ) unchanged. Input 11 0 0 1 1 0 0

1 1

Input 0

1 0 1 0 1 0

1

I2

Control State 0 0 0 0 1 1 1 1

Output O1 Output O2 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1

We will consider quantum channels for states on the symmetric Fock space over a general metric space G equiped with a measure u. In most concrete models G will be the Euclidean space Rd, and u will be the Lebesgue measure on Rd. The splitting rates a,@ must not be constants but arbitrary measurable functions from G to C. satisfying 1a(z)I2 I@(z)12 = 1 for all z E G. It turns out that for this model the whole process can be described in an explicit way. For the output state we get simple expressions. In [26,

+

254 281 the channel is described by a mapping V having on coherent vectors expe@exp, with 6, y E C. the simple form

= exPcre+pr -Be+x,. (1) A similar description is given there for photon number states. In the present paper we get formally the same description but with 6, y being functions from G t o C. So the description given in [26] fits into our context in the case that G consists of one point. The interaction in the control gate is given by an operator of multiplication and seems to be rather an artificial one. With respect to this control gate the results presented in this paper should be considered only as a first step to realize such quantum logical gates. In all calculations we have to assume nothing about the two exponential vectors (coherent states) carrying the information 0 or 1. For instance one can take the vacuum state and a coherent (non-vacuum) state. Another possibility is to choose two exponential vectors generated by functions being orthogonal or with disjoint support. So there is a big choice of representatives one can choose for building the logical gate. The present paper is an enlarged and generalized version of [18, 171. Another model where the action in the control gate is given by beam splitting procedures will be given in [16]. Ve-0

Be-,

2. The General Model Let d be the von Neumann-algebra of all bounded linear operators on some separable Hilbert space. By S(d) we denote the set of normal states on the algebra A. In the following wo E S(d)and 70 E S(d)will denote the input states, w1 E S(d)and r)l E S(d)will be the output states after the first beam splitting, Q E S(d)will be the control state and G; the output after passing the control gate. W; E S(d)and 771 E S(d)will be the inputs for the second beam splitting, and finally w2 E S(d)and r)2 E S(d)are the final outputs.

Definition 2.1. A mapping E' : S(d@d)+ S(d@d) we call a completely positive compound channel if the dual map E : d@d+ d@d given by the relation J(E(C))= E*("

<

(C E A@d, E S(d@d))

is a linear mapping being completely positive, i. e. n

D,*E(C:Cj)Dj 2 0 i,j=l

(Ci E d@d,Di

E d@d,n E

N)

(2)

255 and identity preserving, i. e.

E(P@I[) = I[@I[.

(3)

Obviously, the condition (3) is already necessary to ensure that E* maps states into states. The condition (2) of complete positivity guarantees that normal states are transformed into normal ones. Now, let w g , q g E S(d)be the input states. We want to construct logical gates where the inputs carry the information 0 or l.Therefore, we consider only the case of input and noise being "independent", i.e. the compound state (on d@d) that has to be transformed by E* is just the product state wo@m. For a discussion on general compound states in this context we refer to [29]. A closer study of these channels is given in [27]. After the transformation with respect to the channel €* we obtain a compound state 5 on d@d , i. e. 5 = w0@qg o €. For the concrete splitting model we will consider in this paper it appears that starting with the product state of coherent states wg, q g the compound state 5 will be again a product state wl@ql of coherent ones. Hereby, w1, ql E S(d),and for all A , B E d the following relations hold:

wi(A) = ~ g @ q g0 E(A@I[) q1(B) = wo@qo 0 E ( Z @ B ) 5 = wi@qi(A@B) = w g @ q g 0 E(A@B).

(4)

(5)

(6)

From the above description one sees that one could investigate different types of time evolutions depending on the choice of the next inputs from the preceding ones. An interesting question is to consider the time evolution defined for A E d andnEMby W~+I

( A ) = w,@qg

0

E(A@I[).

(7)

This means that the last output will be the new input and in each step the same noise (second input) will react with the input. However, in this paper we concentrate our attention to explicit calculations of the logical gate, and thus to the behaviour in one cycle. Further, we restrict our considerations to the case that the considered algebra is the algebra M of all bounded linear operators on a general Fock space which plays a basic r61e in many physically relevant applications. We are going to introduce in the subsequent section the algebra M in a way which is convenient for our considerations.

256

3. The Boson Fock Space First we will collect some basic facts concerning the Fock space and certain operators acting on it. Let G be an arbitrary complete separable metric space and 6 its a-algebra of Bore1 sets. Further, let Y be a locally finite diffise measure on [G,@], i.e. u ( K ) < 00 for bounded K E 6 and ~ ( { z } )= 0 for all singletons x E G. Especially we are concerned with the case where G = Rd and Y is the &dimensional Lebesgue measure, but with minor changes of notations also the case of atomic measures like the counting measure on G = Z d fits into our framework. We avoid these changes for the sake of readability. By M we denote the set of all finite integer-valued measures on [G,61, i.e. M is the set of finite counting measures. One can show easily (6. inst. [23, 31 that each 'p E M can be written in the form 'p = S, S, for some n E N and zj E G (where S, denotes the Dirac measure in 2). So, the elements of M can be interpreted as finite (symmetric) configurations in G.

+ + -

a

-

Remark: We also could take MI = ((21,.. . ,zn) : z j E G , n E as the set of configurations which seems to be more natural than the set M of counting measures. However, the elements cp E M are symmetric by definition. So we can avoid all kinds of symmetrization procedures that complicate extremely several calculations if one starts with MI as configuration set. We equip M with its canonical a-algebra M - the smallest a-algebra containing all sets of the form {'p E M : ' p ( K ) = n } , K E 6,n E N. Observe that ' p ( K ) = n means that the configuration 'p has exactly n points in the subset K of the phase space G. On [M,M] we introduce a measure F by setting F(Y)

: = x , ( O ) f C Z 1/ X y n>l

Gn

(2CLj)

l.44zl,...,zn])

(YEtW.

j= 1

(8) Hereby, xy denotes the indicator function of a set Y and o is the empty configuration in M , i.e. o(G) = 0. Observe that F is a a-finite measure. Since Y was assumed to be diffise one easily checks that F is concentrated on the set

M , := {'p E M : 'p({z}) 5 1 for all

2

E G}

of so called simple counting measures (i.e. without multiple points).

(9)

257 Definition 3.1. M := L2(M,m,F) is called the (symmetric) Fock space over G.

Remark: Usually one defines the symmetric Fock space I'(31) over a Hilbert space 31 as the direct sum of the symmetrized tensor products 31$$Lrnof the = ,; 7-lgm As,we . remarked already underlying Hilbert space 31, i.e. I?(%) = @ above we introduced the Fock space in a way adapted to the language of counting measures especially to avoid symmetrization procedures. A further advantage of the definition given above is that the Fock spwe over a L2-space again will be a L2-space. It is very easy to show that I'(7-l) and M are isomorphic (cf. [12]). For details we refer to [13, 14, 191 but also e.g. to [21]where a similar definition of the Fock space is given. Observe that M is again a separable Hilbert space. Now, for each n 2 1 let M" := M@" be the n-fold tensor product of the Hilbert space M . Obviously, M" can be identified with L2(Mn,F"). Basic for the proofs in this paper will be that integration in M n (with respect to Fn)can be replaced by an integration with respect to F.

Lemma 3.1. Let f : M" Then

+ C be integrable with respect to F"

(or 2 0).

Hereby, q3 5 cp means that q3 is a subconfiguration of cp, i.e. cp - q3 E M . The proof of the first part in (10) for the case n = 2 one can find e.g. in [ll 1, in [13 ] or in [22, 241 where the above lemma is called

simple induction shows that (10) is valid for all n (10) is just a reformulation of the first one.

-lemma. A $ 2 2. The second part of

258

Definition 3.2. For a given function g : G M + CC defined by exp,(cp) =

{

+ C the function

exp, :

if cp= o,

1

n g(z) if

=ED

cp

#o

(11)

V(I.))>O

is called exponential vector generated by 9. We make use of the following well-known properties of exponential vectors: Lemma 3.2. Let f and g be functions from G to C and ( p , c p ~ , c p ~be elements from M . Then we have exPf(cpl+

92)

= expf('p1) . expf('p21,

(12)

c

e x P f ( a . exp,(cp - $1, G5v exPf.,(cp) = exPf(cp) * exp,(Cp).

expf+,('p) =

llexpg[l& = el1g1lt,(G,y) (expf, exp,)M = e(f9 9

) ~ 2 ( ~ 8 ~ )

(13)

(14)

(9 E L 2 G 41,

(15)

(f, 9 E L2(G,v)).

(16)

Observe that exp, E M if and only if g E L2(G,v). Further, it is wellknown that the linear span of exponential vectors from M is dense in M . The von-Neumann algebra L ( M ) of all bounded linear operators on the Fock space M we will denote by d and the tensor product d@d of the vonNeumann algebras A by d2.Obviously, we may identify d2and L ( M 2 ) . The identity in d we denote by 11. For g, h E L,(G,v) we denote by exph@exh,i. e. Bg,hQ(cp) =

Bg,h

the integral operator with kernel

J ,F(dg)exP,(cp)exp,(~)Q(~)

(9E M , cp E M ) .

(17) Observe that for g, h E L2(G,v) we have Bg,h 6 A. Immediately from (16) we get B,,hexpf = e(,>f).exph

(1,9, h E L z ( G , v ) ) .

(18) Operators of the type Bg,hdetermine a normal state on A completely, i. e. we have the following result:

259

Lemma 3.3. ([6, 81). Let wj, j = 1, 2 be normal states o n A. Then = w2 if and only if

w1

4. The Quantum Channel

Let a, /3 : G

-+

CC be measurable mappings such that

We define a linear operator and 91, 9 2 E M

Va,p : M 2 + M 2 by setting for CP E M 2

The above defined operator plays a basic r61e in the definition of the quantum channel we want to consider. First we will show that on exponential vectors the operator Va,s has the same form as the operator (1) mentioned in the introduction and describes (with (Y and /3 being constants) the state change of the usual beam splitter model of quantum optics (cf.inst. [2]). The results below in this section are contained at least partially in [8, 7, 9, lo]. However, for completeness and readability of the present paper we will present all proofs.

Proposition 4.1. Let Va,p be the operator defined b y (21) with a and satisfying (20). For all g, h E L2(G,u ) one has

260

Proof: For arbitrary 9, h E &(G, v) we have expg@exphE M 2 . Applying several times Lemma 3.2 we get for pl, 9 2 E M

cc

expa($l)expp(pl - $1) dl591 92592 -exp-~($2)e%(c2 - $2)expg($1 $2)exph(w 9 2 - $1 - $2) (Va,p

e q g @

exph >(91,@) =

+

=

+

expa($l)eqg(dl)expp(pl - $l)exph(ql - $1) dl591 d25'f'Pa 'exp-8($2)expg($2)e~(p2 - $2)exph(@ - $2)

eqcxg($l)expph(pl - $1) exp-Bgenh(p2 - $2) dl591 925v2 = eqag+ph(pl) .eq-?3g+,h(p2) = exPcxg+ph @ exp-8g+zh('P1 ,@) what ends the proof. =

More general splitting models were considered in [7, lo]. Observe that for all 9 E L2(G,v) Va,p exp,@expo = expag €3 e x p q g

(23) Splitting procedures of this type (considered as operators from d to d@d) were studied in detail for instance in [8]. Similar models were considered in [4, 53.

Proposition 4.2. The operator Va,a defined by (21) with 01 and /? satisfying (20) is an isometry. Proof: For 9, h E &(G, v) we get 1IVa,a m

g @ eqh11%f2 =

2

~ ~ e x p a g + ~'hllexp-~g+Ehllh ~~M

= exp{Il~9+/?h1I2}.exP{ll- O 9 + W I 2 } = exp{142119112 + IP1211~112 +w?(9lh)

+d(h,9)}

+ 1~12119112- W ( 9 , h) - olS(h9)} ~exp{l~1211~112 = eq{llgl12 + llhl12} = I l e q g @ exphIl%f2. Since the linear span of all tensor products exp, @ exp, of exponential vectors is dense in M 2 this proves that Va,p is isometric. 0 The operator V& adjoint to Va,p is of the same type as Va,p.

Proposition 4.3. Let Va,p be given by (21) with 01 and /? satisfying (20). Then

v:,p = %,-p

(24)

261

Proof: It is sufficient to show (24) for exponential vectors. For g, h E L2( G, v )we get

%,-avcY,sexpg

@

exph = e w a(ag+Bh)-B(-$g+Zh

-

@

- e~l~lag+ijiSh+lBI'p-~Bh @

= expg

@

e w ( a g + B h ) + a ( -B'g+Zh

expa~g+JB12h-aB'g+lalzh

exph

what ends the proof.

0

+

From (20) we get lE(z)I2 I - P(z)I2 = 1 for all z E G. So we conclude from Proposition 4.2 that V& = V Z , - ~: M 2 + M 2 is an isometry too. Consequently, we obtain

Proposition 4.4. The operator Va,a : M 2 + M 2 with a and (20) is unitary.

satisfying

From Proposition 4.4 we get immediately that the mapping E : C ( M 2 )+ L ( M 2 )defined by

(C E L ( M 2 ) is completely positive and preserves the identity.

Definition 4.1. The completely positive channel €& : S(d2)+S(d2) the dual map of which is given by (25) (i. e. E&<(C) = <(Ea,p(C)) for C E d2, E S(d2)) is called noisy quantum channel with rates a, p.

<

This notion stems from applications where the second input is intepreted as a noise. If there is no noise (i. e. the second input 770 is the vacuum state) the above procedure sometimes is called attenuation process. For more details on beam splittings we refer to [l]and [6, 8, 7, 9, lo], for connections to quantum Markov chains cf. also [20]. We want to characterize the transformation of normal coherent states on

A. Definition 4.2. Let f E Lz(G,v). The coherent state to f is defined by

Gf(A)= (expf, Aexpf)e-llfllz

@f

corresponding

( A E A). (26)

From (16) in Lemma 3.2 and (18)we get that for an operator was defined by (17)) and a coherent state @g one has

@f(Bgsh ) = e(9> f)+(f>h)-llflla*

Bg,h

(Bg,h (27)

262 Now we want t o characterize the transformation of coherent normal input states.

Theorem 4.1. Let

with g, h

and let

be given by (25) wit

satisfying (20). We set (cf. (4) and (5))

Then

Proof: For arbitrary Ae A we get

is just the normalization factor. So we finaaly obtain

Analogously, one shows

We see that coherent signals corresponding to g and h are transformed into coherent ones corresponding to mixtures of these functions with the rates a and p. Since for A, B E A wo@qo O Ea,p(A@B)

= (Va,gexpg@exph,(A@B)Va,gexpg@exph)e-llglla-llhl12 - (emag+ph7 AexpcYg+gh) * (exp-pg+ijih>B e ~ - ~ g + ~ h ) e - ” g ” 2 - l l h l l Z

-

= Ul(A) ql ( B ) we see (as remarked already in Section 2) that for coherent states the ’independence’ of input and noise ’survives’, i. e. (cf. (6)) wl@ql = wo@qo 0 €a$.

(29)

5. Control Gate Let U : M @ M + M @ M be unitary and let the control state normal state on A. We have a ( A ) = wi@@(U*(A@X)U)

(A E A).

e

be a

263

Problems : Which U are reasonable (from technical point of view)? Which e can serve as control states? Which U and Q are so that the truth table will be satisfied? Define U : M @ M c)M @ M by ~ ~ 2 ~( (~ ~( c1 p uq(cpl, cp2) := ( - 1 ) ~ 9 ~ ~ ~,cp2) E~ ~ MI ~ 2 Q E M @ M ) (30) where 191 = cp(G).

Lemma 5.1. U is se2f-adjoint and unitary. For g E L2(G,u) and q E M we have U ( ~ X P ~ @ Q ) ( P I= , V~~Q) ( - ~ ) I ~ ~ I ., Q(c~2)(vI)

(31)

Proof: Since U is just a multiplication operator with a real-valued function it will be self-adjoint. Because of ((-l)m)2 = 1 for all m E N one has U2 = I[. For g E L2(G, u ) and Q E M we get using Lemma 3.2

Now, let e be a pure normal state on A, i.e. there exists a Q E M , Q such that

#0

The output from the control gate will be W; = ul@e(U*((.)@I[)U). Proposition 5.1. Let wo = u9, qo = uh,and 1 Gi(A) = - F(dcp)(9(cp)(2u( - l ) ’ u l ( c r g + B h )

iiw

JM

Q

be given by (32). Then (A)

( A E A).

(33)

Proof: Because of Lemma 3.3 it is enough to show (33) for operators A being of the type B,, ,92. Using (31) we obtain for all 91, cp2 E M , 91, g2, f E L2(GI 4 ( B 9 ~,92@x)

U ( e q f @ Q ) ( v lI 9 2 ) = ( B 9 ~,gn@~)exp(-l)lV21 f(’P1) . Q(cp2) = B,,,,,(exP(-l)lV,If)(cpl)

. Q(cp2)

Consequently, setting f = crg 6 1(

4 1 ,.!I2

+ ph we get for all gl, g2 E L2(G, v)

1

where we used in the last line (27) and 11 f 112 = 11 (-1)l~lf 112 for all

(P

E M.

Denote by Modd resp. Meven the configurations with odd resp. even total number of particles, i.e. Modd = {(P E M : /(PIodd), M even = {(P E M : ](PI even). As a direct consequence of Proposition 5.1 we obtain

Proposition 5.2. Let

Q

satisfy (32). Then

w;( A ) = Xlw

-("g+Dh) + XZW ffg+ph

with

So, if for instance the control state Q is the vacuum state (Q= wO)then LS; = wl = w w+Bh . If Q is an one-particle state (or some state concentrated on odd configurations) then &c = w -("g+ph). The choice of the control state allows us to switch from the coherent state w "g+ph to w -("g+ph).

6. Final Output AEter Second Beam Splitting Let wo = wg, 710 = wh be coherent normal states on A (the inputs Il and I2) and Q = (Q, (-)*)/11Q1(2 a pure state (the control state). Further, let

265 -

XI, A2 be given by (34). Then W; (cf. Proposition 5.2) and q1 = w - P ~ + ~ ~ represent the inputs for the final second beam splitting the rates of which we denote by a1, 81, i. e. a1, PI : G C ) C and Ia1(2)I2 IP1(2)12 = 1. Consequently, the final outputs 01 and 0 2 are given by the states

+

= ~ l @ q l ( ~ ~ ~ , P l ( ( . ) @ ~ ) ~ a l , P lrh ) , = G@v1( G 1 , p lP @ ( . ) ) V a 1 , P 1 ) (35) Directly from Theorem 4.1 we obtain w2 = AlW(-a"l-P1a)g+(-alP+EPl)h + )(2W(UPl-P1p)g+(alP+EPl)h (36) and q2 = A1 ( & - E p ) g + ( B z + a a l ) h + x2 (- az-7Tip)g-f (-p&+acrl) h . (37) w2

We consider two cases. Case 1: Let e be a pure state concentrated on even configurations, i. e. A1 = 0, A2 = 1. For example let e be the vacuum state wo. Then we obtain w2

= ,(ual-P18)g+(alP+~Pl)h

(38)

and q2

= ,(-a&-EP)g+(-PS;+m)h.

(39) Case 2: As second case let us consider e being concentrated on odd configurations, i. e. A1 = 1, A2 = 0. For example let e be an one-particle state: Q(cp) = 0 for cp $! Ad1 = {&. 2 E G}. Then we obtain w2

=,(-aal-P18)g+(-alS+~Pl)h

(40)

and q2

= ,(aE-(.1P)g+(PZ+aal)h

(41) satisfied if

truth tables of our logical gate will be incase 1: w2 = wg and q 2 = wh and in case 2: w2 = wh and q 2 = wg. So the gate has to be constructed in such a way that in the control gate there is the possibility to choose between two control states: The control gate is switched off with a control state el of the form considered in case I and is switched on with a control state e2 of the form considered in case 2. An easy calculation shows that this will be satisfied if and only if the splitting constants a, p, a1, 81 fulfil The

a = -p = p1 =

(42)

266

Let us remark that a1 = 5, = -p are the splitting constants of the operator adjoint to Va,p. So this assumption is already necessary to get the original input back if there is no disturbance in the control gate. Summarizing, if the splitting constants satisfy (42) the full information even represented by the original input states - will be obtained at the end after the second beam splitting. So if the information "0" resp. "1" are represented by coherent vectors g and h and the beam splitting constants fulfill (42) the truth table will be satisfied. All calculations above have been done for arbitrary g and h from L2(G, v). We obtain the right truth values also in the (trivial case) that both vectors are equal. If one does not demand that the output will be the original input 11resp. 12 but only that the output has to belong to a certain class of coherent states we may drop the assumptions about the splitting constants and find more interesting control gates.

References 1. L. Accardi and M. Ohya. Compound channels, transition expectations and liftings. Applied Mathematics & Optimization, 39:33-59, 1999. 2. R.A. Campos, B.E.A. Saleh, and M.C. Teich. Quantum-mechanical lossless beam splitter: m(2) symmetry and photon statistics. Phys.Rew. A , 40(3):1371-1384, 1989. 3. D.J. Daley and D. Vere-Jones. A n Introduction to the Theory of Point Pro-

cesses. Springer Series in Statistics. Springer-Verlag, New York, Berlin, Heidelberg, 1988. 4. Karl-Heinz Fichtner and Masanori Ohya. Quantum teleportation with entangled states given by beam splittings. Comm. Math. Phys., 222:229-247, 2001.

5. Karl-Heinz Fichtner and Masanori Ohya. Quantum teleportation and beam splitting. Comm. Math. Phys., 225:67-89, 2002. 6. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Beam Splittings and Time Evolutions of Boson Systems. Forschungsergebnisse der Fakultit f i r Mathematik und Informatik, Math f Inff 96f 39:105 pages, 1996. 7. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Non-independent Splittings and Gibbs States. Mathematical Notes, 64(3-4):518 - 523, 1998. 8. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Time Evolution and Invariance of Boson Systems Given by Beam Splittings. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(4):511 - 531, 1998. 9. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings. Math. Nachr., 218:25-47, 2000. 10. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. On exchange mechanisms for bosom. to appear in: Random Operators and Stochastic Equations, 2004.

267 11. K.-H. Fichtner and W. Freudenberg. Point processes and normal states of boson systems. Naturwiss.- Technisches Z e n t m m N T Z , Leipzig, 1986. 56 p. 12. K.-H. Fichtner and W. Freudenberg. Point processes and the position distribution of infinite boson systems. J. Stat. Phys., 47:959-978, 1987. 13. K.-H. Fichtner and W. Freudenberg. Characterization of states of infinite boson systems I. On the construction of states of boson systems. Comrnun. Math. Phys., 137:315 - 357, 1991. 14. K.-H. Fichtner and W. Freudenberg. Remarks on stochastic calculus on the Fock space. In L. Accardi, editor, Quantum Probability and Related Topics VII, pages 305 - 323, Singapore, New Jersey, London, Hong Kong, 1991. World Scientific Publishing Co. 15. E. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21:219-253, 1982. 16. W. Freudenberg, M. Ohya, N. Turchyna, and N. Watanabe. Quantum logical gates realized by beam splittings. Preprint, 2004. 18 pages. 17. W. Freudenberg, M. Ohya, and N. Watanabe. On beam splittings and mathematical construction of quantum logical gate. Surikaisekikenkyusho Kokyuroku, 1142:23-35, 2000. Mathematical aspects of quantum information and quantum chaos. 18. W. Freudenberg, M. Ohya, and N. Watanabe. On beam splittings and quantum logical gate on Fock space. Surikaisekikenkyusho Kokyuroku, 1139:113123, 2000. New developments in infinite dimensional analysis and quantum probability theory. 19. W. Freudenberg. Characterization of states of infinite boson systems 11. On the existence of the conditional reduced density matrix. Commun. Math. Phys., 137:461 - 472, 1991. 20. W. Freudenberg. On a class of quantum Markov chains on the Fock space. In L. Accardi, editor, Quantum Probability and Related Topics, volume IX, pages 215-237, Singapore, New Jersey, London, Hong Kong, 1994. World Scientific Publishing Co. 21. A. Guichardet. Symmetric Halbert Spaces and Related Topics, volume 231 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1972. 22. J.M. Lindsay and H. Maassen. An integral kernel approach to noise. In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications 111, volume 1303 of Lecture Notes an Mathematics, pages 192 -208, Heidelberg, 1988. Springer-Verlag. 23. K. Matthes, J. Kerstan, and J. Mecke. Infinitely Divisible Point Processes. Wiley, Chichester, 1978. 24. P.A. Meyer. Quantum Probability for Probabilists, volume 1538 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, 1993. 25. G.J. Milburn. Quantum optical Fredkin gate. Physical Review Letters, 62:2124-2127, 1989. 26. M. Ohya and H. Suyari. An application of lifting theory to optical communication process. Reports on Mathematical Physics, 36:403 - 420, 1995. 27. M. Ohya and N. Watanabe. Construction and analysis of a mathematical

268 model in quantum communication processes. Electron. Commun. Jpn., Part 1, 68(2):29-34, 1985. 28. M. Ohya and N. Watanabe. On mathematical treatment of Fredkin-ToffoliMilburn gate. Physica D, 120:206-213, 1998. 29. M.Ohya. Some aspects of quantum information theory and their applications to irreversible processes. Reports on Mathematical Physics, 27:19 - 47, 1989.

THE CHAOTIC CHAMELEON

RICHARD D. GILL Mathematical Institute, University of Utrecht, Netherlands, €4 E URANDOM, Eindhoven, Netherlands gzllOmath.uu.nl http ://www. math.uu.nl/people/gill Various local hidden variables models for the singlet correlations exploit the detection loophole, or other loopholes connected with post-selection on coincident arrival times. I consider the connection with a probabilistic simulation technique called rejection-sampling, and pose some natural questions concerning what can be achieved and what cannot be achieved with local (or distributed) rejection sampling. In particular a new and more serious loophole, which we call the coincidence loophole, is introduced.

1. Introduction

It has been well known since Pearle (1970) that local realistic models can explain the singlet correlations when these are determined on the basis of post-selected coincidences rather than on pre-selected event pairs. These models are usually felt to be unphysical and conspiratorial, and especially that they simply exploit defects of present day detection apparatus (hence the name “the detection loophole”). However, Accardi, Imafuku and Regoli (2002) (“the chameleon effect”),Thompson and Holstein (2002) (“the the chaotic ball effect”),and others have argued that their models could make physical sense. Further examples are provided by Hess and Philipp (2001a,b, 2004), Kracklauer (2002), Sanctuary (2003), in many cases unwittingly. Already, Gisin and Gisin (1999) show that these models can be simple and elegant, and should not be thought of as being artificial. Accardi et al. (2002) furthermore insist that their work, based on the chameleon effect,has nothing to do with the so-called detection loophole. Rather, they claim that the chameleon model is built on a fundamental legacy of measurement of quantum systems, that there is also indeterminacy in whether or not a particle gets measured at all, and when it gets measured. Furthermore, they focus entirely on perceived defects of the

269

270 landmark paper Bell (1964), where the incompatibility of the singlet correlations with local realism was first established. Now Bell himself became well aware of imperfections in his original work and in Bell (1981) (reprinted in Bell, 1987), taking account of one and a half decades of intense debate, he explicitly elaborated on the experimental protocol which is necessary, before one can conclude from an experimental violation of the Bell-CHSH inequality, that a local realistic explanation of the observed phenomena is impossible. That protocol is not adhered to by Accardi et al. (2002),nor (of course) by any of the previously cited works in which local realistic violations of Bell-CHSH inequalities are obtained. It is a mathematical fact that “chameleon model” of the type proposed by Accardi et al. (2002) can be converted into a “detection loophole model”, and vice-versa. This result has been independently obtained by Takayuki Miyadera and Masanori Ohya, and by the present author (unpublished). In this paper I do not want to continue the philosophical debate, nor address questions of physical legitimacy of these models. Instead I would like to extract a mathematical kernel from this literature, exposing some natural open problems concerning properties of these models. Possibly some answers are already known to experts on Bell-type experiments and on distributed quantum computation. I would especially like to pose these problems to experts in probability theory, since the basic renormalization involved both in the chameleon model (under the name of a “form factor”) and in detection-loophole models, is well known in probability theory under the name of rejection-sampling. From now I will use the language of Applied Probability: simulation, rejection-sampling, and so on; and avoid reference to physics or philosophy. The main new contribution of this paper is the discovery of a new loophole, which we call the coincidence loophole, which occurs when particle pairs are selected on the basis of nearly coincident arrival times. It has recently been shown by Larsson & Gill (2003) that this loophole is in a certain sense twice as serious as the well-known detection loophole. 2. The Problem

Suppose we want to simulate two random variables X, Y from a joint probability distribution depending on two parameters a, b. To fix ideas, let me give two key examples:

X,Y are binary, taking the values f l . The parameters a, b are two directions in real, three dimensional space. Case 1 The Singlet Correlations.

271

W e will represent t h e m with two unit vectors in @I (two points o n the unit sphere S2). The joint density of X, Y (their joint probability mass function) is 1 P r a , b{X =x ,Y = y } = p ( x , y ; a , b ) = -4( l - x y a . b ) ,

(1)

where a . b stands f o r the inner product of the unit vectors a and b and x,y = f l . Note that the marginal laws of X and Y are both Bernoulli (i) on {-1, +l}, and their covariance equals their correlation equals - a . b. In particular, the marginal law of X does not depend o n b nor that of Y o n a. Case 2 The Singlet C o ~ l o t i o n Restricted. s This is identical t o the previous example except that we are only interested in a and b taking values in two particular, possibly different, finite sets of points o n S2.

Next I describe two different protocols for “distributed Monte-Carlo simulation experiments”; the difference is that one allows rejection sampling, the other does not. The idea is that the random variables X and Y are going to be generated on two different computers, and the inputs a, b are only given to each computer separately. The two computers are to generate dependent random variables, so they will start with having some shared randomness between them. The programmer is allowed t o start with any number of random variables, distributed just how he likes, for this purpose. Cognoscenti will realize that it suffices to have just one random variable, uniformly distributed on the interval [0,1], or equivalently, an infinite sequence of fair independent coin tosses. There is no need for the two computers t o have access to further randomness-they may as well share everything they might ever need, separately or together, from the start. The difference between the two protocols, or two tasks, is that the first has to get it right first time, or if you prefer, with probability one. The second protocol is allowed to make mistakes, as long as the mistakes are also “distributed”. Another way to say this, is that we allow “distributed rejection sampling”. Moreover, we allow the second protocol not to be completely accurate. It might be, that the second protocol can be made more and more accurate at the expense of a smaller and smaller acceptance (success) probability. This is precisely what we want to study. Success probability and accuracy can both depend on the parameters a and b so one will probably demand uniformly high success probability, and uniformly good accuracy.

272

Task 1 Perfect Distributed Monte- Carlo. Construct a probability distribution of a random variable 2, and two transformations f and g of 2, each depending o n one of the two parameters a and b, such that

f(Z;a),g(Z;b)

N

X,Y

f o r a l l a , b.

(2 )

The symbol ‘N’ means ‘is jointly distributed as’, and X , Y on the right hand side come from the prespecified (or target) joint law with the given values of the parameters a and b.

Task 2 Imperfect Distributed Rejection Sampling. As before, but there are two further transformations, let me call t h e m D = S(2,a) and E =~ ( 2 b),; such that S and E take values 1 and 0 or if you like, ACCEPT and REJECT, and such that

f(Z;a),g(Z;b)I D = l = E

A

X,Y.

(3)

The symbol ‘1’ stands for ‘conditional on’, and ‘A’ means ‘is approximately distributed as’. The quality of the approximation needs to be quantified; in our case, the supremum over a and b of the variation distance between the two probability laws could be convenient (a low score means high quality). Moreover, one would like t o have a uniformly large chance of acceptance. Thus a further interesting score (high score means high quality) is inf,,bPr{D = 1 = E } .

3. The Solutions By Bell (1964) there is no way to succeed in Task 1 for Case 1. Moreover, there is no way to succeed in Task 1 for Case 2 either, for certain suitably chosen two-point sets of values for a and b. Consider now Task 2, and suppose first of all that there are only two possible different values of a and b each (Case 2). Let the random variable 2 consist of independent coin tosses coding guesses for a and b, and a realization of the pair X , Y drawn from the guessed joint distribution. The transformations 6 and E check if each guess is correct. The transformations f and g simply deliver the already generated X , Y . One obtains perfect accuracy with success probability 1/4. It is known that a much higher success probability is achievable at the expense of more complicated transformations. Now consider Task 2 for Case 1. So there is a continuum of possible values of a and b. Note that the joint law of X , Y depends on the parameters

273

a, b continuously, and the parameters vary in compact sets. So one can partition each of their ranges into a finite number of cells in such a way that the joint law of X , Y does not change much while each parameter varies within one cell of their respective partitions. Moreover, one can get less and less variation at the expense of more and more cells. Pick one representative parameter value in each cell. Now, fix one of these pairs of partitions, and just play the obvious generalization of our guessing game, using the representative parameter values for the guessed cells. If each partition has k cells and the guesses are uniform and independent, our success probability is l / k 2 , uniformly in a and b. We can achieve arbitrarily high accuracy, uniformly in a and b, at the cost of arbitrarily low success probability. However, Gisin and Gisin (1999) show we can do much better in the case of the singlet correlations: Theorem 1 Perfect conditional sirnulation of the singlet cornlations. For Case 1 and Task 2, there exists a perfect simulation with success probability uniformly equal to 112. See Gisin and Gisin (1999) for the very pretty details. Can we do better still? What is the maximum uniformly achievable success probability? The joint laws coming from quantum mechanics always satisfy n o action at a distance (“no Bell telephone”), i.e., the marginal of X does not depend on b nor that of Y on a. This should obviously be favourable t o finding solutions t o our tasks. Does it indeed play a role in making these simulations spectacularly more easy for quantum mechanics, than in general? Does “no action at a distance” ensure that we can find a perfect solution to Task 2 with success probability uniformly bounded away from O? Am I indeed correct in thinking that one find probability distributions p with action at a distance, depending smoothly on parameters a, b, for which one can only achieve perfection in the limit of zero success probability? It would be interesting to study these problems in a wider context: arbitrary biparameterized joint laws p ; extend from pairs to triples; . . . 4. Variant 1: Coincidences

Instead of demanding that S and E in Task 2 are binary, one might allow them t o take on arbitrary real values, and correspondingly allow a more rich acceptance rule. Suggestively changing the notation t o suggest times, define now S = d ( 2 ; a ) and T = ~ ( 2 b).; Instead of conditioning on the

274

separate events D = 1 and E = 1 condition on the event (S-2’1 < c where c is some constant. Obviously the new variant contains the original, so Variant Task 2 is at least as easy as the original. Accardi, Imafuku and Regoli (2002) suggest that they tackle this variant task claiming that it has nothing t o do with detector efficiency, but on the contrary is intrinsic to quantum optics, that one must post-select on coincidences in arrival times of entangled photons. By Heisenberg uncertainty, photons will always have a chance t o arrive (or to be measured) at different times. In those cases their joint state is not the singlet state. Therefore, if we were to collect data on all pairs (supposing 100%detector efficiency) we would not recover the singlet correlations. In actual fact the mathematical model of Accardi et a2. (2002) applies to the original task, not the variant. Still, in many experiments this kind of coincidence post-selection is done. Its effects (in terms of the loophole issue) has never yet been analysed. The common consensus is that it is no worse than the usual detection loophole. I convert this consensus into a conjecture:

Conjecture 1 No improvement from coincidences. There is no gain from Variant Task 2 over the original. Amazingly, this conjecture turns out to be false. In quantitative terms the “coincidence loophole” is about twice as serious as the detection loophole; see Larsson and Gill (2003). Fortunately, modern experimenters are moving (as Bell, 1981, stipulated) toward pulsed experiments and/or to event-ready detectors. In such an experment the detection time windows axe fixed in advance, not determined by the arrival times of the photons themselves. There seems t o be a connection with the work of Massar, Bacon, Cerf and Cleve (2001) on classical simulation of quantum entanglement using classical communication. After all, checking the inequality I S - 2’1 < c is a task which requires communication between the two observers. 5. Variant 2: Demanding More

Instead of making Task 2 easier, as in the previous section, we can try to make it harder by demanding further attractive properties of the simulated joint probability distribution of D ,XI E , Y.For instance, Gisin and Gisin (1999) show how one can achieve nice symmetry and stochastic independence properties at the cost of an only slightly smaller success probability

275 4/9 = (2/3)2. In fact, this solution has even more nice properties, as follows. One might like the simulated X to behave well, when D = 1, whether or not E = 1, and similarly for Y. Suppose we start with a joint law of X , Y depending on a, b as before. Let q be a fixed probability. Modify Task 2 as follows: we require not only that given D = 1 = E , the simulated X , Y have the prespecified joint distribution, but also that conditional on D = 1 and E = 0, the simulated X has the prespecified marginal distribution, and also that, conditional on D = 0 and E = 1, the simulated Y has the prespecified marginal distribution, and also that D and E are independent Bernoulli(q). Another way to describe this is by saying that under the simulated joint probability distribution of X , D, Y ,E , we have statistical independence between D, E , and ( X ,Y ) ,with ( X ,Y ) distributed according to our target distribution and D and E Bernoulli(q), except that we don’t care about X on {D = 0 ) nor about Y on { E = 0} Gisin and Gisin (1999) show that this Variant Task 2 can be achieved for our main example Case 1, with q = 2/3. It is known from considerations of the Clauser and Horne (1974) inequality that it cannot be done with q > 2/(1+ M 0.828. It seems that the precise boundary is unknown. In fact, for some practical applications, achieving this task is more than necessary. A slightly more modest task is to simulate the joint probability distribution just described, conditionally on the complement of the event {D = 0 = E } , i.e. conditional on D = 1 or E = 1. This means to say that we also don’t care what is the simulated probability of {D = 0 = E } . Gisin and Gisin (1999) show that this can be achieved with a variant of the same model, and with success probability 100% (i.e., the simulation never generates an event {D = 0 = E}),and q = 2/3.

a)

Acknowledgments

I am grateful for the warm hospitality and support of the Quantum Probability group at the department of mathematics of the University of Greifswald, Germany, during my sabbatical there, Spring 2003. My research there was supported by European Commission grant HPRN-CT-200200279, RTN QP-Applications. This research has also been supported by project RESQ (IST-2001-37559) of the IST-FET programme of the European Commission.

276 References

L. Accardi, K. Imafuku, and M. Regoli (2002). On the EPR-chameleon experiment. Infinite Dimensional Analysis, Quantum Probability and Related Fields 5,1-20. quant-ph/O112067. J . S. Bell (1964). On the Einstein Podolsky Rosen paradox. Physics I , 195-200. J. S . Bell (1981). Bertlmann’s socks and the nature of reality. Journal de Physique 42, C2 41-61. J. S. Bell (1987). Speakable and Unspeakable in Quantum Theory. Cambridge: Cambridge University Press. J . F. Clauser and M. A. Horne (1974). Experimental consequences of objective local theories. Phys. Rev. D I U , 526-535. N. Gisin and B. Gisin (1999). A local hidden variable model of quantum correlation exploiting the detection loophole. Phys. Lett. A 260,323-327. quant-ph/09905158. K. Hess and W. Philipp (2001a). A possible loophole in the theorem of Bell. Proc. Nat. Acad. Sci. USA 98,14224-14227. quant-ph/0103028. K. Hess and W. Philipp (2001b). Bell’s theorem and the problem of decidability between the views of Einstein and Bohr. Proc. Nat. Acad. Sci. USA 98, 14228-14233. quant-ph/0103028. K. Hess and W. Philipp (2004). Breakdown of Bell’s theorem for certain objective local parameter spaces. Proc. Nat. Acad. Sci. USA 101,17991805. quant-ph/0103028. A. F. Kracklauer (2002). Is ‘entanglement’ always entangled? J. Opt. B: Quantum Semiclass. Opt. 4, S121-Sl26. quant-ph/O108057. J.-A.-Larsson and R.D. Gill (2003). Bell’s inequality and the coincidencetime loophole. To appear in Europhysics Letters. quant-ph/0312035. S. Massar, D.Bacon, N. Cerf and R. Cleve (2001). Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A 63, 052305 (9pp.). qua11t-~h/0009088. P. Pearle (1970). Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418-1425. B. C. Sanctuary (2003). Quantum correlations between separated particles. quant -ph/0304186. C. H. Thompson and H. Holstein (2002). The “Chaotic Ball” model: local realism and the Bell test “detection loophole”. quant-ph/0210150.

ON AN ARGUMENT OF DAVID DEUTSCH

FLICHARD D. GILL Mathematacal Institute, University of Utrecht, Netherlands, & E URANDOM, Eindhouen, Netherlands gill0math.uu.nl http://www.math.uu.nl/people/gill We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born’s law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born’s law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason’s theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using diflerent assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.

1. Introduction: are quantum probabilities fixed by

quantum determinism? Quantum mechanics has two components: a deterministic component, concerned with the time evolution of an isolated quantum system; and a stochastic component, concerned with the random jump which the state of that system makes when it comes into interaction with the outside world, sending at the same time a piece of random information into the outside world. The perceived conflict between these two behaviours is ‘the measurement problem’ as exemplified by Schrodinger’s cat. Here we do not resolve this problem but just address the peaceful coexistence, or possibly even the harmony, between the two behaviours. We will show that some classical deterministic quantum mechanical assumptions, together with the assumption that the outcome of measuring an observable is random, uniquely determines the probability distribution of

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278 the outcome-harmony indeed. More specifically, two generally accepted invariance properties of observables and quantum systems determine the shape of the probability distribution of measured values of an observablenamely, the shape specified by Born’s law. The invariance properties are connected to unitary evolution of a quantum system, and to functional transformation of an observable, respectively. This work was inspired by Deutsch (1999). There it is claimed that a still smaller kernel of deterministic classical quantum theory together with a small part of deterministic decision theory together force a rational decision maker to behave as if the probabilistic predictions of quantum theory are true. In our opinion there are three problems with the paper. The first is methodological: we do not accept that the behaviour of a rational decision maker should play a role in modelling physical systems. We are on the other hand happy to accept a stochastic component (with a frequentist interpretation) in physics, so we translate Deutsch’s axioms and conclusions about the behaviour of a rational decision maker into axioms and conclusions about the relative frequency with which various outcomes of a physical experiment take place. The second problem is that it appears that Deutsch has implicitly made use of a further axiom of unitary invariance alongside his truely minimalistic collection, and needs to greatly strengthen one of the existing assumptions concerning functional invariance, from one-to-one functions also to many-to-one functions. Neither addition nor strengthening is controversial from a classical deterministic quantum physics point of view, but both are very substantial from a mathematical point of view. The third problem is that the strengthening of the functional invariance assumption puts us in the position that we have assumed enough to apply Gleason’s (1957) theorem. Thus at best, Deutsch’s proof is an easy proof of Gleason’s theorem using an extra, heavy, assumption of unitary invariance. The fact that Deutsch’s proof is incomplete has been observed by Barnum et al. (2000). However these authors did not attempt to reconstruct a correct proof. In the concluding section we relate our work to theirs. Wallace (2002, 2003a,b) has also studied Deutsch’s claims at great length and from a rather philosophical point of view. I did not attempt to relate his work to mine. The same goes for Saunders (2002). The paper is organised as follows. In Section 2 we put forward functional and unitary invariance assumptions, which are usually considered consequences of traditional quantum mechanics, but are here to be taken as axioms from which some of the traditional ingredients are to be derived, turning the tables so to speak. One would like to make the axioms

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as modest as possible, while still obtaining the same conclusions. Hence it is important to distinguish between different variants of the assumptions. In particular, we distinguish between (stronger) assumptions about the complete probability law of outcomes of measurements of observables, and (weaker) assumptions about the mean values of those probability laws. An invariance assumption concerning a class of functions, is weaker, if it only demands invariance for a smaller class of functions, and in particular we distinguish between invariance for all functions, including many-to-one functions, and invariance just for one-to-one functions. In Section 3 we prove the required result, Born’s law, for a special state (equal weight superposition of two eigenstates). This case is the central part of Deutsch (1999), who only sketches the generalization to arbitrary states. And already, it seems an impressive result. We prove the result, f o r this special state, in two forms-in law, and in mean v a l u e t h e former being stronger of course; using appropriate variants of our assumptions. Deutsch’s proof is incomplete, since he only appeals to unitary invariance, while it is clear that a functional invariance assumption is also required. The strengthening of the functional invariance assumption can also be used t o derive probabilities as well as mean values, and it is moreover useful from Deutsch’s point of view of rational behaviour, if one wants to extend in a very natural way the class of games being played. Roughly speaking, we extend from the game of buying a lottery ticket t o a game at the roulette table. In the former game the only question is, how much is one ticket worth. In the latter game one may make different kinds of bets, and the question is how much is any bet worth. However, so far we have only been concerned with a rather special state: an equal weight superposition of two eigenstates. As mentioned before Deutsch only sketches the extension to the general case of an arbitrary, possibly mixed, state. He outlined a step-by-step argument of successive generalizations. In Section 4 we follow the same sequence of steps, strengthening the assumptions as seems to be needed. In Section 5, we look back at the various versions of our assumptions, in the light of what can be got from them. We also evaluate the overall result of completing Deutsch’s programme. From a mathematical point of view, it turns out that we have done no more, at the end of the day, than derive the same conclusion as that of Gleason’s theorem, while making stronger assumptions. The payoff has just been a much easier proof. Gleason’s theorem only assumes functional invariance, we have assumed unitary invariance too. We argue that unitary invariance corresponds to a

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natural physical intuition, while functional invariance is something which one could not have expected in advance. It is supported by experiment, and is theoretically supported in special cases (measurements of components of product systems) by locality. We conclude with the conjecture that unitary invariance together with a weakened functional invariance assumption is sufficient to obtain the same conclusion. 2. Assumptions: degeneracy, functional invariance, unitary invariance

Recall that a quantum system in a pure state is described or represented by a unit vector in a Hilbert space, supposed to be infinite-dimensional, and that an observable or physical quantity is described or represented by a self-adjoint (perhaps unbounded) operator X on that space. I shall assume that X has a discrete and nondegenerate spectrum; thus there is a countably infinite collection of real eigenvalues x and eigenstates IX=x), so that one can write X = C,x IX=x )(X=x 1 , while = C A,IX = x ) where A, = (X=xI$). Throughout the paper we make the following background assumption:

I$)

I$)

Assumption 0 Random outcome, in spectrum. The outcome of measuring X is one of its eigenvalues x-which one, is random. Its probability distribution (law) depends o n X and o n

I$).

I write meas+(X) for the random outcome of measuring observable X on state and law(meas+(X)) for its probability distribution, i.e., the collection of probabilities Pr{meas+(X) E B } for all Bore1 sets B of the real line. Deutsch’s paper has the more modest aim just to compute the mean value of this probability law, E(meas+(X)), though as I shall argue before, even under his own terms (computing values of betting games) the whole probability law is of interest. Throughout the paper I will be playing with three main assumptions, though sometimes in stronger and sometimes in weaker forms. Here are the three, in their strongest versions:

I$),

Assumption 1 Degeneracy in eigenstates.

Pr{meas

I-)

(x) = x} = 1.

281 In an eigenstate of an observable, the corresponding eigenvalue is the certain outcome of measurement. Assumption 2 Functional invariance.

Pr{f(meas+(X)) = y ) = Pr{meas+(f(X)) = y).

(2)

Measuring a function f of an observable is operationally indistinguishable from measuring the observable, and then taking the same function of the outcome. Parenthetically remark that this indistinguishability is only as far as the outcome is concerned; as far as the new state of the quantum system is concerned there will be a difference, if the function is many-toone. Parts of Deutsch’s proof only need this assumption for one-to-one functions. In fact he only explicitly used this assumption for the afine functions f(z)= uz + b, but implicitly other functions, including many-toone functions, are involved too. Assumption 3 Unitary inuariance.

We will see that, at first instance, we only require this assumption to hold for a special class of unitary operations U , namely those which permute eigenstates of X . There is then a one-to-one correspondence u on the eigenvalues of X with inverse u* such that UXU* = u(X), U*XU = u*(X), and U I X = z ) = IX=u(z)). In the special case that is a n eigenstate IX =z), Assumption 3 follows from Assumption 1 (degeneracy-ineigenstates). Later we dso need Assumption 3 for unitary operations, diagonal in the basis corresponding to X. Since in the above assumptions, x and y are arbitrary, one could restate the three main assumptions as:

where law denotes the probability law of the random variable in question, so that in particular law(z) denotes the probability distribution degenerate at the point 2. An apparently weaker still set of assumptions would only

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restrict the mean values of the distributions in Assumptions 2 and 3:

As mentioned above, one can weaken the assumptions by restricting the class of functions f or unit&ies U for which the relevant equalities are supposed to hold.

3. The first part of the proof

I return to a discussion of the assumptions after an outline of the proof of my main result:

I will make use of Assumptions 1-3 in their original form, postponing discussion of how one might reach the same conclusion from weaker versions of the assumptions. In this section, following Deutsch, I only prove the result in the special case (a)

for which I am going to obtain the probabilities 1 / 2 for x = z1and x = x2, and zero for all other possibilities. After this, Deutsch attempts to generalize, first (b) to equal weight superpositions of a binary power of eigenstates of X, next (c) to an arbitrary number, then (d) to dyadic rational superpositions, next (e) to arbitrary real superpositions, and finally (f) to arbitrary superpositions. The proofs he gives of these steps are similarly incomplete. I will complete the proof by an alternative and rather short route in the next section, but return to Deutsch’s completion in the section after that. Suppose u, a one-to-one correspondence on the eigenvalues of X , maps 21 to x2 and vice-versa, and, after we have labelled the other eigenvales as x1,, n r i Z,maps x1, to z;+~. Let U denote the unitary which performs the same permutation of the eigenvectors. Let u* denote the inverse of u. Exploiting the relationship between u and U , and their relationship to X

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and $, as well as our other assumptions, we find,

Pr{meas+(X) = q } = Pr{measv+(X) =

21)

= Pr{meas+(U*XU) = q } = Pr{meas+(u(X)) = q }

= Pr{u(meas+(X)) =

XI}

= Pr{meas+(X) = u*(z1)}

(6)

= Pr{meas+(X) = 22).

Replacing z1by an eigenvalue xk, i.e., any other than 2 1 or 22, and running through the same derivation, we see that all other eigenvalues have equal probabilities. Since there are an infinite number of them, and since according t o our background assumption the outcome of measuring X lies in its spectrum, we have obtained the required result: the probabilities of z1and 22 must both equal 1/2, all the other eigenvalues 2; must get zero probability. We used Assumptions 2 and 3 (functional and unitary invariance), not Assumption 1 (degeneracy in an eigenstate). However, this assumption is needed t o deal with the case o f . . . an eigenstate. The proof method allows us t o deal with an equal weight superposition of any positive finite number of eigenstates of X . We only used functional invariance for one-to-one functions. Deutsch was only interested in mean values of the probability distributions of outcomes, since the fair value of the game: measure X on and receive the value of the outcome in euro’s (g), is precisely g E(meas+ (X)). (Here we are assuming that the utility of having some number of euro’s is equal to that number. The reader may replace euro’s by dollars, camels, or whatever else he or she prefers). In a moment I will also add a new game and receive g 1 if the outcome 20 is to the discussion: measure X on found. The value of this game should be g ~ ( Z O 12. Let us assume that the spectrum of X consists of all the integers (negative and non-negative). Then for given z1 and 2 2 there is an affine map u(2) = u z b = 2 1 x2 - 2 which defines a unitary transformation U as above. For these U , X and the same $ as before we rewrite the argument

I$)

I$)

+

+

I$)

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before as E(meas+(X)) = E(measv+(X)) = E(meas+(U*XU)) = E(meas+(W))) = E(4meas+(X)))

+

= z1 z2- E(meas+(X))

(7)

yielding the required equality, E(meas+(X)) = i(z1

+ z2).

(8)

Deutsch’s proof was a cryptic version of the argument I have just given, except that he did not mention the unitary invariance assumption. He writes 21 for value, instead of E. In my opinion, without the extra (unitary invariance) assumption, his proof fails. The degeneracy Assumption 1 is not used at this stage. However one may note that Assumption 1 (degeneracy) implies that Assumption 3 (unitary invariance) holds when the state is an eigenstate of the observable X. One could therefore consider Assumption 3 as a natural interpolation from Assumption 1. I return to this later. As has been shown by de Finetti and by Savage, a rational decision maker who must make choices when outcomes are ‘indeterminate’ (I must avoid all terminology suggestive of probability theory, since the words ‘random’, ‘probability’ and so on, are not allowed to be in our vocabulary) behaves as if he (or she) has a prior probability distribution and indeed updates it according t o Bayes’ law when new information (outcomes) becomes available. Thus it seems to me that whether one starts with utilities and assumes rationality, or with probability and the frequency interpretation, is very much a matter of taste. In my opinion the latter is closer to physical experience and indeed we know that casinos and insurance companies make good money from the frequency interpretation of chance. I consider the many repetitions in the frequency interpretation to be no more and no less than a thought experiment. When one claims that the probability of some event is some number, one is asserting that the situation in question is indistinguishable from a certain roulette game or lottery. This allows me also to talk about probabilities of outcomes of once-off experiments. For instance, a certain physical experiment might have some chance of producing a black hole which would swallow the whole universe. The probability that this would indeed happen, if the devilish experiment were actually carried out, would be computed by doing real

I$)

285 physics in which one would imaginarily set the chain of events into motion, many many times, in which uncontrolled initial conditions would vary in all kinds of ways from repetition to repetition. How they would vary, and what possibilities could be considered equally likely, should be a matter of scientific discussion. This may appear circular reasoning or an infinite regress or just plain subjectivism, but this does not bother me: it works, and it is not subjective, since we may rationally discuss the probability modelling. When I use the mathematical model of probability, I am only claiming an analogy with something familiar, like a casino, lottery, or coin toss. I think that it is the same in the rest of physics, when we talk about mass, electric charge, or magnetic field: we might think or we might hope that we are talking about real things in the real world but we can only be certain that we are talking about ingredients of mathematical models which are anchored t o the real world by analogies with familiar down to earth daily experience. My frequentistic position is perhaps better labelled “Laplacian counterfactual frequentism” and though one might collapse this label to “subjectivism”, I believe it is as instrumentalistic or as operationalistic as anything else in physics. 4. Completing the proof

More can be got out of the functional invariance assumption, by considering other functions f, and most crucially, certain many-to-one functions. In my opinion we must do this anyway, in order to complete the proof on the lines indicated by Deutsch (see next section). It is an open question, whether we can do without. With the choice f = I{z},and writing [X = x] for the projector onto the eigenspace of X corresponding to eigenvalue x (and later also for the eigenspace itself), since I,,}(X) = [ X = z ] ,we read off Pr{meas$(X) = x} = ~ r { m e a s $ ( ( [ ~ = x ]= ) I}.

(9) Indeed, if we only assume the mean value form of the functional invariance assumption, we can read off the same conclusion, since the random variables I{,}(meas~(X)) and meas$([X=x]) are both zero-one valued. Till this point we had dealt with nondegenerate observables and equal weight superpositions of eigenstates. Now we can add to this, also degenerate observables (since these can always be written as functions of nondegenerate observables). Moreover, even if we start with the assumptions in their weaker mean value form, we can still obtain the stronger conclusion about the whole probability law of the outcome.

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In fact, with brute force we arrive now very quickly at the most general result (it remains, namely, to consider arbitrary states). From functional invariance (whether in terms of probability laws or whether in terms of their mean values) we have shown that a probability can be assigned to each closed subspace of our Hilbert space, countably additive over orthogonal subspaces, and equal to 1 on the whole space. Now we can invoke Gleason’s theorem to conclude that the probability of any subspace is of the form tr{pA} for some density matrix p. It remains to show that p = but this follows from our first axiom that measuring an observable on an eigenstate yields with certainty the corresponding eigenvalue: consider the observable X = itself, and subspace A = [$I (the one-dimensional subspace generated by Deutsch’s extension of his results to the most general case (see next section) is very hard to follow. He repeatedly invokes substitutability, whereby an outcome of one game may be replaced by a new game of the same value. He does not say which substitutions are being made. However he is clearly thinking of substitutions, leading to composite games with composite quantum systems, product states, and observables on each subsystem. During these constructions and substitutions, the observables being measured and the states on which they are being measured, keep changing, while the Spartan notation v(x) in which the symbol x refers to an observable, an eigenvalue, and an eigenstate simultaneously, begs confusion. The mere construction of product systems implies that more is being assumed above the structure so far (so far we only spoke of observables and states on one fixed quantum system). As I will indicate below, it appears that the extra assumption of unitary invariance and the strengthened functional invariance assumption involving many-to-one functions as well as one-to-one functions, together with a natural assumption about measuring separate observables on a product system in a product state, enable one to fill the gaps. If the repair job is not too difficult, one finishes with a relatively easy proof of Gleason’s theorem, under the supplementary condition of unitary invariance. The construction of product systems will also help us extend results from infinite-dimensional quantum systems to finite dimensional, including 2-dimensional-the case not covered by Gleason. Functional invariance assumptions on product systems, or more generally, for compatible observables, play a key role in many foundational discussions of quantum mechanics. Recall that observables X , Y commute (or are compatible with one another) if and only if both are functions of a

l$)($l

1$)($1

I$))!

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third 2; and the third can be chosen in such a way (with a minimal set of eigenspaces) t o make the mapping 2 I+ (X, Y)a one-to-one correspondence in the sense that we can write X = f(Z), Y = g ( Z ) , Z = h(X,Y) where h i s the inverse of ( f , g ) . In other words, two (or more) commuting observables can be thought of as components of a vector-valued observable, or equivalently as defining together one ‘ordinary’ observable. Whether one thinks of them together as a vector or as a scalar observable is merely a question of how the eigenspaces are labelled. One can define joint measurement of compatible observables in several equivalent ways. Assuming Liiders’ projection postulate for how a state changes on measurement, the sequential measurements, in any order, of a collection of compatible observables, are operationally indistinguishable from one another. One may therefore think equally well of ‘one-shot ’ measurement of 2, sequential measurement of X then Y ,and sequential measurement of Y then X. This leads to a further extended functional invariance assumption: = law (meas+(f(-f)I), law (f(meas$ (2)))

(10)

where 2 = (XI ,... ,X,) is a vector of mutually compatible observables and f : Rk + Iwm. Apparently weaker is the mean value form of this: E ( f ( m e W 2 ) ) ) = E(meas$/,f(-f)));

(11)

though as I showed above, by playing around with indicator functions, the two are equivalent. We can recover from the assumption the fact that the probability law of a measurement of X alone is the same as the first marginal of the joint law of the two outcomes of a joint measurement of commuting X ,Y.As I have argued in Gi11(1996a,b), these consequences of the standard theory form a crucial though often only implicit ingredient in many of the famous no-go arguments against hidden variables in the literature. Somewhat irreverently I have dubbed (11) ‘the law of the unconscious quantum physicist’. Deutsch’s approach is similar to that of some probabilists, in that he would prefer to make Expectation central, and have Probability a consequence (in fact, he would prefer to do without the word Probability altogether). This is fine, and indeed many probabilists do take this approach (Whittle in his textbook on Probability argues that one should do the same for quantum probability, too). Now in our situation we want to start with hypothesizing existence of mean values, and by making some structural assumptions about them. From this we want to derive the form of the mean values. As I have noted above, since l ~ , ~ ( X is )both an observable itself,

288

and a function of the observable X, it would appear that fixing all mean values of (outcomes of measurements of) all observables, fixes all probability laws of (outcomes of measurements of) all observables. The point I want to make, is that this indeed works, provided we have the functional invariance assumption (for mean values only, if you like, but we must have if for a very large class of functions). Do we need to consider many-toone functions? If our assumptions are only about expectations, I think we do need many-to-one functions. However, with modest distributional input, one need further only consider one-to-one functions, as follows. Suppose we know the mean value of meas+(exp(it arctanx)), and suppose we assume functional invariance, in law, for all one-to-one functions; in particular, the functions f(z)= exp(itarctanx), for each real t. Then we know law(meas+X). It is possible to avoid complex-valued functions, try for instance f(z)= scos($(arctanz+.rr/2)) +tsin(i(arctanz+w/2)) for all r e d s and t. Let me return to the contrast between Deutsch’s and Gleason’s argument. Deutsch’s proof, on completion, seems a little simpler and more direct. His assumptions are much stronger: he needs unitary invariance. His assumptions are more representative of classical quantum mechanicsunitary evolution has to be considered an essential part of this. In the first stages of his argument, deriving mean values for some rather special observables and rather special states, he moreover only needed to consider functional invariance under one-to-one transformations. This assumption is close to tautological (the apparatus for measuring a bX is not going to be essentially different from that for measuring X).However, even from the point of view of deriving fair values of games, probability laws as well as mean values are equally relevant. For instance, what is the fair value of the game: measure X and receive ~91 if the outcome z o is obtained? The easiest way to deal with this game too, is to include functional invariance for the indicator functions too, and then one need not work any more but simply appeal to Gleason’s theorem.

+

5. Discussion

Later in this section I will run through Deutsch’s steps to complete his proof. The aim will be to see whether, with weaker versions of our main assumptions, not strong enough to give us Gleason’s assumptions so easily, we could also arrive at the desired conclusion. (The answer is that at present, I do not know). But first I would like to discuss what grounds one

289

could have for the functional and unitary invariance assumptions, against the background assumptions that measuring an observable yields an eigenvalue, and that in an eigenstate, the outcome is certain. Functional invariance for one-to-one functions seems to me more or less definitional. For many-to-one it is much less definitional, also less empirical, since there will vary rarely truely exist essentially different measurement apparatuses for ‘doing’ X and doing f ( X ) . Just occasionally there will be empirical evidence supporting functional invariance: for instance, when X and Y do not commute, but for some many-to-one functions, one has f ( X ) = g ( Y ) ,there might be empirical (statistical) data supporting it, based on the quite different experiments for measuring X and for measuring Y , and finding the same statistics (or mean values) for f of the outcomes of the first experiment, g of the outcomes of the second. There is one very strong empirical fact supporting the assumption (in its form for vector observables): when we simultaneously measure observables on separate components of a product system (even if in an entangled state) we have the same marginal statistics, as if only one component was being measured. Altogether, the nature of this assumption would seem to me to be: we extend a definitional assumption concerning a smaller class of functions f-the affine functions-to a much larger class, by mathematical analogy, trusting that the world is so elegantly and mathematically put together, that the ‘obvious’sweeping mathematical generalization of an indubitable fact is usually correct; we are supported in this by some empirical (statistical) evidence for some special cases. Similarly the assumption of unitary invariance seems to be largely a leap of faith, since there will be little empirical (statistical) evidence to support it. But again, one might prefer to think of the leap of faith as a natural mathematical generalization. Our first assumption-that measuring an observable on an eigenstate produces the eigenvalue-tells us law (measuG( X ) ) = law (meas+(u*xu)),

(12)

whenever U permutes eigenspaces and II, is an eigenvector! Extending this to arbitrary states can be thought of as an interpolation, in harmony with ideas of wave-particle duality. It seems to me that waveparticle dualitythe very heart of quantum physics-essentially forces probability on us, since it is the only way to get a smooth interpolation between the distinct discrete behaviours at different eigenstates of an observable. We just have to live with smoothness at the statistical level, instead of at the (counterfactual) level of individual outcomes.

290

I would now like to discuss the remaining steps of Deutsch’s proof. As we saw, functional invariance in its strongest form implies the conditions of Gleason’s theorem, which makes all further conditions and further work superfluous. Now the reason functional invariance is so powerful, is that we assumed it to hold for all functions f , in particular, many-to-one functions. In the spirit of the first part of Deutsch’s proof it would make sense to demand it only for one-to-one functions. It seems to me a reasonable conjecture that Deutsch’s theorem is true under the three assumptions: functional invariance for one-to-one functions, unitary invariance, and the degeneracy assumption. As was stated earlier, after (a) the two-eigenstate equal weight superposition, Deutsch extends this (b) to binary powers, (c) to arbitrary whole numbers of equal weight superpositions, (d) to rational superpositions, (e) to real and finally (f) to arbitrary. As we saw, steps (b) and (c) can also be dealt with by his own method for the two-eigenstate case. Deutsch’s argument for (d) involves completely new ingredients and assumptions. He supposes that an auxiliary quantum system can be brought into interaction with the system under study, thus yielding a product space and a product state. The observable of interest X is identified with X 8 1, and this is considered as one of a pair (X8 1 , 1 8 Y )where the observable Y is cleverly chosen, so that in the product system, and with this product observable, we are back in an equal weight superposition of eigenstates. He then makes the assumption: measuring X on the original system is the same as measuring ( X , Y ) on the product sytem and discarding the outcome of Y. Uncontroversial though this may be, we are greatly expanding on the background assumptions. Moreover we are actually assuming functional invariance for a many-to-one function: namely, the function which delivers the 2-component of a pair (2,~). By the way, Deutsch’s proofs of steps (b) and (c) similarly involve such constructions. Step (e) is an approximation argument which can presumably be made rigorous, though perhaps differently to how Deutsch does it. Step (f) as presented by Deutsch involves yet another new assumption: measuring an observable can be represented as a unitary transformation on a suitable product system, so that after a new unitary transformation mapping 12) to e+4 12) one can remove complex phases from a superposition of eigenstates. This argument seems to be unnecessarily complicated. Our unitary invariance assumption together with the unitary transformation just described, takes care of extending results from real to complex superpositions. The work of Deutsch has been strongly criticised by Finkelstein (1999)

291

and by Barnum et a1 (1999). They also point out that the first step of Deutsch’s proof is incorrect, however, do not recognise that it can be repaired by a supplementary, natural, condition. They also point out that Gleason’s theorem does the same job as Deutsch purports to do, but do not see the very close connection between Gleason’s and Deutsch’s assumptions. They point out also that the later steps of Deutsch’s proof depend on various appeals to the substitutability principle, without stating which games were to be substituted for which. I must admit that it took me a long email correspondence with David Deutsch, before I was able for myself to fill in all the gaps. Finally they also point out that the work of de Finetti and Savage implies that rational behaviour under uncertainty implies behaviour as if probability is there. It is therefore just a question of taste whether or not one adds a probability interpretation to the ‘values of games’ derived by Deutsch. My conclusion is that Deutsch’s proof as it stands is valid, though the author is implicitly using unitary as well as functional invariance. All his assumptions together imply the assumptions of Gleason’s theorem, and much more. Consequently the proof as given does not have a great deal of mathematical interest. However the fact that distributional conclusions could already be drawn for some states and some observables, at a point at which only functional invariance for one-to-one functions had been used, and in my opinion, with a most elegant argument, justifies the conjecture I have already mentioned:

Conjecture 1 Deutsch’s theorem is true under the three assumptions: functional invariance f o r one-to-one functions, unitay invariance, and the degeneracy assumption. Unitary invariance alone tells us that the law of the outcome of a measurement of X only depends on the absolute innerproducts l ( ~ l. So the task is to determine the form of the dependence.

l$~)

Acknowledgments I am grateful for the warm hospitality and support of the Quantum Probability group at the department of mathematics of the University of Greifswald, Germany, during my sabbatical there, Spring 2002. My research there was supported by European Commission grant HPRN-CT-200200279, RTN QP-Applications. This research has also been supported by

292

project RESQ (IST-2001-37559) of the IST-FET programme of the European Commission. References

H. Barnum, C. M. Caves, J. Finkelstein, C. A. F’uchs and R. Schack (2000), Quantum Probability from Decision Theory? Proc. Roy. SOC. Lond. Ser. A 456, 1175-1182. D. Deutsch (1999), Quantum Theory of Probability and Decisions, Proc. Roy. SOC.Lond. Ser. A 455, 3129-3137. R.D. Gill (1996a), Discrete Quantum Systems, www.math.uu.nl/people/gill/Preprints/chapter2.pdf.

R.D. Gill (1996b), Hidden Variables and Locality, www.math.uu.nl/people/gill/Preprints/chapterl4.pdf. A. Gleason (1957), Measures on closed subsets of a Hilbert space, J. Math. Mech. 6 , 885-894. S . Saunders (2002), Derivation of the Born Rule from Operational Assumptions, quant-ph/0211138. D. Wallace (2002), Quantum Probability and Decision Theory, Revisited, quant-ph/0211104. D. Wallace (2003a), Everettian Rationality: defending Deutsch’s approach to probability in the Everett interpretation, quant-ph/0303050.To appear in Studies in the History and Philosophy of Modern Physics, under the title “Quantum Probability and Decision Theory, Revisited”. D. Wallace (2003b) ,Quantum Probability from Subjective Likelihood: improving on Deutsch’s proof of the probability rule, q~ant-ph/0312157.

VOLTERRA REPRESENTATIONS OF GAUSSIAN PROCESSES WITH AN INFINITE-DIMENSIONAL ORTHOGONAL COMPLEMENT

W J I HIBINO Faculty of Science and Engineering, Saga University, 84 0-8502, Saga, JAPAN e-mail: hibinoyOcc.saga-u.ac.jp

HIROSHI MURAOKA Research Center of Computational Mechanacs, Inc., 142-004I , Tokyo, JAPAN e-mad: muraokaOrccm.co.jp

We consider whether the noncanonical Volterra representation may have an infinite-dimensional orthogonal complement or not by the use of the method of the stationary processes.

Keywords: Gaussian processes; Noncanonical representations; Volterra representations. AMS Subject Classification: 60G15,60G10

1. Introduction

-

It is easy to see that, for a given Brownian motion B = { B ( t ) t; 2 0}, a Gaussian process B, = {E,(t);t 2 0} defined by &(t) =

I” (--+ 2q

9

1 UQ IY

’) dB(u)

-iq

is again a Brownian motion for any nonzero q > -1/2. Moreover, P. L6vy [9] pointed out that there were infinitely many polynomials P so that

Z(t) =

Jd”

P(u/t)dB(u)

293

294

is again a Brownian motion. Among these many representations of a Brownian motion, there is the only one special representation: so-called a canonical representation [5].

For a Gaussian process X = { X ( t ) ;t 2 0) defined by

the representation (2) is said to be canonical with respect to B, if & ( X ) = & ( B ) for each t , where & ( X ) is the o-field generated by { X ( s ) ; s5 t}. As & ( X ) can be interpreted as past information of X , we can say that in canonical representation theory past information of B can be completely acquired by that of X . It is noted that, in a Gaussian case, & ( X ) = & ( B ) is equivalent to H t ( X ) = H t ( B ) , where H t ( X ) is a closed linear hull of { X ( s ) ;s It>. For example, the representation (1) is noncanonical since B, satisfies the property

{

H t ( B ) = H t ( E q )a3 LS I ' u ' d B ( u ) }

,

where LS{. . .} is a linear span of {. . .}. Let g1,92,. . . ,QN E L&,[O,cm) be linearly independent in (0, t ) for each t > 0. In the joint work [3],the authors have found how to construct the noncanonical representation of a Brownian motion having the N-dimensional orthogonal complement whose basis is g = ( 9 1 ~ 9 2 ,...,g ~ } :

is a Brownian motion, and is noncanonical with respect to B satisfying

H t ( B ) = H&)

@ LS

S j ( U ) d B ( U ) , j = 1 , 2 , . ..

where

We remark that the Grammian matrix r(t)is invertible since g is linearly independent in (0, t).

295 The form (3) is a Volterra representation. (This terminology is due to [2].) Volterra representations are closely related to the famous innovation theorem, and so on. (see e.g. Hida and Hitsuda [S]) In this article, we shall consider whether the noncanonical Volterra representation may have an infinite-dimensional orthogonal complement by the use of the knowledge of the stationary processes. In Section 2, we review canonical representation theory for stationary processes, and then we give the result in [4] that there exists a noncanonical representation having an infinite-dimensional orthogonal complement. In Section 3, we discuss various conditions for Volterra representation to have an infinite-dimensional orthogonal complement. 2. Stationary processes

Suppose that F ( . ,.) in the representation (2) is a homogeneous function of degree cr i.e. F(at,au) = aQF(t,u)for any positive a. By using the transformation

the process X is transformed into the stationary process Y ( s )=

L

F(1,e-2(8-"))e-(8-")dW(u), s E R,

where dW(u) = $e-"dB(e2") is a Wiener measure. In these representations, each canonical property corresponds as follows: X is canonical with respect to B , if and only if Y is canonical with respect to W ;on the other hand, X is noncanonical with respect to B satisfying

I'

H t ( X )I

ugdB(u), t > 0,

for q > -1/2, if and only if Y is noncanonical with respect to W satisfying

H 8 ( Y )I

+

/'

ePudW(u), s E R,

-W

for p = 2q 1 > 0. Concerning stationary processes, canonical representation theory is well developed in deep connections with functional analysis. In this section, we shall give a brief review of the theory. Let a stationary process Y = { Y ( s ) ;s E R} be represented as

G(s- u)dW(u).

(5)

296

Due to the Paley-Wiener theorem [l],the Fourier transform 6 of G lives in the Hardy class H2+ in the upper half-plane C+ = { z E C ;Sz > 0) since G belongs to L2[0,00). It is also known that c E H2+ has a unique decomposition: c(X) = CCO(X)CI(X) with c ~ ( x= ) &exp cr(X) = n(X)exp

* 1+wX

d,B(w)

+ ihX} ,

where C is a constant of a unit modulus, II is a Blaschke product, f is a spectral density function of the process Y ,h is a nonnegative constant, and p is a nondecreasing function of bounded variation whose derivative vanishes almost everywhere. Here, co and CI are called an outer function and an inner function, respectively. Both are analytic in C + . The outer function never vanishes in C+.All the zero-points there are undertaken by the inner function. Though the representation kernel G satisfies 1 -16(X)lz = f(X), X E R, 2w the outer function co also satisfies 1

-1c0(X))~ = f (A), X E R. 21r Thus the inner function has a unit modulus on the real-axis. Related to canonical representation, the following fact in [8] is important: The representation (5) is canonical with respect to W , if and only if the inner function of 6 is absent. This tells us that the outer function determines the law of the process and that the inner function causes the noncanonical property.

Example 2.1. For the representation (1) of a Brownian motion, using the transformation (4) we obtain a stationary Brownian motion (OrnsteinUhlenbeck process) Y having the representation (5) with

The Fourier transform c of G is c(X) =

( 1 (=X)' +) , where p = 2q + 1. 1-iX ip

297 The first factor is the outer function, and the second is the inner one. As we have remarked, it is noncanonical satisfying

/'

H,(Y) I

ePudW(u), s E R.

-m

The zero-points in C+ are related to the orthogonal complement of the noncanonical representation:

J-00

if and only if the Fourier transform of G in (5) has a zero-point XO in C+. As the authors have pointed out in [4], there exists a noncanonical representation having an infinite-dimensional orthogonal complement: The stationary process Y defined by the representation ( 5 ) with

where 00

< ca and pn > 0,

(7)

n=l

satisfies

H,(Y)I

{ 100 ePnudW(u); n

EN

1

in H , ( W ) ,

sE

R.

(8)

The condition (7) guarantees the convergence of the infinite product in (6). By using the inverse transform X ( t ) = &?Y(log&) of (4), we can prove the following theorem. Theorem 2.1. There exists a noncanonical representation of a Gaussian process X satisfying

for the sequence

satisfying (7).

If we take co(X) = 1/(1- i X ) in (6),we can easily see that there exists a noncanonical representation of a Brownian motion having an infinitedimensional orthogonal complement.

298

3. Volterra representations We call a Volterra representation for the representation

where k ( s , .) belongs to L2(0,s) for any s

I" (la

k ( s , u)2du)

> 0, satisfying ds < 00.

It is well-known in [7] that X is equivalent to B (i.e. the distributions of X and of B are mutually absolutely continuous), if and only if X admits a Volterra representation with k E L2((0,t ) 2 )for any t > 0, namely,

I' la

k(s,u)2duds< oo for any t

> 0.

In this case, the representation (10) is canonical with respect to B . As we have seen in Section 1, the noncanonicd representation (3) of a Brownian motion is always a Volterra representation. Therefore, the Volterra kernel

is not square-integrable. Needless to say, we can check it by a direct calculation. In order that the representation kernel of (10) is a homogeneous function of degree zero, we shall restrict to the case of k ( s , u ) = (l/s)cp( u / s ) , where cp belongs to L2(0,1). Then the representation (10) turns to

By using the transformation (4) the stationary process Y is obtained as in the form

Y ( s )=

/'

--co

e-(8-u) (1 -

)

e"$(v)du dW(u).

(12)

Here we put $(v) = 2e-"cp(eM2") E L2(0,oo), for short.

Proposition 3.1. If a stationary process Y of the form (12) satisfies ( 8 ) f o r p , > 0, then supp, < 00.

299

Proof. If the property

H 8 ( Y )I/' eP"dW(u) -cQ

is satisfied, then

for any s E R. It is reduced to fcQ

By using the Schwarz inequality,

fiL lI+llL2(0,rn)

<

Thus, we have proved the desired statement. By noting that zero-points of an inner function accumulate either infinity or zero when they are pure imaginary, it is enough to consider whether zero-points may tend to zero.

Theorem 3.1. If a stationary process Y is of the form (12), then, f o r any sequence 0 < pl < pz < . . . , the process Y never has the property ( 8 ) . Proof. That Y satisfies (8) is equivalent to that has infinitely many zeropoints {ip,) in C+. However, is analytic there. So the cluster points of its zero-points should be on the boundary. Thus the zero-points can tend to only infinity. Nevertheless, the zero-points cannot tend to infinity because of the proposition above. Therefore, the proof is finished. 0 The following lemma is obvious, because an inner function is analytic [l].

Lemma 3.1. If the modulus of a n innerfunction is continuous in the closed then the cluster point of zero-points there of the inner upper half-plane function is only point at infinity, if any.

c,

Theorem 3.2. If a stationary Brownian motion Y is of the form (12) with @ E L1(O,m), then Y never satisfies ( 8 ) .

Proof. The Fourier transform of the representation kernel

G ( t ) = e-t (1 - L e " @ ( v ) d u ) , t 2 0 ,

300 of (12) is

1 G(X) = 1 -zx (1 -$(A)) h

.

Therefore the inner function of (12) is 1 - $(A), since the outer function of a stationary Brownian motion is 1/(1 - i X ) . For T,!J belongs to L1(O,oo), its Fourier transform is continuous in By assuming that Y has the property (8),the zero-points {ip,} can tend only to infinity because of the lemma above. However, { p , } cannot tend to infinity thanks to Proposition 3.1. 0

c.

Noting that

we can prove the following theorem.

Theorem 3.3. If a Brownian motion X is of the form (11) with & q ( z ) E L1(O,l),then X never satisfies (9). 4. Concluding remark

(I) We have considered only the case where zero-points of an inner function are pure imaginary. According to Proposition 3.1, the imaginary parts of zero-points of an inner function cannot tend to infinity; on the other hand, according to Theorem 3.2, under some conditions they cannot tend to zero. However, even if we restrict the process Y of (12) to be real-valued, the inner function may have zero-points in C+ besides pure imaginary. The condition for a stationary process Y to be real-valued is that 6 is symmetric with respect to the imaginary-axis i.e. 6 ( z )= 6(-z*). For example, the sequence {Cn} = {ktn iq} can be the zero-points of an inner function. Indeed,

+

can be an inner function if 00

n=l

'

O.

301

In this case, the orthogonal complement of H , ( Y ) in H , ( W ) contains

4L

evU cos(&u)dW(u),

L

evu sin(&u)dW(u); n E N

for s E R. (11) Thanks to the innovation theorem, if

I

I'

Ht(x) 1

k(t,u)dB(u)in Ht(B), t > 0 ,

then X of the form (10) is a Brownian motion. Especially if X is of the form (11) with cp(z) = CnEN unzQnand satisfies (9), then X is a Brownian motion. The condition for X of a Volterra representation (10) to be a Brownian motion is expressed

k(t,u) = J, k(t,w)k(u,w)dw, vt

> vu > 0.

For X of the form ( l l ) , it is reduced to

In terms of

$J

in (12), this corresponds to

qqs) =

jo +(s + u)$J(u)du, vs > 0.

Finally we point out that if cp is of the form cp(z) = EnEN unzqn,the condition is =1, VmEN. nEN

References 1. H. Dym and H. P. McKean; Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press (1976). 2. H. Follmer, C-T. Wu and M. Yor; On weak Brownian motions of arbitrary order. Ann. Inst. H. Poincax6 Probab. Statist. 36 110.4(2000), 447-487. 3. Y.Hibino, M. Hitsuda and H. Muraoka, Construction of noncanonical representations of a Brownian motion. Hiroshima Math. J. 27 (1997),439-448. 4. Y. Hibino, M. Hitsuda and H. Muraoka; Remarks o n a noncanonical representation for a stationary Gaussian process. Recent developments in infinitedimensional analysis and quantum probability. Acta Appl. Math. 63 (2000), 137-139.

302 5. T. Hida; Canonical representations of Gaussian processes and their applications. Mem. Coll. Sci. Univ. Kyoto 33 (1960),109-155. 6. T.Hida and M. Hitsuda; Gaussian Processes, Representation and Applications, Amer. Math. SOC. (1993). 7. M.Hitsuda; Representations of Gaussian processes equivalent to Wiener process, Osaka J. Math. 5 (1968),299-312. 8. K.Karhunen; Uber die StruMur stationarer zufalliger finktionen, Ark. Mat. 1 (1950), 141-160. 9. P.LBvy; Fonctaons altatoires (i corrtlation lintaire, Illinois J. Math. 1 (1957), 217-258.

THE METHOD OF DOUBLE PRODUCT INTEGRALS IN QUANTISATION OF LIE BIALGEBRAS.

R.L. HUDSON School of Computing and Mathematics Nottingham Dent University Burton Street Nottingham N G I 4BU Great Britain

++-

n

+

(1 d ~ [ h ]are ) defined as formal Ordered double product integrals such as power series with coefficients in the space of tensors over a Lie algebra L in which the Lie bracket is got by taking commutators in an associative algebra, where ~ [ h ] is a formal power series with coefficients in L @ 1: and vanishing zero order term. They are characterised by quasitriangularity identities. A necessary and sufficient condition for such a product integral to satisfy the quantum Yang-Baxter equation is applied to the quantisation problem for Lie bialgebras.

1. Introduction. Let ( L ,[., .I, 6) be a Lie bialgebra, that is a Lie algebra ( L ,[.,.]) equipped with a Lie cobracket 8,which is a linear map 6 : L +L @ L whose range lies in the subspace of skew-symmetric elements of C @ L and which satisfies the co-Jacobi and cocycle identities

(6 8 idr)6

+ 7(2,3,1)(6 @ id r )S + 7(3,1,2)(6@ idr)S = 0,

Here we introduce the following notations which will be used throughout the paper. Given vector spaces V1,V2,...,V , and a permutation 7r of V1@V2 8 -..@V,, 7 ,,is the linear map from V1@V28 . .@V, t o V,I @V,Z @. . -@Vfln which appropriately permutes components of product tensors. Also we use the place notation that K 1 and K 2 indicate that K is to be placed in the first and second copies respectively of a tensor product space. The quantisation problem for Lie bialgebras is the following. Given the Lie bialgebra (C, [., .I, 6), find a deformation Hopf algebra of the universal enveloping algebra of the Lie algebra ( L ,[., .I) (that is, in one parlance, a

-

303

304 quantum group) of which 6 is the infinitesimal of the coproduct A[h],that is

hS(L)= A[h](L)- A[h]'PP(L)+ o(h2) where A[h]'pp is the opposite coproduct ~ ( 2 , l ) A [ hThe ] . quantisation problem was first solved in complete generality by Etingof and K a ~ h d a nTheir .~ method is long and involved. It depends on the construction of a general associator based on the monodromy of the Knizhnik-Zamolodchikov system of holomorphic differential equations. It seemed strange that a purely algebraic result could only be proved in this analytic way. An approach towards a possible alternative solution of the quantisation problem came in the work of Enriquez.2 The basis of Enriquez's approach was to seek the deformation in the so called shuffle Hopf algebra over the vector space L, got by equipping the space 7 ( C ) of tensors over L with the well-known shuffle product and a natural coproduct. My own approach to quantisation over the last seven years was originally motivated by quantum stochastic calculus and in particular the concepts, originally derived in that context but later seen to be independent thereof, of simple product integral and particularly double product integral. The relevant notion of simple product can be realised directly in universal enveloping algebras and is characterised by a group-like property with respect to the c o p r ~ d u c t .But ~ attempts to form a corresponding double product in universal enveloping algebra^^^^^^ involve a symmetrisation procedure which destroys grouplike properties. Instead, to move away from the context of quantum stochastic calculus, double product integrals must constructed in the full space 7 ( L ) of tensors, not just its symmetric subspace. This is equipped with a noncommutative deformation of the shuffle product which is essentially an abstraction of the multiplication formula for iterated integrals in quantum stochastic calculus, by making the basic assumption that L is itself an associative algebra, abstracting the algebra of It6 differentials, and the Lie bracket in L is got by taking commutators. It turns out that the natural coproduct for the shuffle product remains a coproduct for the deformed multiplication leading to a noncommutative and noncocommutative Hopf algebra called the It6 Hopf algebra. We are then able to construct double products which are characterised by a double group-like property. It is this property, in the form of the quasitriangularity relations, which forms the first stage of Enriquez's quantisation programme. To accomplish the next stage we find a necessary and sufficient condition on a double product integral for it to satisfy the quantum Yang-Baxter equa-

305 tion. This condition leads to the hierarchy of equations of Ref. 19 for the coefficients of the generator of the double product, also found by Enriquez2 in a different context. The conjecture of Ref. 19 that this hierarchy always possesses solutions can be answered positively, but only by invoking the Etingof-Kazhdan quantisation procedure. Quantum stochastic product integrals featured in some precursors and early versions of quantum stochastic calculus." The notion arose first in Ref. 10 where the forward product integral f f ( p d Q - qdP) provided an intuitive way of thinking about the second quantised time-orthogonal unitary dilation. A more systematic use of the idea can be found in Ref. 12 where the quantum It0 formula was perceived as a groupoidal multiplication rule for stochastic product integrals generalising the Weyl form of the canonical commutation relations. Since that time the product integral notion has received relatively little attention in the quantum probability community other than in the work of Holev0.~>~1~ Though they were originally defined in a different way,15 double product integrals can be described as iterated simple products with initial or system algebra. There are two ways of doing this; that they produce the same result can be regarded as a kind of multiplicative Fubini theorem or as a continuous analog of the equality

which holds whenever the X j , k have the property that Xj,k commutes with xjlkl whenever both j # j ' and k # k', but not necessarily for example when j = j' but k # k'. Quantum stochastic double product integrals sometimes have rather poor convergence properties. For example, for the double product

where A is the usual number process20 and z is a complex parameter, it = dA, that the exponential can be shown,15 using the multiplication

306

matrix element

where 21 = j1g1 and 2 2 = S,’j.g, and pm,n is the number of m x n incidence matrices such that every row and every column contains at least one entry 1. Since pm,n can be approximated by 2mn for large m,n, since a random choice of 1’s and 0’s will usually have a 1 in each row and each column, it can be seen that this double series diverges when z = 1 far dl nonzero 2 1 and 22. A mare painstaking argument’ shows that the corresponding power series in z has radius of convergence zero. Thus it is fortunate that, not only can the theory of double products be abstracted away from quantum stochastic calculus, but also, in applications to quantisation, only formal power series occur. 2. The It6 Hopf algebra.

Let C be a complex vector space. The vector space T ( C ) = @,“==, (Bn L) of all tensors over L becomes a commutative unital associative algebra when equipped with the shufle product defined by linear extension of the rule that, for arbitrary L1, L2,. . . ,Lm+n E L,

(L1@3L2@*. .@Lm)(L1@L2@‘. . *@Ln) =

c

~ ( 1@)~ 7 r ( 2 @. )

*

*gL7r(m+n)

XES,,,

where Sm, is the set of (m,n)-shufles,that is permutations T of {1,2,. . . ,m n } such that ~ ( 1 < ) ~ ( 2 )< . . < ~ ( mand ) ~ ( m1) < ~ ( m2) < ... < ~ ( mn). Equivalently, for arbitrary a,B E T(L), a/3 = y, where the homogeneous components of y are given in terms of those of a and ,B by

+

+

+

+

c

AuB={1,2,...,n},AnB=B

ahl

Here and elsewhere we use the place notation that, for example, denotes that the [A[-thrank homogeneous component al~lof a is placed in the tensor product of the \A\ copies of L within @n L labelled by the elements all of A. BEl is defined analogously, so that in the combination

afil@,l

307

C aren occupied exactly once. The unit element is ~ T ( L = ) ( 1 @ , 0 , 0 ., ..) E 7 ( C ) . The commutative shuffle product algebra T ( C ) becomes a noncocommutative Hopf algebra when equipped with the coproduct defined by linear extension of n copies of

m

A ( & @ L2 @ .. .L,) = c ( L i @L2 @ ...L j ) @ ( L j + l @Lj+2

...L,)

j=O

where the summand is regarded as an element of T ( C ) @ 7 ( C ) . The counit is defined by 4Q:o,a1,a2,. . .)

(& C1 @ (Bm-j' C 1 C

= Qo,

and the antipode is the linear extension of

S(L1 @ L 2 @ . . . L m = ) (-I)m(Lm@Lm-l @ . * . L 1 ) . Now let 2 be a not necessarily unital complex associative algebra. Then we may define the It6 shufle product in T ( C ) by the equivalent formulas

( L I@L2@ * * * Lm)(L1@L2 8. * Ln) -,=(PI

c

L,, @ L P 2 @ * * * L P L

,P2>...Pb)EPm,n

AUB={1,2,...,n} Here, in the first formula,13 Pm,, is the set of It6 shufles (sticky shuffle would perhaps be a more descriptive term) consisting of partitions p = ( p 1 , p 2 , . . .p k ) of { 1 , 2 , . . . ,m n } in which each p j is either a pair (s, t ) with 1 5 s 5 m 5 t 5 m n or a singleton and in which the natural orders of the subsets { 1 , 2 , . . .,m} and {m 1 , m 2 , . . . ,m n } are preserved in the ordered set ( p l, p 2 , ... p k ) . Each LPj is defined to be L , if p j = {s} and the product L,Lt if p j = (s,t ) . In the second formula we use place notation as before and reduce the double occupancies of C which occur when A n B # 0 by using the product in C . The It6 shuffle product makes T ( C ) into a unital associative algebra, with the same unit element as before, which is noncommutative if C is noncommutative. When C is the algebra of It6 differentials of quantum stochastic calculus then the map which sends each ct E 7 ( C ) into the sum of iterated stochastic integral processes whose integrators are the homogeneous components of Q: is multiplicative.

+

+

+

+

+

308 The map L 3L (0, L, 0, 0, .. .) E T(L)is a Lie algebra homomorphism when both associative algebras are equipped with the commutator Lie bracket. It can be shown16 that the universal extension to the universal enveloping algebra U ( L ) of the Lie algebra L is an isomorphism of unital associative algebras from U ( L ) onto the subalgebra S(L)of T(L)formed by the symmetric tensors. Remarkably17, the coproduct A and counit E introduced above for the shuffle product algebra T(L)remain multiplicative for the It6 shuffle algebra T(L)and equip it with a Hopf algebra structure in which the antipode is a deformation4 of that for the shuffle product of the form L1

€3 L2 €3

. . . L,

* (-l),(L,

€3 L,-1

€3

.L1) + terms of lower rank.

When the associative algebra L is noncommutative, T(L)is thus a noncommutative and noncocommutative Hopf algebra containing a cocommutative sub-Hopf algebra S(L)isomorphic to U ( L ) . We call it the It6 Hopf algebra.

3. Calculus in the It6 Hopf algebra.

d

We define the right and left differential maps : T(L)+ T(L)€3 L and t d : T(L)+ L @ T(L)by linear extension of the actions on homogeneous product tensors

d(L18L2 8.. .L,)

= (L1 €3 ’ . . @ L,-1)

€3

L,,

t

d(L1€3L2€3.*.L,) =L1€3(L2€3.-.Lm).

Alternatively for arbitrary a E T ( L ) ,

a4

= (idT(L) €3

t

d ( a )=

(Q

@)

( A ( 4 - 1T(L)€3 a ) ,

€3 idT(L)) (N4 -

€3 1 T(L))

where Q is the associative algebra homomorphism a + a1 from the ideal T(L),in T(L)consisting of elements whose zero rank homo eneous components vanish, equivalently the kernel of E , to L. and d satisfy the Leibniz-It0 formulas

d

&QB) t

= &4B

8

+ J(P) + d(a)&B),

+ ad(D)+ %(a)%@)

d (aB) = %(a)B

in which T(L)€3 L and L €3 T(L)are regarded as two-sided T(L)-modules using the multiplicative right and left actions of T(L)on itself as well as associative algebras using the tensor product multiplication.

309

By comparing actions on homogeneous product tensors we see that the coproduct A is recovered from either of the differential maps as 00

M

where we make the natural identifications W

00

7 ( L )€4 T ( L )= @ (7(L)€4C?ln ( L ) )= @ ( @ y L )€4 7 ( L ) ) n=O

n=O

and the iterated differential maps are defined by t t = d ('1 = d , and for n > 1

d(') 2,

den) = (2€4 id Bn-IL)

%ten)

$(n-'),

d(O)= td

= (idBn-lL €4

(O)

= idT(L),

2)

%(n-l).

Alternatively we may describe A as the unique solution of either of the differential equations (idT(L) @

2)A = (A @ idL) 2,(idT(q @

E)

A = idT(L),

which exhibit A in each case as a flow of which the corresponding differential map is the generator. Combining the counit properties ( E €4 idT(L)) A = id T(L) = (idT(q 8 E ) A with (2) we obtain the right and left TaylorMaclaurin expansions idqq =

(E

@ idT(L)) n=O

4. Simple product integrals with system algebra.

Let d be a n associative algebra. When equipped with the convolution product / M

\

/ w

\

/ w

N

the space d[[h]] of formal power series in an indeterminate h with coefficients in d becomes an associative algebra, which is unital if d is so. We will find the following well known theorems useful (for proofs, see Ref. 14 for example).

31 0

Theorem 4.1. Let a[h]E hd[[h]] be a formal power series in which the zero-order coeficient vanishes. Then there exists a unique two-sided quasiinverse for a[h]in hd[[h]], that is an element a'[h] such that

+

a[h] a"h] -ta[h]a"h]= a"h]

+ a[h]+ a'[h]a[h]= 0.

Theorem 4.2. Let d be unital and let A[h] E d [ [ h ] be ] a formal power series in which the zero-order coeBcient is 1 A. Then there exists a multiplicative inverse A-l[h] which is of the same form. Let 7 ( C ) be the It8 Hopf algebra over the associative algebra C and let A be a unital associative algebra which we call the system algebra. Let Z[h]= Zj[h]€3 L(j) E h ( A €3 C)[[h]]. The infinite sums

'&

m

m

Y [ h ]=

C C...,

n=Ojl,j2,

Zj,

[h]Zj,-l[h]. . .Zj, [h]€3 L(jl) €3 L(j2)€3 . . . €3 L(jn) (4)

j.,

can be rearranged algebraicly as well-defined formal power series belonging which respectively satisfy the algebraic identities to (d 63 T ( C ) )[[h]] (idd 8 A) X[h]= X[h]132X[h]1'3, (ida €3 E ) X [ h ]= 1 A , (idd €3 A) Y [ h ]= Y[h]113y[h]1'2, (idd €3 E ) Y [ h ]= 1 A. Here the coproduct and counit act on formal power series coefficient-wise and we likewise apply place notation to coefficients in (-4 63 7 ( L )€3 7 ( C ) )[[h]]. It can be shown" that all solutions of these identities are of this form for some Z[h]E h (-4 €3 L ) [[h]]. Also, X [ h ]and Y [ h ]are solutions, unique in each case, of the differential equations

(ids €3 2)x [ h ]= x[h]"2z[h]1'3, (idd 63 (idd €3 (idd €31'd X[h]= Z[h]122X[h]1*3, (idd 63 2)Y [ h ]= l[h]113Y[h]1j2, (idA 63

E)

X[h]= 1 A ,

E)

X [ h ]= 1 a,

E)

Y [ h ]= 1 a,

31 1

We call X [ h ]and Y [ h ]the right and left directed product integrals generated by l[h]and denote them by

+

t

+

X [ h ]=d n(1 dl[h]),Y [ h ]=d n(1+ d [ h ] ) respectively, where the subscript indicates that the system algebra A is laced to the left of T(L)in each case. Product integrals dl[h]), Ifid(1 dl[h])in which the system algebra is on the right, generated by an element l[h]of h (C €3 -4) [[h]], are defined analogously and characterised algebraically as solutions of

+

+

(A €3 idd) X [ h ]= X[h]1’3X[h]2’3, ( E €3 idd) X[h]= 1 A ,

(A @ idd) Y [ h ]= Y[h]2’3Y[h]1’3, ( E @ idA) Y [ h ]= 1A , respectively, and differentially as solutions of

(2€3 idd) X [ h ]= X[h]1’3i[h]2’3,€3 idd) x [ h ]= 1A, (E

(2€3 idd) X [ h ]= l[h]1’3x[h]2’3,@ idd) x [ h ]= 1 A, (E

(2€3 idd) Y [ h ]= l[h]1’2Y[h]2’3,€3 idd) Y [ h ]= 1A. (E

-

When the system algebra is nonunital we define decapitated product h

h

integals A f f ( l + d [ h ] )A, f i ( 1 + d [ h ] )ffd(1 , +dl[h])and f i A ( 1 +dl[h])by omiting the initial unital term in the expansions (3) and (4). For example 03

+dl[h])=

lj, [h]2j2 [h].. .lj,

A n (1

[h]@L(jl)@L(j2) €3 * . .€3 L(jn)

n=l j 1 , j 2 , ...,j,,

and such a decapitated integral is characterised algebraicly as a solution of

+

(idd €3 A) X[h]= x[h]1’21%(L) x[h]’j31 ?j-(L)

+ x[h]112X[h]193,

(idd €3 E ) k [ h ]= 0 A , and differentially as that of (idd €3

2)x [ h ]= x[h]’121[h]173+ 1+(Lll[h]1’3,(id

€3 E ) x [ h ]= 0 A,

31 2

or of

as is easily proved by adjoining a unit to A.

5. Double product integrals. Now let r[h]be a n element of h (C@J C)[[h]]. Regarding the first copy of C in C €4 C as a left system algebra we may form the decapitated directed product integral ~ z ( +1 dl[h]).This is an element of h (C€9 ‘T(C))[[h]]. Hence, regarding 7(C) as a unital right system algebra we may form the directed product integral

f i ~ ((1~+)d ( ~ 3 (+1 dr[h])))which is

an element of (7-(C) @ 7 ( C ) )[[h]].Similarly we may form the element

~ ( , q f(1i

+ d ( E L ( l + dr[h]) of ( 7 ( C )€9 7 ( C ) )[[h]]. It is proved in Ref. 15

that

We define the common value to be the forward-backward directed double

n

-+e

+

product integral (1 dr[h])generated by r[h].By reversing arrows we can also define the backward-forward directed double product integral

+-+

n

F

(1 + dr[hl) =

1 + 4 L r - p + dr[hI))

Two other directed double products, forward-forward and backwardbackward, can be defined similarly but seem to be of less interest. It follows from the algebraic characterisations of directed simple product integrals that the double product integrals

=n

R[h]

(1

+ dr[h]),R”h]

=n + (1

dr[h])

satisfy

(A €4 idd) R[h]= R[h]193R[h]233, (idd €9 A) R[h]= R[h]1z3R[h]112, (5) (A €9 idd) R’[h]= R’[h]293R’[h]1,3, (idA 63 A) R’[h]= R’[h]192R[h]133,(6)

313

together with

It is not difficult t o show that these relations characterise forward-backward and backward-forward directed double product integrals (see Ref. 17 for the forward-backward case). (5) and (7) are known as the quasitriangularity satisfies the quasitrianrelations. Thus an element of (7(L)8 7 ( L ) )[[h]] gularity relations if and only if it is a forward-backward directed double product integral. The first stage of Enriquez’ quantisation procedure for Lie bialgebras is to construct solutions of the quasitriangularity relations. Theorem 5.1. Let r[h] and r’[h] be mutually quasiinverse elements of

n

-+t

(1 + dr[h])and h ( L @ L) [[h]].Then verses in ( T ( L )@ 7 ( L ) )[[h]].

n (1 + dr’[h])are mutual in-

t+

Proof. Setting

we have

where

Using the Leibniz-It6 formula, it follows that

314

Since

( z @ i d ~ ) q [ h= ] (z@idr)

-

(n,( + 1

dr' [h ] )

using differential characterisations of decapitated product integrals, again using the Leibniz-It6 formulawe have

since r[h]and r'[h]are mutually quasiinverse. Also by multiplicativity of E,

by characterisation of decapitated products. It follows that p[h] p[h]q[h]= 0 and hence by (8) that ( i d q ~@ )

2)(P[hlQ[hl) = 0.

+ q[h]+

31 5

Since by multiplicativity of E and characterisation of directed simple product integrals,

id^(^) €3 E ) (P[hlQ[hl)= id^(^:) €3 E) WI (&-(L)

@E)

(Q[hl)

= 1 T(L)

it follows that P[h]Q[h]=

~ T ( L ) ~ T ( Lthat ),

is

n

-tt

dr'[h])= 1 T ( L ) ~ ~ ( LAC )similar . argument shows that ( 1 + W h l ) = 1T ( L ) @ T ( L ) 0 -

+ dr[h]) n (1 + n ( 1 + dr'[h]) n t-i

(1

t-t

-it

A corollary t o Theorem 3 is that, for mutually quasiinverse r [ h ] ,r'[h] E h ( L €3 L ) [ [ h ] ]the , map J[h]from 7 ( L )€3 7 ( L )to ( T ( L )8 7 ( L ) )[[h]]given by

is multiplicative. Since A is multiplicative, it follows that the map A[h] = J [ h ] Afrom 7 ( L )t o ( T ( L )€3 T ( L ) )[[h]]is also multiplicative. Notice also that, for arbitrary E T ( L ) ,by multiplicativity of E , characterisation of product integrals and counitality of E ,

<

(id T(L)@ E ) ( A ( < ) )

= 1 T ( L ) J 1 T ( L ) = <,

so that (idT(q €3 E ) A[h] = idT(L). Similarly ( E €3 idT(L)) A[h] = idT(L), so E is counital for the multiplicative map A[h].In the next section we shall consider conditions under which A[h]is coassociative. 6. The quantum Yang-Baxter equation.

For A[h] t o be coassociative we require

31 6

Here and in what follows maps which are formal power series are composed by convolution, so that, with A[h] = hNAN where each AN maps T ( L )to T ( L )€3 T ( L ) ,the coassociativity condition (9) becomes

c;=,

N

N

(Aj€3 idT(L)) AN-j = j=O

(idT(L) €3 Aj) AN-j j=O

as maps from T ( L )to T(L)€3 T ( C ) €3 T ( C ) . Since A[hl(J) = J[hl(A(t)) = ~ [ h I A ( W [ h l

n

-+t

for 6 E T ( C ) , with R[h]= have

(“I

+ dr[h]) and R’[h] = n

t-i

(1

(1

+ dr’[h]), we

€3 idT(L)) “KO

.’[W2

= “112 {(A €3 ~ ~ T ( L ) ~ ~ ~ ~ ~ l ~ ~ ~ ~ ~ ” ~ l l } = R[h]12{(A €3 i d ~ ( ~ ) ) R [ h l €3 ( AidqL))A(t)(A €3 i d ~ ( ~ ) ) R ’ [ hR’[h112 ]} = R[h]12{R[h]”R[h]”(A €3 id~(~))A(J)R’[hl~~R’[hl~~) R’[h]12 = {R[h]12R[h]13R[h]23} (A €3 idT(L))A(J) {R’[h]23R’[h]13R’[h]12}

using the quasitriangularity relations (5) and (6). A similar argument shows that (idT(L) €3 WI)A[hI(t) = { R[h]23R[h]13R[h]12}( i d q t ) @ A)A(t) { R’[h]12R’[h]13R‘[h]23} . Since A is coassociative, we see that a sufficient condition for A[hl _ _ to -++ (1 clr[h]) (or equivalently its inverse be coassociative is that R[h] =

n

R’[h]

+

=‘n‘(1 + dr’[h])) satisfy the quantum Yang-Baxter equation G-b

R[h]12R[h]13~[h123 = R[h]23R[h]13R[h] 12. The following theorem is proved in Ref. 18. Theorem 6.1. A necessay and suficient condition that the forward-++

n

+

backward directed double product integral R[h] = (1 dr[h]) satisfy the quantum Yang-Bazter equation in (‘T(C) €3 T ( L )€3 ‘T(C))[[h]] is that the generator r[h] satisfy the equation

r [h]1% [h]13 + r [h]1%[hi23 + 7- [h]% [hi23 + T [h]1%[h]13r[hi23

+

= r[h]13r[h]12 r[h]”r[h]’2

in (L 8 C @ L)[[h]].

+ r[h]”r[h]’3 + r[h]”r[h]’3r[h]’2

(10)

31 7 The condition (10) occurs in a different context in Ref. 2. In Ref. 19 it was claimed erroneously that (10) was necessary and sufficient for the symmetrised double product integral to satisfy the quantum Yang-Baxter equation. hNrN and comparing coefficients of powers Substituting r[h] = of h in (10) we find, for h2,

xgzl

,.12,.13 12 23 1 1 + T l r1

=

13 23

+ r , r1

that is, the lowest order coefficient equation [T:2,T:3~

13 12 23 12 p1 +r1 r1

r1

TI

23 13 r1 7

+r,

satisfies the classical Yang-Baxter

+ [T112,T1231 + [rl13,T1231 = 0.

For higher powers hn+l we find s+t=n+l

s+t+u=n+1

s+t=n+l

s+t+u=n+1

where the sums are over ordered pairs and triples of natural numbers. Isolating the terms involving T,, this becomes

+

c

s+t+u=n+l

23 13 12

(rs Tt ru

-

12 13 23

rs

rt

ru

)

Assuming r 1 , ~ 2... , rn-l have been found this gives an equation for T,,. In Ref. 19 it wm conjectured that the resulting hierarchy of inhomogeneous linear equations always possesses solutions for a given solution TI of the classical Yang-Baxter equation. In fact the general argument of Enriquez2 shows that the conjecture is true. However Enriquez’s argument depends on the existence of a quantisation as proved in general by Etingof and Kazhdan3 using the KnizhnikZamolodchikov associator. Thus we do not yet have an alternative to the existence proof of quantizations of Lie bialgebras of Etingof and Kazhdan. Assuming that ~ [ hsatisfying ] (10) has been found, let us compute the classical limit of the deformation Hopf algebra got by equipping the algebra

31 8

T(L)with the corresponding deformed coproduct A[h]. T h e Lie cobracket is given by (1). We have, using the multiplicativity of T ( ~ , ~ ) ,

A[h](L) - A[h]OPP(L) = R[~lA(L)R"hl- 7(2,1) (R[hlA(~)R"hl) = R[h]( L €3 1 18L ) R"h] - 7(2,1) (R[h]( L €3 1 1€3 L ) R"h])

+ + = R[h] ( L €3 1+ 1 8 L ) R"h] - 7(2,1)(R[h])( L€3 1 + 1 8 L ) 7(2,1)(Rf[h]). Now1' R[h]= 17(qe7(~)+r[h]+o(h2) and its inverse R'[h]= 1T(L)@T(L) ~ [ h ]o(h2), hence also

+

7(2,1)(R[hl)= 1T ( t ) @ T ( L+) 7(2,1)4hI+ 4 h 2 ) , 7(2,1)(R"hI) = 1 T(L)@T(L) - 7-(2,1)T[h]+ o(h2>. Substituting in (l),we find t h a t

6(L) = [Tl

- 7(2,1)T1,

( L €3 1

+ 1 8L ) ] .

Thus we have a coboundary Lie algebra4 determined by the skew-symmetric part of the tensor T I . Thus commutator Lie bialgebras of this type can be quantised by t h e method of double product integrals.

References 1. R Bacher, private communication. 2. B Enriquez, Quantisation of Lie bialgebras and shuffle algebras of Lie algebras, Selecta Math. (N.S) 7, 321-407 (2001). 3. P Etingof and D Kazhdan, Quantization of Lie Bialgebras I, Selecta Math. 2, 1-41 (1996), 11, ibid 4, 213-231 (1998), 111, ibid 4, 233-269 (1998). 4. P Etingof and 0 SchifFman, Lectures on quantum groups, International Press (1998). 5. A S Holevo, Time-ordered exponentials in quantum stochastic calculus, pp 175-182 in Quantum Probability VII, ed L Accardi et al, World Scientific 1992. 6. A S Holevo, Exponential formulas in quantum stochastic calculus, Proc Royal Society of Edinburgh 126A,375-389 (1996). 7. A S Holevo, An analogue of the It8 decomposition for multiplicative processes with values in a Lie group, Sankya A 53, 158-161 (1991). 8. R L Hudson, Algebraic stochastic differential equations and a Fubini theorem for symmetrised quantum stochastic double product integrals, pp 75-87 in Quantum Information 111, ed T Hida et all World Scientific (2001). 9. R L Hudson, Calculus in enveloping algebras, Jour. London Math. SOC(2) 65, 361-380 (20020

31 9 10. R L Hudson, P D F Ion and K R Parthasarathy, Timeorthogonal unitary dilations and noncommutative Feynman-Kac formulas, Commun. Math. Phys. 83, 261-280 (1982). 11. R L Hudson and K R Parthasarathy, Quantum Ito formula and stochastic calculus, Commun. Math. Phys. 93, 301-323 (1984). 12. R L Hudson and K R Parthasarathy, Construction of quantum diffusions, pp173-198 in Quantum Probability I, ed L Accardi et all Springer LNM 1055 (1984). 13. R L Hudson and K R Parthasarathy, The Casimir chaos map for U ( N ) ,Tatra Mountains Math. Jour. 3, 1-9 (1994). 14. R L Hudson, K R Parthasarathy and S Pulmannovd, The method of formal power series inquantum stochastic calculus, IDAQP 3, 387-401 (2000). 15. R L Hudson and S Pulmannovd, Symmetrized double quantum stochastic product integrals, J Mathematical Phys 41,8249-8262 (2000). 16. R L Hudson and S Pulmannovd, Chaotic expansions of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, Proc. London Math. SOC.(3) 77, 462-480 (1998). 17. R L Hudson and S PulmannovB, Double product integrals and Enriquez quantisation of Lie bialgebras I, to appear in J Mathematical Phys. 18. R L Hudson and S Pulmannovd, Double product integrals and Enriquez quantisation of Lie bialgebras 11, Nottingham Trent preprint, submitted to Lett. Mathematical Phys. (2003). 19. R L Hudson and S Pulmannovb, Explicit universal solutions of the quantum Yang-Baxter equation constructed as double product integrals, pp 289-296 in ICMP X I I I (2000) Proceedings, ed A Grigoryan et all International Press (2002) 20. K R Parthasarathy, An introduction to quantum stochastic calculus, Birkhauser (1992). SCHOOL OF COMPUTING AND

UNIVERSITY, BURTONSTREET, BRITAIN .

MATHEMATICS, NOTTINCHAM TRENT NOTTINGHAM NG1 4BU, GREAT

ON NONCOMMUTATIVE INDEPENDENCE *

ROMUALD LENCZEWSKI Institute of Mathematics Wroctaw University of Technology Wybrzeie Wyspiariskiego 27 50-370 Wroctaw, Poland E-mail: 1enczewOim.pwr.wroc.pl

We examine the main notions of noncommutative independence, namely tensor, free, boolean and monotone independence. We collect the results on unification of these notions from the point of view of reducing them to tensor independence. We also show how to reduce A-freeness to tensor independence and demonstrate that in a similar way one can construct analogous mixtures of tensor independence and boolean independence as well as tensor independence and monotone independence.

1. Introduction

In noncommutative probability there is no single notion of independence. In addition to the so-called tensor independence, which has similar features as classical independence, there are other, more noncommutative notions of independence like, for instance, freeness, boolean independence and monotone independence (or, equivalent to it, anti-monotone independence) each associated with some new type of probability theory. The axiomatic theory, see Refs. 1-2, singles out these notions of independence as such which are associated with the so-called 'natural' products of states satisfying a given set of axioms. On the other hand, it has been shown in Ref. 3 that there exists a discrete interpolation between boolean independence and freeness. If (dl, q 5 1 ) l ) l E ~ is a family of noncommutative probability spaces, then the hierarchy of freeness is a sequence of noncommutative probability spaces (A("),d m ) ) m -> l , such that

*This work is supported by KBN grant No 2P03A00723 and by the EU Network QPApplications Contract No. HPRN-CT-2002-00279.

320

321 in the sense of convergence of mixed moments, where j(") :UlELdl

+ A(")

are suitable *-homomorphisms. Here, U I E L d l denotes the free product of d l ' s without identification of units and * I ~ L + Iis the free product of 41's. (units are identified in the limit). Thus, the states dm) o j ( " ) , called mfree products, approximate the free product of states in the weak sense. What is important, the whole sequence (A("), dm))m>l - is embedded in a noncommutative probability space (.&$) of the form

IEL

IEL

and therefore the construction reduces the free products of states to the tensor products of states in the weak sense. Moreover, the first-order approximation m = 1 gives the boolean product of states. Next, F'ranz observed in Ref. 4 that a construction similar to that for the boolean product of states can be done to recover the monotone product of states. Finally, in Ref. 5 , it was shown that the free product of states can be reduced to a tensor product of states in the strong sense. We have constructed a suitable 'closure' of the tensor product of unital *-algebras 31 by adding certain 'affiliated' operators called monotone closed operators. Then we can embed free random variables represented by infinite series of simple tensors in a noncommutative probability space (A, $), where now

IEL

[EL

with ?3 denoting the 'closed' tensor product called monotone tensor product, where the procedure of taking the closure is similar to that in the von Neumann algebra tensor product. These results lead to the following theorem. Theorem 1.1. Let ( d l , 4 1 ) be ~ noncommutative ~~ probability spaces. For every 'natural' product of states 0 1 ~ ~ 4there 1 , exist noncommutative probability spaces ( d l , $ ~ ) L E Land a (in general, non-unital) *-homomorphic embedding A

h

j : UIELdl

+A

(1.3) h

such that J o j = 0 1 ~ ~ and 4 1 $ o j l d ~= 41 for every 1 E L, where d and are given by ( I . 2).

A

322 In other words, every ‘natural’ product is equivalent to a restriction of j ( U I E L d l ) , which means that every ‘natural’ product of states can be reduced to a tensor product of states. In this context, it would be interesting to determine whether a similar treatment is possible for a larger class of models, especially those related to the Gaussianization on the interacting Fock spaces studied in Ref. 7 by Accardi and Boiejko. For each of the ‘natural’ products we show in this paper how to explicitly realize this reduction. Thus we collect the results of Refs. 3-5, where we refer the reader for details. We also show that the A-free product of states introduced in Ref. 6 can be reduced to a tensor product of states. Finally, we introduce new ‘mixed’ products, which may be called A-boolean and A-monotone. If it suffices to use algebraic tensor products, we use the notation of (l.l),whereas if we need the monotone tensor product, we use the notation of (1.2). In all theorems we refer to the setting of Theorem 1.1 and, for notational simplicity, we assume that L = N.

2to the *-subalgebra

2. Boolean Independence

Assume that ( d i ) i E L is a family of unital *-algebras and ($l)lE= are states on these algebras.

Definition 2.1. By the boolean product of states state on U i E L d l given by the recursion

3

where XI, E d l ( k ) and Z(1) # Z(2) # ... variables come from different algebras.

$1

we understand the

# Z(n), thus the neighboring

2

In fact, one can show that the functional obtained in this way is a state, i.e. is positive and normalized. For the proof of positivity in the more general case of the conditionally free product of states, see Ref. 8. The mathematical roots of the boolean product of states come from the regular free product of functions on discrete groups of Boiejko given in Ref. 9. The basic tool which enables us to reduce the boolean product of states to the tensor product is that of the boolean extension (see Ref. 3 ) of a state $ on a unital *-algebra A. Namely, let’s extend d freely by a single projection P to get

323

and then extend $ to a state

$ on x by the recursive formulas

&wPv) = &J)IJ(v), & P ) = 1

7

for any v ,w E Then is called the boolean extension of 4 and 3 is called the boolean extension of A. In other words, is the boolean product of $ and the state h on C [ P ]given by the linear extension of h ( 1 ) = h ( P ) = 1. Note that the projection P plays the role of the separator of words from A. Namely

2.

6

&P*X1PX2P.. . PXnPP) = $ ( X l ) $ ( X 2 ) .. . $ ( X n ) where a,/3 E ( 0 , l ) and X I , . . . ,X , E A. In the easiest case of two noncommutative probability spaces, ( A , $ ) and (f?,$), it is not hard to see that the mapping

j:AUB+x@# where

x = A * @[PIand # = 23 * @[P’],given by j ( X )= x

@ P’,

j(Y)= P

€3Y

where X E A, Y E B,is a *-homomorphism such that ($8$) o j agrees with the boolean product of $ and $. It is also instructive to observe that one can associate a *-bialgebra structure with a unital free *-algebra A and produce boolean independent copies of random variables from A by iterating the coproduct. For instance, if A = @[XI,the coproduct is given by

A ( X )= X @ P + P @ X where the projection P is group-like, i.e. A(P) = P @ P. Succesive iterations of the coproduct give a finite number of boolean independent random variables. Namely, the sequence

-, ,

X i = P @ . . @ P @ X @ P @ . .@ P 1 < 1 < N , N-1 times

1-1 times

is a sequence of boolean independent random variables w.r.t. the state $’N. The general case is now straightforward. To conform with the statement =2 1 and 81 = 51. For simplicty we of Theorem 1.1, it is enough to put take L = N. For details, see Ref. 3.

A^1

Theorem 2.1. Let j : U n E N d n + BnEN An be the *-homomorphism given by

-

A

j ( X ) = P @ .. . @ P @ X @ P 8.. . n-1 times

324

where X E An. Then

3

0

j agrees with the boolean product of states

&.

Proof. It is an easy consequence of the definition of the boolean extension of a state. For instance, in the case of the boolean product of two states, $1 and I&, we have ( J o j ) ( x Y ..) . = &(XI? ..)&( PY...) = d l ( X )$1

(P.. .) $2 (Y.. .)

=+l(X)(&Jj)(Y...) for any X E A1 and Y E A2, which conforms with Definition 2.1.

0

3. Monotone Independence

In this Section we show how to realize the statement of Theorem 1.1 for monotone independence in the sense of Muraki and Lu, see Refs. 10-11.

Definition 3.1. By the monotone product of states the state on UIELAl given by the recursion

3

( ~ I ) I ~ weL understand

&xlx2.--xn) = ~ l ( k ) ( X k ) 3 ( X l . . . X k - l X k +... l

xn)

(3.1)

whenever Z(k) is a local maximum in the tuple (Z(l), Z(2), . . .,Z(n)), i.e. l k - 1 < l k and Ik+l < l k for Some 1 5 k 5 72, where xk E d l ( k ) and we adopt the convention that if k = 1 or k = n, then only one of these inequalities holds. Note that our definition is equivalent to that given by Muraki in Ref. 10. In a similar way we define monotone independent subalgebras of a given algebra. It was observed in Ref. 4 that one can reproduce monotone independent random variables by taking the coproduct which is ‘half-boolean’ and ‘halfclassical’. Namely, if we take the coproduct on A given by

A(X)= X @ P + I @ X

--

where X E A, with P being group-like, then the variables obtained by iterating this coproduct, namely

X n = l @... @ l @ X @ P @... @ P , 1 L n S N n-1 times

N-1 times

are monotone independent w.r.t. the state J B N . It is then straightforward to extend this result to any linearly ordered index set L. For notational

325 A

simplicity, we take L = N below. As in the boolean case, we set A

4n =

A,

2,

=

-

4n.

+ BnEN A, be the *-homomorphism given A

-

Theorem 3.1. Let j : UnENdn

by

j ( X ) = 1 8 . ..@I 1 €3x€3 P €3.. . n-1 times

where X E A,. Then

o j agrees with the monotone product of states.

Proof. Suppose Z(k) is a local maximum, i.e. Z(k) Z(k + 1) < Z(k). Then

(60 j ) ( ...Xk-lXkXk+l

> Z(k + 1)

and

-

...) = ...l#ll(k)(... P X k P...)... = f$[(k)(Xk) x ...&)(... PP...)...

= 4 l ( k )( X k where Xl E Ai for Z = k - 1, k,k shown in the calculation.

)(6 j )(...Xk-l 0

Xk+l. ..I

+ 1 and only the most relevant &(k)

is 0

Let us give another realization of the monotone product of states which will turn out useful in Section 5 . For each unital *-algebra dl we create free products of copies dl(k)of Al, extend these free products by an increasing sequence of projections (q,), i.e. such that Qmqn = qmhn and q: = Qn (for simplicity, we use (qm) for each algebra dl) and then take

-& = k N - A ( k ) * c [ q i ,~ 2 , 4 3 7 .. -I/ J (we identify the empty word in the product Uk,Nd(k) with the unit of the algebra @[q1,q 2 , 4 3 , . . .I), where J is the two-sided ideal generated by commutation relations

q m X ( n ) = X ( n ) q m for m

> n,

with X ( n ) denoting the n-th copy of X E di. Let A b e then given by (l.l), where, for simplicity we assume L = N.

2, of the above form, let j *-homomorphism given by

Theorem 3.2. For

-

: UnENdn

j(X)=qn+l €3**.€3qn+1€3X(n)@qn+1€3.-n-1 times

+ A^

be the

326

where X E An. Then there exist states with the monotone product of states &.

&,

Proof. It is enough to construct states

on

i n such that 4o j

agrees

&, which satisfy the conditions h

w ( q m x ( n ) q m - 4n(X)qm)v E kerdn for m In

w - q1wqi E ker&, for any X E An and w,'u E

An, together with ?n(qm) = 1

for any m E N. Note that these conditions can be satisfied by the states lifted from the tensor product of Boolean extensions of &'s of the form

Jn=

lJ$"

oq

where q : i n 3 22" is the *-homomorphism given by

q ( x ( n ) )= I@(~-') 8 x 8 I@" e(n-1) p @ ~ qI(qn) = 1 for any X E A and n E N. This finishes the proof.

0

Example 3.1. We illustrate the theorem with a diagram. The vertical

1

2

3

4

5

6

7

Figure 1. A diagram for monotone independence.

axis corresponds to n and thick lines are associated with projections qn+l.

327 Solid arcs represent connections between elements from the same algebra which produce joint moments, whereas dotted arcs represent connections between elements from the same algebra which produce factorized moments since there is a projection between the elements which acts as a filter (for a general framework with projections acting as filters, see Ref. 12). The diagram in Figure 2 corresponds to monotone independent random variables for Z(1) = Z(3) = Z(6) = 2, Z(2) = Z(5) = 4 and Z(4) = Z(7) = 1. The corresponding moment is given by J o j ( x l . -.x7) = 91(x4x7)92(x1x3)92(x6)94(x2)44(x5)

since the connections 3 - 6 and 2 - 5 cannot be realized due to the fact that the projections associated with X, and X4, namely 43 and q2, respectively, make the moments on level 2 2 factorize. The same result is obtained by using Definition 3.1 and succesive elimination of singletons associated with local maxima. 4. F’reeness

A reduction of the free product of states of Avitzour and Voiculescu, see Refs. 13-14, to a tensor product of states (or, equivalently, a reduction of freeness to tensor independence) is more complicated. In Ref. 3 we have done it in the weak sense of convergence of mixed moments by constructing the hierarchy of freeness and in Ref. 5 we have strenghtened this result as stated in Theorem 1.1. For limit theorems and the GNS construction, see Refs. 15-16. For notational simplicity we now describe the most important points of the construction of Ref. 5 for two algebras. Definition 4.1. By the free product of states state on * l E L d l given by the recursion

6

J(XIX2 *

a .

& a )

(41)lE~ we

= #l(l)(XI)J(X2* . . X,)

understand the

(4.1)

whenever XI E dl(l)and Xk E Alp) n ker$l(k) for 2 5 k 5 n with Z(1) Z(2) # .. . # Z(n).

#

A discrete interpolation between the boolean product of states and the free product of states is given by the hierarchy of m-free products of states given in Ref. 3, whose definition is given below. Note that for each m E N the m-free product of states has to be defined on the free product of dl’s without identification of units.

328

Definition 4.2. By the m-free product of states ( ~ ! J I )we ~ ~understand L the state $ on U I E L d l given by the recursion (4.1) under the assumption that x k E d l ( k ) for all k and x k E ker$l(k) for 2 5 k 5 m. This implies that mixed moments of the associated m-free random variables agree with the moments of free random variables if n _< 2m. In particular, if we take two noncommutative probability spaces ( d , 4) and (a,$), then the m-free random variables can be written as sums

k= 1 where X(k)’s and Y ( k ) ’ s are copies of X E d and Y E L? which are tensor independent w.r.t. some tensor product state $8 with 4, being extensions of 4 and $, and (Pk)’S, wk)’s are suitably defined orthogonal projections, some of which do not commute with X ( k ) ’ s and Y ( k ) ’ s . If m = 1, we obtain the boolean product of states. In turn, in the limit m + co,the mixed moments of the variables j(”)(X) and j(”)(Y) tend to the moments of free random variables. Thus the hierarchy of mfree products gives a discrete interpolation between the boolean product and the free product. Nevertheless, it is desirable to represent free random variables as series of the above forms, with m replaced by m. This requires us to introduce a suitable closure on a *-algebra level. We sketch below how this can be done. For details, see Ref. 5 . There are three essential steps in the construction. Step 1. For given unital *-algebras d and f3,we construct countable free products (without identification of units) of their copies, namely k=l

6,

%(A) = UkEd(k),

. . A

$J

%(a)= UkENa(k)

(we allow empty words which can be viewed as units) and then extend each such product freely by a sequence of increasing projections, namely %o(d) =%(A) *@[ Q I , ~ ~ , E I , - . . ] %iio(f3)

=GCi(a)*@[Q:,Q:,Q~,...]

where the empty words mentioned above are identified with the units of q q t , q 2 , 4 3 , . . .] and @[a:,qi, q$,. . .I, respectively. Note that we have qmqn = qm/\n and q; = qm (similar equations hold for the q;). Step 2. We want now the sequences (qm) and (qA) to play the role of approximate units in our products. For this purpose, we form quotients

3to(d) = Go(d)/I,3to(L?)

Go(B)/J’

329

where I and I’ are two-sided ideals generated by *-relations of the form

X ( k ) = q,X(k), k

< rn and Y ( k ) = q L Y ( k ) , k < m

respectively, where X E A and Y E a. In other words, for given k-th copies of A and f3 there exist in Xo(A) and Xo(f3) large enough projections which act as units on the given copies of A and f3, respectively, and all ‘earlier’ copies. Step 3. Finally, we construct the ’closures’ of Xo(A) and X o ( f 3 )w.r.t. the sequences (am) and (4;) and denote them by %(A)and X ( D ) , respectively. Namely, %(A) is the unital *-algebra of equivalence classes

[X,, em], where Xnem = Xme, and X;e, = X A e , for m < n where (em) = (f&(m)) is a subsequence of (qm) such that ( k ( m ) )t co (we construct “(a) in a similar way). We can say that the closures are taken with respect (4,) and (&), respectively. Moreover, sequences (X,) and (Y,) come from increasing sequences of *-subalgebras

X m E U k < m A ( k ) * c[41,4 2 , 4 3 7 . * . ] / I

These equivalence classes are called monotone closed operators (for details of this construction, modelled on the algebra of closed operators affiliated with *-algebras, see Ref. 5). What is more important, this concept of closure allows us to consider infinite series on the level of tensor products, except that the usual tensor product of *-algebras has to be replaced by the monotone tensor product. We refer the reader to Ref. 5 for details. Let us only mention here that the definition of 3 is similar to the von Neumann algebra tensor product and the closure is taken w.r.t. the sequence (qm 8 4;). Thus, we can define a unital *-homomorphism j :A * B

+ X(A)SX(B)

by the formulas

where X ( k ) ’ s and Y ( k ) ’ sdenote copies of X E A and Y E f3 in A(k) and B ( k ) , respectively and (&), 03:) are sequences of orthogonal projections given by pk = qk - Q k - 1 , pi = 4; - qLV1 with Qo = 4; = 0.

330

Theorem 4.1. For given states 4 and $ on A and 23, respectively, there exist states and on %(A) and %(B) and a unital *-homomorphism

4

3

j : d * 23 + %(d)B%(B)

--- --

such that (4@$) o j agrees with the free product of states 4 * $. Proof. The homomorphism j is given by (4.3). The states be defined by the following conditions: w(qnX(n)qn

- 4(X)qn)v,

3 and 4can

w - Q l W l E ker3,

and

for every with

together

for every m E N. That such states exist can be shown by lifting tensor products of boolean extensions of 4 and $ to %(A)and %(B),respectively, of the type used in the proof of Theorem 3.2. For details, see Ref. 5. This result can be generalized to an arbitrary family of unital *-algebras.

Theorem 4.2. For given states $1 on unital *-algebras dl there exist states & on %(Al) and a unital *-homomorphism j : *ladl + %%(Ai) 1€L

such that the state

3o j

agrees with the free product of states

*zE~c$i.

Proof. For notational simplicty we assume that L = N and that dl = A for every 2 E L. In that case we can use permutations 7r : XBm + given by m

k=l

M

k=l

and their extensions to the monotone closures. Denote by ~ 1 the, trans~ position which exchanges 1 and n (if n = 1, we get the identity). Then the unital *-homomorphism j can be written as

331

where X E A, and

_ _

Pk

=qk@qk@.

..- qk-l@'qk-l%.

.

I

with q(0) = 0. Roughly speaking, this formula replaces Pk = Qk - q k - 1 and p i = q6 from Eq.(4.3) - now the 'free action' of a variable from a given algebra is extended on all algebras (by abuse of notation we use (qm) 0 at all tensor sites). A detailed proof can be found in Ref. 5 .

Remark. It is worth noting that in this formulation it is very natural to associate with each algebra a sequence of states. Since states are obtained by lifting the tensor product states Tyw from to 31(dl), one can lift the tensor products of boolean extensions of different states @ k > l & , k , where $l,k'S are for each E L posssibly different states on dl. In particular, if we take

xy"

&

z

$1

if k = 1

we obtain the so-called conditional (or, $-) independence of Ref. 8. The case of general sequences ( $ l , k ) k > l is a special case of the model of freeness for seqeuneces of states, see Ref-17. 5. Mixed types of independence

This section is motivated by the model of A-freeness of Mlotkowski, see Ref. 6, which is a mixture of tensor independence and freeness. In particular, we show that A-freeness can be reduced to tensor independence. In an analogous fashion we introduce mixtures of tensor independence and boolean independence which we call A- boolean independence, and a mixture of tensor independence and monotone independence - A-monotone independence. In view of the fact that the corresponding product functionals are restrictions of the tensor products states, we immediately conclude that these functionals are states. The idea of producing such mixtures is based on placing projections onto the cyclic vectors at tensor sites corresponding to appropriate copies of the algebra d and keeping the identity at the remaining sites. For simplicity assume that L = N and let A C N x N. We again form free products of copies of each algebra, except that we extend them by sequences of projections related to A. Let

A = U k c W d l ( k ) * C[Qi Q 2 , Q 3 , - - .]/JA 7

332 where (Qn) is a sequence of commuting projections and JA is the two-sided ideal generated by commutation relations

Q n X ( k ) = X(k)Qn for { k , n } E A where X ( k ) is the k-th copy of X E di. We can now construct a state on the free product of An’s, which may be called the A-boolean product of states since the algebras An and A, commute if {n,m } E A, and otherwise they are boolean independent. The associated notion of independence may be called ‘A-boolean independence’.

Theorem 5.1. Let j : UnENdn 3 2 be the unital *-homomorphism given by

-

. j ( X ) = Qn 8... @ Qn @ X ( n )@ Qn8.. n-1 times

where X E An. Then, there exist states &n o n Xn such that X E A, and Y E A, commute w.r.t. $ o j if { m , n } E A, and otherwise they are boolean independent. Proof. The states & are obtained by lifting tensor products of boolean extensions. We let

where

q ( X ( n ) ) = I@(~-’) @ x @ 1mrn q(Qn) 1

QD

pj @

QD

li

i$A(n)

jEA(n)

for any X E dl and A(n) = { m : { n , m } $ A}, where Pj = P coincide with the projection by which we extend the algebra An to get its boolean extension. Let

-

j ( X ) = Qn @ . .. @ Q n @ X ( n )@ Qn 8.. . n-1 times

j ( Y )= Qm @ .. . @

Qm

@Y(m) Q m @ . .

m-1 times

where X E A, and Y E Am and m # n. By definition, if { n , m } E A, then j ( X ) and j ( Y ) commute. Otherwise, j ( X ) and j ( Y ) are boolean independent w.r.t. the state since

8

QnY(m)Qn- +m(y)Qn E ker&, if m E A(n)

333

QmX(n)Qm - 4n(X)Qm E ker4n if n E A(m)

+.

which follows directly from the definition of Thus, if {n,m} $ A, they are boolean independent w.r.t. the state This completes the proof. 0

4.

Example 5.1. Let L = { 1,2,3} and suppose A consist of one pair only: { 1,3}. Instead of infinite tensor product it is enough to take = Al €4 2 2 € 4 & a n d & = & @ ~ & @ &L~e t X ~ d l , Y ~ d 2 a n d Z ~ Note d3. that if we take

A

j ( x ) = x(1)8 QA 8 QA j ( y ) = QB 8 y(2) 8 qB j ( Z ) = QC 8 QC €4 2(3)

where A = {2}, B = {1,3}, C = {2}, then we get qAz(3) = z(3)QA QCX(1) = X(1)QC which implies that j(X) and j ( 2 ) commute. Moreover, we can identify QBX(1)QB= 4 1 W Q B qAx(2)qA = 42(x)qA QBZ(3)QB= 4 3 ( a n 3 QcX(2)qc = 42(X)QC in the weak sense, which implies that the pairs {j(X),j(Y)} and 0 {j(Y),j ( Z ) } are boolean independent w.r.t. the state

4.

In an analogous fashion one can construct a mixture of tensor independence and monotone independence. The state obtained in the theorem below may be called the A-monotone product of states since it has the following properties: if { n , m } E A, then d, and d,,, commute and otherwise they are monotone independent with respect to the product state. The associated notion of independence may be called A-monotone independence.

-

Theorem 5.2. Let j : U n E N d n

+ A be the *-homomorphism given by

j ( X ) = Qn € 4 . . . € 4

Qn

@X(n)8 Qn 8 . ..

n-1 times

where X E An. Then, there ezkt states & on An such that X E An and Y 6 dmcommute u.r.t. 0 j if { m ,n} E A, and otherwise, they are monotone independent.

4

334

Proof. The proof is similar to that of Theorem 5.1. The only change is that in the definition of q(Qn) we replace A(n) by A+(n) = (m > n : b m ) $! A). 0. Example 5.2. Consider the same setting as in Example 5.1, the only difference being that we require now j ( X ) and j ( 2 ) to commute, whereas the pairs ( j ( X ) , j ( Z ) )and ( j ( Y )j,( 2 ) )to be monotone independent (order is important). Note that if we take

j ( x )= x(1)€9 QA €9 QA j ( y )= QB €9 y(2)8 QB j ( Z ) = QC €9 qc c 32 ( 3 ) where A = { 2 ) , B = ( 3 ) and C = 0, then q c X ( 1 ) = X(1)qc qAz(3)

Z(3)QA

since 1 $! C and 3 $! A , hence j ( X ) and j ( 2 ) commute, and we can identify qAY(2)qA = 4 2 ( y ) q A

q B X ( 3 ) q B = 43(z)qB

in the weak sense, which implies that the pairs ( j ( X ) , j ( Y ) )and ( j ( Y ) , j ( Z )are ) monotone independent with respect to the state &. Let us finally reproduce the A-free product of states. For simplicity we assume that L = N. For a given subset A of the set of pairs from N,we can define

XA(dI) = UrnEMdl(m) *C[Qm;mE M]/IA where M = N x

N and IA is the two-sided ideal generated by relations Q n x ( m ) = x ( m ) Q n if { m l , n l ) E A

X ( m ) = Q n X ( m ) if { m l , n l } 4 A and m2

< 122

where n = (721,712) and m = (ml,m2). Then we can form the monotone closure of % A ( d , ) as in the free case and we denote it % A ( d n ) . We thus let i l = % A ( d , ) in ( 1 . 2 ) .

Theorem 5.3. For given states 4% on unital *-algebras An there exist on X(d1) and a unital *-homomorphism states

&,

j : *nENdn -+

&A(dn) nEN

335

such that the state

3=

3 0j , agrees with the A-free product of states 41 where

BnCNJn.

Proof. We define the states Jn

Jn

by

=(@

&,m)

ov

mEM

where

&,m

= & for every m and

The homomorphism j

Where

is defined by where

for

P(n,k)

= Q ( n , k ) G Q ( n , k ) G ** *

- Q(n,k-l)@Q(n,k-l)@.

*.

Note that this is a dilation of the representation of free random variables. Thus j agrees with the A-free product of states. The details are rather technical and are omitted.

30

It is worth pointing out that in our approach, positivity of the product states associated with A-boolean independence, A-monotone independence and A-freeness is an immediate consequence of the positivity of 4. A

References 1. A. B. Ghorbal, M. Schurmann, “Non-commutativenotions of stochastic independence” Math. Proc. Camb. Phil. SOC.133 (ZOOZ), 531-561. 2. N. Muraki, ‘‘The five independences as natural products”, Inf. Dim. Anal. Quant. Probab. Rel. Topics 6 (2003), 337-371. 3. R. Lenczewski, “Unification of independence in quantum probability”, Inf. Dim. Anal. Quant. Probab. Rel. Topics 1 (1998), 383-405. 4. U. fianz, “Unification of boolean, monotone, anti-monotone and tensor independence and LBvy processes”, Mat. Zeit. 243 (2003), 779-816. 5. R. Lenczewski, “Reduction of free independence to tensor independence”, Inf. Dim. Anal. Quant. Probab. Rel. Topics, to appear.

6. W. Mlotkowski, “A-free probability” , Inf. Dim. Anal. Quant. Probab. Rel. Topics 7 (2004), 1-15. 7. L. Accardi, M. Boiejko, “Interacting Fock spaces and Gaussianization of probability measures” , Centro Vito Volterra preprint No. 321 (1998). 8. M. Boiejko, R. Speicher, “$-independent and symmetrized white noises”, in Quantum Probability and Related Topics VI, Ed. L. Accardi, World Scientific, Singapore, 1991, 170-186. 9. M. Boiejko, “Uniformly bounded representations of free groups”, J. Reine Angew. Math. 377 (1987), 170-186. 10. N. Muraki, “Monotonic independence, monotonic central limit theorem and monotonic law of large numbers”,Inf. Dam. Anal. Quant. Probab. Rel. Topics 4 (2001), 39-58. 11. Y . G. Lu, “On the interacting Fock space and the deformed W i p e r law”, Nagoya Math. J . 145 (1997), 1-28. 12. R. Lenczewski, “Filtered random variables, bialgebras and convolutions”, J. Math. Phys. 42 (2001), 5876-5903. 13. D. Avitzour, “Free products of C*- algebras”, Trans. Amer. Math. SOC.271 (1982), 423-465. 14. D. Voiculescu, “Symmetries of some reduced free product C*-algebras”, in Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer, Berlin, 1985, 556-588. 15. U. Franz, R. Lenczewski, “Limit theorems for the hierarchy of freeness”, Prob. Math. Stat. 19 (1999),23-41. 16. U. Franz, R. Lenczewski, M. Schiirmann, “The GNS construction for the hierarchy of freeness” , Preprint No. 9/98, Wroclaw University of Technology, 1998. 17. T. Cabanal-Duvillard, “Variation quantique sur l’independence: la a-independence” , preprint, 1993.

LEVY PROCESSES AND JACOB1 FIELDS

E. LYTVYNOV Department of Mathematics University of Wales Swansea Singleton Park Swansea SA2 8PP U.K . E-mail: e.lytvynov0swansea.ac.uk

We review the recent results on the Jacobi field of a (real-valued) LBvy process defined on a Riemannian manifold. In the case where the LQvy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock s p x e . This representation is inspired by a result by Accardi, Franz, and Skeide’ .

1. Introduction

This paper is devoted to study of the Jacobi field of a real-valued LCvy process defined on a Riemannian manifold X. We recall that a LCvy process in this case is defined as a generalized stochastic process with independent values in the space V’-the dual of the space V of all smooth, compactly supported functions on X (cf. Ref. 16, see also Ref. 31). The notion of a Jacobi field in the Fock space first appeared in the works by Berezansky and Koshrnanenko7$*,devoted to the axiomatic quantum field theory, and then was further developed by Bruning (e.g. Ref. 14). These works, however, did not contain any relations with probability measures. A detailed study of general commutative Jacobi fields in the Fock space was carried out in a serious of works by Berezansky, see e.g. Refs. 3, 4 and the references therein. We start with recalling, in Section 2, the classical results on the Jacobi matrix and its spectral measure, which is a probability measure on R As examples, we discuss the Jacobi matrices which correspond to the orthogonal polynomials of Meixner’s type27. In Section 3, we discuss the chaotic decomposition for Gaussian and

337

338

Poisson process and corresponding Jacobi fields, which act in the Fock space. We recall that the construction of the unitary isomorphism between the Gaussian, respectively Poisson space and the Fock space through the multiple stochastic integrals is essentially due to It618i19(one also has to add the names of Wiener and Segal in the Gaussian case). The Jacobi field of the Gaussian measure is the classical free-field in the quantum field theory, and the Jacobi field of the Poisson measure was independently discovered by Hudson and Parthasarathyl’ and Surgailis30, though these authors did not use the term Jacobi field. In this paper, we present a generalization of the results of Refs. 9, 10, 21, by introducing a parameter X E R connecting the Gaussian case (A = 1) and the Poisson case X = 1 (see also Refs. 25, 26). In Section 4, we review the recent results on the Jacobi field of a general LCvy process on X, Refs. 23, 11. Now, the corresponding Jacobi field acts in the so-called extended Fock space, which indeed extends the usual Fock space in a natural way. In Section 5, we study the special case of LCvy processes of Meixmer’s type, i.e., gamma, Pascal, and Meixner processes, Refs. 22, 20, see also Ref. 5. We characterize these process as those LCvy process which respect the set of finite, smooth vectors in the extended Fock space. We also show that the Jacobi field of such a process has a much simpler form than in the general case. Finally, in Section 6, we again consider the general case of a LCvy process. We first recall the unitary isomorphism between the L2-space of the L6vy process on X and the L2-space of a Poisson random process on R x X. Using this isomorphism, we construct a unitary equivalent representation of the Jacobi field of a LCvy process in the usual Fock space over L2(R x X,v 8 o),where Y is the LCvy measure of the process and o is its intensity. This representation is inspired by a result by Accardi, F’ranz, and Skeidel . The corresponding creation, neutral, and annihilation operators now have a much simpler representation than in the extended Fock space. However, a drawback of this representation is that the n-particle subspaces of the usual Fock space do not correspond to the orthogonal chaoses of the LBvy process.

339 2. Jacobi Matrix and its Spectral Measure

Let us consider the Hilbert space with e,=(O

,...,0,

l2

spanned by the orthonormal basis 1

v

,&(I...).

n-th place

(Jn,m)cm=O

An infinite matrix J = is called a Jacobi matrix if Jn,n=:an E R for n E Z+, Jn,n-l = Jn.-l,n=:bn > 0 for n E N, and = 0 for In - ml > 1. Thus, the matrix J is symmetric and has non-zero elements only on the three central diagonals. We denote by l 2 , o the dense subset of l2 consisting of all finite vectors, i.e.,

Z+such that

3N

t!2,0:={(f(n))r=0 : E

f(") = 0 for all n

2N}.

Each Jacobi matrix J determines a linear symmetric operator in domain e2,O by the following formula:

J e , = bn+len+l

+ anen + bnenPl,

L2

n E Z+, ePl:=O.

with (2.1)

We denote by J the closure of J , which evidently exists since the operator J is symmetric. Under some appropriate condition on the behavior of the coefficients a,, b, at infinity, the operator J can be shown to be self-adjoint (see e.g. Ref. 2 for details). We have2:

Theorem 2.1. Assume the operator is self-adjoint. Then, there edsts a unique probability measure p on (R,B(R)) (B(R) denoting the Bore1 B algebra on R) and a unique unitary operator I : e,

+L2($p)

such that Ieo = 1 and, under I , the operator 3 goes over into the operator of multiplication by the variable, i.e.,

(~J~-lf)= ( zxf(x), )

f

E I(Dom(J)) =

{g

E

L 2 ( R , p ) : L x 2 g ( x ) 2p(&)

<

m) .

The measure p in Theorem 2.1 is called the spectral measure of the Jacobi matrix J . Since eo E Dom(Jn) for each n E N,the measure p has all moments finite, i.e.,

340 Thus, we can implement the procedure of orthogonalization of monomials P ,n E Z+, to obtain a sequence Q n ( x ) , n E Z+, of normalized orthogonal polynomials. As easily seen, I e , = Q,, n E Z+, and by virtue of (2.1), we

get the following recursion formula for the polynomials ( Q , ( X ) ) ~ = ~ :

xQn(z)= bn+lQn+l(x) + anQn(x) + bnQn-l(x), TI.

E Z+, Q-i(x):=O, QO(x) = 1.

(2.2)

We will also deal with orthogonal polynomials with leading coefficient 1. Such a polynomial of order n will be denoted by : x " : ~It . is easy to see :z,:~is nothing but the projection of the function Z" in the Hilbert space A2(& p) onto its subspace consisting of all polynomials of order up to n. Since the leading coefficient of Q n ( z ) , n E N,is equal to bm)-l, we get from (2.2) the following recursion relation for

(nz=,

(:~":~)r=~:

2 . : 2 y p= : p + l : p

n E Z+,

+ a,, :z,:~ + b:

:x,-':~,

: Z - ~ : ~ : = O:,z':~= 1.

Let us now consider the inverse problem. Let p be an arbitrary probability measure on ($ B(R)) having all moments finite and such that the set of all polynomials is dense in L 2 ( & p ) . Additionally, we suppose that the support of p has an infinite number of points. Via the orthogonalization procedure, one may always construct a sequence of normalized orthogonal ~, will satisfy the recursion relation (2.2) with polynomials ( Q , ( Z ) ) ~ =which some a , E R and b, > 0. We have:2 Theorem 2.2. Let p be a probability measure on (&B(Iw)) as described above. Then, p is the spectral measure of the Jacobi matrix J having the elements a, on the main diagonal and the elements b, on the 08-diagonals. (In particular, the corresponding closed operator j is self-adjoint.)

Let us consider some examples of Jacobi matrices and their respective spectral measures. Example 1. Let a, = 0, b, = f i . Then, the spectral measure p is standard Gaussian: p(&)

= (27r-'I2 exp (-x2/2) dx.

It has the Fourier transform F

ezzyp(dz) = exp JR

(-y2/2)

,

yER

34 1

The orthogonal polynomials ( : x ~ : ~ ) : are = ~ Hermite polynomials having the generating function G p ( t ,x):=

O0

n=O

tn

- :x'Yp= exp(ta: - t 2 / 2 ) ,

t ,x E R.

n!

Example 2. Let a, = An with X > 0 and b, = (centered) Poisson measure of the form

fi.Then,

p is the

00

p ( h ) = exp(-1/X2)

C X-2n(l/n!)6 x n - l / x ( h ) . n=O

It has the Fourier transform

The orthogonal polynomials (:x~:~);=~ are Charlier polynomials having the generating function

c

Gp(t,x):=

O0

n=O

tn

-:

n!

x ~=:exp ~ (zA-l log(1

+ A t ) + X-2(log(l + A t ) - A t ) )

for x E Iw and t from a neighborhood of zero in R. Example 3. Let a, = 2n and b, = d m a (centered) gamma measure:

) ,H > 0. Then, p is

It has the Fourier transform e-iyx

Y E R, IYI < 1. (1- iy)" The corresponding orthogonal polynomials (:x~:~):=~ are Laguerre polynomials having the generating function eixyp(dx) =

tn C~ ((x+l)t/(t+l)), n! : x ~=: (l+t)-"exp O0

Gp(t,x):=

a: E R, t

> -1.

n=O

d m ) ,

H

Example 4. Let a, = An, A 2 0 , X # 2, and bn = > 0. Then, for A > 2, p is a (centered) Pascal distribution:

342

+

- + k - l),k E N, and for 0 5 X < 2,

where ( x ) ~ : = l ,(x)k:=x(x 1 ) . ( x p is a (centered) Meixner distribution:

x exp ( - (z

+ X/2)2(4 - X2)-1/2

arctan (X(4 - X2)-ll2)) dx.

Define a,P E C through the equation 1

+ Xz + z2 = (1 - a z ) ( l -

PZ).

(2.3)

Then, in both cases, the Fourier transform of the measure p is given by the following formula, which holds for y from a neighborhood of zero in R

The corresponding orthogonal polynomials (:z~:~);?~are the Meixner polynomials of the first kind for X > 2 and the Meixner polynomials of the second kind, or the Meixner-Pollaczek polynomials, for 0 5 X < 2. In both cases, the generating function has the following form for t and z from a neighborhood of zero in I k O0

G,(t, z):= n=O

tn -:

n!

x ~=: ~

Examples 1-4 essentially form the complete solution to the following problem, which was formulated and solved by M e i ~ n e in r ~1934. ~ Suppose that functions f(z) and q ( z ) can be expanded in a formal power series of z E C and suppose that f(0) = 1, q ( 0 ) = 0, and W(0) = 1. Then, the equation 03

G(t,z):= exp(z!P(t))f(t) = n=O

pn(z) t n -

n!

generates a system of polynomials P,(z),n E Z+, with leading coefficient 1. (These polynomials are now called Sheffer polynomials.) Find all polynomials of such type which are orthogonal with respect to some probability measure p on R The complete solution to this problem is given by Examples 14,as well as by the measures and respective orthogonal polynomials that are given by linear transformations of B,i.e., by transformations of the form z e uz b,

+

U,bE$

af0.

343 3. Chaos Expansion for Gaussian and Poisson Process We proceed to consider the infinite-dimensional case. Let X be a complete, connected, oriented C” (non-compact) Riemannian manifold and let B ( X ) be the Bore1 a-algebra on X. Let a be a Radon measure on (X, B(X))that is non-atomic, i.e., a ( { x ) ) = 0 for every x E X and non-degenerate, i.e., a ( 0 ) > 0 for any open set 0 c X. Note that .(A) < 00 for each A E O,(X)-the set of all open sets in X with compact closure. We denote by D the space C r ( X ) of all real-valued infinite differentiable functions on X with compact support. This space may be naturally endowed with a topology of a nuclear space, see e.g. Ref. 13 for the case X = Rd and e.g. Ref. 15 for the c s e of a general Riemannian manifold. We recall that

D = proj lim 3tT. T ET

(3.1)

Here, T denotes the set of all pairs ( q , 7 - 2 ) with 7 1 E Z+ and 7-2 E C w ( X ) , 7 - 2 ( 5 ) 2 1 for all x E X, and 3tT = is the Sobolev space on X of ‘ ,l denoted order 7-1 weighted by the function 72,i.e., the scalar product in E by (., . ) T , is given by

where V idenotes the i-th (covariant) gradient, and dx is the volume measure on X. For 7 , ~ E ’ T, we will write 7’ 2 7- if 7-i 2 7 1 and r;(z) 2 72(x) for all z E X. The space D is densely and continuously embedded into L 2 ( X ,Q). As easily seen, there always exists 7-0 E T such that X, is continuously embedded into L2(X,a). We denote T’:={T E T : 7- 2 7-0) and (3.1) holds with T replaced by TI.Let us just write T instead of T’. Let 3t-, denote the dual space of 3tT with respect to the zero space 31:=L2(X,a). Then

D’= ind lim 3t-, TET

is the dual of D with respect to 3t,and we thus get the standard triple

D‘ 3 3t 3 D. The dual pairing between any w E 2)‘ and t E 2) will be denoted by ( w , [ ) . We denote by C(D’) the cylinder a-algebra on D1.

344

For each n E N,we denote by F(n)(L2(X, a))the n-th symmetric tensor power of L 2 ( X , a ) .That is,

P) (L2 ( X ,a)):=L2(X,a)

&I,

where

63 stands for symmetric tensor product. Next, we set FW(D):=Db,

F.(n)(2)1):=2)+

Let also F(0)( L 2 ( X ,a))= F(0)(D) = F(0)(D'):=R

The space 00

F ( L 2 ( X ,a)):=@ F(n)(L2(X, a))n! n=O

is called the symmetric Fock space over L 2 ( X , a ) . Thus, F(L2(X,a)) is the Hilbert space whose elements are of the form (f(")),"&, f(") E F(")(L2(X, a)),with Dc)

=

II(f(n))~=oII:(L2(x,,))

C IIf(n)II:cn,(L2(x,u))n! <

00.

n=O

Let Ffin(D) denote the subset of F ( L 2 ( X a)) , consisting of all sequences such that f(") E F(")(D), n = 0,. . . ,m, and m E Z+. As easily seen, Ffin(D) is dense in F ( L 2 ( X a)). , For each E D,let a + ( t ) denote the standard creation operator on

<

F f i n (2)) : U+([)f'"'

= (63f'"),

f'"'

E F ( q D ) ,n E

z+.

The adjoint operator of a+(() in .F(L2(X,o)) restricted to Ffin(D)is the standard annihilation operator a - ( [ ) given by

f'"'

E

F(qD).

We also define on Ffin(D) the neutral operator a O ( t ) t, E 27,as follows: (a0([)f("')(.1,.

where

( 0 ) "

. . ,.n)

= n([(zl)f(n)(21,. . * ,a%))-,

denotes symmetrization of a function.

f'"'

EF(qD),

345 '

Now, we fix a parameter X E [0, 00) and define operators ax(<):=a+(<)

+ Xu0(() + u - ( t ) ,

t E 2).

These operators are symmetric and let ii~(t), 6 E 73, denote the closure of a ~ ( < ) Furthermore, . the operators fix(<), t E D,may be shown to be self-adjoint.21Furthermore, since the operators ax (t)have a three-diagonal form with respect to the orthogonal structure of the Fock space, (ii,([))tE~ can be thought of as a Jacobi field in F ( L 2 ( X0, ) ) . Theorem 3.1. For each X 2 0 , there exists a unique probability measure p~ o n (D',C(D')) and a unique unitary operator

44, : W 2 ( X , d )+ L2(D',PA) such that Ip,O = 1, where fl:=(l,O,O, ...) is the vacuum vector in E D,under I p A ,the operator i i ~ (goes t ) over i.e., into the operator of multiplication by (.,t),

.F(L2(X,o)), and for each t

( I P A w ) I ; ; m J ) = ( w ,t ) F ( t ) ,

{

F E I p X (Dom(Cx(t))) = G E L2(D',PA):

L,

(w, t)2G(w)2 PA(&)

< m} .

For X = 0, PO is the standard Gaussian white noise measure, having the Fourier transform:

For X > 0, is the (centered) Poisson white noise measure having the Fourier transform

Let P(n)denote the set of all continuous polynomials o n D' of order up to n, n E Z+, i.e., functions o n 'D' of the f o r m

Let

and

denote the closure of

in

Let

346

For f(") E F(")(V)denote by :(w@", f(n)):pA the projection of the monomial (w@", f'")) onto Then, px-almost everywhere we have:

'J3k).

:(w@n,f(n)):pA = ( : W @ ? p A , f'"'),

f'"' E F(")(V),

where f o r each w E V', : w @ " : ~E~ F(")(V')i s given by the recurrence relation :W@(n+l). .PA

- :W@(n+l). . p a (213 -

-n(:w

N

* *

.,z n + ~ = ) (:wBn:pA (21 . . .,zn)w(zn+1)) 9

@(n-1)..px(21,...,~n-1)~(zn+1 -zn))-

- Xn(:w@n:pA(zl,. . . ,zn)6(z,+1 - x , ) ) ~ , :w@o:pA= 1, :w@l:pA= w . Arthermore, we have, for any

f(")

n E N,

E F(")(V):

( I p A f ( " ) ) ( w ) = ( : w @ " : ~f(")) ~,

px-a.e.

The generating function of the orthogonal polynomials is given by

cp E

v,w E a',

f o r X = 0, and b y

GpA(9,w):=

c

O01 -i n. ( : w @ " : ~ ~ ,

n=O

for X > 0 where, in the latter formula, w E V' and cp is from a neighborhood of zero in V which depends o n w . The proof of this theorem in the case X = 0 , l may be found in Ref. 21, see also Refs. 6, 9, 10. The case X > 0, X # 1, may be derived quite analogously (compare also with Refs. 25, 26). For each f(") E F(")(L2(X,o)), we now define ( : w @ " : ~f'")) ~ , as the element of L2(VD', PA)given by IpA f ("1. One then easily sees that, for each A E O,(X), we have = P " ( ( : W : ~ ~px-a.e., ~ A ) ) where P,(.) is a Hermite, respectively Charlier polynomial on IR of n-th order. We also note that the constructed unitary isomorphism between the can be derived by using multiple Wiener-It6 Fock space and L2(VD',p,)

(:~@":~~,ly)

347

stochastic integrals, see Ref. 21 for details. More exactly, we have, for each f'"' E F(n'(L2(X, 0)):

f'"') =

(:W@'n:pA,

L.

where the random measure

f'"'(z1,. .. ,z,)X ( d z 1 ) . . . X ( d z n ) ,

X on X is defined by

A E O,(X). X ( A ) = X(A)(U):=(:~:~~,~A), We also note that choosing the parameter X to be strictly negative would really mean the transformation of the space D' of the form w r-) -w, which is why we have omitted this choice. 4. Jacobi field of a LQvy process

Our next aim is t o find the Jacobi field of a general LCvy process, or a LCvy white noise measure. So, let us first fix the class of LCvy processes we are going to deal with. Let R:=R \ (0). We endow R with the relative topology of R and let B ( R ) denote the Bore1 a-algebra on R. Let v be a Radon measure on (R, B(R)), whose support contains an infinite number of points. Let

D (ds):=s2v( ds).

(4.1)

We suppose that D is a finite measure on (R,B(R)), and furthermore, there exists E > 0 such that L e Y p (+I)

fi(ds) <

(4.2)

By (4.2), the Laplace transform of the measure D is well defined in a neighborhood of zero and may be extended to a n analytic function on ( z E @. : 121 < E } . Therefore, the measure P has all moments finite, and moreover, the set of all polynomials is dense in L2(R,P). We now define a centered L6vy process as a generalized process on 2)' whose law is the probability measure pu,,, on (D', C(D')) given by its Fourier transform

cp €2).

(4.3)

The existence of pu,,, follows from the Bochner-Minlos theorem. By (4.3), v is the LCvy measure of the process and 0 is its intensity.

348 In what follows, without loss of generality we can suppose that B is a probability measure on R. (Indeed, if this is not the case, define Y ' : = c - ~ Y and a':=ca, where c:=fi(R).) By using (4.2), (4.3), and Sec. 11 of Ref. 29, we easily conclude that the set of all continuous polynomials on 2)' is dense in L2(D', py,),. Therefore, analogously to Theorem 3.1, we can set Pk10 to be the closure of P(n)in L2(D',pv,,) and let

We evidently get the orthogonal decomposition m

n=O

Next, for f(") E F(")(D), we set : ( u B n ,f(n)):pu, to be the orthogonal projection of the monomial (uBn,f'")) onto It is straightforward that, for a fixed n E Z+, the set of all such projections is dense in For any f("),g(") E F ( n ) ( D ) we , define a scalar product

!j32!

d .

1 :=z s,,:(uBLn,

yg;-.

f(n)):pY,O:(~~'n,g(n)):pu,O Pv,u(dw).

(4.5)

The sense of the notation FLi, .(L2(X, a)) will become clear later on, but now we note that, if on the right hand side of (4.5) we had the Gaussian or Poisson measure instead of the LQvywhite noise measure pv,,, , then this expression would just be equal to the scalar product of f(") and g(n) in F(")(L2(X, a ) ) . (D) We define the Hilbert space FLi, .(L2(X, a)) as completion of F(") with respect to the norm generated by the scalar product (4.5) (as will be shown below, the F2i,v(L2(X,a))-norm of any f(") E F(n)(D), f(") # 0, is positive). Let also

n=O

By construction, Ffin(D) is a dense subset of the Hilbert space FExt,v(L2(X,a))* By virtue of (4.4)-(4.6), we get a unitary isomorphism I p u , O : FExt,v(L2(x,a))

L2(D',pv,u)

349

by setting

IpJW=

f'"'

:(w@'n,f(n)):/+,

E.F(qD),

and then extending Ipu, by linearity and continuity. Our next aim is to explicitly identify the scalar product in .Fg;,"(L2(X,o)). To this end, denote by Zy,othe set of all sequences a of the form

,...,a n , O , O ,...), a i ~ Z + n, E N . IaJ:=CEl ai. For any f(") E .F(,)(D), n E N,and any a E Zy,osuch a=(al,a2

Let that

la1

we define a function D,f(,)

+ 2 a 2 + 3a3 + . . . = n, on XIaI by

--

(&f(%%. .., q a l ) :=f(%,.. .,~a1,~,,+1,~a1+1,~,1+2,2,1+2,. . .72a,+a2,2a,+a2, 2 times

2 times

~a,+a2+1,~,l+a2+1,,~al+a2+1,~~

*

2 times

(4.7)

3 times

We have (cf. Ref. 23): Theorem 4.1. Let b,, n E N, be the elements of the off-diagonals of the Jacobi matrix J whose spectral measure is fi (see Theorem 2.2). For any f("),g(") E .F(n)(D), we have:

where

Though the preceding theorem gives a complete answer to the problem of explicit determination of the scalar product of two elements from .F(")(D) in the . F ~ ~ , u ( L 2 ( X , o ) ) - s c aproduct, lar we have not yet identified which elements belong to 3E*, ,,(L2(x, u)) after completing it from F(,)(DD).

350

To this end, we define, for each a! E

ZTo,the Hilbert space

L2,(Xlal,a@lQl):=L2(X, a)bal

@

P ( X ,a)&az @ . . . .

Define a mapping

@

up): F(")(D)+

Lt(X'a1,a @ l q K ay,

aEZy,o:la1+2az+...=n

by setting, for each f ( " ) ( D ) ,the L i ( X l a l , a@lal)-coordinateof UP'f'")to be Daf(") (see (4.7)). By virtue of Theorem 4.1, Up)may be extended by continuity to an isometric mapping of " ( L ~ ( xa)) , into

FL~, a3

L2,(XlOl, a@lal)K,,".

U E Z ~ ,la1+2ua+...=n ~ :

Furthermore, we have (cf. Refs. 12, 23): Theorem 4.2. The mapping

up): Fki, " ( L 2 ( X ,a))+

@

L2,(Xlal,a@l~l)Ka, y

aEZy,o: la1+2az+...=n

is "onto, " and hence UP)is a unitary opertator. In what follows, having the unitary isomorphism Up)in mind, we will identify FrA, . ( L 2 ( X , a))with the space

@

L2,(X1al,a@'l"l)Ka,,. aEZ~,0:la1+2a!2+~~~=n Taking (4.6) into account, we will, therefore, identify F

@ L2,(XJaJ,a~JuJ)K~,"(la!l + 2a2 +

E~ " (, L 2 ( Xa)) , with *.

.)!.

aEZ7,O

Furthermore, for any f(") E

+

-

F& .(L2(X, a))and for any a! E Zy,osuch

that la1 2a2 + . . = n, we will denote by coordinate of f'"). We note that for a = (n,O,O,. . .), we have

fp)the L2,(Xlal,a@l*1)-

L2,(Xlal,o@'"')Ka, y = F(n) ( L 2 ( X ,o)), and hence we may think of F & i , y ( L 2 ( X , a ) ) as an extension of F ( n ) ( L 2 ( Xa)), , and of F E ~ " (~L 2, ( Xa)) , as an extension of ,T(L2(X,a)).

351

Having constructed the orthogonal expansion of the space L2(D', pv,o) given by the unitary operator Ipu,uand Theorems 4.1, 4.2, we can now ask ourselves: What is the explicit structure of the Jacobi field of the measure pv,o? More exactly, for any ( E D,denote by M ( ( ) the operator of multiplication by () in L2(D',pv,u): (a,

( M ( O F ) ( w )= kJJ,t)F(w),

{

F E Dom(M(0) = G E L 2 P ' , p v l b ):

1 V'

(w, ()2G(w)2p v , 6 ( h )< m}

,

and let

~(():=~~u~uM(()Ipu,u.

<

Evidently, Ffin(D) c Dom(J(()) for each E D. Furthermore, one can show that the restriction of .?(() to Ffin(D),denoted by J ( ( ) , is an essentially self-adjoint operator, and hence j(() is indeed the closure of J(E) (cf. Ref. 11). So, what is the explicit action of the operators J ( & ( E V? We have the following t h e ~ r e m . ~ ~ > l l Theorem 4.3. For each ( E V ,we have

J(0= J+ (0+ JO (0+ J - (0. Here, J+(() is the zlsual creation operator:

J+(()f'"' = (hf'"',

Neb,f o r each f(")

f'"' E 3(")(D). (4.9) E 3(n)(V), f)(~)f(~) E FLi,v(L2(X,o)) and ( J 0 ( 0 f ( " ) ) a ( ~ l *,

*

,2101)

00

=

,. . . ,21al))

a k a k - l s a (((Zai+...+ab)(Daf(n))(21 k=l

[email protected]., a E Z T ~ ,la1 + 2a2 -+ ... = n,

(4.10)

J - ( ( ) f ( " ) = 0 if n = 0, J - ( ( ) f ( " ) E FF
, ,q a l ) * * *

= nsa( SxE(I)(D.+llf'"')(2,~l,.. .,.Ial)4dZ)) f

ak-lb~-lSa(E(zal+...+ab)(Da-lb

-l+lbf(n))(21,' ' . , 2 1 ~ 1 ) )

k12

[email protected]., a E zq0, lal + 2a2 + ... = n - 1.

(4.11)

352

I n formulas (4.10) and (4.11), we denoted by S, the orthogonal projection ofL2(Xl,l,o@l,1)onto L ~ ( x ~ ~ ~ , ~ @ ~ ~ ~ ) ,

,...,c y , , - l , ~ ~ ~ , , f l , c y , , + ~ ,...),

cyfl,,:=(al

CYEZ?,, n E N ,

and by ak, bk the respective elements of the Jacobi matrix J whose spectral measure is D.

+

Let us also note that, for each cy E Zg, such that lal 2az+ . . . = n, n E N, and for each f?) E Lt(XIaI,alal),the random variable Ipu,mf?) is a multiple stochastic integral constructed with respect to the so-called orthogonalized centered power jump p r o c e ~ s . ~ ~ ~ ~ ~ 5. Process of Meixner's type

Comparing the results of Sections 3 and 4, we see that the Jacobi field of a Gaussian, or Poisson process respects the set Ffin(D),while in the case of a Lkvy process this is, in general, not true. Still, we can ask ourselves whether there are any Lkvy processes whose Jacobi fields do respect Ffin(D). In the latter case, we can hope for a simplified formula for the action of the Jacobi field on .Ffin(D). In fact, we have the following t h e ~ r e m . ~ ~ ~ ~ ~ ~ ~ ~ Theorem 5.1. Suppose that, for each [ E D,

J([)Ffin(D)C Ffin(D). Then, the elements an, bn of the Jacobi matrix J whose spectral measure is D have the form: a,, = X(n

+ l ) , bn = S

J m .

(5.1)

Here, X E IR and H > 0 are arbitrarily chosen parameters. firthemnore, we have in this case, for each f(") E F ( n ) ( D ) ,n E Z+:

( J O ( [ ) ~ ( " ) ) ( Z ~.., ., Z n ) = ~ n ( ~ ( ~ l ) f (. .~,x), ,() ~) ~1, ,aBn-a.e., . so that J o ( [ ) = Xuo([), where a o ( [ ) is the usual neutral operator, and

Thus,

353

where a;(() is the standard annihilation operator, and (az(()f(”))(xl,...,z,-l)

=n(n-

l ) ( ~ ( ~ l ) f ( n ) ( z l , x,x~,...,x,l,~2 1))

N

is a n annihilation operator of a new type, which is connected with the nonL2-scalar product. Let us consider in detail the case of a LBvy measure as in Theorem 5.1, i.e., a measure pv,u for which fi is the spectral measure of the Jacobi matrix having elements (5.1) on the central diagonals. By Examples 3, 4 in Section 2, we see that D is a Pascal measure if X > 2, a gamma measure if X = 2 and a Meixner measure if 0 5 X < 2. In what follows, for simplicity of notations we will suppose that the parameter 6 is equal to 1 and X 2 0. Let us denote by K , the K,,” constant given by formula (4.8) in our case (notice that this constant is, indeed, independent of the choice of the parameter A), and let px denote the pv,u measure corresponding to the parameter A. We will also skip u in the notation F E ~ ” (,L 2 ( Xa)). , We see that the constant K , has the following form:

Let us give a combinatoric interpretation of this number. Under a loop K. connecting points X I ,. . . ,x m , m 2 2, we understand a class of ordered sets ( ~ ~ ( ~.1. ,,x. ~ ( ~ where ) ) , 7r is a permutation of ( 1 , . ..,m}, which coincide up to a cyclic permutation. Let us also interpret a set {x} as a “one-point” loop K., i.e., a loop that comes out of x. Let 0, = { K . ~ ., . ., K . I B , ( } be a collection of l0,l loops ~ . that j connect points from the set {xl,.. . ,z,} so that every point xi E ($1,. . .,z,} goes into one loop ~ . = j K . ~ ( ~from ) 6,. Then, for a E ZT0, la1 2a2 ... = n, K , is the number of all different collections of loops connecting points from the set {zl,. . . ,z,} and containing a1 one-point loops, a2 two-point loops, etc. We have (cf. Ref. 22, see also Ref. 20):

+

+

Theorem 5.2. For each X 2 0, px is a unique probability measure o n (D’,C(D’)) and 1 p :~ F E x t ( L 2 ( X , g ) )

L2(D’,pA)

a unique unitary operator such that IpxR = 1, where R = (1,0,0,. . .) is the ‘uacuum vector in F E x t ( L 2 ( X , a ) ) , and for each E V , under I p x the operator jx(() goes over into the operator of multiplication by (.,() in

<

354

L2(V',px).Here, jx(() is defined as the closure of the operator J(<) given as in Theorem 5.1 and corresponding to the parameter A. The Fourier transform of the measure @A is given, in a neighborhood of zero in D,by the following formula:

f o r X = 2 and by

f o r X # 2, where the parameters a,P E C are defined by (2.3). Thus, for X = 2, px is a gamma white noise measure, f o r X > 2, px is a Pascal white noise measure, and for 0 5 X < 2, px is a M e h e r white noise measure. In particular, for any A E O,(X), the random variable :(-,l A ) : p x has a gamma, respectively Pascal, respectively Meixner distribution corresponding t o the parameter X and IC = a(A) (see Examples 3,4 of Section 2). Furthermore, px-almost everywhere we have: :(w@'n,f(n)):px= (:w@*:px, f'"'),

f'"' E F ' q D ) ,

where f o r each w E 0, : w @ " : ~E~ F(")(V') is given by the recurrence relation :,@(n+l). 'PA

=

:w@(n+l) -

(:W@'n:px( 2 1

- n (:U@(n-l). .PA(%. - n(n - 1)( : U @ ( n - l ).PA(% -

*

: p x (217 . .

. ,%+1)

,. . . ,zn)w(2,+1)) * *

12n-l)q&a+1

-GL))N

* . ,z"-l)q% - 2n-1)+,+1

-

- X n ( : w @ n ) : p x ( x l ., .. ,xn)d(zn+l - z ~ ) ) ~ ,n E N, :w@o:px= 1, : w @ l : p x= w. The generating function of the orthogonal polynomials is given by

f o r X = 2, and by

355

for X # 2 . Formulas (5.2), (5.3) hold for each w E V’ and for cp from a neighborhood of zero i n V which depends on w . For each f(”) E FLi(L2(X,o)), we now define ( : w @ ? ~ ~f‘”)) , as the element of L2(D’,px)given by IpAf‘”). For each A E O,(X ) consider lf” as the element of Fgi(L2(X,o)) defined as the Fgi(L2(X,a))-limit of a sequence {cppn}iE~ in F(,,)(V)such that cpi(z) + l ~ ( 2as ) i + 00 for each z E X , the functions cpi are uniformly bounded, and the supports of pi are uniformly bounded (it can be easily shown that this limit is independent of the choice of a sequence { c p i } i E ~ in D). One then easily sees that ( : w @ ~~ : ~p~”= ,) P , , ( ( : W : ~ ~ ,I A ) ) ,

px-a.e.,

where P,, is a Laguerre, respectively Meixner polynomial on JR of n-th order. 6. The usual Fock space representation of the Jacobi field of a LQvy process

As we saw in Section 4, the Jacobi field of a general LQvyprocess has a quite complicated form in the extended Fock space. In this section, we will give a unitary equivalent description of the Jacobi field realized in a usual Fock space. Though the n-particle subspaces of the latter Fock space do the not correspond to the orthogonal chaoses (i.e., to the subspaces corresponding realization of creation, neutral, and annihilation operators will have a much simpler form. We first need to recall the Poisson space realization of a LQvyprocess, 6. Refs. 19, 23. Let r R x x denote the configuration space over R x X defined as follows:

f@c!e),

raxx:={y c R x

x : #(y n (1st 2 E ) x A) < 00 for each E

> 0 and A E O,(X)}.

Here, g(A) denotes the cardinality of a set A. Each y E r R x x may be identified with the positive Radon measure

where d(s,z) denotes the Dirac measure with mass at (s,z),

c

(s,z)EB

S(B,Z):=zero measure,

356 and M + ( R x X ) denotes the set of all positive Radon measures on the Bore1 a-algebra B(R x X ) . We endow the space r R x X with the relative topology as a subset of the space M + ( R x X ) with the vague topology. Let r I T ,denote ~ a the Poisson measure on (rRxx,B(r,.x)) with intensity Y 8 a. This measure can be characterized by its Fourier transform

J,,,,e i ( r J )

(eif(siz)- 1) v(ds) o(dx)],

r v B u (dy)= exp

K X X

f E Co(R x X). Since the Poisson measure rvBu possesses the chaotic decomposition property, we have the unitary operator

F(L~(R x X , Y @ u)) 3 = (gn);& 05

c) I

g : = C I(”)(gn)E L2(raxx,ryao).(6.1) n=O

On the other hand, we have the following p r o p ~ s i t i o n . ~ ~ Proposition 6.1. W e m a y define a unitary operator

u : L2(rRxX,r”B316) + L2(D’,LL”,U) by setting ~1 = I ,

u ( z ( ” ( c ~ ~ ) . . . =z ((*~, ~)1()c. . ~. ( ~. , ()~)n ) , cpi, . .. V n E V , n E N,

and then extending this mapping by linearity and continuity t o the whole L2(J?Rxx,rvBu).Here, f o r cp E V , Z(l)(cp) denotes the first oder Wiener-It6 stochastic integral of the function f (5, x) = scp(x) in the space L2(rRxX,rvBu), i.e., z(’) (p) = ~ ( l(sp(x)). ) By Proposition 6.1 and (6.1), we get the following unitary operator:

U(l):=UI : F(L2(R x

x,Y @ a)) + L2(V’,p”,cr).

Next, by (4.1), the operator

L2(R,fi) 3 f(s) c) ( u ( 2 ) f ) ( S ) : = f ( S ) S E L2(R,Y ) is unitary, Furthermore, the set of all of all polynomials is dense in L2(R,fi) = L2($ fi). Therefore, by Theorem 2.2, there exists a unique Jacobi matrix J defining a self-adjoint operator J in C2 such that fi is the spectral measure of the Jacobi matrix J. Let u(3): C2

+ L2(R,fi)

357 be the corresponding unitary operator, i.e., the operator under which j goes over into the operator of multiplication by the variable s and U(3)eo= 1. Hence,

+ P ( R ,v)

u(4):=u(2)u(3) : C2

is a unitary operator, (U(4)eo)(s)= s, and under U(4) the operator j goes over into the operator of multiplication by s. Since evidently

L2(R x

x ,v €3 a ) = L2(R,v) €3 L 2 ( X ,a ) ,

we get the unitary operator

u(5): c 2 €3 L y x , a ) 4 L2(R,v) €3 L2(X,a ) given bby

where id denotes the identity operator in L 2 ( X , a ) . We now naturally extend U ( 5 )to the unitary operator u(6): F(C2 €3

L (x,a ) ) + F(L2(R, v) €9 P ( X ,a ) ) .

Finally, setting

we get the unitary operator

u : F(C2 @ L 2 ( X , a )+) L2(D’,PV,b). Denote by 5 the linear subspace of F(C2 €9 L 2 ( X ,a ) ) that is the linear span of the vacuum vector and vectors of the form ((@cp)*n, where ( E &,o, cp E D,and n E N. The set 5 is evidently a dense subset of F(C2€3L2(X, a)). Denote by J+,3, J- the creation, neutral, and annihilation operators in C2.0 which form the Jacobi matrix J , i.e.,

358 where a,, b, are the corresponding elements of J (see Section 2). Now, for each cp,$ E V and E &,o, we set

<

(P)&3(t €3$I0, + n((J+t)€3 ((P$))&(t 8 AO((P)(t 8 $)@ln:=n((JOt) €3 ((P$))&(t €3 cp)@(,-l),

A+(cp)(t8 $Pn:=(e0

€3

cp)@(n-l)l

A-(cp)(t 8 $)*,:=n(E, eo)(cp,$)(t €4 cp)@(,-l)

+ n ( ( J - t )8 (cp$))&(t €3 cp)@(,-').

(6.2)

Thus,

A+(cp) = a+(eo 8 cp)

+ ao(J+ €3 cp),

AO(cp)= aO(JO8 cp), A-(cp) = a-(eo €3 cp)

+ ao(J- 8 cp),

where a+(-), a'(.), a - ( - ) are the creation, neutral, and annihilation o p erators in F(l2 8 L 2 ( X , o ) ) . (In fact, we should explain that under e.g. ao(Js 8 cp) we understand the differential second quantization of the operator J+ 8 cp in l 2 8 L 2 ( X ,(T),which, in turn, is the tensor product of the operator J+ in l 2 defined on l z , and ~ the operator of multiplication by cp in L 2 ( X ,(T) defines on 'D.) Note also that

A(cp):=A+(cp) + AO(cp)+ A-(cp) =a+(eo €3 cp) u'((J+ P J-) 8 cp) =a+(eo 8 cp) a ' ( ~€3 cp) a-(eo 8 cp).

+ +

+ + +

+ a-(eo 8 cp)

It is now trivial to see that, under the unitary isomorphism U ( 6 ) the , operator A(cp) goes over into the operator

+

a(scp(z))= a+(scp(z)>+ a O ( s W > a-(scp(z))

(6.3)

defined on the corresponding subset of F(L2(R, v) 8 L2(X, 0 ) ) .From here, using Theorem 3.1 and Proposition 6.1, one easily concludes the following proposition (compare with Sec. 4, in particular, Theorem 4.1 of Ref. 1). Proposition 6.2. The operators A(cp), cp E V , are essentially self-adjoint and, under the the unitary U,each A(cp),cp E V ,goes over into the operator of multiplication by cp) in the space L2('D',P " , ~ ) ,while Ueo 8 1 = 1. (e,

Let us compare the above result with the results of Section 4. We thus have the unitary operator U between the spaces F(lz 8 L2(X,o))

359

and L2(Vt,pu,,,) and the unitary operator IPU,-between the spaces . T E ~ ~ , ~ (0)) L ~and ( X L2(Dt,j+,). , Define the unitary operator U:=u-lIpv,e : FExt,v(L2(X,0)) F(& @ L 2 ( x ,0))We then have

UR = eo 8 1, and for each cp E D,the operator J(cp) acting in F E ~ , , , ( L ~ 0)) ( X ,goes over acting in F(t2 8 L 2 ( X , 0 ) ) .Furthermore, for each into the operator cp E I), we have the following decomposition of J(p)-the restriction of ,?(cp) to Ffin(D) (recall that the closure of J(cp) is indeed j(cp)):

a(cp)

J(cp) = J+(cp>

+ J O ( ( P ) + J-(cp)*

(6.4)

On the other hand, we have the decomposition

A(cp) = A+ ( 9 )+ A0 (cp)

+ A- (9)

(6.5)

defined 5. We would like to show that the decompositions (6.4) and (6.5) are unitary equivalent under U. However, it is easy to see that UTfin(D) # 5. Therefore, we take the closed operators J+(cp), ~ ( ( C P ?(cp), ), A+(cp), Ao(cp),A-(cp), which evidently exist since their adjoint operators are densely defined. Theorem 6.1. For each cp E D,under the u n i t a y U, the operators j+(cp), P (cp), .F(9) go over into the operators A+(cp), A" (9), 8- (p), respectively.

The proof of this theorem is based on formulas (4.9)-(4.11), (6.2) and analogous to the proof of Theorem 3.1 in Ref. 23 (see also Sec. 2 of Ref. 24). Acknowledgments

The author is grateful to Yu. Berezansky for numerous fruitful discussions during the years. The author would also like to thank L. Accardi and U. Franz for useful discussions connected with the last section of this work. References 1. L. Accardi. U. F'ranz and M . Skeide, Renormalized squares of white noise and other other non-Gaussian noises as L h y processes on real Lie algebras, Comm. Math. Phys. 228 (2002) 123-150. 2. Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators (Amer. Math. SOC.,1968).

360 3. Yu. M. Berezansky, Commutative Jacobi fields in Fock space, Integral Equations Operator Theory 30 (1998) 163-190. 4. Yu. M. Berezansky, Direct and inverse spectral problems for a Jacobi field, St. Petersburg Math. J. 9 (1998) 1053-1071. 5. Yu. M. Berezansky, Pascal measure on generalized functions and the corresponding generalized Meixner polynomials, Methods f i n c t . Anal. Topology 8 (ZOOZ), no. 1, 1-13. 6. Yu. M. Berezansky and Yu. G . Kondratiev, Spectral Methods in Inflnite Dimensional Analysis (Kluwer Acad. Publ., 1994). 7. Yu. M. Berezansky and V. D. Koshmanenko, A n asymptotic field theory in terms of operator Jacobian matrices, Soviet Physics Dokl. 14 (1969/1970) 1064-1066. 8. Yu. M. Berezansky and V. D. Koshmanenko, Axiomatic field theory an terms of operator Jacobi matrices, Teoret. Mat. Fiz. 8 (1971), 75-191 (Russian) 9. Yu. M. Berezansky, V. 0. Livinsky and E. W. Lytvynov, Spectral approach to white noise analysis, Ukrain. Math. J. 46 (1993) 177-197. 10. Yu. M. Berezansky, V. 0. Livinsky and E. W. Lytvynov, A generalization of Gaussian white noise analysis, Meth. f i n c t . Anal. and Topol. 1 (1995) 28-55. 11. Yu. M. Berezansky, E. Lytvynov and D. A. Mierzejewski, The Jacobi field of a L 6 y process, Ukrain. Math. J., to appear. 12. Yu. M. Berezansky and D. A. Mierzejewski, The structure of the extended symmetric Foek space, Methods Funct. Anal. Topology 6 (ZOOO), no. 4, 1-13. 13. Yu. M. Berezansky, Z. G. Sheftel, and G . F. Us, Functional analysis. Vol. 11, Operator Theory: Advances and Applications, Vol. 86 (Birkhauser Verlag, 1996). 14. E. Briining, When i s a field a Jacobi field? A characterization of states on tensor algebras, Publ. Res. Inst. Math. Sci. 22 (1986) 209-246. 15. J. DieudonnB, Treatise on Analysis, Vol. 3 (Academic Press, 1972). 16. I. M. Gel’fand and N. Ya. Vielenkin, Generalized Functions, Vol. 4. Applications of Harmonic Analysis (Academic Press, 1964). 17. R. L. Hudson and K . R. Parthasarathy, Quantum I t 6 3 formula and stochastic evolutions, Comm. Math, Phys. 93 (1984) 301-323. 18. K. It6, Multiple Wiener integrals, J. Math. SOC.Japan 3 (1951) 157-169. 19. K. It6, Spectral type of the shift transformation of differential processes with stationary increments, naris. Amer. Math. Sac. 81 (1956) 253-266. 20. Y. Kondrtatiev and E. Lytvynov, Operators of gamma white noise calculus, Infin. Damen. Anal. Quant. Prob. Rel. Top. 3 (2000) 303-335. 21. E. Lytvynov, Multiple Wiener integrals and non-Gaussian white noises: a Jacobi field approach, Meth. f i n c t . Anal. and Topol. 1 (1995) 61-85. 22. E. Lytvynov, Polynomials of Meixner’s type in infinite dimensions-Jacobi fields and orthogonality measures, J. f i n c t . Anal. 200 (2003) 118-149. 23. E. Lytvynov, Orthogonal decompositions for Ldvy processes with an application to the Gamma, Pascal, and Meixner processes, Infin. Dimen. Anal. Quant. Prob. Rel. Top. 6 (2003) 73-102. 24. E. Lytvynov, The square of white noise as a Jacobi field, Preprint, 2004.

361 25. E. Lytvynov, A. L. Rebenko and G. V. Schchepan’uk, W i c k theorems in non-Gaussian white noise calculus, Rep. Math. Phys. 39 (1997) 217-232. 26. E. Lytvynov and G . F. Us, Dual Appell systems in non-Gaussian white noise calculzLs, Meth. f i n c t . Anal. and Topol. 2 (1996) 70-85. 27. J. Meixner, Orthogonale Polynomsysteme mit einem besonderen Gestalt der erzeugenden f i n k t i o n , J. London Math. SOC.9 (1934) 6-13. 28. D. Nualart and W. Schoutens, Chaotic and predictable representations f o r LCvy processes, Stochastic Process. Appl. 90 (2000) 109-122. 29. A. V. Skorohod, Integration in Hilbert Space (Springer-Verlag, 1974). 30. D. Surgailis, O n multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984) 217-239. 31. N. Tsilevich, A. Vershik, and M. Yor, An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process, J . f i n c t . Anal. 185 (2001) 274-296.

k-DECOMPOSABILITY OF POSITIVE MAPS*

WADYSIAW A. MAJEWSKI Institute of Theoretical Physics and Astrophysics Gdadsk University Wita Stwosza 57 80-952 Gdadsk, Poland E-mail:fizwam0univ.gda.pl MARCIN MARCINIAK Institute of Mathematics Gdahsk University Wita Stwosza 57 80-952 Gdadsk, Poland E-mail: motrnmOuniv.gdo.pl

1. Introduction For any C*-algebra A let A+ denote the set of all positive elements in A. A state on a unital C*-algebra A is a linear functional w : A + C such that w ( a ) 2 0 for every a E A+ and w(1) = 1 where II is the unit of A. By S ( A ) we will denote the set of all states on A. For any Hilbert space H we denote by B ( H ) the set of all bounded linear operators on H . A linear map cp : A + B between C*-algebras is called positive if cp(A+) C B+. For k E N we consider a map (Pk : Mh(A) + Mk(B) where n/rk(A)and kfk(B)are the algebras of k x k matrices with coefficients from A and B respectively, and cpk([aij])= [ ~ ( a i j ) We ] . say that cp is k-positive if the map vk is positive. The map cp is said to be completely positive when it is k-positive for every k E N. A Jordan morphism between C*-algebras A and B is a linear map p : A + B which respects the Jordan structures of algebras A and B , i.e. p(ab ba) = p ( a ) p ( b ) p ( b ) p ( a ) for every a, b E A. Let us recall that every Jordan morphism is a positive map but it need not be a completely positive

+

+

'Supported by RTN HPRN-CT-2002-00279

362

363

one (in fact it need not even be 2-positive). It is commonly known21 that every Jordan morphism p : A + B ( H ) is a sum of a *-morphism and a *-antimorphism. The Stinespring theorem states that every completely positive map cp : A + D ( H ) has the form 'p(a) = W*r(a)W,where x is a *-representation of A on some Hilbert space K , and W is a bounded operator from H to K . Following S t ~ r m e we r ~ say ~ that a map cp : A + B ( H ) is decomposable if there are a Hilbert space K , a Jordan morphism p : A + B ( K ) ,and a bounded linear operator W from H to K such that cp(a) = W*p(a)Wfor every a 6 A. By B ( H ) 3 b I+ bt E B ( H ) we denote the transposition map (for details see Section 2). We say that a linear map cp : A + B ( H ) is k-copositive (resp. completely copositive) if the map a H ~ ( ais) k-positive ~ (resp. completely positive). The following theorem22 characterizes decomposable maps in the spirit of Stinespring's theorem:

Theorem 1.1. Let cp : A conditions are equivalent:

+ B ( H ) be a linear

map. T h e n the following

'p is decomposable; (2) f o r every natural number k and f o r every matrix [aij]E M k ( A )such that both [aij] and [uji] belong t o Mk(A)+ the m a t r h [cp(aij)]is in Mk(B(H))+; (3) there are maps c p 1 , c p ~: A + B ( H ) such that cpl is completely positive and 9 2 completely copositive, with cp = 'pl 9 2 .

(1)

+

The classification of decomposable maps is still not complete even in the case when A and H are finite dimensional, i.e. A = B ( P ) and H = Cn. The most important step was done by S t ~ r m e r Choi4l5 ~ ~ , and W o r o n o ~ i c z ~S~t.~ r m e rand Woronowicz proved that if m = n = 2 or m = 2 , n = 3 then every positive map is decomposable. The first examples of nondecomposable maps was given by Choi (in the case m = n = 3) and Woronowicz (in the case m = 2, n = 4). It seems that very general positive maps (so not of the CP class) and hence possibly non-decomposable ones, are crucial for an analysis of nontrivial quantum correlations, i.e. for an analysis of genuine quantum maps23i19,7,14,15,16. Having that motivation in mind in our last paperlo we presented a step toward a canonical prescription for the construction of decomposable and non-decomposable maps. Namely, we studied the notion of k-decomposability and proved an analog of Theorem 1.1. Moreover it turned out that it is possible to describe the

364

notion of k-decomposabilty in the dual picture. More precisely, the analog of Tomita-Takesaki construction for the transposition map on the algebra B ( H ) can be formulated (Section 2). Application of this scheme provides us with a new characterization of decomposability on the Hilbert space level (see Section 3). Thus, it can be said that we are using the equivalence of the Schrodinger and Heisenberg pictures in the sense of Kadison8, Connes6 and Alfsen, Shultz'. Section 4 provides a detailed exposition of two dimensional case and establishes the relation between Starmer construction of local decomposability and decomposability fo distinguished subsets of positive maps. 2. Tomita-Takesaki scheme for transposition

Let H be a finite dimensional (say n-dimensional) Hilbert space. Define w E B ( H ) ; , , as w(a) = Pea, where e is an invertible density matrix, i.e. the state w is a faithful one. Denote by (H,, T ,R) the GNS triple associated with ( B ( H ) ,w). Then, one can identify the Hilbert space H , with B ( H ) where the inner product (-,.) defined as (a,b) = P a * b for a,b E B ( H ) . With the above identification one has R = @'I2 and n ( a ) n = a0 for a E B ( H ) . In this setting one can simply express the modular conjugation Jm as the hermitian involution, i.e. J,,,ae1/2 = e1i2a*for a E B ( H ) . Similarly, the modular operator A is equal to the map p . e-'; Let {xi}i=~,...,~ be the orthonormal basis of H consisted of eigenvectors of e. Then we can define

for every f E H . The map Jc is a conjugation on H . So, we can define the transposition on B ( H ) as the map a H at 5 Jca*Jc where a E B ( H ) . By r we will denote the map induced on H , by the transposition, i.e.

where a E B ( H ) . Let Eij = lxi)(xjI for i, j = 1, . . . ,n. Obviously, { Eij} is an orthonormal basis in H,. Hence, similarly to (2.1) one can can define a conjugation J on H,

JaQ1l2 =

C

(Eij,aQ1l2)Eij

ij

where a E B ( H ) . We have the following

(2.3)

365

Proposition 2.1. Let a E B ( H ) and

5 E H,.

Then

at< = Ja*J(. Now, define the unitary operator U on H , by

u=

c

IEji)(EijI

ij

Clearly, UEij = Eji. The properties of U , introduced above conjugation J and modular conjugation J , is are described by

Proposition 2.2. One has: (1) U 2 = I a n d U = U * (2) J=UJ,; (3) J , J,,, and U mutually commute; (4) J commutes with the modular operator A .

The following theorem justifies using the term scheme” for transposition

”

Tomita-Takesaki

Theorem 2.1. If r is the map introduced in (2.2), then r = UA1/2.

Moreover one has the following properties: ( UA = A-IU; (1) 2) If a is the automorphism of B(H,) implemented by U , i.e. a(.) = UxU* for x E B(H,), then ct maps n ( B ( H ) ) onto its commutant r(B(H))’; (3) If Vp denotes the cone {Aflae1/2: a E B ( H ) + } C H , (cf. 2, for each ,f? E [0,1/2] then U maps Vp onto v 1 / 2 ) - p . I n particular, the is invariant with respect to U. natural cone P =

v/4

Corollary 2.1. UA112 maps Vo into itself. Summarizing, this section establishes a close relationship between the Tomita-Takesaki scheme and transposition. Moreover, we have the following :

Proposition 2.3. Let 5 H uc be the h o m e o m o r p h i ~ r nbetween ~ ~ ~ the natural cone P and the set of normal states on n ( B ( H ) )such that u d a ) = (51 4

1

a

E B(H).

366

c

For every state w define w‘(a) = w(at) where a E B ( H ) . If E P then the unique vector in P mapped into the state w i by the homeomorphism described above, is equal to Uc In the sequel, we will need the following construction: Suppose that we have a C*-algebra A equipped with a faithful state W A and consider the tensor product A@BB(H),where H is the same as above. Then W A @w is a faithful state on A@BB(H).So, we can perform GNS constructions for both (A,W A ) and ( A 8 B ( H ) ,W A 8 w ) and obtain representations ( H A ,T A , 0,) and (H,,x,,a,) respectively. We observe that we can make the following identifications: (1) H@ = H A 8 HT, (2) T@ = x A ‘8x3

(3) a @ = = A @ ’ R where (ITm, x,0) is the GNS triple described in the begining of this section. With these identifications we have J, = JA @ J, and A, = AA @ A where J,, JA, J, are modular conjugations and A,, AA, A are modular operators for (n,(A @ B(H))”,a,), (TA(A)”, a ~ )( x, ( B ( H ) ) ”a) , respectively. Since RA and R are separating vectors, we will write aaA and bR instead of T A ( ~ ) and ~ A n(b)R for a E A and b E B ( H ) . The natural ~ o n eP ’,~ ,for ~ (T,(A@B(H))“, 0,) is defined as the closure of the set

where &(.) = J, . J, is the modular morphism on x,(A 8 B ( H ) ) ” = TA(A)”@ x ( B ( H ) ) ” . Motivated by Proposition 2.3 we introduce the ”transposed cone” P& = (I8 U)P, where U is the unitary defined in (2.4). Elements of this cone are in 1-1 correspondence with the set of partial transpositions 4 o (id @ t ) for all states 4 on A 8 B ( H ) . It can be easily calculated that we have the following Theorem 2.2. The transposed cone

P& is the closure

of the set

367

where a i s the automorphism introduced in Theorem 2.1(2). Consequently, P&, = P& where P& is the natural cone f o r ( T A ( A )€3 7@v))',Q@).

3. k-decomposability at the Hilbert-space level

Let A be a unital C*-algebra, H be a Hilbert space and let cp : A 4B ( H ) be a linear map. We introduce" the notion of k-decomposability of the map cp was studied.

Definition 3.1. (1) We say that cp is k-decomposable if there are maps cp1,cp~ : A + B ( H ) such that 'pl is k-positive, cp2 is k-copositive and cp = cpi +cp2. (2) We say that cp is weakly k-decomposable if there is a C*-algebra E, a unital Jordan morphism p : A + E , and a positive map $J : E + B ( H ) such that $ J l p ( ~ ) is k-positive and cp = $J 0 p.

The connection between k-decomposability, weak k-decomposability and the Stprrmer conditionZ0 is the following

Theorem 3.1. For any linear m a p cp : A conditions:

+ B ( H ) consider

the following

(Dk) cp is k-decomposable; ( w k ) cp is weakly k-decomposable; ( S k ) for every matrix [aij] E Mk(A) such that both [aij] and [uji]are in Mk(A)+ the matrix [cp(aij)]i s positive in M k ( B ( H ) ) ; T h e n we have the following implications: (Dk)

+ ( w k ) # (sk).

The results of Section 2 strongly suggest that a more complete theory of k-decomposable maps may be obtained in Hilbert-space terms. To examine that question we will study the description of positivity in the dual approach to that given in in the above theorem, i.e. we will be concerned with the approach on the Hilbert space level. Let us resrict to the case cp : M +M where M C B ( H M )is a concrete von Neumann algebra with a cyclic and separating vector O M . When used, W M will denote the vector state W M ( . ) = ( O M , . ~ M ) The . natural cone (modular operator) associated with ( M ,O M ) will be denoted by PM (AM respectively). We assume that that cp satisfy Detailed Balance IT1*, i.e. there is a positive unital map cpfi such that w(a*cp(b)) = w(cpfi(a*)b)for a , b E M .

368

In this case cp induces a bounded operator Tv on HUM which commutes strongly with AM and satisfies T,*(PM)c P M . Let B ( V ) 3 a I+ at E B ( P ) denotes the usual transposition map on the algebra of n x n-matrices. Let u be a faithful state on B ( P ) and let P, denote the natural cone for ( M €3 B ( P ) , u M@ u). From Theorem 3.1 it follows that to develop the theory of decomposability on the Hilbert space level we should examine the action of the map I@T, on the transposed cone PA = (I @ U)Pndescribed in the previous section, where the operator U on B ( P )was introduced in the previous section (for some orthonormal basis { e i } of eigenvectors of Q U O ) . It can be deduced from Proposition 2.5.26 in the book of Bratteli and Robinson3 and the results of previous section that the cones Pn and PA have the following forms:

Pn = AA'4{[aij]nn

: [aij]E

Mn(M)+},

P: = An1'4 {[ajilnn : [aij]E Mn(M)+}. It turns out that the adaptation of Lemma 4.10 in the paper of Majewski12 leads to the following characterization of k-positivity and k-copositivity Lemma 3.1. The m a p cp : M + M is k-positive (k-copositive) i f and only if (Tv @ lI)*(P,) c Pn (respectively (Tv @ lI)*(Pn)C PA) f o r every n = 1, ...,k. By m(7) we denote the closed convex hall of the subset 7. Now, we are in position to give promised result.

Theorem 3.2. Consider the following two conditions o n cp: (1) cp is weakly k-decomposable; (2) (Tv €3 I)*(Pn)C m(Pn UP;) f o r every n = 1,.. .,k.

Then, in general, the property (2) implies (1). If, in addition, the cone Pn n P,' is equal to the closure of the set {Ak'4[aij]G : [aij],[aji] E Mn(M)+}

then (2) follows from ( I ) . In particular in the finite-dimensional case the two conditions are equivalent. Remark 3.1. It can be easily showed that W(PnU PA) and P, n P,' are dual cones. It is still an open question whether the equality

369 holds in general. 4. Application of local decomposability to low dimensional

cases

In this section we indicate how the discussed techniques relied on decomposition may be used for a characterization of linear positive unital maps cp : Mm(C) + Mn(C) in the case of low dimensions m and n. S t ~ r m e proved r ~ ~ that each positive map cp : Mm(C) + Mn(C)is locally decomposable, i.e. for every non-zero vector r] E cc" there exist a Hilbert space K,,, a linear map V, on K,, into cc",such that IlV,,(l 5 M for all r ] , and a Jordan *-homomorphism p,, of Mm(C) such that cp(a)ll= V,p,,(a)V,r]

(4.1)

for all u E Mm(C).For the readers convenience we remind the construction of S t ~ r m e rin details. Given a vector r] E cc", llr]ll = 1, we consider the state w,, on Mn(C) defined as %(a> = (r],cp(a)r]), a E Mn(C).

(4.2)

Let C = { a E Mn(C) : w,,(a*a) = 0) and R = { u E Mn(C) : w,,(aa*) = 0). Observe that C is a left ideal in Mn(C)while R is a right ideal. By Kl and K, we denote the quotient spaces M n ( C ) / L and M,(C)/R respectively. For any a E Mn(C)we write [all and [a], the abstract classes of a in Kl and K, respectively. Next, let K,, = K l @K, and define the scalar product ((.,.)) on K,,

For simplicity we will write [a]instead of [all @ [a], for a E Mn(C). By G we denote the subspace of K,, consisted of every such elements and by G' its orthogonal complement. Finally, V, and p, are given by p(a) ( [ h ] l @ [ b 2 ] r ) = [ a b l ]@ ~ [ b a ] r , a, b1, b2 E

V,k =

Mn(C);

if k = [u]for some a E Mn(C), if k E G'.

(4.4) (4-5)

The crucial point in our considerations is the characterization of face structure of the set of unital positive maps between matrix algebras which was done by Kyeg. Recall that if C is a convex set then a convex subset

370

c C is called a face if for any z,y E implication holds:

F

XZ

C and 0 < X < 1 the following

+ (1 - X)y E F +

Z,Y E

F.

Kye proved that any maximal face of the set of unital positive maps from Mm(C) into Mn(C) is of the following form

F€,q= {cp : cp(J0 = 1,cp(lO(t1h = 0)

(4.6)

for some [ E Cm and q E cc". In the sequel we describe the case m = n = 2. As we mentioned in the introduction, Stprrmer20proved that every positive map cp : M2(C) + M2(C) is (globaly!) decomposable. The next proposition indicates the relationship between this phenomenon and the notion of local decomposability for cp in a maximal face. We need the following notations. If [ and q are arbitrary unit vectors in C? then let be an orthonormal basis in C? such that 5 1 = t and similarly ql,q2 be a basis such that ql = q. By eij we denote the operator I&)(&l for i,j = 1,2.

Proposition 4.1. Suppose cp E Fc,,,. L e t K,,, V,, and p,, be as in (4.1). Then

d.1

= V,,P,(4V,*,

a E M2(0

(4.7)

if and only if Trcp(e12) = Trv(e21) = 0 ,

ncp(e22)

= 1,

(4.8)

Proof. From the definition (4.6) of Fc,,, it follows that the projection ell is an element of C and R. On the other hand both C, and R are proper ideals in M2(C) because cp is unital. Consequently L = M2(C)e11 and R = e11M2(C) and the Hilbert space K = M2(C)/M2 (C)ell @M2(C)/e11M2(C) is four dimensional. By direct computations it can be checked that the elements 1c1

= &[el21 = h [ e 1 2 ] ,

+ &[e121p

k2

= &[e211 = &[e21]l+

&[e2ilr

k3

= [ e ~=] [e22]1

Ic4

= [ e ~ ]-l [e22Ir

+ [e22]r

371

form an orthonormal basis in K . Moreover, from (4.5) and (4.6) we get the following equalities

v,ki = JZcp(e12)rli = h(17i7cp(e12)q1)171+ h(172,cp(ei2)rli)q2 = a712 where a = fi(q2,(p(eI2)q1). The last equality is due to the fact that e12 E ellM2(C) C kerw, (cf. (4.2)). Similarly we check that

Vqk2 = P172 where P = fi(q2,cp(ez1)q1). The definition (4.6) of Ft,, and the fact that cp is unital imply

V,k3 = cp(e22)ql = cp(1)ql- cp(e11)qi = 71. Finally, from the dafinition (4.5) of V, and from the fact that k4 is orthogonal t o G it follows that

V,k4 = 0. Hence, the matrix of the operator V, : K and {ql,q2} has the form vq=

+

in the bases {kl, k2, k3, k4)

0010 [d300]*

(4.10)

In order to prove sufficiency in the statement of the theorem, assume that cp fulfils conditions (4.8) and (4.9). We will show that q(a) = V,p,(a)V; for any a E M2(C). Observe that from local decomposability (4.1) we have

cp(a)171 = VqP,(a)V,'l.

So, it remains t o prove that da>ll2 = V,p,(a)V,*7?2

(4.11)

~ a system of matrix units in for every a E M2(C). As { ~ . j } i , j = I ,forms M2(C) it is enough to show (4.11) for a = eij where i,j = 1,2. Let a = e12. Following the assumption and the fact that el2 E e11M2(C) C kerw, we have

0 = Trcp(e12) = (171,cp(e12)171)+ (7?2,cP(e12)172)= (172,'P(e12)172).

Hence,

de12)712 = (717 cp(e12)7?2)7?1+ (172, (P(e12)7?2)172= (71,'P(e21)7?2)7?1 '

372

On the other hand, application of (4.10) and (4.4) yields

= 2V, ((771, v(e2i)r?2H(ei 2 )[ei 2 ] + <»7i,y>(ei2)»72)A,(ei 2 )[e 2 i]) [en]j + [e22]r) (fc3 - fc4) So, we get ¥3(612)772 = By simillar computations we check that Now, let a = e22 . We have 1 = Trv5(e22) = (e22)r72 = (771,^(622)772)771 + (T72,y(e22)772)772 = (^(622)771,^) = (771,%) = 0. Moreover, = 0.

The last equality follows from the fact that ei2 6 enM2(C) = K and 621 eM 2 (Qe n =£. Finally, let a = e\\,. Then

= 2 (1(772, ¥5(e 21 )r7i)| 2 + |(772, yp(e 12 )r?i)| 2 ) 7?2. As (f € F^t1J then Try>(en) = (771,^(611)771) + (772,^(611)772) = (772,^(611)772), hence ¥>(eu)7?2 = (7?i,v(eii)r7 2 )77i+(77 2 ,^(eii)77 2 )r7 2 = (7?2, y(eii)772}r72 Prom (4.9) we conclude that y(eii)7? 2 = Vr,pn(eu)V*r)2 and the proof of sufficiency is finished. It is easy to observe that in order to prove necessity one should repeat the same computations in the converse direction. D

373 References 1. E.M. Alfsen and F.W. Shultz, State spaces of operator algebras, Birkhauser, Boston, 2001. 2. H. Araki, Some properties of modular conjugation operator of a von Neumann algebra and a non-commutative Radon-Nikodym theorem with a chain rule, Pac. J. Math. 50 (1974),309-354. 3. 0.Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I : Second Edition, Springer-Verlag, New York, 1987. 4. M.-D. Choi, Completely positive maps on complex matrices, Lin. A@. Appl. 10 (1975),285-290. 5. M.-D. Choi, Positive semidefinite biquadratic forms, Lin. Alg. Appl. 12 (1975),95-100. 6. A. Connes, Charactkrisation des espaces vectoriels ordonnhs sous-jacents aux algkbres de von Neumann, Ann. Inst. Fourier, Grenoble 24 (1974),121-155 7. M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett A 223, 1-8 (1996) 8. R.V. Kadison, Transformations of states in operator theory and dynamics, Topology 3 (1965) 177-198 9. S.-H. Kye, Facial structures for the positive linear maps between matrix algebras, Canad. Math. Bull. 39(1) (1996),74-82. 10. L.E. Labuschagne, W. A. Majewski and M. Marciniak, On k-decomposability of positive maps, preprint, math-ph/0306017. 11. L. E. Labuschagne, W. A. Majewski and M. Marciniak, On decomposition of positive maps, in preparation. 12. W. A. Majewski, Transformations between quantum states, Rep. Math. Phys. 8 (1975),295-307. 13. W.A. Majewski, Dynamical Semigroups in the Algebraic Formulation of Statistical Mechanics, Fortschr. Phys. 32(1984)1, 89-133. 14. W. A. Majewski, Separable and entangled states of composite quantum systems; Rigorous description, Open Systems & Information Dynamics, 6, 79-88 (1999). 15. W. A. Majewski, “Quantum Stochastic Dynamical Semigroup” , in Dynamics of Dissipations, eds P. Garbaczewski and R. Olkiewicz, Lecture Notes in Physics, vol. 597, pp. 305-316,Springer (2002). 16. W. A. Majewski, On quantum correlations and positive maps, Lett. Math. P h y ~ 67 . (2004),125-132. 17. W. A. Majewski and M. Marciniak, On a characterization of positive maps, J . Phys. A: Math. Gen. 34 (2001),5863-5874. 18. W.A. Majewski and R. Streater, Detailed balance and quantum dynamical maps, J. Phys. A : Math. Gen. 31 (1998),7981-7995. 19. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett 77, 1413 (1996) 20. E. Strarmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. 21. E. Strarmer, On the Jordan structure of C*-algebras, Trans. Amer. Math.

374 Soc 120 (1965),438-447. 22. E.Starmer, Decomposable positive maps on C*-algebras,Proc. Amer. Math. SOC.86 (1982),402-404. 23. G.Wittstock, Ordered Nomed Tensor Products in "Foundationsof Quantum Mechanics and Ordered Linear Spaces" (Advanced Study Institute held in Marburg) A. Hartkkmper and H. Neumann eds. Lecture Notes in Physics vol. 29, Springer Verlag 1974. 24. S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. P h y ~ 10 . (1976),165-183.

AN INTRODUCTION TO LEVY PROCESSES IN LIE GROUPS

MING LIAO Department of Mathematics Auburn University Auburn, AL 36849, USA Email: [email protected]

A LBvy process in a Lie group is a process that possesses independent and stationary multiplicative increments. The theory of such processes is not merely an extension of the theory of Ldvy processes in Euclidean spaces. Because of the unique structures possessed by the non-commutative Lie groups, these processes exhibit certain interesting properties which are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behavior of the stochastic processes and the underlying structures of the Lie groups. In this article, we will provide an introduction to LBvy processes in Lie groups, and present some old and new results in this area. The topics include, besides some general theory, LBvy processes in compact Lie groups, invariant Markov processes in homogeneous spaces, limiting properties of LBvy processes in semi-simple Lie groups of noncompact type, and dynamical aspects of such processes. Most of these results have been or will be published elsewhere, therefore, little proof will be given. See also Applebaum' for some other results in this area not mentioned here.

1. Some general theory

Let G be a Lie group with identity element e. A chdlhg (right continuous with left limits) process gt in G is called a L6vy process if it possesses independent and stationary multiplicative increments, that is, for s < t , the increment g;lgt is independent of a { g , ; 0 5 u 5 s}, the a-algebra generated by the process up to time s, and has the same distribution as g ; l g t - s . Let g t = g O 1 g t . Then gf is a LBvy process in G starting at e and is independent of g o . In this paper, a measure on a topological space X is always defined on the Bore1 a-algebra B(X)of X unless explicitly stated otherwise. Let B(X)+denote the set of non-negative B(X)-measurable functions. The convolution of two measures p and Y on G is the measure p * Y on G

375

376 defined by p * ~ ( f=) J f(gh)p(dg)v(dh)for f E B(G)+. Let pt be the distribution of 9:. Then { p i ; t E R+ = [0, m)} is a convolution semigroup of probability measures on G, that is, pt+, = pt * p,, and it is continuous in the sense that pt + po = 6, weakly as t + 0. Conversely, given a continuous convolution semigroup of probability measures pt on G, there is a unique (in the sense of distribution) Levy process gt starting at e with distribution p t . It is easy t o show that a LCvy process gt is a Markov process with transition semigroup Pt given by

P t f ( 9 )= E [ f ( g 9 ; ) ] . for f E B(G)+. This is a Feller semigroup, that is, if f E Co(G), then Ptf E Co(G) and Ptf + f uniformly on G as t + 0, where Co(G) is the space of continuous functions on G vanishing at infinity (under the onepoint compactification topology). The process gt is left invariant in the sense that its semigroup Pt is left invariant, that is, Pt(f o Zg) = ( P t f ) o 1, for any g E G, where Zg: G 3 h I+ gh E G is the left translation. Conversely, a left invariant Feller process gt in G is a LCvy process. The LCvy process defined above may be called a left LCvy process. Similarly, one may define a right LCvy process using the increment g t g r l instead of g y ' g t . Then it is a Feller process in G with transition semigroup P t f ( g ) = E [ f ( g z g ) ] ,where g; = gtg;', and is invariant under the right translation r g : G 3 h I+ hg E G . The left and right Levy processes are in natural duality. In the following, a Levy process will mean a left LCvy process unless explicitly stated otherwise. As a Feller process, the distribution of a LCvy process gt is completely determined by its generator L. Hunt15 obtained an explicit formula for L. Let g be the Lie algebra of G, which may be regarded as the tangent space T,G of G at e . For X E 8, let X' and X p be respectively the left and right invariant vector fields on G induced by X , that is, X l ( g ) = DZg(X) and X r ( g ) = D r g ( X ) for g E G, where the D preceding a smooth map denotes its differential. Let Ci9'(G)be the space of functions f E Co(G) such that X 6 f E Co(G) and X6YYlfE Co(G) for any X , Y E 6 . Similarly, the function space C,">'(G)is defined replacing X' by X p . Let { X l , . . . , X d } be a basis of 6. A set of functions 2 1 , . . . ,X d E CF(G) (the space of smooth functions on G with compact supports) will be called coordinate functions associated to the above basis if xi(e) = 0 and Xixj = d 6ij. Let 1xI2 = CiXl x:. Theorem 1 . (Hunt) Let gt be a Ldvy process in a Lie group G . Then the

377

domain D ( L ) of its generator L contains C,"l'(G) and f o r f E Ci9'(G),

where aij are some constants forming a non-negative definite symmetric matrix, X o E G and II is a measure o n G satisfying

f o r any neighborhood U of e. Conversely, given aij, X O and II as above, there is a unique (in the sense of distribution) Lkvy process in G whose generator restricted t o C,""(G) is given by (1). Any measure II on G satisfying (2) is called a LCvy measure. In the case of Theorem 1, 11 is called the LCvy measure of the LCvy process gt and is the characteristic measure of the Poisson random measure N on IR+ x G that counts the jumps of gt:

N([O,t] x B ) = #{s E (0, t ] ; g,--'gs E B } .

(3)

Hence, gt is continuous if and only if II = 0. The continuous L6vy processes include the (Riemannian) Brownian motion in G defined with respect a left invariant Riemannian metric on G, see Section 2.3 in \Ref. 21 for more details. Applebaum-Kunita2 shown that the Levy process gt can be characterized as the unique solution of a stochastic integral equation driven by a Brownian motion Bt and the Poisson random measure N , corresponding to the L6vy -It6 representation in Euclidean case. Let fi be the compensated random measure of N defined by f i ( d t d g ) = N ( d t dg) - dtII(dg). Theorem 2. (Applebaum-Kunita) There i s a d-dim Brownian motion Bt with covariance matrix { a i j } and independent of N such that f o r

378

Conversely, given a triple of independent G-valued random variable go, Brownian motion Bt and Poisson random measure N on lR+ x G with characteristic measure II being a Ldvy measure, there is a unique cddldg process gt in G satisfying (4) for any f E G,">l(G)and it is a Ldvy process. Note that the first integral in (4) is a Stratonovich stochastic integral, and the integral with respect to N exists and is finite because of (2). The t = J(o,tl is used here. convention If the Lkvy measure II has a finite first moment, that is, if J 1x1 mZ < 00, then both (1) and (4) simplify as

so

and

(6)

xfZl

where YO= X O (Jx i d I ) X i . If II is finite, then there are iid (independent and identically distributed) stopping times 7, of exponential distribution of mean l/II(G), and an independent sequence of iid G-valued random variables on of distribution II/II(G), such that the Lkvy process gt may be obtained by solving the stochastic differential equation d

dgt = c X i ( g t ) 0 dBf

+ Y,'(gt)dt

i= 1

together with jump conditions g(T,) = g ( T , - ) o , ,

where T, =

(7)

cy=l

~ i .

379 All these results hold also for a right LQvyprocess with suitable changes, X' and gh in (1) should be replaced by C,""(G),X' for example, Ci9'(G), and hg. Because a general Lie group G does not have a natural linear structure, the stochastic integral equation for the LQvy process gt can only be given in a functional form as in (4)and (6). If G is a matrix group, then it is possible to write down a stochastic integral equation directly for the process g t . Now let G = GL(d,R),the group of the d x d real invertible matrices. This is a &-dimensional Lie group and its Lie algebra g is the space gI(d, R) of the d x d real matrices. We may identify g = g [ ( d ,R) with the Euclidean space Rd2 and G = GL(d,R) with a dense open subset of Rd2. For any X = { X i j } E Rd2, let = (& X $ ) l I 2 be its Euclidean norm. Let Eij be the matrix that has 1 at place ( i , j ) and 0 elsewhere. Then E i j , i , j = 1 , 2 , . . .,d , form a basis of g. A set 2 = {zij} of associated coordinate functions may be chosen such that z(g) = g - I d in matrix form for g close to e = I d (the d x d identity matrix). For g E G = GL(d,R),the tangent space T,G can be identified with Rd2, therefore, any element X of T,G can be represented by a d x d real d X i j ( a / a g i j ) f ( g ) ,where g i j are matrix { X i j } in the sense that X f = &=l the standard coordinates on Rd" It can be shown that for g , h E GL(d,R) and X E T,G = gI(d, R),DZ,oDrh(X)is represented by the matrix product gXh, where X is identified with its matrix representation {Xij}. Therefore, we may write gXh for DZ, o D r h ( X ) . Thus, X ' ( g ) = gX and X r ( g ) = X g . Let gt be a left LQvy process in G = GL(d,R). Then it satisfies the stochastic integral equation (4)for any f E C,">'(G). Let f be the matrixvalued function on G defined by f ( g ) = g for g E G. Although f is not contained in C,"*'(G),at least formally, (4)leads to the following stochastic integral equation in matrix form:

1x1

where X O E g = gI(d, R), Bt = { B Y } is a &-dim Brownian motion and N is an independent Poisson random measure on R+. x G with characteristic measure Il being a L6vy measure. See Section 1.5 in Ref. 21 for the proof of the following result.

380 Theorem 3. Assume

E[lgo12]< 00

and

Ih - Idl2I2(dh)< 00.

(9)

Then there is a unique cbdldg process gt in G = G L ( d , R) that satisfies the equation (8). Moreover, gt is a L t v y process and f o r any t > 0 ,

Conversely, any L t v y process gt in G satisfying (9) is the unique solution of a stochastic integral equation of the form (8).

2. LQvyprocesses in compact Lie groups

Let G be a compact Lie group. A Lie group homomorphism U from G into the group U ( n ) of n x n unitary matrices is called a unitary representation of G. It is called irreducible if it has no nontrivial invariant subspace of U? . The set Irr(G)+ of equivalence classes of non-trivial irreducible unitary representations of G is countable. For S E Irr(G)+, let U6 E S with dimension da. For any matrix A , let A’ denote its transpose and A its complex conjugate, and write A* = A’. The following standard result can be found in Sections 11.4 and 111.3 in Brocker and tom Died5. Let p~ denote the normalized Haar measure on G. Theorem 4. (Peter- Weyl) The set of matrix elements U$, multiplied by

G, f o r m a complete orthonormal system in L 2 ( p ~ ) Therefore, . f

d6 nace(&U6)

= P G ( f ) -k 66 Irr(G)+

in L2 sense f o r f E L 2 ( p ~ )where , A6 = pG(f Ua*) = J f ( g ) u 6 * ( g ) p G ( d g ) . In this section, we will assume that gt is a L6vy process in a compact connected Lie group G starting at e with distribution p i . We will obtain a Fourier series expansion of the distribution density pt = dpt/dpG, when it exists, and from it to obtain the exponential convergence of pt to p~ as t + 00 under total variation norm. The existence of pt is also established in certain cases. The results of this section are taken from Liao20. More generally, Fourier method has been applied to the convolution products of probability measures on locally compact groups in Heyer14 and Siebert.27 It has also been used to study the convergence of random walks

381

to Haar measure on finite and some special compact groups in Diaconis' and Rosenthal.26 Let { X I ,. . . ,X d } be a basis of g and let { a i j } be the coefficients in (1). The Ldvy process gt is called non-degenerate if { a i j } is non-degenerate and it is called hypo-elliptic if the space

generates the Lie algebra 6 . The hypo-elliptic condition is weaker than the non-degeneracy. A continuous Ldvy process satisfying this condition is a hypo-elliptic diffusion process in the usual sense. It is well known that such a process possesses a smooth distribution density pt = dpt/dpG for t > 0. For a discontinuous Levy process g t , it is shown in Ref. 20 that if it is non-degenerate and has a finite L6vy measure, then it has a n L2 distribution density pt for t > 0. We will see that the existence of pt holds also under hypo-ellipticity if the Ldvy process gt possesses additional invariance property. Theorem 5. Let gt be a L t v y process with generator L and go = e. Assume it has a n L2 distribution density pt = d,ut/dpG f o r t > 0 . Then f o r g E G,

The series converges absolutely and uniformly f o r ( t , g ) in [ E , m) x G for any E > 0, hence, ( t , g ) e p t ( g ) is continuous. Moreover, i f gt is hypoelliptic, then the eigenvalues of the matrix L(U6* ) ( e ) have negative real parts. Consequently, pt -+ 1 uniformly o n G as t + 00. Let I I f 112 and 11 f I Ioo be respectively the usual L2-norm and the Loo-norm of a function f (under P G ) . The total variation norm of a signed measure v is defined by llvllt,, = s u ~ l ~Iv(f)l. l < ~ By Theorem 5 , if gt is hypo-elliptic, then its distribution pt converges to the normalized Haar measure p~ under the total variation norm, that is, (Ipt - PGlltv -+ 0, as t -+ 00. Theorem 6. Assume gt is a hypo-elliptic L t v y process with go = e and its distribution pt is invariant under the inverse map o n G. Then it has an L2 distribution density pt for t > 0. Moreover, the eigenvalues of L(U6 *)(e), S E I v ( G ) + , are negative and possess a negative upper bound -A, and for

382

any E

> 0, there are constants C > c > 0 such that f o r t > E , ce-xt 5 llpt - 1112 5 Ce-xt llpt - 111, 5 Ce-xt, ce-xt 5 Ilpt - PGlltv 5 Ce-xt.

and (11)

Let L i i ( p ~ be ) the space of conjugate invariant functions in L 2 ( p ~ )The . character xa of an irreducible unitary representation U 6 , defined by xa = Trace(U6), is conjugate invariant. A version of Peter-Weyl theorem says that the irreducible characters xa form an orthonormal basis of L Z i ( p ~ ) . Thus, for f E LZi(p~),

c

f=PG(f)+

PG(fXa)Xa

6E Irr(G)+

in L2-sense. It can be shown that the character that

xa

is positive definite in the sense

k i,j=l

for any finite set of gi E G and & E CC From this it is easy to show that lxal 5 x a ( e ) = da. Let $6 = Xa/da (normalized character). Theorem 7. Let gt be a hypo-elliptic Lbvy process with 90 = e and with conjugate invariant distribution pt . Then it has an L2 distribution density pt f o r t > O a n d f o r g E G , dae t G ( e ) x a ( g ) . Pt(9) = 1 +

c

6E Irr(G)+

The series converges absolutely and uniformly for (t,9 ) in [ E , m) x G for any > 0. Moreover, (11) hold with X = infa[-(l/2) uijX:Xj&(e) + J(1- Re$a)dII] > 0.

E

c&=,

Example: Let G = SU(2), the group of 2 x 2 unitary matrices of determinant 1. Any matrix in SU(2) has eigenvalues e f e i for some real 6'. Therefore, any conjugate invariant function on G may be regarded as a function of 6' E [0, T ] . The irreducible characters (including the trivial one) are given by x,(6') = sin(n8)/ sin 6' for n = 1 , 2 , . . .. If gt is a hypo-elliptic continuous Lkvy process in SU(2) with 90 = e and with conjugate invariant distribution, then its distribution density pt for t > 0 is given by

383

where c is some positive constant.

3. Invariant Markov processes in homogeneous spaces Let M be a manifold and let G be a connected Lie group acting on M . A Markov process xt in M with transition semigroup Qt is called G-invariant if &t is G-invariant in the sense that Qt(f o g ) = ( Q t f ) 0 g for f E C o ( M ) and g E G. The main results in this section are taken from Section 2.2 in Ref. 21. Let gt be a LQvyprocess in G with go = e. For p E M , the process xt = gtp is called the one-point motion of gt from p . Let K be the isotropy subgroup of G at p , that is, K = { k E G; kp = k } . Recall the semigroup Pt of g t is left invariant. If Pt is also right Kinvariant, that is, if P t ( f o T k ) = (Ptf)O T for ~ k E K and f E Co(M),then it is easy to show that xt = gtp is a Markov process in M with semigroup Qt given by

Vf E Ca(M),

(Qtf)

0

x = pt(f 0 r),

(12)

where A: G -+ M is the map defined by g I+ gp. Moreover, xt is Ginvariant. Assume the G-action on M is transitive. Then M may be regarded as the homogeneous space G / K and A: G + G/K as the natural projection. Assume also K is compact. Then it can be shown that xt = gtp is a Ginvariant Feller process in M . Later we will see that any G-invariant Feller process in M starting at p is the one-point motion of a right K-invariant LQvy process gt in G with go = e from p . In the rest of this section, we will assume that K is compact and xt is a G-invariant Feller process in M = G / K starting at p = A(.). A measure p on M is called K-invariant if kp = p for any k E K, where kp denote the measure defined by kp(B)= p ( k - l ( B ) ) for B E B ( M ) or equivalently by k p ( f ) = p(f o k) for f E B ( M ) + . It is clear that the distribution ut of xt is K-invariant. A section on M is a map S: G + M = G / K satisfying A o S = idM (the identity map on M ) . There may not be a continuous section defined globally on M , but there is always a (Borel) measurable one. For two K-invariant measures p and Y on M , their convolution is the measure defined by

384

where S: G + M is a measurable section. By the K-invariance of p and u, p * u is independent of the choice of S and is K-invariant. It is easy to show that the distributions ut of xi, t E & , form a continuous convolution semigroup of K-invariant probability measures on M . Conversely, given such a convolution semigroup ut on M ,there is a unique (in the sense of distribution) G-invariant Feller process xt in M starting at p with distribution ut. Moreover, its transition semigroup Qt is given by

Since the convolution of measures on M defined here reduces to the ccnvolution of measures on G defined in Section 1 when G = M with K = {e}, the notion of G-invariant Feller processes in M generalizes that of L h y processes in G. We have mentioned Hunt's result on the generators of L6vy processes in a Lie group. We will state an analog of this result for G-invariant Feller processes, proved also by Hunt," after some preparation. For g E G, let Ad(g) = DZ, o DT,-I: g + g. The Lie group G acts on its Lie algebra g via the map: G x g 3 (9,X) I-) Ad(g)X E 8,called the adjoint action of G on g and denoted by Ad(G). The restriction of this action to K is denoted by Ad,(K) or simply by Ad(K). Since K is compact, there is an Ad(K)-invariant inner product on g. Fix such an inner product. Let f be the Lie algebra of K and let p be the orthogonal complement off in g. Choose an orthonormal basis {XI,. . . ,Xd} of g such that X I , . . . ,Xn form a basis of p and Xn+l,. . . ,Xd form a basis of f. A differential operator T on M is called G-invariant if T(fog) = (Tf) og for f E C m ( M ) and g E G. Such an operator is completely determined by T f ( p ) for f E Cm(M).It will be called a G-invariant diffusion generator if it is the generator of a G-invariant diffusion process (i.e., a G-invariant continuous Feller process) in M . It is easy to show that given constants bij and b i , there is a unique G-invariant diffusion generator T on M such that for f E Cm(M),

T f b )=

2 i,j=l

bijX,'Xi(f

o

r)(e)

+

biXf(f o r)(e),

(14)

i=l

provided that bij form a non-negative definite symmetric matrix, and together with bi satisfy n bij

n

bp[Ad(k)]ip for any k E K,

bpq[Ad(k)]ip[Ad(k)]jqand bi =

=

P4=1

p=l

385

where [Ad(k)] is the matrix representation of Ad(k) given by Ad(k)Xj = Cy=l[Ad(k)]ijXi. Note that in this paper, a stochastic process is always assumed to have an infinite life time unless explicitly stated otherwise. In particular, no killing of the process can occur, and hence, T cannot contain a constant term. Y i X i ) E M . ReConsider the map Rn 3 y = (yl,. . . ,yn) H r(ex;=i stricted to a sufficiently small neighborhood of 0 in Rn, the map is a diffeomorphism and hence y 1 , . . .,yn may be used as local coordinates on a neighborhood V of p in M . The yi may be extended to be functions in C,00(M). It is easy to show that Cy=lyi(x) Ad(k)Xi = Cy=lyi(kx) Xi for any k E K and 2 E V . We may assume that this holds for all x E M by suitably modifying the yi outside V . Thus modified, the yi will be called the canonical coordinate functions on M (associated to the basis {XI,.. . ,Xn} of PIWe are now ready to state Hunt's result on the generators of G-invariant Feller processes in M . A slightly different version is presented here.

z

Theorem 8. Let be the generator of a G-invariant Feller process xt in M = G / K starting at p = r ( e ) . Then its domain D ( z ) contains C F ( M ) and for any f E C F ( M ) ,

where is a G-invariant difision generator on M and measure on M satisfying

fi(c <

fi is a K-invariant

n

fI({o}) = 0,

y:)

00

and

fI(Uc)< 00

i=l

for any neighborhood U of 0. Conversely, given T and I?, there is a unique (in the sense of distribution) G-invariant Feller process xt in M such that its generator at point p , restricted t o C r ( M ) , is given b y (15).

z

This theorem can be used to prove the converse of a fact mentioned earlier, that is, the one-point motion of a right K-invariant LCvy process gt in G with go = e from p is a G-invariant Feller process.

Theorem 9. If xt is a G-invariant Feller process in M = G / K starting at p = r ( e ) , then there is a right K-invariant Lkvy process gt in G with 90 = e such that xt = gtp (in the sense of distribution).

386

The Fourier analysis may be applied to study the infinite divisional laws, and hence the distributions of a G-invariant Markov processes, on a symmetric space M = G/K, see Gangolli.” 4. Limiting properties of LQvyprocesses

Perhaps, one of the most interesting discoveries in the probabilistic connection with Lie groups is the limiting properties of Brownian motions and random walks in semi-simple Lie groups of noncompact type, both of which are examples of LBvy processes. Dynkin8 in 1961 studied the Brownian motion xt in the space X of Hermitian matrices of determinant one in connection with Martin boundary. He found that as t + 00, xt + 00 only along certain directions and at non-random exponential rates. Since X = SL(d,C ) / S U ( d ) ,where SL(d,C) is the group of d x d complex matrices of determinant one and SU(d) is the subgroup of unitary matrices, the limiting properties of xt in X may be regarded as those of the Brownian motion gt in SL(d,C) that projects to xt. The group SL(d,C) belongs to a class of Lie groups called semi-simple Lie groups of non-compact type and the space SL(d,C)/SU(d) is an example of symmetric spaces. Such a space possesses a polar decomposition under which any point can be represented by “radial” and “angular” components (both maybe multi-dimensional). Dynkin’s result says that the Brownian motion in X has a limiting angular component and its radial component converges to 00 at non-random exponential rates. The convergence of the angular component was extended by 0 r i h a 1 - a ~ ~ to a general symmetric space. A complete result on a general semi-simple Lie group was obtained by Malliavin.22. See also Norris, Rogers and Williams,23 T a ~ l o r ,Babillot; ~ ~ ’ ~ ~ and Liao“ for some of more recent related study. In a different direction, Furstenberg and KestenlO in 1960 studied the limiting properties of products of iid matrix or Lie group valued random variables. Such processes may be regarded as random walks or discrete time LBvy processes in Lie groups. This study was continued in Fur~tenberg,~ T ~ t u b a l i nVirtser31 ,~~ and R a ~ g iIn . ~Guivarc’h ~ and Raugi,12 the limiting properties of random walks on semi-simple Lie groups of non-compact type were established under a very general condition. These methods could be extended to a general LBvy process. This extension was made in Liao17 and was applied to study the dynamical properties of LBvy processes in Lie groups viewed as stochastic flows on certain homogeneous spaces.

387

We now briefly describe the basic theory of semi-simple Lie groups of noncompact type. The reader is referred to a standard text such as Helgason13 for more detail. A Lie algebra g is called semi-simple if it does not contain any abelian ideal except ( 0 ) and a Lie group is called semi-simple if its Lie algebra is. It can be shown that g is semisimple if and only if its Killing form, a bilinear form on g defined by B(X,Y) = Trace[ad(X)ad(Y)], is non-degenerate, where ad(X) is the linear endomorphism on g given by ad(X)Y = [X,Y] (Lie bracket). Let G be a Lie group with Lie algebra 9. A Cartan involution on G is a Lie group automorphism 0 on G such that 0 # id c and O2 = idG. Let 8: g + g be its differential. Then 8 has exactly two eigenvalues: 1 and -1. Let t and p be respectively the eigenspaces of 8 corresponding to the eigenvalues 1 and -1. Then g = t @Ip is a direct sum and it is easy to show that [t,t] c t, [t,PI c P and [P,PI c t. If B is negative definite on g, then G is compact and is said to be of compact type. If B is negative definite on t and positive definite on p, then G is noncompact and is said to be of noncompact type. In either case, g is semi-simple because B is non-degenerate. The subset K = (9 E G; 0 ( g ) = g} of G fixed by 0 is a closed subgroup of G with Lie algebra t. The homogeneous space G / K is called a symmetric space, which plays an important role in differential geometry. In the rest of this paper, we will assume that G is a connected semisimple Lie group of noncompact type with Lie algebra 9 and Cartan involution 0 unless explicitly stated otherwise. In this case, the Killing form B induces an inner product (.,-) on g given by (X,Y) = -B(X,BY), under which g = t @ p is an orthogonal decomposition. We will also assume that G has a finite center. Then K is compact and is a maximal compact subgroup of G. Let a be a maximal abelian subspace of p. A linear functional a on a is called a root if the space ga = {X E g; ad(H)X = a ( H ) X for H E a}, called the root space of a,is nonzero. We have an orthogonal direct sum decomposition g = go @I C, ga, where go = a @ u with u = ( 2 E t; a d ( 2 ) H = 0 for H E a}. The subspaces of a determined by the equations a = 0 divide a into several open convex conic regions, called the Weyl chambers. Fix a Weyl chamber a+. A root a is called positive if a > 0 on a+. Any root a is either positive or negative, that is, equal to -a for some positive root a. Let n+ = Ca,O ga and n- = ‘&,o g-a, where the summations are taken over all positive roots. Both are nilpotent Lie algebras. Let A , N +

388

and N - be respectively the subgroups of G generated by a, n+ and n-. Let A+ = exp(a+) and let A+ be its closure. Let U = {k E K ; Ad(k)H = H €or H E a} and U' = {k E K; Ad(k)a C a}, called respectively the centralizer and the normalizer of a in K . The former is a normal subgroup of the latter, but has the same Lie algebra u. The quotient group W = U'/U is finite and is called the Weyl group. It acts on a via the map: W x a 3 (5, H ) I+ sH = Ad(k,)H E a with s = kJJ E W for k, E U'. The group G possesses the Cardan decomposition G = K&K in the sense that any g E G may be written as g = [a+q with <,q E K and a unique a+ E &. Although the choices for (<,q) are not unique, when a+ E A+, they are given by ([u,u-lq) for u E U. An element g of G is called regular if a+ E A+. Regular elements form an open dense subset of G. The group G also has the Iwasawa decomposition G = N-AK in the sense that the map: N - x A x K 3 ( n , a , k ) I+ g = nak E G is a diffeomorphism. There are other versions of Iwasawa decompositions such as G = KAN + . A typical example of a semi-simple Lie group of noncompact type is G = SL(d,Il%),the group of d x d real matrices of determinant one, with the Lie algebra g = s[(d,R), the space of d x d traceless real matrices, and the Cartan involution given by O(g) = g'-l. Then K is the special orthogonal group SO(d) of d x d orthogonal matrices of determinant one with Lie algebra t = o(d), the space of d x d skew-symmetric matrices, and p is the space of d x d traceless real symmetric matrices. In this case, the subspace a of p formed by traceless diagonal matrices is a maximal abelian subspace of p. The roots are aij given by aij(H) = Hi - Hj for H = diag(H1,. . . , H d ) E a and i # j . The root space gaij is one-dimensional and is spanned by the matrix Eij that has 1 at place (i,j) and 0 elsewhere. One may take a+ = (diag(H1,. . . ,Hd) E a; H I > Hz > . . > Hd} to be the chosen Weyl chamber. Then positive roots are aij with i < j . The nilpotent Lie algebras n+ and n- are respectively the spaces of upper triangular and lower triangular matrices of zero diagonal. The groups A, N + and N - are respectively the subgroups of G consisting of diagonal matrices of positive diagonal, upper triangular matrices of unit diagonal, and lower triangular matrices of unit diagonal. The group U is discrete with u = {0} and contains only the diagonal matrices with f l along diagonal, the group U' is formed by permutation matrices, and the Weyl group W = U'/U acts on a by permuting the diagonal elements of H E a.

389

For G = SL(d, R), the Cartan decomposition G = K Z K can be proved by diagonalizing the symmetric matrix gg' for g E SL(d,R). The Iwasawa decomposition G = N-AK (resp. G = KAN+) is a consequence of GramSchmidt orthogonalization applied to the rows (resp. columns) of a matrix in SL(d,R). Now return t o a general G. Let gt be a LCvy process in G and let pt be the distribution of gt = go'gt. Let T, be the closed semigroup generated by supp(pt), the supports of measures pt, for t E ,&I that is, T, is the closed subset of G containing all supp(pt) and satisfying x ,y E T p xy E T,. Let G, be the closed subgroup of G generated by supp(,ut) for t E .&I Following Guivarc'h and Raugi,12 a subset H of G will be called totally irreducible if there do not exist g l,. . . ,gk,x E G such that H c U:==,gi(N-UAN+)"z, where the superscript c denotes the complement in G. It will be called totally right irreducible if the union here is replaced by U;=lx(N-UAN+)Cgi. Note that N-UAN+ is an open subset of G whose complement is lower dimensional, that is, the complement is contained in a union of finitely many lower dimensional sub-manifolds of G. Therefore, if H is not a lower dimensional subset of G, then it is totally (right) irreducible. A sequence gj in G is called contracting if in the Cartan decomposition gj = & a i q j , a(.:) + 00 as j -+ 00 for any positive root a. Since the Lie group exponential map exp: a + A is a bijection, its inverse log: A + a is well defined.

Theorem 10. Let gt be a Ldvy process in G with the Cartan decomposition gt = Qa$qt and the Iwasawa decomposition gt = ntatlct of G = N-A K . Assume G, is totally irreducible and T, contains a contracting sequence. T h e n almost surely, (tU converges in K / U and nt converges in N- as t + 00, and 1 1 H+ = lim - log a t = lim - log at ttcc

t

t-tcc

t

exists, is non-random and is contained in a+. The convergence of ( l / t ) l o g a t and &U in the above theorem correspond respectively to the radial and angular convergence on the symmetric space G / K mentioned earlier. The same result for random walks in G was proved by Guivarc'h and Raugi.12 The ideas can be used to prove for a general LCvy process, see Lia0.l7l2l See Section 6.6 in Ref. 21 for some sufficient conditions which guarantee the hypotheses of Theorem 10.

390

Theorem 10 holds also for a right LBvy process gt in G with the following changes: the total irreducibility should be replaced by the total right irreducibility, the convergence of
The limiting properties of Ldvy processes may also be studied from a dynamic point of view. Let R be the underlying probability space. A collection of maps Bt: R + R, t E rW,, is called a semigroup of time-shift operators if each Bt preserves the probability measure P on R, and they together form semigroup in t in the sense that BtBs = Os+t and $0 = idn. Let M be a (smooth) manifold and let Diff(M) be the group of the diffeomorphsims: M + M . A dynamical system, or a stochastic flow, on M is a stochastic process $t in Diff(M) with $0 = idM together with a semigroup of time shift operators 8t such that the following co-cycle property holds: Vs,t E & and w E 0,

#s+t(u)

= $s(etw)#t(w).

See Arnold3 for a comprehensivetreatment of the dynamical theory of such systems. We now describe the Lyapunov exponents and the associated stable manifolds of a stochastic flow $t on a manifold M equipped with a Riemannian metric (11 112; z E M}. The family of maps at: M x R + M x R given by (z,w ) I+ ( $ t ( w ) z ,Btw), t E rW,, is a semigroup in t and is called the skew-product flow associated to $t. A probability measure u on M is called a stationary measure of #t if E ( # t u ) = u. Under a certain condition, there is a unique stationary measure u. Moreover, there are constants A1 > A2 > ... > A,, a subset 'I of M x R invariant under at, in the sense that)?I(';@ = I?, with u x P ( r ) = 1, and for any (2,w ) E I?, the subspaces of the tangent space TxM:

-

TzM = K(z,w)

2 V2(z,~)2 ... 2 Vr(z,u) 2 Vr+l(zc,w) = (0)

such that

for i = 1,2,. . . ,T , where x(z,w ) - & + l ( z , w ) is the set difference. The numbers X i are called the Lyapunov exponents. The random subspace

391

V ,(2, w ) is called the subspace of T,M associated to the exponent X i and di = dim[q(z,w) ( z , w ) ] (independent of ( z , w ) ) is called multiplicity of Xi. The Lyapunov exponents X i are the limiting exponential rates at which the lengths of tangent vectors on M are stretched or contracted under the stochastic flow $ t , and together with K(z,w), they are independent of the Riemannian metric on a compact manifold M . A connected sub-manifold M’ of M is called a stable manifold of a negative Lyapunov exponent X i at ( z , w ) E if M’ C {y E M ; (y,w) E r}, T, M’ = V ,(2, w ) and 1 Vy E M’, limsup - log dist($t(w)z,$t(w)y) 5 Xi, t+m

t

where dist denotes the Riemannian distance on M . Roughly speaking, the distance between any two points in M’ tends to zero exponentially fast at the negative exponential rate X i under the stochastic flow $ t . A stable manifold of X i at ( z , w ) is called maximal if it contains any stable manifold of X i at (z,w). The local existence of stable manifolds is proved in Carverhill.6 Intuitively, one would expect that the maximal stable manifolds of a negative exponent form a foliation of a random open dense subset of M , but such a global theory under a general setting can be quite complicated, see Chapter 7 in Ref. 3. If a Lie group G acts on a manifold, then a right L6vy process gt in G with 90 = e may be regarded as a stochastic flow on M for a suitable choice of R and 6t. Let G be a semi-simple Lie group of noncompact type with a finite center. We will continue to use the notation introduced in the previous section. Let Q be a closed subgroup of G with Lie algebra q. Assume Q contains A N + . Then the homogeneous space M = G/Q is compact. For G = SL(d,R), such homogeneous spaces include the sphere Sd-l, the special orthogonal group SO(d) and several other interesting spaces. Let gt be a right L4vy process in G with 90 = e. We will describe explicitly, in terms of the group structure, the Lyapunov exponents and the associated stable manifolds of gt regarded as a stochastic flow on M = G/Q, or as well as a clustering pattern of this stochastic flow. See Chapter 8 in Liao21 for more detail. Let gt = & a t q t = lctatnt be respectively the Cartan decomposition G = K&K and the Iwasawa decomposition G = K A N + of gt, and for g E G, let gtg = t!a;+qi = lc!a!n! be the corresponding decompositions of gig. Assume the process gt satisfies the hypotheses in the version of Theorem 10 for right L6vy processes. Then for any g E G, almost surely,

392

Uvf converge as t

+ 00,

and the limit H + in (16) hold with u t and at replaced by u:' and a: respectively. Moreover, the non-random H t E a+ is independent of g E G. By choosing the K-components in the Cartan decomposition of gtg properly, one may assume that $ converges nf and

ast+oo.

For H C M x $2, let H ( z ) = {w E R; (z, w ) E H } , called the z-section of H at x E M , and let H ( w ) = {z E M ; (2, w ) E H } , called the w-section of H at w E fl. Let T : G + M = G / Q be the natural projection. For g E G , X E g and v E T,M, we may write g X for D l , ( X ) E TgG and gv for Dg(v) E Tg,M. It can be shown that there is a subset of M x R invariant under the skew-product flow associated to the stochastic flow gt such that P(I'(z)) = 1 for all z E M and

r ( w ) = gn&(w)-'.rr(N-UAN+)

(17)

for (2,w ) E I' and g E ~ - ' ( z ) . Note that r ( w ) is a dense open subset of M.

Theorem 11. Let cr be a negative root or zero. For any ( z , w ) E T-W and y E Ad(n",w)-l)[ga - (ga n 411, 1 t-tm lim -log t IlD~t(w)DT(gY)II~,(,= ) , ++),

r, g

E

where [ga - (ga n q)] is the set dzfference. Consequently, the Lyapunov exponents of the stochastic flow q5t on M = G / Q are given b y C Y ( H + ) , where CY ranges over all negative roots and zero with ga $Z q. Therefore, all the exponents are non-positive, and they are all negative i f and only i f

ucq. Let X1 > XZ nents. Define

> . . - > A,

be the set of all the distinct Lyapupov expo-

ni =

ga and Ni = exp(ni).

(18)

a(H+)<Xi

Theorem 12. Let ( z , w ) E I? and g E r - ' ( z ) . Then f o r 1 5 i 5 r , & ( x , w ) = D ~ [ g n & , ( w ) - l n i is ] the subspace of T,M associated to the exponent Xi, and i f X i is negative, then

M ~ ( z , w )= ~ [ g n & ( ~ ) - ~ N i ] is the maximal stable manifold of X i at

(2, w ) .

393

A family of sub-manifolds { H,} of a manifold H , each of dimension Ic, is said t o be a foliation of H if any x E H has a coordinate neighborhood V with coordinates X I , . . . ,x d such that each subset of V determined by

xk+l = c1, xk+2 = c2,

. ., xd = Cd-k

is equal to H,nV for some 0,where c l , c2,. . .,Cd-k are arbitrary constants. The sub-manifolds H , form a disjoint union of H and are called the leaves of the foliation.

Theorem 13. Let X i < 0 . Then the family of stable manifolds M i ( x ,w ) of X i is a foliation of r ( w ) . Moreover, if i < r , then each Mi(x,w) is foliated by {Mi+l(y,w); y E Mi(x,w)}, the family of the stable manifolds of the exponent X i + l contained in M i ( x ,w ) . Any X E g induces a vector field X * on M defined by X * f ( x ) = ( d / d t ) f ( e t x x ) It=O for f E C 1 ( M ) . Since K is compact, the Riemannian metric on M may be chosen under which K acts isometrically on M . The limiting property under the Cartan decomposition gt = t t a f q t implies that for large t , the stochastic flow gt is approximately composed of the following three transformations: a fixed random isometric transformation qm, the non-random flow of the vector field (H+)* and a “moving” random isometric transformation &. Because an isometric transformation preserves the geometry on M , the asymptotic behavior of the stochastic flow gt is largely determined by the flow of the single vector field (H+)*. In general, a point x E M is called a stationary of a vector field X on M if X ( x ) = 0. This is equivalent t o saying that x is a fixed point of the flow $t of X . A stationary point x will be said to attract a subset W of M if V y E W , $ t ( y ) + x as t + 00. A subset W of M is called invariant under the flow $t if $t(W) C W . A stationary point x of X will be called attracting if there is an open neighborhood V of x that is a disjoint union of positive dimensional sub-manifolds V, such that each V, is invariant under $t and contains exactly one stationary point that attracts V,. The stochastic flow gt on M exhibits the following clustering pattern at large time t: gt(w) sweeps r ( w ) , an open dense subset of M , into a collection of “moving” points, and these points form a subset of M that is an isometric image of the set the attracting stationary points of (H+)*. Therefore, it is important to know the set of attracting stationary points of (H+)*. This information is provided below.

Theorem 14. The set of stationary points of (H+)* on M is r ( U ’ ) and the set of attracting stationary points is .(U).

394 References 1. Applebaum, D. (2000) “Ldvy processes in stochastic differential geometry”, in Ldvy Processes: Theory and Applications, ed. by 0. BarnsdorE-Nielsen, T. Mikosch and S. Resnick, pp 111-139, Birkhauser. 2. Applebaum, D. and Kunita, H. (1993) “Ldvy flows on manifolds and Levy processes on Lie groups”, J. Math. Kyoto Univ. 33, pp 1105-1125. 3. Arnold, L. (1998) “Random dynamical systems”, Springer-Verlag. 4. Babillot, M. (1991) “Comportement asymptotique due mouvement Brownien sur une varidtd homogene A courbure negative ou nulle”, Ann. Inst. H. Poincard (prob et Stat) 27, pp 61-90. 5. Brocker, T. and Dieck, T. (1985) “Representations of compact Lie groups”, Springer-Verlag. 6. Carverhill, A.P. (1985) “Flows of stochastic dynamical systems: ergodic theory”, Stochastics 14, pp 273-317. 7. Diaconis, P. (1988) “Group representations in probability and statistics”, IMS, Hayward, CA. 8. Dynkin, E.B. (1961) “Nonnegative eigenfunctions of the Laplace-Betrami operators and Brownian motion in certain symmetric spaces”, Dokl. A M . Nauk SSSR 141, pp 1433-1426. 9. Furstenberg, H. (1963) “Noncommuting random products” , Trans. Am. Math. SOC.108, pp 377-428. 10. Furstenberg, H. and Kesten, H. (1960) “Products of random matrices”, Ann. Math. Statist. 31, pp 457-469. 11. Gangolli, R. (1964) “Isotropic infinitely divisible measures on symmetric spaces”, Acta Math. 111, pp 213-246 (1964). 12. Guivarc’h, Y . and Raugi, A. (1985) “RontiBre de firstenberg, proprietds de contraction et convergence” , Z. Wahr. Gebiete 68, pp 187-242. 13. Helgason, S. (1984) “Differential geometry, Lie groups, and symmetric spaces”, Academic Press (1978). 14. Heyer, H. (1977) “Probability measures on locally compact groups”, Springer-Verlag. 15. Hunt, G.A. (1956) “Semigroups of measures on Lie groups”, Trans. Am. Math. SOC.81, pp 264-293. 16. Lim, M. (1994) “The Brownian motion and the canonical stochastic flow on a symmetric space”, Transactions AMS 341, pp 253-274. 17. Lim, M. (1998) “Ldvy processes in semi-simple Lie groups and stability of stochastic flows”, Trans. Am. Math. SOC.350, pp 501-522. 18. Lim, M. (2001) “Stable manifolds of stochastic flows”, Proc. London Math. SOC.83, pp 493-512. 19. Lim, M. (2002) “Dynamical properties of Ldvy processes in Lie groups”, Stochastics Dynam. 2, pp 1-23. 20. Lim, M. (2003) “Ldvy processes and Fourier analysis on compact Lie groups”, to appear in Ann. Probab. 21. Lim, M. (2004) “Ldvy processes in Lie group”, book to be published by Cambridge Univ. Press.

395 22. Malliavin, M.P. and Malliavin, P. (1974) “Factorizations et lois limites de la diffusion horizontale audessus d’un espace Riemannien symmetrique” , Lecture Notes Math. 404, pp 164-271. 23. Norris, J.R., Rogers, L.C.G. and Williams, D. (1986) “Brownian motion of ellipsoids”, Trans. Am. Math. SOC.294, pp 757-765. 24. Orihara, A. (1970) “On random ellipsoids”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17, pp 73-85. 25. Raugi, A. (1997) “Fonctions harmoniques et thborkmes limites pour les marches gleatoires sur les groupes”, Bull. SOC. Math. France, memoire 54. 26. Rosenthal, J.S. (1994) “Random rotations: characters and random walks on SO(N)” , Ann. of Probab. 22, pp 398-423. 27. Siebert, E. (1981) “Fourier analysis and limit theorems for convolution semigroups on a locally compact groups”, Adv. in Math. 39, pp 111-154. 28. Taylor, J.C. (1988) “The Iwasawa decomposition and the limiting behavior of Brownian motion on a symmetric space of non-compact type’’, in Geometry of random motion, ed. by R. Durrett and M.A. Pinsky, Contemp. Math. 73, Am. Math. SOC.,pp 303-332. 29. Taylor, J.C. (1991) “Brownian motion on a symmetric space of non-compact type: asymptotic behavior in polar coordinates”, Can. J. Math. 43, pp 10651085. 30. Tutubalin, V.N. (1965) “On limit theorems for a product of random matrices”,Theory Probab. Appl. 10, pp 25-27. 31. Virtser, A.D. (1970) “Central limit theorem for semi-simple Lie groups”, Theory Probab. Appl. 15, pp 667-687.

DUALITY OF W*-CORRESPONDENCES AND APPLICATIONS

PAUL S. MUHLY* Department of Mathematics University of Iowa Iowa City, IA 52242

muhlyOmath.uiowa.edu

BARUCH SOLEL+ Department of Mathematics Technion

32000 Haifa Israel mabaruch8techunix.technion.ac.il

1. Introduction

In [21] Pimsner used C*-correspondences to construct and study a rich class of C*-algebras. In our work [ll]we introduced and studied a class of nonselfadjoint operator algebras, called tensor algebras, that are constructed from C*-correspondences and are subalgebras of Pimsner’s C*-algebras. Later, in [16], we studied the Hardy algebras (which are the ultraweak closures of tensor algebras associated with correspondences over von Neumann algebras). Together they form a rich class of operator algebras containing a large variety of algebras (such as analytic crossed products [9, 201, noncommutative disc algebras [24], free semigroup algebras [ 5 ] , quiver or free semigroupoid algebras [12, 71 and others). The definition of a C*correspondence and the construction of the tensor algebra associated with it will be presented in Section 2. *Supported by grants from the U S . National Science Foundation and from the U.S.Israel Binational Science Foundation. tSupported by the US.-Israel Binational Science Foundation and by the h n d for the Promotion of Research at the Technion.

396

397 The simplest example of a tensor algebra is the classical disc algebra A(D) (associated with the correspondence C over the algebra C). The study of its representation theory amounts to the study of contraction operators (up t o unitary equivalence). It turns out that many ingredients of that theory can be generalized to the study of the representation theory of general tensor or Hardy algebras. In this paper we shall deal mostly with the Hardy algebras. The simplest one is the classical Hm(T). Model theory was originally formulated by Sz-Nagy and Foias and others (see [19]) t o study contraction operators on Hilbert space. In Section 4 we describe how one can define canonical models for certain representations of the Hardy algebra. This is mainly an exposition of some of the results in [17]. It generalizes some of the classical results as well as the results of Popescu [23] for row contractions. Our goal is to establish a bijective correspondence between the completely non coisometric representations of the Hardy algebra and characteristic operator functions. In the classical theory the characteristic operator function defining the model is a Schur multiplier; i.e. an H m function defined on D taking values in B ( € I , & )(for a pair of Hilbert spaces E l , & ) . When Hm(ID) is replaced by a general Hardy algebra (denoted H w ( E ) , where E is a correspondence over a von Neumann algebra M ) the “Schur multipliers” are elements of another Hardy algebra. That Hardy algebra is associated with a correspondence that is “dual” to E in a sense that we shall make precise in Definition 3.1. This is not the first occurance of the concept of duality for W*correspondences. The origins of the construction may be traced back at least to Arveson [2] where he associated a Hilbert space (i.e. a correspondence over C) to an endomorphism Q of B ( H ) ( equivalently, to the correspondence ,B(H) associated with the endomorphism). Arveson’s construction was generalized by us [14] to yield a corresponence Ee associated to a completely positive map 0 on a general von Neumann algebra M . The correspondence Ee is the dual of the correspondence studied in [22], [l] and [lo]. In fact, it can be shown that the techniques of [14] that yield endomorphic dilations for semigroups of completely positive maps are, in a sense that may be made precise, dual to the techniques used by Bhat and Skiede in [4] t o achieve their dilation result. A discussion of the relation between [14] and [4] appears in [27] and will be developed further in [18]. Duality of correspondences has also proved useful in the analysis of “curvature” for completely positive maps [15] and in our work on the Hardy algebra of a W*-correspondence [16]. It is in study of Hardy algebras where

398

duality first appeared in our thinking even before the appearance of [14] (although [16] was completed much later). In Section 3 we define and discuss duality for W*-correspondencesand present the duality theorem (Theorem 3.1). In fact, we shall generalize here some results (and definitions) of [16] to the context of W'-correspondences from one von Neumann algebra M to another, N . (In [16] the main results were proved under the assumption that M = N ) . This will allow us to present and sketch the proof of Theorem 3.2 using duality to give necessary and sufficient conditions for two correspondences to be Morita equivalent (in the sense of [13]). Full details will be developed in [18] In the next section we introduce the notation and constructions used throughout the paper.

2. Correspondences and operator algebras

We start by introducing the basic definitions and constructions. We shall follow Lance [S] for the general theory of Hilbert C'-modules that we shall use. Let A be a C*-algebra and E be a right module over A endowed with a bi-additive map (., .) : E x E + A (refered to as an A-valued inner product) such that, for 1,q E E and a E A, ( t , ~ = ()t , q ) a , (t,q)* = ( q , O ,and (t,t)2 0, with (I,() = 0 only when t = 0. Also, E is assumed to be We write L ( E ) for the space of complete in the norm 11<11 := l (<,<)l1 /2. continuous, adjointable, A-module maps on E. It is known to be a C'algebra. If M is a von Neumann algebra and if E is a Hilbert C'-module over M, then E is said to be self-dual in case every continuous M-module map from E to M is given by an inner product with an element of E. Let A and B be C'-algebras. A C'-correspondence from A to B is a Hilbert C'-module E over B endowed with a structure of a left module over A via a nondegenerate *-homomorphism cp : A 4 L(E). When dealing with a specific C'-correspondence, E , from a C*-algebra A to a C'-algebra B, it will be convenient to suppress the cp in formulas involving the left action and simply write a t or a . t for cp(a)t. This should cause no confusion in context. C'-correspondences should be viewed as generalized C*-homomorphisms. Indeed, the collection of C'-algebras together with (isomorphism classes of) C*-correspondences is a category that contains (contravariantly) the category of C*-algebras and (conjugacy classes of) C'-homomorphisms. Of course, for this to make sense, one has to have a notion of composition of correspondences and a precise notion of isomor-

399

phism. The notion of isomorphism is the obvious one: a bijective, bimodule map that preserves inner products. Composition is “tensoring”: If E is a C*-correspondence from A to B and if F is a correspondence from B to C, then the balanced tensor product, E € 3 F~ is an A, C-bimodule that carries the inner product defined by the formula (El

€3 111712 @ 112)EBBF := ( r l l , c p ( ( G r E 2 ) E ) 1 1 2 ) F

The Hausdorff completion of this bimodule is again denoted by E @ B F and is called the composition of E and F . At the level of correspondences, composition is not associative. However, if we pass to isomorphism classes, it is. That is, we only have an isomorphism ( E €3 F ) €3 G N E @ ( F €3 G). It is worthwhile to emphasize here that while it often is safe to ignore the distinction between correspondences and their isomorphism classes, at times the distinction is of critical importance. In this paper we deal mostly with correspondences over von Neumann algebras that satisfy some natural additional properties as indicated in the following definition.

Definition 2.1. Let M and N be von Neumann algebras and let E be a Hilbert C*-module over N . Then E is called a Hilbert W*-module over N in case X is self-dual. The module E is called a W*-correspondencefrom M to N in case E is a self-dual C*-correspondence from M to N such that the *-homomorphism cp : M + L ( E ) giving the left module structure on X is normal. If M = N we shall say that E is a W*-correspondence over M . Remark 2.1. An isomorphism of a W*-correspondence El from M I to N1 and a W*-correspondence E2 from M2 to N2 is a triple (a, !€J, T) where a : M I + M2 and T : N1 + N2 are isomorphisms of von Neumann algebras, !€J : El + E2 is a vector space isomorphism preserving the a-topology and for e , f E El and a E M1,b E N1, we have Q ( a e f ) = a(u)Q(e)T(b) and (\E ( e ) ,!€J (f)) = T(( e ,f)). When dealing with correspondences over M and over N (i.e. when Mi = Ni, i = 1,2), we shall require that a = T (unless we say otherwise).

It is evident that the composition of W*-correspondencesis again a W*correspondence. Note also that a W*-correspondence from a von Neumann algebra N to C is a Hilbert space H equipped with a (normal) representation of N . If E is a W*-correspondence from M to N then E € 3 H~ is defined as above and is a Hilbert space equipped with a normal representation of M . If CT is the representation of N on H we shall also write E @, H

400

for this tensor product and we note that, given an operator X E L ( E ) and an operator S E o ( N ) ' , the map E @ h I+ X<@ Sh defines a bounded operator on E @g H denoted by X 8 S. Observe that if E is a W*-correspondence over a von Neumann algebra M, then each of the tensor powers of E , viewed as a C*-correspondence over M in the usual way, is in fact a W*-correspondence over M and so, too, is the full Fock space F ( E ) , which is defined to be the direct sum M @ E @ E@'@.. . , with its obvious structure as a right Hilbert module over M and left action given by the map cpm, defined by the formula cpw(a):= diag{a, cp(a),y.d2)(a),cp(3)(a), * }, where for all n, cp(n)(a)(&@&8..),( = (cp(a)&) €3 <2 @ . . .(, (1 €3t 2 @ * . . E [email protected] tensor algebra over E , denoted T+(E),is defined to be the norm-closed subalgebra of L ( F ( E ) ) generated by cpw(M) and the creation operators Tc, E E , defined by the formula Tsq = 6 @ q, q E 3 ( E ) . We refer the reader to [ll]for the basic facts about T+(E).

<

Definition 2.2. Given a W*-correspondence E over the von Neumann algebra M, the ultraweak closure of the tensor algebra of E , T+(E),in L ( F ( E ) ) ,will be called the Hardy Algebra of E , and will be denoted H'"(E). Example 2.1. If M = E = U2 then F ( E ) can be identified with H 2 ( T ) . The tensor algebra then is isomorphic to the disc algebra A(D) and the Hardy algebra is the classical Hardy algebra Hm(T). Example 2.2. If M = C and E = P then 3 ( E ) can be identified with the space Z2(@) where @ is the free semigroup on n generators. The tensor algebra then is what Popescu refered to as the %on commutative disc algebra" A, and the Hardy algebra is its w*-closure. It was studied by Popescu [24] and by Davidson and Pitts who denoted it by L, [5]. Example 2.3. Let M be a von Neumann algebra and a be an injective normal *-endomorphism on M. The correspondence E associated with Q! is equal t o M as a vector space. The right action is by multiplication, the M-valued inner product is (a,b) = a*b and the left action is given by a; i.e. cp(a)b = a(a)b. We write ,M for E. It is easy to check that E@'"is isomorphic to M. The Hardy algebra in this case will be refered to as the non selfadjoint crossed product of M by a and is related to the algebras studied in [9]and [20]. Example 2.4. For a = i d , the correspondence defined above will be called

401

the identity correspondence over M . Example 2.5. Suppose now that 0 is a normal, contractive, completely positive map on a von Neumann algebra M. Then we can assiciate with it the correspondence M @eM obtained by defining on the algebraic tenson product M €3 M the M-valued inner product (a €3 b, c €3 d) = b*6(a*c)dand completing. (The bimodule structure is by left and right multiplications). This correspondence was used by Popa [22], Mingo [lo], AnantharamDelarouche [l] and others to study the map 0. (In [4] it is refered to as the GNS-module). If 0 is an endomorphism this correspondence is the one described in Example 2.3. In most respects, the representation theory of H w ( E ) follows the lines of the representation theory of T+(E).However, there are some differences that will be important here. To help illuminate these, we need to review some of the basic ideas from [ l l , 12, 141.

Definition 2.3. Let E be a W*-correspondence over a von Neumann algebra M . Then: (1) A completely contractive covariant representation of E on a Hilbert space H is a pair (T,a),where

(a) a is a normal *-representation of N in B ( H ) . (b) T is a linear, completely contractive map from E to B ( H ) that is continuous in the a-topology of [3] on E and the ultraweak topology on B ( H ) . (c) T is a bimodule map in the sense that T(S6R) = a ( S ) T ( ( ) a ( R )6, E E , and S,R E M. (2) A completely contractive covariant representation (!!',a) of E in B ( H ) is called isometric in case

for all It should be noted that the operator space structure of E which Definition 2.3 refers to is that which E inherits when viewed as a subspace of its linking algebra. Also, we shall refer to an isometric, completely contractive, covariant representation simply as an isometric covariant representation. There is no problem with doing this because it is easy to see that if one has a pair (T,a) satisfying all the conditions of part 1 of Definition 2.3, except possibly the complete contractivity assumption, but which is isometric in

402

the sense of equation (l),then necessarily T is completely contractive. (See

P11.1 As we showed in [ll, Lemmas 3.4-3.61 and in [16], if a completely contractive covariant representation, ( T , a ) ,of E in B ( H ) is given, then it determines a contraction 5? : E @, H -b H defined by the formula 5?(q €3 h) := T(q)h,r] @ h E E BU H . The operator 5? intertwines the representation 0 on H and the induced representation aE := cp(.) @ IH on E BCrH ; i.e.

F(cp(.)€3 I ) = a(.)T. In fact we have the following lemma from [16, Lemma 2.161. Lemma 2.1. T h e map (T,a) + p is a bijection between all completely contractive covariant representations (T,a ) of E o n the Hilbert space H and contractive operators 5? : E B UH + H that satisfy equation (2). Given such a 5? satisfying this equation, T , defined by the formula T(<)h:= 5?([ @ h ) , together with a is a completely contractive covariant representation of E o n H . Further, (T,a ) is isometric i f and only if 5? is a n isometry. Associated with (T,o)we also have maps Tn : EBn @ H + H defined by Pn(t1 @ t 2 . * * @ t n @ h) = T(&)T((2)* . .T((n)hIn the next theorem we give a a complete description of the representations of the tensor algebra. The main ingredient in the proof of this theorem is the construction, to a given completely contractive covariant representation (T,a ) of E , of an isometric representation (V,T ) that dilates it and is the minimal isometric dilation. Since we shall later need the notation set in this construction, we briefly describe it. (For full detailes see [ll). Given (T,a) on H we set A = ( I - 5?*5!)1/2 (in B ( E BU H ) ) , A, = ( I - p5?*)'j2(in B ( H ) ) , D = A(E BU H ) and D* = A , ( H ) . Also let Lc : H + E@,H be the map Lch = (@h an d D(()= AoLc : H + E@,,H. Note that T ( ( )= P o Lt. The representation space K of (V,T)is

where al(a)is the restriction to 2, of cp(a) 8 I H . The representation x can be defined by ?r = d i a g ( a , a l , o 2 , . ..) where Ok+l(u) = Vk(u) 63 1.0. The

403 map V : E

+ B ( K ) is defined by

where Lg here is the obvious map from E s m €3 V to Es(m+l)€3 V. It turns out that (V, T) is a covariant isometric representation of E, it dilates (",a) in the sense that, for J E E and a E M , V ( J ) *and .(a) leave H invariant and their restrictions to H are equal to T ( t ) *and o(u) respectively. It is also a minimal dilation (in an obvious sense) and can be shown to be the unique (up to unitary equivalence) minimal isometric dilation of ( T ,a). We now write QOfor the space of all vectors perpendicular to every vector of the form V(J)k, ( E E , k E K . Note that &O is .rr(M)-invariant and, for every 51,6, . . . J m E E and kl ,k2 in Qo , (V (G) V(52) * * .V (Jm) kl ,k2) = 0. Such a subspace of K is said to be wandering. It is easy to see that V is also a wandering subspace. Whenever M g K is a wandering subspace, there is a unitary operator, denoted W ( M ) ,from T ( E )@*IM M onto the V(E)-invariant subspace of K generated by M (denoted L w ( M ) ) .We also note that there is an isometry u from &O onto V, that commutes with ~ ( u ) for a E M.(See [17]). Theorem 2.1. Let E be a W*-correspondence over a von N e u m a n n algebra M . To every completely contractive covariant representation, (T,o), of E there is a unique completely contractive representation p of the tensor algebra T+(E) that satisfies

P(T0 = T ( t )

5E E

and p(cpw(a)) = d a ) a E M .

*

The map (T,o) p is a bijection between the set of all completely contractive covariant representations of E and all completely contractive (algebra) representations of T+( E ) whose restrictions t o cpm(M) are continuous with respect t o the ultraweak topology on L(.F(E)). Definition 2.4. If (T,o) is a completely contractive covariant representation of a W*-correspondence E over a von Neumann algebra M , we call the

404

representation p of T+(E)described in Theorem 2.1 the integrated form of ( T , a )and write p = a x T . Remark 2.2. One of the principal difficulties one faces in dealing with 7+(E) and H w ( E ) is to decide when the integrated form, a x T, of a completely contractive covariant representation (T,a) extends from T+(E) to H m ( E ) . This problem arises already in the simplest situation, vis. when M = C = E . In this setting, T is given by a single contraction operator on a Hilbert space, T+(E) "is" the disc algebra and H w ( E ) "is" the space of bounded analytic functions on the disc. The representation a x T extends from the disc algebra to H w ( E ) precisely when there is no singular part to the spectral measure of the minimal unitary dilation of T . We are not aware of a comparable result in our general context but we have some sufficient conditions. One of them is given in the following lemma. It is not necessary in general. Lemma 2.2. [16, Corollary 2.141 If ((f'l( < 1 then a x T extends t o a a-weakly continuous representation of Hw(E) .

3. Duality of W*-correspondences

In this section we discuss the concept of duality for W*-correspondences. As we noted in the introduction, it was first used, implicitly at least, in [2] to construct the product system associated t o an Eo-semigroup on B ( H ) and a bit more explicitly in [14] where the construction was extended to Eo-semigroups on a general von Neumann algebra. Definition 3.1. Let E be a W*-correspondence from M to N . Let a : M + B ( H ) and T : N + B ( K ) be normal representations of the von Neumann algebras M and N . Then the r-a-dual of E, denoted Er>",is defined t o be

(11 E B ( H , E @r K ) I v ( a ) = (cp(a)@ 11% a E MI. An important feature of the dual Er%" is that it is a W*-correspondence. Proposition 3.1. With respect to the actions of a(M)' and r ( N ) ' and the a(M)'-valued inner product defined as follows, ET+' becomes a W*correspondence from T ( N ) ' t o a ( M ) ' : For Y E a ( M ) ' , X E r(N)' and T E ETtu,X . T . Y := ( I @ X ) T Y and , for T ,S E Erlu,(T,S),,(M)~ := T * S . Remark 3.1. When M = N (i.e. E is a W*-correspondence over M ) and 7 = a,we write E" (in place of E'*u). The importance of this space is that

405

it is closely related to the representations of E . In fact, the operators in E" whose norm does not exceed 1 are precisely the adjoints of the operators of the form for a covariant pair (T,IT).In particular, every q in the open unit ball of E" (written D(E")) gives rise to a covariant pair (T,IT) (with q = f*)such that IT x T is a representation of H"(E). Given X E H w ( E ) we can apply this representation (associated to q) to it. The resulting operator in B ( H ) will be denoted by X ( q * ) . In this way, we view every element in the Hardy algebra as a (B(H)-valued) function on D(E"). This point of view is exploited in [16] to deal with interpolation problems.

Example 3.1. Suppose M = E = C and IT the representation of C on some Hilbert space H . Then it is easy to check that E" is isomorphic to B ( H ) . Fix an X E H w ( E ) . As we mentioned above, this Hardy algebra is the classical H"(T) and we can identify X with a function f E Hw(T). Given S E Eu = B ( H ) ,it is not hard to check that X ( S * ) ,as defined above is the Hw-functional calculus f (S*). Example 3.2. If 0 is a contractive, normal, completely positive map on a von Neumann algebra M and E = M M (see Example 2.5 ) then, for every faithful representation IT of M on H , the 0-dual is the space of all bounded operators mapping H into the Stinespring space K (associated with 0 as a map from M to B ( H ) )that intertwine the representation IT (on H ) and the Stinespring representation 7r (on K ) . This correspondence has proved very useful in the study of completely positive maps. (See [14] and [15]). If M = B ( H ) this is a Hilbert space and was studied by Arveson [2]. Note also that, if 0 is an endomorphism, then this dual correspondence is the space of all operators on H intertwining IT and IT o 0. We now return to discuss the general case (where M is not necessarily equal to N ) . The term "dual" that we use is justified by the following result, which is proved as Theorem 3.6 in [16] under the assumption that M = N . See also [28]. The full result along with its applications to Morita theory (in particular Theorem 3.2) will be proved in [18].

Theorem 3.1. Let E be a W*-correspondence f r o m M to N and let IT, T be faithful, normal representations of M (on H ) and N (on K ) respectively. If we write LI f o r the identity representation of u(M)' (on H ) and 12 f o r the identity representation of r ( N ) ' (on K ) , then one can form the 11-12-dual of E's" and we have (ET>")LIILZ

E.

406

The isomorphism in the theorem is ( ~ , Q , T from ) E onto (ETru)'l*'2 where !I! is defined by

where 5 E E , q E ETiuand h E H . Note that q is a map from H to E @ Kso that L;q(h) lies in K and the equation above defines a map from Ergu @ H to K whose adjoint can be shown to have the intertwining property required from an element of (ETru)'1*'2. In order to prove that 9 is onto one uses the following lemma. It was proved in [16,Lemma 3.51 for the case u = r. The proof in the general case requires just a minor adjustment.

Lemma 3.1. W h e n E is as above and u and r are faithful representations of M and N respectively, we have

V { X ( H ): X E ET*u} =E

@T

K.

The following two lemmas show that the operation of taking duals "behaves nicely" with respect to direct sums and tensor products.

Lemma 3.2. Given W*-correspondences El and EZ f r o m M to N and faithful representations u (of M o n H ) and r (of N o n K) we have (El CB E2)TluE E?' @ Ei"'. Lemma 3.3. Let E be a W*-correspondence f r o m M to N and F be a W*-correspondence f r o m N t o Q. Let cr, r and T be normal faithful representations of M , N and Q respectively. T h e n the m a p X @ Y I+ (IE8 X ) Y (for X f F")' and Y E ETiu) defines a n isomorphism F"*T@ ETiu rU ( E €3 F),sU. Given a W*-correspondence 2 over M and two faithful representations cr and r (of M ) , the dual correspondences 2" and 2' are, in general, non isomorphic but as we shall show bellow they are Morita equivalent. We will also show that the converse holds. For this, we first recall the definition of Morita equivalence for W *-correspondences. In [13 we discussed Morita equivalence for C*-correspondences. For W*-correspondencesthis concept is defined similarly (with minor changes). Recall first that an M-N equivalence bimodule is an M-N bimodule X that is endowed with M - and N-valued inner products, M ( . , .) and (., . ) N , making X a full and selfdual (right) Hilbert W*-module over N and a full and selfdual (left) Hilbert W*-moduleover M such that ~ ( J , q ) = c J(q, for J , q and in X . By definition, the von Neumann algebras M and N are

c

c ) ~

407

strongly Morita equivalent in case there is an M-N equivalence bimodule [25.

Definition 3.2. (cf. [13]) W*-correspondences E and F , over M and N respectively, are said to be (strongly) Morita equivalent if there is a n MN equivalence bimodule X such that E @ M X E X @ N F (where the isomorphism here is a triple ( i d , Q, i d ) for some map q). In the statement of the following theorem, the isomorphisms are in the sense indicated at the end of Remark 2.1. Theorem 3.2. Let E and F be W*-correspondences over a-finite von Nue-

m a n n algebras M and N respectively. T h e n the following conditions are equivalent. ( I ) There is a W*-correspondence Y (over some von N e u m a n n algebra Q ) and two faithful representations T I and ~2 of Q such that E E Y"l and F ?% Y T 2 . (2) E and F are strongly Morita equivalent. (3) There are faith@ representations a and T , of M and N respectively, such that E" CY F r . Proof. We shall only sketch the proof. For a detailed proof one needs to be careful about the maps involved in each of the isomorphisms below (see Remark 2.1). To prove (3) implies (1) , write 11, for the isomorphism from a(M)' onto T ( N ) ' (implied by the assumption that E" E F T ) ,write LI and ~2 for the identity representations of a(M)' and T(N)' respectively and set Y = E". Then it follows from duality that E E Y L 1and F E YL2*. To prove (1) implies (2) let Z be the identity correspondence Q and set X = Z"l'"2. Then E @ X ynl @Zfll,RZ ( Q @ Y ) " l t " 2 E ( y @ Q ) " 1 9 " 2 E @ Y"2 2 X @ F . For the last part, assume E and F are Morita equivalent and X is the equivalence bimodule implementing the equivalence. Assume N B(K) and write H for X @ N K . Write a for the identity representation of N' (on K ) and T for the representation of N' on H defined by .(a) = Ix @ a. Note that T(N')' = C ( X ) @ IK and this algebra is isomorphic to M (since M 2 C ( X ) for an equivalence bimodule X).Write II, for this isomorphism (from M t o T ( N ' ) ' ) . Let 11 be the identity representation of T(N')' on H and 12 be the identity representation of N on K . Write Z for the identity correspondence of N'. Given x E X , define S ( x ) to be the map from K to 2 aTH given by S ( z ) ( k ) = I Br (x €3 k). It is easy to check that S(z) Z"12"2

408 lies in In fact, the triple It then follows from duality that

is an isomorphism of X and Z

4. Applications of duality: Commutants and Canonical

models In this section, E will be a W*-correspondence over a von Neumann algebra M . As was mentioned in the introduction, there are several applications for the dual correspondences. Here we concentrate on using the Hardy algebra associated with a dual correspondence in order to study the Hardy algebra associated with the original correspondence E. The first result along these lines is the identification of the commutant of H m ( E ) (given in some induced representation). The other is the development of canonical models to study representations of H”(E). As we shall see, the characteristic functions can be identified with elements of H m ( E T )for some dual correspondence E‘ . Although H m ( E ) was defined as a subalgebra of L ( 3 ( E ) )it is often useful t o consider a (faithful) representation of it on a Hilbert space. Given a faithful, normal, representation a of M on H we can “induce” it to a representation of the Hardy algebra. To do this, we form the Hilbert space 3 ( E )Bu H and write

I n d ( a ) ( X )= X @ I , X E H m ( E ) . (Note that I n d ( a ) ( X ) is a well defined bounded operator on 3 ( E )&,I H for every X in L ( F ( E ) ) .Such representations were studied by M. Rieffel in [26]). Ind(a) is a faithful representation and is a homeomorphism with respect t o the a-weak topologies. Similarly one defines I n d ( ~ (where ) L is the identity representation of a(M)’ on H ) , a representation of H m ( E u ) . The following theorem shows that, roughly speaking, the algebras H m ( E ) and H m ( E u )are the commutant of each other. For the proof, see [16, Theorem 3.91.

Theorem 4.1. [16] Let E be a W*-correspondence over M and o be a faithful normal representation of M o n H . T h e n there exists a unitary operator U : 3(E‘) B H + 3 ( E )@ H such that

U*( I d (L ) (H” (E‘)))U = ( I n d ( a )(Hm (E)))’

409

and, consequently, ( I n d ( o )( H m(E)))”= I n d ( o )( H w ( E ) ) . Example 4.1. Given an n x n matrix C in Mn(Z+). One can associate with it a W*-correspondence E ( C )over the algebra D, of all diagonal n x n matrices. (See [12 for details). The tensor algebra associated with E ( C ) is called the quiver algebra or the path algebra (associated with the directed graph defined by C). It is a subalgebra of the Cuntz-Krieger C*-algebra Oc. If u is the identity representation of D, (on C?), then E(C)“= E ( C ) (where Ct is the transpose matrix). Thus, Theorem 4.1 gives another proof of [12, Proposition 5.41 and of [7, Theorem 5.81. ( Note that a semigroupoid algebra of [7] is the image, under an induced representation, of a quiver algebra.)

Another way in which the Hardy algebra of the dual correspondence plays a role in studying H m ( E ) is through canonical models. Here, roughly, the elements of the Hardy algebra of the dual play the role that “Schur multipliers” play in the classical theory. The canonical models are used to study certain representations of the Hardy algebra. The representations of H”(E) that we shall study here are the completely noncoisometric (abbreviated c.n.c ) and the (7.0representations. We first recall the definitions. (See [16] for more details). Recall that, given a covariant representation (T,u) of E on H , it has a unique minimal isometric dilation (V,T ) on K . It was described in Section 2 and we use here the notation introduced there. Definition 4.1.

(1) A completely contractive covariant representation ( T , a ) of E is called a C.0-representation if, for every h E H , Ilr?l,.hll + 0. Equivalently, if L,(&o) = K . (2) A completely contractive covariant representation (T,o ) of E is said to be completely noncoisometric if &(a)V L,(Qo) = K .

Clearly, every C.0-representation is completely non coisometric. Theorem 4.2. [16] If ( T , a ) is a c.n.c representation then T x o extends t o a w*-continuous representation of H m ( E ) .

We now define characteristic operator functions in this context. It generalizes the classical case [19] and the case studied by Popescu [23]. Here

410

we shall present the constructions and the main results. Full details will appear in [17].

Definition 4.2. Given a von Neumann algebra M and a W*correspondence E over M , A characteristic operator function is a tuple (0,&I, &,TI, 72) such that

Ii) For i = 1,2, &i is a Hilbert space and ri is a representation of M on €i

.

(ii) 0 : 7(E)

€1

+T(E)

@T2

is a contraction satisfying

&2

(cpw(a) @ I&*)@ = @(cpw(a) €3 I&1), a

EM

(4)

and = 0(T<€3 I&l).

(TC€3

(5)

(iii) There is no non zero vector 2 E €1 such that 2 = P & , ~ * P E ~ O ~ . (We say that 0 is purely contractave). (iv) A e ( T ( E )€3 &I) = A e ( ( T ( E )@€I) 8 €1) (where A e = (I 0" 0 )112).

If, in addition, 0 is an isometry then it will be called an inner characteristic operator function. (In this case, (iv) holds automatically).

&i

We shall often refer to 0 as the characteristic operator function (when and ~i are assumed to be known). Note that it follows from (4) and (5) that, if we write € for €1 @ €2

and let

T

be the representation

TI

then the matrix

@ 72,

(i:)

(viewed

as an element of 3 ( E )@ € ) lies in the commutant of I n d ( 7 ) ( H W ( E ) )If. 7 is faithful we will be able to use Theorem 4.1 to conclude that the matrix defines an element of H w ( E T ) .If T is not faithful, one can form T' = T @ T O that is a faithful representation (on a larger space) and consider the 3 x 3 matrix with 0 in the 2 , l entry instead of the 2 x 2 matrix above. Since this is just a technical point, we shall ignore it here and use Theorem 4.1 to define 6 E H w ( E T ) by

Ind(@) = u The left hand side will also be written G*6is the projection onto €1.

(

00

o) u*.

6 @ I&.

Note that, if 0 is inner,

41 1

Next we construct, for a fixed characteristic operator function 0 a covariant representation associated to it. For this, we write A , = ( I F ( E ) @ E~ 0*0)1/2 E B ( F ( E )@ €1) and set

K ( O ) = ( F ( E )63 €2) @ A e ( F ( E )@ E l ) 2 F ( E )@ E and

H ( O ) = ( ( F ( E )@ E 2 ) @ A e ( F ( E )@ €1)) 8 {Ox @ Aex : x E F ( E ) @ €1). Note that, if 0 is inner, we get 0'0 = U*(ql @ IE)U = IF(E)@ q1 and A e = 0. Thus, in this case, K ( O )= F ( E )@ € 2 and H ( O ) = ( F ( E )@ € 2 ) 8 O(F(E)8 €1). We shall also write Pe for the projection from K ( O ) onto H ( O ) . We have the following. Theorem 4.3. Let O be a characteristic operator jknction and let K ( O ) and H ( O ) be as above. For every a E M and E E we define the operators Se(t) and $e(a) o n A e ( F ( E )@ €1) by

<

S e ( t ) A e g = Ae(Tc @ I E , )9~E ,F ( E ) @ €1 and

Also, we define o n K ( O ) the operators

and

Then

(i) (Se,$e) and (Ve,p e ) define isometric covariant representations of E o n A e ( F ( E )@ El) and K ( O ) respectively. (ii) The space K ( O ) 8 H ( O ) i s invariant f o r (Ve,pe) and, thus, the compression of (V.,pe) to H ( O ) , which we denote by (Te,oe),is a completely contractive covariant representation of E . Explicitely,

T e ( t )= PeVe(t)lH(@) , tEE and

412

(iii) The representation (Te,ae)is completely n o n coisometric. It is a (2.0-representation if and only if 0 is inner. The converse of the theorem above also holds; that is, every c.n.c representation of E gives rise to a characteristic operator function. To construct it we now fix such a representation and let V,A, K , KO,V and V * be as in Section 2. Also, let p1 be the restriction of A to V and p2 be the restriction of A (or 0 ) t o D,. (Again, we shall assume that p = p l @ p 2 is faithful. Otherwise a minor technical correction is needed). We also write

G=D@V,. Associated with the wandering subspaces Qo and V of K we have the unitary operators W ( Q o )and W ( V )defined above (see the discussion preceeding Theorem 2.1). We write Qw for the projection of K onto L,(Qo) (=the range of W ( Q 0 ) ) . Also recall that u is an isometry from Q o onto V, that commutes with A ( . ) for a E M . It induces an isometry, written I F ( E ) 8 u from F ( E )8 Qo onto F ( E )@ D,. We now write

OT = ( I F ( E @ ) u ) W ( Q ~ ) * Q ~ :W F ((E~))@ V + F ( E )8 V,.

(6)

Theorem 4.4. Given a c.n.c representation (T,0 ) of E o n a Hilbert space H , the tuple ( O T ,D,V,,p1, p z ) , defined above, is a characteristic operator function in the sense of Definition 4.2. The representation is a C.0representation i f and only i f the characteristic operator function is inner. Moreover, the representation constructed from OT as in Theorem 4.3 is unitarily equivalent to (T,a ) . We also have the following. Theorem 4.5. Suppose we start with a characteristic operator function (0,€1, &,TI,7 2 ) and write (Te, ae) for the c.n.c representation constructed in Theorem 4.3. With this representation we can associate the characteristic operator function (OT, V ,V,,p1, p 2 ) as in Theorem 4.4. Then, the two characteristic operator functions are unitarily equivalent in the sense that there are unitary operators W1 : € 1 + V (intertwining r1 and p l ) and W2 : € 2 + V, (intertwining 7 2 and p 2 ) such that

OT = ( I F ( E ) 8 W2)@(IF(E)8 w:). The above results show that the characteristic operator functions (or the elements 6 obtained from these) serve as complete invariants for c.n.c representations of H w ( E ) .

41 3

We also state the following result which we know only for C.0representations. In this result we refer to compositions 0 = 0 1 0 2 where 0 is the inner characteristic operator function associated with a C.0representation and Oil i = 1,2, is an inner characteristic operator function but is not necessarily purely contractive. Two such compositions 0 = 0 1 0 2 = 0;0/,are said to be equivalent if 0; = &(I 8 &) (and 0; = ( I 8 V<)02) for some unitary operator VO. Theorem 4.6. Let ( T , a ) a (7.0-representation of E o n H (with u x T

the associated representation of H m ( E ) ) . Let 0 be the inner characteristic operator function of this representation. T h e n there is a bijection between the subspaces of H that are Hm(E)-invariant and (equivalence classes o f ) factorizations 0 = 0 1 0 2 of 0 as a composition of two inner characteristic operator functions (that are not necessarily purely contaractive). References 1. C. Anantharaman-Delaroche, O n completely positive maps defined by a n irreducible correspondence, Canad. Math. Bull. 33 (1990), 434-441. 2. W.B. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC. 80 (1989). 3. M. Baillet, Y . Denizeau and J.-F. Havet, Indice d’une esperance conditionelle, Comp. Math. 66 (1988), 199-236. 4. B. V. R. Bhat and M. Skiede, Tensor product systems of Hilbert modules

5. 6.

7. 8. 9. 10.

11. 12. 13.

and dilations of completely positive semigroupps, Inf. Dim. Anal. Quantum Prob. and Rel. Topics 3 (2000), 519-575. K. Davidson and D. Pitts, T h e algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275-303. R.G. Douglas, Canonical models, in “Topics in operator theory”, Math. Surveys 13 (Ed. C. Pearcy) (1974), 161-218. D. Kribs and S. Power, Free semigroupoid algebras, Preprint. E.C. Lance , Hilbert C*-modules, A toolkit f o r operator algebraists, London Math. SOC.Lecture Notes series 210 (1995). Cambridge Univ. Press. M. McAsey and P.S. Muhly, Representations of non-self-adjoint crossed products, Proc. London Math. SOC.47 (1983), 128-144. J. Mingo, T h e correspondence associated t o a n inner completely positive map, Math. Ann. 284 (1989), 121-135. P.S. Muhly and B. Solel, Tensor algebras over C*-correspondences (Representations, dilations and C*-envelopes), J. Funct. Anal. 158 (1998), 389-457. P.S. Muhly and B. Solel , Tensor algebras, induced representations, and the Wold decomposition, Canad. J. Math. 51 (1999), 850-880. P.S. Muhly and B. Solel, O n the Morita equivalence of tensor algebras, Proc. London Math. SOC.81 (2000), 361-377.

414

14. P.S. Muhly and B. Solel, Quantum Markov processes (correspondences and dilations), Int. J. Math. 13 (2002), 863-906. 15. P.S. Muhly and B. Solel, T h e curvature and index of completely positive maps, Proc. London Math. SOC.87 (2003), 748-778. 16. P.S. Muhly and B. Solel, Hardy algebras, W*-correspondences and interpolation theory, Preprint. 17. P.S. Muhly and B. Solel, O n canonical models for representations of Hardy algebras. In preparation. 18. P.S. Muhly, M. Skeide and B. Solel, Representations of BQ(E),commutants of won Neumann bimodules, and product systems of Hilbert modules. In preparation. 19. B. Sz-Nagy and C. Foias, Analyse H a m o n i q u e des Operateurs de L’espace de Hilbert, Akademiai Kiado (1966). 20. J. Peters, Semi-crossed products of C*-algebras, 3. F’unct. Anal. 59 (1984), 498-534. 21. M. Pimsner, A class of C*-algebras generalyzing both Cuntz-Krieger algebras and crossed products by Z, in Pree Probability Theory, D. Voiculescu, Ed., Fields Institute Comm. 12, 189-212, Amer. Math. SOC.,Providence, 1997. 22. S. Popa, Correspondences, Preprint (1986). 23. G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Oper. Theory 22 (1989), 51-71. 24. G. Popescu, won Neumann inequality for B(3tn)l,Math. Scand. 68 (1991), 292-304. 25. M.A. Rieffel, Morita equivalence for C* -algebras and W* -algebras, J. Pure Appl. Alg. 5 (1974), 51-96. 26. M.A. Rieffel, Induced representations of C*-algebras, Adv. in Math. 13 (1974), 176-257. 27. M. Skeide, Commutants of von Neumann modules, representations of B4(E) and other topics related to product systems of Hilbert modules, to appear in the proceedings of the AMS Joint Summer Research Conference “Advances in Quantum Dynamics” , held at Mt. Holyoke College in the summer of 2002. 28. M. Skeide, won Neumann modules, intertwiners and self-duality. Preprint 2003.

EXTENDABILITY OF GENERALIZED QUANTUM MARKOV STATES

H.OHNO Graduate School of Infomataon Sciences, Tohoku University, Aoba-ku, Sendai 980-8579,Japan E-mail: [email protected] The aims of this paper are to introduce the notion of generalized quantum Markov states, which extends translation-invariantquantum Markov states and C*-finitely correlated states in the setting on a UHF algebra to those on an AF C*-system with a shift homomorphism,and to show the extendability of generalized quantum Markov states on gauge-invariant parts of UHF algebras.

1. Introduction The notion of quantum Markov chains was introduced by Accardi in [I], and as a special case of quantum Markov chains, the notion of quantum Markov states was defined by Accardi and F’rigerio in [2], and the notion of C*-finitely correlated states was discussed by Fannes, Nachtergaele and Werner [7]. Further discussions on quantum Markov states are found in [3], [8] and [9] for example. In the present paper we introduce the notion of generalized quantum Markov states, which extends translation-invariant quantum Markov states and C*-finitely correlated states in the setting on a UHF algebra to those on an AF C*-system with a shift homomorphism. Among typical examples of such AF C*-systems are gauge-invariant parts of UHF algebras with the usual right shift and the Temperley-Lieb algebras with the Jones shift. A generalized quantum Markov state is defined by an initial state and a sequence of completely positive maps with Markov property. In this paper we concentrate our discussions on generalized quantum Markov states on gauge-invariant parts of UHF algebras, and our most interest is in the extendability problem of extending those generalized quantum Markov states to Markov states on ambient UHF algebras. It is worth noting that this kind of extendability problem has the same feature as in [4], where the extension of KMS states was treated in a similar situation.

41 5

41 6

In Section 2 we first introduce generalized quantum Markov states, and show that any sequence of completely positive maps defining a generalized quantum Markov state on a UHF algebra is determined by a single completely positive map so that the generalized quantum Markov states on UHF algebras are nothing but C*-finitely correlated states. In Section 3 let A be the gauge-invariant subalgebra of the UHF algebra B = Md by a unitary subgroup G in the d x d matrix algebra Md, that is, A = U, A, with A, = the fixed point subalgebra under the product action by G. We prove that if G is abelian, then any generalized quantum Markov state on A extends to a Markov state on B. Furthermore, this extendability remains true in some non-abelian cases of G being a symmetric group or an infinite dihedral group.

@Fl

2. Definition and some examples

In this section, we define generalized Markov states and consider some examples. Let

U121%$?4?4&*"' be an inductive system of finite-dimensional C*-algebras M, (n E w) with embeddings j, : M, + %,+I, where embeddings are injective unital homomorphisms. Moreover we assume that there exists a sequence of injective unital homomorphisms on : a, + %,+I (n E w) such that j n + l 0 on

= on+1 0 j ,

(1)

for all n E N. Then we have the inductive C*-algebra M = lima, (= 4

U,"==, 24,)

and the inductive limit injective homomorphism o = limo, on ---t U. We call the homomorphism o the shift on U.

Definition 2.1. Let E, : %,+I maps such that

+ M,

(n E

w) be unital completely positive

E n + l o o n + l = on 0 E n , En+10 j n + l o

j n =jn,

nE

(2)

N

(3)

Moreover, let p be an initial state on U1 such that

A E 81.

p(A) = @ l o

Then a state

+

o,(A)), on U can be defined by + ( A )= p ( E l

o Ez 0

-

* *

0

En 0 j n ( A ) )

(4)

41 7

for

and

In fact, this is well defined since

for all A E 8,. Furthermore, Q is a-invariant because

= p(E1

oE1 o E2 o..*o En o j n ( A ) ) = p(E1 0 E2 0 0 En o j n ( A ) ) 001

*

*

a

= $(A)

an.

for all A E We call the state 4 on 8 defined above the generalized Markov state on (a,a) associated with { E n } and p.

( n E N) are different embeddings with the same If j n : 8, + Bratteli diagrams as j n % , then we can choose unitaries Un E 8, such that

jn

= AdUn+l o j n 0 AdU:

for n E N with Ul = I . Here, the inductive limit C*-algebra & is isomorphic In this situation, if we set to 8 by the isomorphism cy = limAdU,. --+ 6n = AdUn+lo 0 AdU:, En = AdU, o En o AdU:,,,

nE

N,

then it is immediate to check that {jn}, {&n}l {An}and p satisfy (1)-(4). The corresponding generalized Markov state 4 on & satisfies

iJ(AdUn(A))= Q(A)

an.

for all A E In this way, we observe that (&, 8,b) is conjugate to (8,O, Q) via the isomorphism a.

Example 2.1. Let 8, = @:=, Md, where Md is the d x d matrix algebra, with canonical inclusion 8, + an+1.Then 8 = be the UHF algebra Md. Let a be the usual right shift on 8. For each unital

@gl

u;=,Mn

418

completely positive map E : Md 8 Md such that

+ Md

p(E(IMd €3 A ) ) = p ( A ) , a 0-invariant state 4 on M is defined by

and for each state p on

hfd

A E Md,

~ ( A8I * * €3 An) = p(E(A1€3E(A2 8 * * * 8 E(An 8 I M ~ *)*

a)))

for n E N and A l , . . . , A n E Md. I f we set En = id%,-, 8 E , then this is a generalized Markov state associated with { E n } and p. This example shows that the generalized Markov states are the generalization of translation-invariant quantum Markov states introduced by Accardi and Fkigerio [2], and C*-finitely correlated states due to [7].

Theorem 2.1. If 4 i s a generalized Markov state o n a UHF algebra t h e n it i s a Markov state introduced in E x a m p l e 2.1.

a,

are fixed by En, we have

Proof. Since the elements of

En(XY)= XEn(Y) for X E 8,-1

and Y E Mn+1 ([5]). Therefore, by using (2), we get

En(X 8 A ) = X E , ( I M ~ @ ~8- 'A ) = X 8 El(A) for X E Mn-l and A E Md €3 Md. Hence, we obtain En = id%,-, €3 El.

Example 2.2. Let '13 be the UHF algebra @,"=, Md, and G be a subgroup of the unitary group of order d. We set

=

n

n

i=l

i=l

(8 Md)G = { A E 8Md I g@"Ag*@'

= A , for all g E G),

Up==,

the fixed point algebra under the product action by G, and Q = !21n that is the gauge-invariant part BGof '13 ([lo]). Let CJ be the restriction of the usual right shift to Q. A unital completely positive map &' : Md 8Md + Md and a state 3, on Md are taken as in Example 2.1. Here, we assume further that E satisfies the G-covariance condition @(Ad(g €3 g ) ( A 8 B ) )= Adg&(A €3 B) for all A , B E Md and all g E G. Then for each n E N,we can define E, : %,+I + I#, by En = &lMn+l with = idBy-l Md €3 &, and we set ,=l p = p1M1. Let be a Markov state on '13 associated with {&} and p , and 4 be the restriction of to a. Then is a generalized Markov state on (%, o) associated with {En} and p.

6

6

41 9

3. Extendability of generalized Markov states on gauge-invariant parts

(@El

Md)G be a gauge-invariant part of the Definition 3.1. Let M = UHF algebra Md as in Example 2.2. A generalized Markov state q5 on M is said t o be extendable if there exists a Markov state on Md such that 41% = 4.

@El

4

@zl

In this section, we consider the extendability of generalized Markov states on gauge-invariant parts. The next theorem is our main result. Theorem 3.1. Let G be an abelian subgroup of the unitary group of order d . Then every generalized Markov state on the gauge-invariant part M = Md)G is extendable.

(@El

To prove this theorem, we have some preparations. When {eij}lli,jSd is a system of matrix units of Md, we write (il . in,j l .jn) for the matrix unit eiljl @ . @ ei,,j,, of Md.

- -

-

-

Lemma 3.1. If G is an abelian subgroup of the unitary group of order d , there exists a system {eij}lsi,j
g@yil. . .in,jl. . .jn)g*@n= gil - -

*

. . .g3;yi1 .. ‘ Z., , J 1.

ga,gj,l

* *

.jn).

This implies that

. .

M, = span((i1. * ’2,,J1*

. j n ) I gil

*ging;l.. .g;l = 1, for all g = diag(gl, . . .9,) E G}.

*

Hence, we get matrix units of U, as desired.

0

In the following, we fix a system of matrix units of Md as in Lemma 3.1 for a given abelian unitary subgroup G. Remark 3.1. If the element (il -in,j l -

a

j,) satisfies

# { k I i k = m }= # { Z I j l = m } ,

1 5 W A5 d,

420

-

-

then (il . in, jl . j n ) E M, from the proof of Lemma 3.1. In particular, (il . . in, il . in) is contained in Mn for all 1 5 i l , . . . ,in 5 d. Moreover, if then (il . .in+l,jl. -jn+l)is contained in

-

9

-

-

.*

&(il = (il

*.

*

jl * * .jn+l) il . . i n - l ) . E,(il

in+l,

-

.

*-.in+l,jl

.. . & + I )

* ( jl* * * A - 1 , jl . . * jn-1) - (il...i,-1,jl...j,-1)€3T

for some T E

Md.

For convenience, we write I, J , K and L for the words il . . .in, j1 kl . . .k, and I1 . - l,, respectively. Proof of Theorem 3.1. For ( l i k , J j l ) E 21n+2, there exists that E,+1

* *

j,,

T E Md such

(Iik,JjZ) = (I,J ) €3 T.

First, we prove that we can define the map E : Md €3 Md + Md by E(ik,jZ) = T independently of the choice of I , J . To do so assume (Kit, LjZ) E 21m+2, and we write E,+1

(Kik,LjZ)= ( K ,L ) €3 s

for some S E Md. Notice that (KikIik,LjlJjl) E M,+m+4.

Then

En+,+3(KikIik,LjZ JjZ) = (Kik, L j l ) €3 E,+l(lik,JjZ) = (Kik, LjZ) €3 (I,J ) 8 T and it also equals

Hence, we get s = T. F’urthermore, for arbitrary ( i k , j l ) E Md €3Md, the element (jZik,ikjZ)is contained in ad. Therefore, we can define the map E on Md €3 Md as stated above, and E satisfies En = id@?-l M d €3 EI2in+1. Now, it remains to prove that E is a completely positive mAil. For the words ij (1 5 i,j 5 d), we define the words I i j = 1- ..(i - l)(i 1) . d l ( j -

+ -

42 1

+

-

l)(j 1) . . d, and consider the matrix (Ill11, Ill 11)

(&Id&,

111 11)

..

*

(11 111, I d d d d )

. . . (Id&,

I d d W

From the complete positivity of E 2 d 2 ~ it~ ,follows that E(11,ll)

* * *

E(l1,dd)

E(dd,11) . * * E ( d d , dcl) is positive. This implies ( [ S ] ) that E is a completely positive map.

0

Example 3.1. In the case that d= 2 and

the form of E can be written by the positive matrix

with the property 41,ll

+4

2 1

+ a?2,12 + @2,22

=1

for i = 1,2. For a general non-abelian group G, we do not know whether the generalized Markov states on ( @ E l Md)G are extendable or not. In the following, we show that this is true for two special non-abelian groups.

422 Proposition 3.1. Let G be the infinite dihedral group

Then every generalized Markov state on

---

(@ElM z ) is~ extendable.

-

Proof. The element (il in, jl. .j , ) is invariant under the automorphisms Ad(diag(z-l, I)@,) if and only if it satisfies #{k 1 ik = m} = #{ 1 1 j , = m} for m = 1 , 2 . Moreover, since

and

we get

..

an= span((i1. "in,jl .jn)+ ( i l . . ' i n , j l . -3,) I *.

#{k

A

A

*

I ik = m} = #{Z I j , = m},for m = 1 , 2 } ,

where i = 2 and d = 1. Note that the above set spanning I#, is just a linear basis and is not necessarily a system of matrix units. First, we show that for (Iilc, JjZ) (Iik,JjZ) E Mn+2 there exists T E Md such that

+

*A&

En+i((Iik, JjZ) where

1

=

21

,..*A

+ (ilk,jji))= ( I ,J ) @ T + (P, j ) 8 T ,

and f' is defined by linearly extending (il---in,jl...jn) . Indeed,

---in

( i l - - - i n , j l .-.jn)=

A

*

A

E,+l((Iik,JjZ) + (iii,jji)) = ( ( I , I ) + ( i , f ).E,+1((Iik,JjZ)+(iik,jji)).((J,J) ) +(j,j)) = ( I ,J) @ T + (i, j ) @ T ( I ,j ) @ R + (f,J) @ R

+

for some T ,R E M d . Now, we have

E,+l((Iik,JjZ) + (Iik,JjZ)) = IM., @ ((I,J ) @ T + (i,j ) @ T + (I,3) @ R + (f,J) @ R) AAA

A ..

0 0

+ ( 2 1 , 2 J ) 8 T + (li,1s)@ T + ( 2 f , 2 4 8 F + (11,l j )@ R + ( 2 1 , 2 j ) @ R + (li,1 J ) @ R + ( 2 i , 2 J ) @ R = (lI,lJ)8 T

423 and from (2) it also equals

+

AAA

A ....

En+2 0 a ( ( I i k ,JjZ) (Iik,JjZ)) = (II,lJ)€3 A + ( 2 i , 2 j ) €3 A + (21,2J) €3 B + (li,l j ) €3 B + ( l I , 2 3 ) €3 c ( 2 1 , l J ) €3 ( 2 1 , l J ) c3 D ( 1 1 , 2 j ) €3 fi

e+

+

for some A , B , C , D € Md. Hence, we get Next, we assume 1 5 n 5 m and

+

R = C = D = 0 and T = A = B .

+ (R,i)€3 3

Em+l ((Kik, LjZ)+ ( R 2 ,iji))= ( K ,L ) €3 s for some S €

We prove S = T. Indeed,

Md.

( ( I L K ,L K I ) + (iiR,iBi))

+ ( ~ i kJ,j l ) ) ) A,..

B En+1 ((~ik, Jjz)

( I M ~' 2 m

*

+

A,.,.

( ( L K J ,J K L ) (iRj, Ski)) = ( I L K ,J K L ) 8 T + (iM, j k L ) €3 'f *

and it also equals

+

1 , .

..*A

A

-

A,.,.

E2m+n+1((ILKik,JKLjZ) (ILKik,JKLjZ)) = ( ( I L K ,I L K ) + (iik, jilt))

+ ( K i k ,LjZ))) A,.,.

.(IMd

€3 Em+l ( ( K i k ,L j l )

AA,.

((ILL, J K L ) + (iti, jRi)) = ( I L K ,J K L ) €3 s (it& M i )€3 3, *

+

implying S = T . Hence, we can define E(ik,jZ)= T. This definition gives En = idB?-l Md €3 E121n+1 for n 2 1. In the case El : 242 + 2 4 1 , ,=1

+ (ik,j l ) ) = I M d c3 El ((it%,$) + (ik,jZ)) = I M d €3 (S + 3) I . .,.A

a 0 El ((ik,jZ)

for some S E

Md

,.A

A,.

and from (2) it also equals

+ (ik,j 2 ) ) = I M €3~ ( E ( i k ,jZ) + E(ik,jZ)), A A

E2 o a((ik,jZ)

A

1

so that

+ (ik,jZ)) = E(ik,jZ) + E(ik,jZ) = E ( i k ,jZ) + E ( i k ,jZ). L A

El ((ik,jZ)

A -

* A

,.A

Therefore, E satisfies El = E(M2 as well. Finally, we can prove the complete positivity of E in the same way as in the proof of Theorem 3.1 but with some complication. 0 Proposition 3.2. If G is a symmetric group Markov state on ( @ F l Md)G is extendable.

Sd,

then every generalized

424

Proof. For any cr E

sd,

we define CY : a,

. in, jl .j,))

cr((i1

* *

An element A E element B E

* *

@r=l

Md

+ a,

by linearly extending

= ( c r ( i 1 ) . . * a(&),cr(j1)

M d belongs to such that

Qn

. - a&)). *

if and only if there exists some

Therefore, we get Q, =span{

c

a ( ( & . . i , , j l . * .jn)) 11 5 i l , . . . ,i,,,jl,. . . , j , 5 n } .

aESd

First, we prove that for any such that

xaESd cr(lik, JjZ) E

Mn+2,

there exists X E

M d

Indeed,

We

where

for some

have

where

is the set of all words with length n, and from (2) it also equals

,!& E M d . Let N = 1 2 - e . d E m d . Since P(W) = y(W) = N for some X implies ,B = y,we see that (Ncr(I),N P ( J ) )@ X a o # 0 (or X,p # 0) implies that ( a ( l ) , P ( J ) = ) (y(l),y(J)) for some y E s d . Hence, we can write E,+1(

c

aESd

cr(lik, J j l ) ) =

c

aESd

4 1 ,J ) €4 x a

425

xa

for some E Md. But this element is of the form CaESd a(B)for some BE Md, and we easily see that it can be written as CaESd ~ ( ( 1J)€3 , X ) for some X E Md. Next, assume that a ( l )# I and a ( J )# J for d l 01 E s d except identity. We show that the map E : Md€3Md -+ Md can be defined by E(ik,jZ) = X if

my=','

c

&+1(

(YESd

41% J j l ) ) =

c

a((1,J) €3 XI.

aESd

To do so assume

where

for some Y

and

dor all

except

identity. We obtain

and it also equals

implying Y = X. Hence, we can define E (ik, jl) = X. Furthermore, for each

Since

426

and

we get For general I and J, we define

ssubgroup of G. We obtain

for some

Hence, we get

This implies that

and it also equals

427

p

where is an equivalent class of S ~ I H I JWe . get En = id@?-i Md@E124n+l. Finally, we can prove the complete positivity of E in the'&ne way as in the proof of Theorem 3.1 but with some complication. 0 In these two examples, the proofs depend on a special structure of some basis of 24,. Hence, in the case where a basis is more complicated, for example G is the unitary group U ( 2 ) ,we can not argue in a similar way. References 1. L. Accaxdi, Proc. School of Math. Phys. Camerino, 268 (1974). 2. L. Accardi and A. F'rigerio, Proc. Roy. Irish. Acad. 83A(2), 251 (1983). 3. L. Accaxdi and V. Liebscher, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2, 645 (1999). 4. H. Araki, R. Haag, D. Kastler and M. Takesaki, Comm. Math. Phys. 53, 97 (1977). 5. M. D. Choi, Illinois J . Math. 18, 565 (1974). 6. M. D. Choi, Linear Alg. Appl. 10, 285 (1975). 7. M. Fannes, B. Nachtergaele and R. F. Werner, Commun. Math. Phys. 144, 443 (1992). 8. V. Y . Golodets and G. N. Zholtkevich, Teoret. Mat. Fiz. 56(1), 80 (1983). 9. H. Ohno, Interdisc. Inf. Sci. 10(1), 53 (2004). 10. G. Price, J . F h c t . Anal. 49, 145 (1982).

WHITE NOISE APPROACH TO THE LOW DENSITY LIMIT

A.N. PECHEN* Centro Vato Volterra, Unaversata da Roma Tor Vergata 00133, Roma, Italy

The white noise approach to the investigation of the dynamics of a quantum test particle interacting with a dilute Bose gas is presented. In this approach one proves that the appropriate operators of the Bose gas converge, in the sense of correlators, to operators constructed from some quantum white noise. The limiting dynamics is described by a quantum white noise equation or an equivalent quantum stochastic equation driven by a quantum Poisson process. These equations are applied to derivation of a quantum Langevin equation and a linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper is devoted to the approach which was developed by L. Accardi, I. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. In the second part the development of the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fork-antiFock representation, is described. This simplifies the derivation and allows t o express the strength of the quantum number process directly in terms of the one-particle S-matrix.

1. Introduction

The fundamental equations in quantum theory are the Heisenberg and Schrodinger equations. However, it is a very difficult problem to solve explicitly these equations for realistic physical models and one uses various approximations or limiting procedures such as weak coupling and low density limits. These scaling limits describe the long time behavior of physical systems in different physical regimes. One of the powerful methods to study the long time behavior in quantum theory is the stochastic limit method, which was developed by Accardi, Lu and Volovich in the book1, devoted mainly to the weak coupling regime, where one studies the long time dynamics of a quantum open system weakly interacting with a reservoir. The dynamics of the total system in this limit *permanent address: Steklov Mathematical Institute of Russian Academy of Sciences, Gubkin str. 8, 119991, Moscow, Russia. E-mail: pechenQmi.ras.ru

428

429

is described by the solution of a quantum stochastic differential equation driven by a quantum Brownian motion; the reduced dynamics of the system is described by a quantum Markovian master equation. Also in the low density regime the dynamics is described by a quantum stochastic differential equation. In this regime one considers the long time quantum dynamics of a test particle interacting with a dilute Bose gas in the case the interaction is not weak but the density n of particles of the gas is small. The low density limit is the limit as n + 0 , t + co,nt = Const (t stands for time) of certain quantities like matrix elements of the evolution operator and the reduced density matrix. In this limit the reduced time evolution for the test particle will be Markovian, since the characteristic time ts for appreciable action of the gas on the test particle (time between collisions) is much larger than the characteristic time t R for relaxation of correlations in the gas. Accardi and Lu2-* and later Rudnicki, Alicki, and Sadowski5 proved that the matrix elements of the evolution operator (in the ”collective vectors”) converge in the low density limit to matrix elements of the solution of a quantum stochastic differential equation driven by a quantum Poisson process. The quantum Poisson process, introduced by Hudson and Parthasarathy6, should arise naturally in the low density limit, as conjectured by fiigerio and Maassen’. The stochastic golden rule for the low density limit, that is a set of simple rules for the derivation of the limiting quantum white noise and stochastic differential equations, was developed in Refs 8-10, where the case of a discrete spectrum of the free test particle’s Hamiltonian was considered. This method is based on the white noise approach and uses the stochastic limit technique1 (in Ref. 11 the method is generalized to the case of a continuous spectrum). As the main result, the normally ordered quantum white noise and stochastic differential equations were derived. Then these equations were applied to derivation of the quantum Langevin and master equations describing the evolution of test particle’s observables. The idea of the white noise approach is based on the fact that the rescaled free evolution of appropriate operators of the Bose gas converges, in the sense of convergence of correlation functions, to operators constructed from some quantum white noise operators. Then, using these limiting operators which are called master fields, one can derive the quantum white noise equation for the limiting dynamics and put this equation to the normally ordered form which is equivalent to a quantum stochastic differential equation.

430

The white noise approach to the derivation of the stochastic equations in Refs. 2-4, 8, 9 uses the Fock-antiFock representation for the canonical commutation relations (CCR) algebra of the Bose gas, which is unitary equivalent to Gel'fand-Naimark-Segal (GNS) representation. The white noise approach of Ref. 10 considerably simplifies the calculations by not using the Fock-antiFock representation. A useful tool is the energy representation introduced in Refs. 8,9,where the case of orthogonal formfactors was considered. This was extended, in Ref. 10, to the case of arbitrary formfactors and arbitrary, not necessarily equilibrium, quasifree low density states of the gas. In Ref. 10 to each initial low density state of the Bose gas in the low density limit one associates a special "state", which is called a causal state, on the limiting master field algebra. The time-ordered (or causal) correlators of the initial Bose field converge to the causal correlators of master fields, which are number operators constructed from some white noise operators. This approach allows to express the intensity of the quantum Poisson process directly in terms of the one-particle S-matrix. In this case the algebra of the master fields (20), the limiting equation (28), and the quantum It0 table (30)do not depend on the initial state of the Bose gas (see Sect. 4 for details). Instead, the information on the initial state of the gas is contained in the limiting state ' p ~ of the master field [defined by (23)-(26)], which is now not the vacuum. Here L is determined by the initial density of particles of the gas; if the gas is in equilibrium at inverse temperature p, then L = e-PH1 (see the next section). To get the correct master equation one has to take the conditional expectation determined by the state ( P L . The dynamics in the low density limit is given by the solution of the quantum white noise equation (28), which is equivalent to the following quantum stochastic equation dUt = dNt(S - 1)Ut

(1)

where Ut is the evolution operator at time t , S the one-particle S matrix describing scattering of the test particle on one particle of the gas, and Nt(S - 1) the quantum number process with strength S - 1. In order to describe these objects let us introduce two Hilbert spaces Xs and XI, which are called in this context the system and one-particle reservoir Hilbert over the Hilbert space of squarespaces, and the Fock space r(L2(&; XI)) integrable measurable vector-valued functions from & = [0, co) to XI. Then the solution of the equation is a family of operators U t ; t 2 0 in XS@l?(L2(R+;3c1))(adapted process); S is a unitary operator in 3ts @XI.

431

Let us define the number process. Let X be a self-adjoint operator in a Hilbert space Ic; for any f E Ic let Q(f) be the normalized coherent vector in the Fock space r(Ic). The number operator N ( X ) is the generator of the one-parameter unitary group r(eitx) characterized by r(eitx)Q(f) = Q ( e i t x f ) ;t E R The number operator is characterized by the property ( Q ( f ) , N ( X ) Q ( g )= ) ( f , X g ) ( Q ( f )Q , ( g ) ) . The definition N ( X ) is extended by complex linearity to any bounded operator X on of Ic. Let us consider Ic of the form L2(lR+;Xl)E L2(lR+)€3I XI.For any bounded operators X o E B ( X s ) , X I E B ( X 1 ) and for any t 2 0 define Nt(X0 €3I X I ) := X O€3I N(X[o,t]@ X I )and extend this definition by linearity to any bounded operator K in X S €3I X I . The family { N t ( K ) } t ? o of operators in X S €31r(L2(R+;Xl)) is called qzlantum number process with strength K. Equations (1) and (16) describe the dynamics of the total system and can be applied, in particular, to derivation of the irreversible quantum linear Boltzmann equation for the reduced density matrix of the test particle. This equation can be easily obtained from the quantum Langevin equationg. The reduced dynamics of the test particle in the low density limit with methods, based on a quantum Bogoliubov-Born-Green-KirkwoodYvon (BBGKY) hierarchy, has been investigated by Diimcke12, where it is proved that, under some conditions, the reduced dynamics is given by a quantum Markovian semigroup. In the white noise approach the reduced dynamics can be easily derived from the solution of the limiting quantum stochastic differential equation. Namely, the limiting evolution operator Ut and the limiting state (PL determine the reduced dynamics by

Z ( X ) = (PL(U,+(X€3I l)Ut),

(2)

where X is any observable of the test particle, ( P L ( - )denotes the conditional expectation, and T t is the limiting semigroup. This equality shows that Ut is a stochastic dilation of the limiting Markovian semigroup. Using the quantum Ito table for stochastic differential dNt one can derive a quantum Langevin equation for the quantity @ ( X €3I 1)Ut. Then, taking partial expectation, one gets an equation for T t ( X )and obtains the generator of the semigroup (see end of Sect. 4). This is a general feature of the white noise approach: one first obtains the Langevin equation and then gets the reduced dynamics of the test particle. Let us note that although the quantum stochastic equation (16), which was derived in Refs. 2,9, is different from (1) it gives the same reduced dynamics.

432

The low density limit can be applied to the model of a test particle moving through an environment of randomly placed, infinitely heavy scatterers (Lorentz gas) (see the review of Spohn13). In the Boltzmann-Grad limit successive collisions become independent and the averaged over the positions of the scatterers the position and velocity distribution of the particle converges to the solution of the linear Boltzmann equation. An advantage of the stochastic limit method is that it allows us to derive equations not only for averaged over reservoir degrees of freedom dynamics of the test particle but for the total system+reservoir. The convergence results and derivation of the linear Boltzmann equation for a quantum Lorentz gas in the low density and weak coupling limits are presented in Refs. 14, 15. The Coulomb gas at low density is considered in Ref. 16. The structure of the paper is the following. In Sec. 2 a test particle interacting with a Bose gas is considered. In Sec. 3 the white noise approach developed by L. Accardi, I. Volovich and the author in Refs. 8, 9 is presented. Sec. 4 devoted to the white noise approach developed in Ref. 10. The main results are: the causally normally ordered quantum white noise equations (16), (27) and equivalent quantum stochastic equation (34) for the limiting evolution operator; the quantum Langevin equation (17) for the evolution of any test particle’s observable; the linear Boltzmann equation for the reduced density matrix (Theorem 3.4). 2. Test Particle Interacting with a Dilute Bose Gas

Consider two non-relativistic particles, with masses M and m, which are called the test particle and a particle of the gas. Suppose the particles interact by a pair potential U ( R - T ) , where R and T denote positions of particles. Then the classical dynamics of the particles is determined by the Hamiltonian Hcl = P2/2M +p2/2m U ( R - T ) , where P and p are momentums of the particles. The quantum Hamiltonian of such a system is obtained by identification of P with the momentum operator ?, R with the position operator Q and, if instead of one particle of mass m there is gas of these particles, by second quantization of the particles of the gas. This Hamiltonian has the form

+

H = HS

+

+ H R + Hint

where H s H R =: Ho is the free Hamiltonian with H s = P2/2M, H R = ~ w ( p ) a + ( p ) a ( p ) d pw,( p ) = p2/2m; the interaction Hamiltonian is Hint = J U ( Q - T ) u + ( T ) u ( T ) ~ T . The free Hamiltonian of one particle of the gas H I

433

is the multiplication operator by the function w . Boson annihilation and creation operators u ( k ) ,u+(p) satisfy the canonical commutation relations

M k ) ,.+@)I

= 6(k - P)

The coordinate representation for these operators is introduced as u (r) = J ei"u(k)dk. The interaction Hamiltonian can be written using the Fourier transform of the interaction potential O(p) := J U(r)eiPrdras Hint

=

s

dkdpO(p)eipQ8 ~ ' ( kp)~(k)

(3)

This Hamiltonian acts in the Hilbert space 31s 8 I'(311), where 31s = 311 = L2(R3 ) . Important features of this system are that the interaction Hamiltonian Hint quadratic in creation and annihilation operators and commutes with the number operator, i.e., it preserves the number of particles of the gas. We will consider, instead of (3), a different interaction Hamiltonian, which nevertheless keeps its basic properties. More precisely, we consider Hamiltonians of the form Hint

= D 8A+(go)A(gl)

+ D+ 8 A + ( g l ) A ( g o )

where D is a bounded operator in 31s; go, g1 E 311 are two form-factors, and A ( g 0 ) = J d k g , * ( k ) u ( k )is the smeared annihilation operator. This Hamiltonian is also quadratic in creation and annihilation operators and preserves the number of particles of the gas. In the present paper we consider the case of a discrete spectrum for the free test particle's Hamiltonian, so that n

where E~ is an eigenvalue and Pn is the corresponding projector. This corresponds to the situation when the test particle is confined in some spatial region. Density of particles of the gas is encoded in the state of the gas, which is chosen to be either the Gibbs state at inverse temperature 8, chemical potential p , and fugacity E = e P p , or more general non-equilibrium Gaussian state. That is a gauge invariant mean zero Gaussian state with two point correlation function

434

Here ( > 0 is a small positive number and L is a bounded positive operator in commuting with the one-particle free evolution St = eitHl (the multiplication operator by a function L ( k ) ) . In the case L = e-PH1, so that L(k) = e-Pw(k),the state ( P L , is ~ just the Gibbs state with the two point correlation function

m,c(a+(k)a(k’))= n ( k ) W - k’) where n(k) is the density of particles of the gas with momentum k:

Notice that in the limit J + 0 the density goes to zero. Therefore the limit ( 0 is equivalent to the limit n(k) 0. The dynamics of the total system is determined by the evolution operator which in interaction representation has the form V(t) := eitHf==e-itHtot. The evolution operator satisfies the differential equation

+

+

where Hint(t) = eitHfr- Hinte-itHfree is the free evolution of the interaction Hamiltonian. The iterated series for the evolution operator is

With the notation D(t) := eitHsDe-itHS the evolved interaction becomes flint(t) := D(t) 8 A+(Stgo)A(Stg1)

+ o + ( t ) 8 A+(stgl)A(Stgo).

Using the spectral decomposition (4)and introducing the set of all Bohr frequencies B, that is the spectrum of the free test particle’s Liouvillean i[Hs, -1, one can write the free evolution of D as D(t) =

c

Duewitw;

D, =

c

PkDPm

The reduced dynamics of any test particle’s observable X in the low density limit is defined as the limit

where ( P L ,(-) ~ denotes partial expectation. The reduced density matrix p(t) is defined through the duality Tr (p(O)Tt(X)) = Tr (p(t)X). As it was

435

mentioned in the Introduction, in the white noise approach the generator of the limiting semigroup can be easily derived from the quantum white noise equation.

3. The White Noise Approach In Refs. 2, 3, 9 the dynamics of the total system is constructed in the FockantiFock representation for the CCR algebra, which is unitary equivalent to the GNS representation. It is defined as follows. Denote by 3-1: the conjugate of XI,i.e. 3-1: is identified to 3-11 as a set and the identity operator L : 3-11 -+ ?Ilis antilinear: Vf E 3-11, c E C 4Cf)

= C*L(f),

(L(f),

&))'

= ( 9 , f)

Then, 3-1; is a Hilbert space and, if the vectors of 3-11 are thought as ketvectors I(), then the vectors of 31: can be thought as bra-vectors ([I. The corresponding Fock space r(3-1:)is called the anti-Fock space. In this section we assume that for any t E R (go, Stg1) = 0. In this case it was shown in Ref. 2 that the dynamics of the total system is given by 8 r(3-1;)which satisfy the family of unitary operators U,(') in 3-1s8 r(3-11) the Schrodinger equation:

atup = -iH<(t)U,('),

u p = 1.

Here the part of the modified Hamiltonian, which gives a nontrivial contribution in the low density limit, has the form

+

+&[A(Stg1) 8 A(Ste-sH1/2go) A+(Stgo)8 A + ( S t e ~ @ ~ ~ /+~h.c. gl)]}

+ t / [ the evolution

After the time rescaling t equation

operator satisfies the

where we introduce for each n, rn = 0 , l and w E B the rescaled fields: 1 N n , m , ~ ( ~:=, t-e-itwf'A+ ) (St/cgn)A(St/,grn)8 1 (7)

E

1

.

Bn,m,c(w,t ) := -eEtW''A(St/tgn) @ A(St/te-BH1/2grn) ( 8 )

./T

and B : r n , E ( t~),is the adjoint of Bn,rn,~(w, t).

436

3.1. Master Field It is convenient to use the energy representation for the investigation of the limit as £ -» 0 of the rescaled fields (7), (8). It is defined in terms of the projections PE := o(H, - E)

which satisfy the properties PEPE, = 8(E - E')PE,

St= I dEPEeitE

PB = PE,

Define the energy representation for the fields (7), (8) as it(Ei-Ea-w)/S

1 7=V?

l

n

2

'

*

g

m

(9) ) (10)

and let B+tmtf(E1,E2,u,t) be the adjoint of Bn!m^(Ei,E2,oj,t). The operators (7), (8) can be expressed in terms of (9), (10) as ,t) = f ,t) = f

Let Q be the set of all linear combinations of the Bohr frequencies with integer coefficients: ft = {w | cj = ^2k nkuk withnj; 6 Z, wfc e B}. Extend the definition of the fields (9), (10) to arbitrary w € ft. The limit as £ -)• 0 of the fields (9), (10) was found in Ref. 9 and is given by the following theorem. Theorem 3.1. The limits of the rescaled fields E2,w,t),

X = B,B+,N

exist in the sense of convergence of correlators and satisfy the commutation relations <5riX(5m,m/(5(t/ -t) E3)S(E2 - E4)S(El -E2- ^)(gn,PElgn)(gm,PE^-^gm) (11) [Bnim(E1,E2,ui,t),Nn,<m,(E3,E4,uil,t')}

= 2v6n,n,6(t -t)

x6(Ei-Ea)S(Ei-E2-u){gn,PE1gn)Bm,tm(E4,E2,u-ul,t) [Nntm(El,E2,u,t),Nnltml(E3,E4,u',t')} x{Sm,nl6(E2-E3)6(E3-E1

(12)

= 2irS(f-t)

+u){gm,PBa9m)Nntm,(Ei,E4,u+ul,t)

-E4)S(E3 -El -w ; )<Sn, J PE 1 5«)^',m(-E 3! -E2 I w + a;

437

The causal commutation relations of the master field are obtained replacing in (11)-(13) the factor d(t' - t ) b y d+(t' - t ) , where the causal &function 6+(t' - t ) is defined in Ref. 1, section (8.4); 27rS(E1 - E2 - w ) b y (i(E1E2 - w - iO))-l and 2nh(E3 - El fw ) by (i(E3- El f w - i O ) ) - ' . 3.2. Fock Representation of the Master Field

The next step is to realize the algebra of the master field by operators acting in a Hilbert space. We realize it in the Fock space, which is constructed as follows. Let K be a vector space of finite rank operators acting on the oneparticle Hilbert space 3c1 with the property that for any w E R; X , Y € K

s =

( x , Y >: = ,

2n

dtTr (e-PHIX*StYS,*)e-iwt = -

s

d

~ (e-PHIX*PEYPE-,) ~ r

1

dtTr (e-p"lx*Yt) e-iwt

< 00

where Yt = StYS,* the free evolution of Y . It was shown in Ref. 3 that the space K is non empty and (., .), defines a prescalar product on K . Let K , be the Hilbert space with inner product (., obtained as completion of the quotient of K by the zero (., .),-norm elements. Denote K: := @ K,. W E Q

Consider the Fock space

W2(WK J o = r ( @ L 2 ( J w L ) ) = @ r ( L 2 ( J b , K w ) )

(14)

W E Q

W E 0

where the last infinite tensor product is referred to the vacuum vectors. Let b;,(X), bt,,(X) be the white noise creation and annihilation operators in this Fock space. These operators satisfy the commutation relations

Each white noise operator bt,,(.) acts as usual annihilation operator in r ( L 2 ( J bK,)) , and as identity operator in other subspaces. The representation of the algebra (11)-(13) can be constructed in the Fock space (14) by the identification

,E2 w , t ) = bt,w (IPEl gn) (PE29m1)

%,,(El

7

The number operator is defined as

c c k (El

.E=O,l W l E R

Nn,m(E1,E2, Q,t ) =

btwl ( 1gn)

- w1)

(%I

I

-WI % ) bt ,w1 --w

(1 PE2 g m ) (PEI-

WI

gE

I

438 with p E ( E ):= (gE,PEe-flH1g,)-l. One easily checks that these operators satisfy the commutation relations (11)-(13). Denote Nn,m (w, t ) :=

1

dE1 dE2N73,m(El, E2

W,

t)

Bn,rn(E,u,t) := /dE’Bn,m(E’,E,u,t) The limiting white noise Hamiltonian acts in

{

Dw8 No,I(w,t )

H(t) =

Xs €3 r ( L 2 ( R +€3) K) as

+I

}

dE[Bl,o(E, -u,t )+ B & ( E , u , t ) ] +h.c.

W E B

3.3. Quantum White Noise, Langevin and Boltzmann Equations

The white noise Schrodinger equation for the evolution operator in the low density limit is

&Ut = -iH(t)Ut

(15)

Following the general theory of white noise equations, in order to give a precise meaning to this equation we will put it in the causally normally ordered form, in which all annihilators are put on the right hand side of the evolution operator and all creators are on the left hand side. This procedure gives a normally ordered quantum white noise equation, which is equivalent to a quantum stochastic differential equation. For any n , m E (0, l},u E R, E E Iw, let R:;Y/(E) be operators in 3ts which are explicitly defined in Sec. 7, Ref. 9. The following theorem was proved in Ref. 9.

Theorem 3.2. The normally ordered f o r m of equation (15) as

+ c (R::r(E)BZ,m(g, u , t ) u t + &yAn(E)UtBn,m(E, u , t ) ) +q:r( E )(gn PEe-6 gm> ut] W

H1

(16)

The normally ordered equation (16) can be applied to derivation of the quantum Langevin equation for test particle’s observables. Let X be any observable of the test particle. The Langevin equation is the equation satisfied by the stochastic flow j t defined by jt(X) X t := U,’XUt.

=

439

Theorem 3.3. The quantity Xt satisfies the quantum Langevin equation: Xt=

E

n,m=0,l

(17)

w/iere tfie siructere maps are Q^12(X) := XR^(E) + R+™£(E)X + 2 E Re% (E + u)R+*£(E)XR^ (E) . Equation (17) can be written in terms of the stochastic differentials: n,m

where Z^'^^E) is a certain operator in £, which is explicitly defined in Ref. 9; £ is a quantum Markovian generator, which has the form of a generator of a quantum dynamical semigroup17:

C(X) = *(X) -

{*(!), X } + i(HeS, X]

Here = 27T E

dE{9e,PEe-^gE){g£lPB+ug£,)R'£(E)X(E)

is

a completely positive map and the effective Hamiltonian Hes := Y.JdE(ge,PEe-^gs)(^e(E)-Rl%(E))l'2i is selfadjoint. The following theorem is a direct consequence of the quantum Langevin equation (17), which was found in Ref. 9. Theorem 3.4. The reduced density matrix satisfies the quantum linear Boltzmann equation dp(t) ~ fjlll

where the generator £, is the dual to the generator £. The generator £» is the sum of its dissipative and Hamiltonian parts, £*(p) = £diss(/9) — i[HeSip}- The explicit form can be obtained as follows. Let T be the one-particle T-operator for the scattering of the test particle and one particle of the gas and Tnsn>(k,k') — (n,k\T\n',k') be its generic

440

matrix element, where In) is an eigenvector of H s with eigenvalue cn and k the momentum of one particle of the gas. Denote ~ u ( k , k ’:= )

C

Tm,n(k,k’)lm)(nl

m,n:E.,-E,,=W

Density of particles of the gas is determined by the function L(k) in (5). For the Gibbs state L(k) = e-flw(k).Other forms of the density correspond to non-equilibrium states of the gas and can be controlled, for example, by filtering. In these notations the dissipative part of the generator is

In the case the gas is in equilibrium this generator coincides with the generator of the quantum linear Boltzmann equation obtained in Ref. 12. 4. White Noise Approach without Fock-antiFock Representation

The approach to derivation of the quantum white noise equations directly in terms of the correlation functions, without use of the Fock-antiFock representation, was developed in Ref. 10. In this approach one introduces the notion of causal state and causal time-energy quantum white noise and proves the convergence of chronological correlation functions of operators 1 (18) N f , 9 , € (4 = -A+ (St/€f)A(St/cd

t

acting in I’(311) t o correlation functions of the time-energy quantum white noise (Theorem 4.1). This time-energy quantum white noise is a family of creation and annihilation operators, with commutator proportional to S-function of time and energy [see (20)]. These operators act in a Fock space which, in difference with (14), does not depend on the initial state of the gas ( P L , ~ . Suppose for simplicity that D ( t ) = D . Then the evolution operator V ( t / t )after the time rescaling t + t/t satisfies the equation

d Udt (t/t) =

-w €3

N90,9I,€(t)

+ D+ €3 ~ 9 1 , 9 0 , € ( t ) ) ~ ( t / t )

(19)

Theorem 4.1 and the causal commutation relations (21) are used to show that the limit as 6 + 0 of the rescaled evolution operator satisfies the causally normally ordered equation (28). That equation is equivalent to the

441

quantum stochastic equation (29) which can be written in Hilbert module notations as (32) and then in terms of the one-particle S-matrix as (34). As i t was stated in the Introduction, in this approach the algebra of the time-energy quantum white noise (20), the quantum Ito table (30) and the quantum stochastic equation for the limiting evolution (34) do not depend on the initial state of the gas. This is different from the approach of Sect. 3, where the commutation relations (11) for the master field, the Hilbert space representation (subsection 3.2) and hence the limiting equation (16) depend on the initial state (through the factor , - P H I ) . Instead, the dependence on the initial state of the gas now is contained in the limiting state ‘ p ~ (in the equilibrium L = e-PH1) (in the approach of Sect. 3 the state of the master field is the vacuum state). Considering the limiting state (PL as the conditional expectation [with the property (31)], one can derive the quantum master equation for the reduced dynamics of the test particle which coincides, when restricted t o the same model, with the analogous equation following from (17).

4.1. Causal Time-Energy Quantum White Noise Define the Hilbert space X N ~ , Has~the completion of the quotient of the set

{F

:&

S.t. llF112 := 27f

s

dE(F(E),PEF(E))< 00

1

with respect to the zero-norm elements. The inner product in X R ~ , H ~ is (F,G) = ~ ~ J ~ E ( F ( E ) , P E G ( Let E ) )Bf(E,t), . B,(E’,t’) be the creation and annihilation operators acting in the symmetric Fock space r(L2(&, X N , , H ~ ) over ) the Hilbert space L2(&, X x 1 p I ) of square integrable functions f : & + Xx,, H ~ .These operators (operator-valued distributions) satisfy the canonical commutation relations

[B,(E,t),Bf(E’,t’)] = 27rd(t’ - t)d(E’ - E)(g,PEf)

(20)

and causal commutation relations

[B,(E,t), Bf(E’,t’)l = d+(t’ - W E ’- E)-Y,,f(E)

(21)

where S+(t’-t) is the causal &function and y g , f ( E )= m !J dt(g, S t f ) e - ” E . The meaning of two different commutators (20) and (21) for the same operators is explained in Ref. 1, Sect. 7. These operators are called time-energy quantum white noise due to the presence of d(t’ - t)d(E’ - E ) in (20).

442

Define the white noise number operators as (22) For any positive bounded operator L in 3tl define the causal gaugeinvariant mean-zero Gaussian state ( P L by the properties (23)-(26): for n = 2 k

(PL(B:~...B~) = C ( P L ( B ~ ~ ’ B. .~.: (’ P ) ~ ( B ~ ~(23) ~B~:~)

where the sum is taken over all permutations of the set ( 1 , . . . ,2k) such k ; :=B;‘“(E,,t,)for thati, < j , , c r = l , ...,k a n d i l < i ~ < - - - < iBZ m = 1 , . . . ,n are timeenergy quantum white noise operators with causal commutation relations (21), and E , means either creation or annihilation operator; for n = 2k

+1

(PL(B;’. . . B z ) = 0

(PL(Bf(E,t)Bg(E’,t’)) = (PL(Bf(E,t)BS+(E’,t’)) =0

(24) (25)

‘ P L ( B f ( E , t ) ~ S ( E , t=’ )X[o,t](t’)(g,PELf) ) (26) The ”state” ( P L does not satisfy the positivity condition. This is a wellknown situation for the weak coupling limit (see Ref. 1) and is due to the fact that we work with time-ordered, or causal correlators. Therefore it is natural to call such ”states” causal states. Definition 4.1. Causal time-energy quantum white noise is a pair ( B f + ( Et ), ,( P L ) , where B f ( E ,t ) satisfy the causal commutation relations (21) and (PL is a causal gauge-invariant mean-zero Gaussian state.

Theorem 4.1. For any n E N in the sense of distributions over simplex tl 2 t2 2 . 2 tn 2 0 one has the limit l i m ( ~ ~ , t ( N f l , g l , ~...Nfn,gn,t(tn)) (tl) = (PL(Nfl,gl(tl) ...N fn,gn(tn))

€+O

This theorem was proved in Ref. 10.

Remark 4.1. This convergence is called convergence in the sense of timeordered correlators. The fact that we use the distributions over simplex is motivated by iterated series (6) for the evolution operator. Remark 4.2. The proof is based on the fact that for any n E N only one connected diagram survives in the limit. This can be interpreted as emergence of a new statistics (different from Bose) in the low density limit. For a discussion of new statistic arising in the weak coupling limit see Ref. 1.

443

The following theorem is important for investigation of the limiting white noise equation for the evolution operator.

Theorem 4.2. The limit state Y J L has the following factorization property: Qn E N

where the equality is understood in the sense of distributions over simple

Theorem 4.1 allows us to calculate the partial expectation of the evolution operator and Heisenberg evolution of any system observable in the low density limit. In fact, partial expectation of the n-th term of the iterated series (6) (or equivalent series for Heisenberg evolution of a system observable) after time rescaling t + t / r includes the quantity t

0

0

<

(where fa, ga are equal to go or 91). The limit as + 0 of this quantity can be calculated using Theorem 4.1. For example, the contribution of the connected diagram is equal to t

1

tl

d t l s dtzS+(tz - t l )

0

1 t2

0

0

dt36+(t3 - t z ) . . .

s

in-1

0

dt,S+(t,

- t,-I)

x S d ~ ( g , , ~ ~ ~ f i ) ? g l , ...?gn-ljtn f 2 ( ~ ) (E)

=t

s

d'(g,l'E'fl)ygl,fa(E)

...~ g n - l,f n ( E )

Similarly one can calculate the contribution of nonconnected diagrams (they give terms proportional to higher powers o f t ) . Summation over all orders of the iterated series gives the reduced dynamics of the test particle. An advantage of the white noise approach is that it allows to get the limiting dynamics in a nonperturbative way, without direct summation of the iterated series. This procedure includes derivation of the causally normally ordered white noise equation for the limiting evolution operator. After that the reduced dynamics of the test particle can be easily found.

444

4.2. The White Noise and Quantum Stochastic Equations

The limiting evolution operator satisfies the white noise Schrodinger equation dUt uo = 1 (27) + D+ El Ng,,go(t))Ut, dt The next step is to put this equation to the causally normally ordered form, i.e., to put all annihilation operators, appearing in Nf,g(t), on the right side of the evolution operator and all creation operators on the left side. Assume that for each E E R the following inverse operators exist - = -i(D €3 N g 0 , g N

To(E) := (1

+ ,Yga,g1 ( E P + - ,Yg1,9a(E)D + (,Yga,ga7gl,gl

TlW

+ ,Yga,91(E)D+

:= (1

- 3$1,ga(E)D

-1

- ,Yg1,sa,Yga,g1)(E)~D+)

+ (Ygo,90,Yg1,g1 - Yg1,g0Yga,s1)(E)D+D)

-1

Denote

Ro,o(E) := ^191,gl(E)m(E)D+, Ro,l(E):= -DT1(E)(1+ r g O , g l ( m + ) R1,d-q := 'Y90,9O(E)D+TO(E)D, Rl,O(E) := D+TO(E)(l - Tg1,go(E)D) Theorem 4.3. The causally normally ordered form of equation (27) is

This theorem was proved in Ref. 10.

Remark 4.3. An immediate consequence of Theorem 4.2 is the following factorization property of the limiting state (PL: (PdBf(E,t)UtBg(E,t)) = (PL(Bf(E,t)Bg(E,t))(PL(Ut) This property of the state (PL similar to the factorization property of the state determined by a coherent vector Q, 11Q11 = 1: ( ~ , B f ( E , t ) U t B g ( E , t )= ~ )(Q',Bf(E,t)B,(E,t)Q)(Q,UtQ) which is usually used to define quantum stochastic differential equations. Normally ordered white noise equation (28) equivalent, through identiU the t , quantum fication BA(E,t)UtBn(E, t)dt = 2 ~ d N t ( l P ~ g m ) ( P ~ g n I )to stochastic differential equation

C

dUt = - 2 ~ n,rn=O,l

/

dERrn,n(E)dNt(lPE9rn)(PEgnI)Ut (29)

445

where Nt is the quantum number process in r(L2(&)@3-11).The stochastic differential dNt satisfies the quantum Ito table dNt (X)dNt ( Y )= d Nt (XU) where X, Y are operators in 3-11. The limiting state '

(30) p has ~

the property

Equation (29) can be written in Hilbert module notations as

[dER,,,(E)

dUt = dNt (-27~

€9 IPEgrn)(PEgnl) Ut

(32)

n,m=O,l

The one-particle S-matrix for scattering of the test particle on one particle of the gas has the form S = 1- 2n n,m=O,l

/

dERrn,,(E) €9

IPmm)(Pmnl

(33)

This is a unitary operator: S+S = SS+ = 1. An immediate conclusion from (32) and (33) is the following theorem which was proved in Ref. 10 and is one of the main results of the paper. Theorem 4.4. The evolution operator in the low density limit satisfies the quantum stochastic equation driven b y the quantum number process with

strength S - 1: dUt = dNt(S - 1)Ut

(34)

This equation can be applied to derivation of the quantum master equation for the reduced dynamics. Let X E B(3-l~) be an observable of the test particle. Its time evolution Xt = UtXUt satisfies the equation dXt = dU2XUt

+ UtXdUt + dUtXdUt

(we identify X and X €9 1). Now, since d U t = U$dNt(S+ - 1) (we use the Hilbert module notations, hence dNt(S+ -1) does not commute with U 2 ) and using the quantum It0 table (30) one gets dXt = U?dNt(S+ - 1)XUt

+ U t X d N t ( S - l)Ut

+UtdNt(S+ - 1)XdNt(S - 1)Ut = UtdNt(O(X))Ut where the map 0 : B(7-l~) + B(7-l~€9 3-11) has the form O(X) = (S+ I ) X ( S - 1)+ (S+- l ) X X ( S - 1) S+XS - X . Simple computations,

+

446

together with the explicit form (33) for the S-matrix and the property PEPE' = 6(E of the projector, give the expression

@ ( X I = 2~

/ d E @ 2 m ( X )@ I P E ~ ~ ) ( P E ~ ~ I n,m=O,l

X&,n(E). Therefore dXt = 27r

~dEd~t(~PE9,)(PEgn~)~~@~m(X)Ut n,m=O,l

zt

The reduced dynamics := (pL(Xt) E B('Hs) is obtained by taking conditional expectation of both sides of this equation in the state ' p ~and using (31):

When restricted to the case of orthogonal form-factors go, 91,this master equation coincides with the one of Ref. 9. The linear Boltzmann equation for the reduced density matrix is the dual to this equation.

Acknowledgments The first part of the paper (Sect. 3) is based on the joint work of the author with Professor L. Accardi and Professor I. Volovich. The author is grateful t o L. Accardi for kind hospitality in the Centro Vito Volterra; t o L. Accardi, Y.G. Lu, and I.V. Volovich for many useful and stimulating discussions. This work is partially supported by a NATO-CNR Fellowship and Grant RFFI 02-01-01084.

References

Y.G.Lu, and I.V. Volovich, Quantum Theory and Its Stochastic Limit (Springer, Berlin, 2002). L. Accardi and Y.G.Lu, J. Phys. A 24, 3483 (1991). L. Accardi and Y.G.Lu, Comm. Math. Phys. 141,9 (1991). Y.G.Lu, J. Math. Phys. 36, 142 (1995). S. Rudnicki, R. Alicki, and S. Sadowski, J. Math. Phys. 33,2607 (1992). R. Hudson and K.R. Parthasarathy, Comm. Math. Phys. 93,301 (1984). A. Frigerio and H. Maassen, Prob. Th. Rel. Fields 83,489 (1989). L. Accardi, A.N. Pechen and I.V. Volovich, J. Phys. A 35,4889 (2002).

1. L. Accardi,

2.

3. 4.

5. 6.

7. 8.

447

9. L. Accardi, A.N. Pechen and I.V. Volovich, Infin. Dimens. Anal. Quant. Probab. and Relat. Topacs 6, 431 (2003). 10. A.N. Pechen, J . Math. Phys. 45, 400 (2004). 11. Y.G. Lu and A.N. Pechen, in preparation. 12. R. Dumcke, Comm. Moth. Phys. 97, 331 (1985). 13. H. Spohn, Rev. Mod. Phys. 52, 569 (1980). 14. R.Esposito, M.Pulvirenti and A. Teta, Comm. Math. Phys. 204, 619 (1999). 15. L. Erdos and H.-T. Yau, Contemp. Math 217, 137 (1997). 16. J. Conlon, E.H. Lieb and H.T. Yau, Comm. Math. Phys. 125, 153 (1989). 17. V. Gorini, A. Kossakowski and E.C.G. Sudarshan, J . Math. Phys. 17, 821 (1976);G. Lindblad, Comm. Math. Phys. 48, 119 (1976).

ASYMPTOTICS OF LARGE TRUNCATED HAAR UNITARY MATRICES*

J. REFFY Department f o r Mathematical Analysis, Budapest University of Technology and Economics, H-1521 Budapest XI., Hungary E-mail: [email protected]

The random variable Urn which takes its values uniformly on the set U ( m ) of n x n unitary matrices (i.e. its distribution is the Haar measure of U ( m ) )is the so-called Haar unitary. If we truncate the last rn - n last columns and m - n bottom rows then an m x m random matrix is obtained. K. Zyczkowski and H-J. Sommers examined the properties of the truncated Haar unitary, and they computed the joint eigenvalue distribution of UL,,,]. Since this is a contraction, all the eigenvalues lie on the unit disc. Now we consider the case m = 2n, and we prove that the large deviation principle holds for the empirical eigenvalue distribution of U[an,,las n + co. We determine the rate function from the logarithmic energy and by using results from potential and the joint eigenvalue distribution of U[zn,,l, theory, we minimalize the rate function in order to get the limit distribution.

1. Introduction

The empirical eigenvalue distribution of an n x n random matrix is the sample mean of the Dirac measures concentrated in the eigenvalues of the matrix. Since the eigenvalues axe random variables, the empirical eigenvalue distribution is a random measure, and its distribution is a probability measure on a class of probability measures supported on some subset of the complex plane, depending on the distribution of the random matrix. If we have an n x n random matrix A , for all n E N with the same distribution, then the sequence (A,) defines a sequence of random probability measures, which under some conditions converges to a deterministic probability measure. The first convergence theorem was proved by Wigner” for self-adjoint Gaussian matrices with independent entries on and above the diagonal. ‘This work was partially supported by otka t032662.

448

449

The eigenvalues of the matrices are real, and the limit distribution is the semicircle law. Similar results were obtained for non-selfadjoint Gaussian matrices3, for Wishart matrices8 and for unitary random matrices. Clearly, the limit distribution is supported on R+ in the case of Wishart matrices, and on the unit circle in the case of unitary matrices. If we have a sequence of random variables with non-random limit, the large deviation theorems state exponential rate of convergence. The first large deviation theorem was made by Crambr in 1938 for the sample means of independent, identically distributed random variables. The next important theorem was made by Sanov, who proved the exponential rate of convergence for the sequence of random measures given by the empirical distribution of independent, identically distributed random variables XI, . . . X,. It is a well-known theorem in statistics, that the limit measure is the distribution of XI. The empirical eigenvalue distribution of a random matrix defines a similar sequence, but the eigenvalues are not independent. The first large deviation theorem for random matrices was proved by Ben Arous and Guionnet2 for self-adjoint Gaussian matrices. Then for other Gaussian matrices, for Wishart matrices and for unitary random matrices the large deviaton theorem was proved by Hiai and Petz5s7@. These theorems can be found in their book4. In this article we prove the large deviation theorem for another class of random matrices: the truncations of Haar unitary matrices, where Haar unitary means, that the distribution is uniform on the set of n x n unitaries. We will show the convergence of the empirical eigenvalue distribution by proving the large deviation theorem.

2. The meaning of large deviation

The simplest example for a sequence of random variables with non-random limit is given by the law of large numbers. Let XI,. . .X,,. .. a sequence of real valued independent identically distributed random variables, with mean m. Then the law of large numbers claims that the sequence of the arithmetic means of (X,) converges to the number m as n + 00. In other words if p, denotes the distribution of the random variable Cy=lXi,then p, n3 ,S where 6, is the Dirac-measure concentrated in the point m. This means that for any G c R which does not contain m, p,(G) "3 0. The large deviation principle (LDP) holds, if the rate of the above convergence is exponential. More precisely, if there exists a f : R + Iw+ lower

450 semicontinous function, and a such that for all G pn(G)

e-L(n)infm6G

cR

f(X)

(1)

&.

then we say that the large deviation principle holds in the scale of Here L(n)n3 00, and the order of magnitude of the function L is given by the degree of freedom of the random variables. The function f is called the rate function. Now recall the definition of the large deviation principle1. Definition 1. Let X be a topological space, and Pn a sequence of probability measures on X. The large deviation principle holds with rate function I in the scale L(n)-l if 1 liminf -log Pn(G) 2 - inf I ( z ) n+m L(n) XEG

for all G

c X open set, and

1 limsup -logPn(F) 5 - inf I(z) n+w L(n) xEF for all F c X closed set.

(3)

For example, in the Cram& theorem L(n) = n, and the rate function is the convex conjugate of the logarithmic moment generating function of the random variables. This means, that if X I ,X2,. . . are independent, identically distributed real valued random variables, and pundenotes the distribution of CZ1Xi, then for the function

f(z):= sup {Xz - log (lE(exp(XXi))) : X E R}

(4)

and for all measurable I? C R

- inf f(z)5 liminflogpn(r) 5 limsuplogp,(r) 5 xEint r n-bw n+w

inf f(z). (5) xar We note, that if we have a sequence of diagonal random matrices with independent, identically distributed diagonal elements, then Sanov theorem implies the large deviation theorem for the empirical eigenvalue distributions. In the case of random matrices mentioned in Section 1. the limit distributions were also known, and the rate function could be obtained from the joint eigenvalue density of the random matrix. The degree of freedom is n2, since a random matrix consists of n2 random variables, so L(n) = n2. The limit measure is the unique minimizer of the rate function. In our case, the situation is different, we first prove the large deviation theorem for a sequence of truncations of Haar unitaries with the rate -

451

function obtained from the joint eigenvalue density, and the next step is to minimize the rate function, in order to obtain the limit distribution.

3. Main result Denote U(m)the set of m x m unitary matrices. If the random variable U,,, takes its values from U(m)uniformly, i.e. for any V E U(m),and H c Wm), Prob(U, E H ) = Prob(U, E V H ) ,

(6)

then Urn is a Haar unitary random matrix, since its distribution is the Haar measure on U(m) By truncating m - n bottom rows and m - n last columns, we get an n x n matrix Ulrn,4. This matrix is not unitary, but a contraction, so the eigenvalues are on the unit disc. K. Zyczkowski, H.-J. Sommers12 found the Y [ , , , , ~joint ] eigenvalue distribution of U[,,,,+],which is given by the density n

on the unit disc V n . The constant Z[,,,]

Z[,,,l

was obtained by Petz and RQfFyg,

:= rnn!n f i l r - n ; j - l j=O

1 m-n+j'

(8)

The following large deviation theoremlo gives the limit of empirical eigenvalue distribution of U[,,,,n~.

Theorem 3.1. Let Pn the empirical eigenvalue distribution of UI,,,,~],i.e. .

n

where A1 (U[m,nl). . . A n (U[m,nl) are the eigenvalues of U[,,,,n~. Then Pn is a measure on M ( V ) ,where M ( V ) denotes the probability measures o n the unit disc V ,and the sequence (P,) satisfies the L D P in the scale n-2 where znn , , g > 1 .

with rate function

(10)

452

for p E M ( V ) , where

B

:=

1 lim -logZ[,,,]

n+w 122

Moreover there exists a unique probability measure pu, such that I&) given by the density

= 0,

on the disc { z : 1x1 5 l/fi}.

1

Density of

p2

We give a sketch of proof in the case of m = 2n, and for the sake of simplicity, we will use the notations 2, := Zp,,,~, and v, := vpn,,].

The computation of B is simple, since according to Eq. (8) 1 B = lim - log& n2

n+j

j=O

n-1-i n-l

1 ,--l =-lim----C ni-wn - 1

1%

n-l+i i

i=l

which is the Riemannian sum of the integral

-1

1

(1-2$1og(%)

1 da:=2-210g2

For the remaining part of the proof we use the function

F ( z , w ) = -log Iz - W I - - (log(1 - 1212) + log(1 - 1w12)), 1

2

and for a > 0 denote

Fa(z,w) := min(F(z,w),a). The function

is continous in the weak* topology, so the functional I(P) =

F ( z ,w)4 4 z ) dP(W) + B = SUP a>O

11

Fab,w)

&(w)

+B

(19) is lower semicontinous. We know' that the large deviation principle is equivalent to the conditions,

and

for all

E M ( D ) ,since M ( D ) is compact.

454

For the proof of Eq. (20) and Eq. (21) we define a subset of V”

c

1 ”

C = (
S(Ci) E G } ,

i=l

so that we can write from Eq. (7)

We can see, that by taking the logarithm, then dividing both sides by n2 and taking the limit, the result is similar to the wanted inequalities. Finally, t o minimize the rate function, we use the following theorem of potential theory.

Theorem 3.2. Let Q : V + (--00, m], such that Q ( z ) = &(1~1), for all z E V , and suppose that Q is diferentiable o n ( 0 , l ) with absolute continuous

dem’vative bounded below, moreover rlim -tl

TQ’(T)

increasing o n ( 0 , l ) and

rQ’(r) = -00.

(24)

Let r0 2 0 be the smallest number f o r which Q ’ ( T ) > 0 f o r all T > T O , and we set & be the smallest solution of &Q’(&) = 1. Clearly 0 5 TO < & < 1. T h e n the functional

attains its minimum at a measure p~ supported o n the annulus SQ = (2 : TO

5 IzI 5 &},

(26)

and the density of p~ is given by 1 d p ~ ( z= ) -2n (~Q’(~))’drdq,

z = rei9.

(27)

If we apply the above theorem for 1 Q ( z ) := --log (1 - 1 ~ 1 ,~ ) (28) 2 then we get the minimizer py. Since M ( V ) is compact, the large deviation theorem implies the convergence of the sequence of the empirical eigenvalue distributions. Finally we show the density function in the case of y = 5, and y = 1,2.

455

Density of p5

Density of p1,2

456 References 1. A. DEMBOand 0. ZEITOUNI,Large Deviations Techiques and Applications, Jones and Bartlett, Boston, 1993. Large deviation for Wigner’s law and 2. G. BEN AROUSand A. GUIONNET, Voiculescu’s noncommutative entropy, Probab. Theory Related Fields 108 (1997), 517-542. 3. V. L. GIRKOElliptic law, Theory Probab. Appl. 30. (1986), 677490. 4. F. HIAI and D. PETZ:The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, Vol. 77, American Mathematical Society, 2000. 5. F. HIAI and D. PETZ, A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices, Ann. Inst. Henri PoincarB, Probabilith et Statistiques 36, 71-85 (2000). 6. F. HIAI and D. PETZEigenvalue density of the Wishart matrix and large deviations, Infinite Dimensional Anal. Quantum Prob. 1,633-646 (1998) 7. D. PETZand F. HIAI,Logarithmic energy as entropy functional, in Advances in Differential Equations and Mathematical Physics (eds: E. Carlen, E.M. Hmell, M. Loss), Contemporary Math. 217(1998), 205-221 Some limit theorems for the eigenvalues of a sample covariance 8. D. JONNSON, matrix, J. Multivariate Anal. 12, 1-38 (1982) 9. D. PETZand J. REFFY,Asymptotics of large Haax unitary matrices, to be published in Periodica Math. Hungar. 10. D. PETZand J. RBFFY,Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices, to be published 11. E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. 6 7 , 325-327 (1958) Truncation of random unitary matrices, 12. K. ZYCZKOWSKIand H-J. SOMMERS, J. Phys. A: Math. Gen. 33, 2045-2057 (2000).

QUANTUM OPTICAL SCENARIOS FOR STOCHASTIC RESONANCE

VYACHESLAV SHATOKHIN B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Fr. Skaryna Awe. 70, B YE-820072 Minsk, E-mail: v.shatokhinOdragon. bas-net. b y THOMAS WELLENS Institut Nonline'aire de Nice, 1361, route des Lucioles, F-06560 Valbonne, E-mail: Thomas. WellensQinln.cnrs.fr ANDREAS BUCHLEITNER Max-Planck-Znstitut fur Physik komplexer Systeme, Nothnitzer Str. 38, D-01187 Dresden, E-mail: [email protected] We are traditionally educated to consider noise a nuisance, hindering the transmission and detection of signals. Recently, however, the paradigm of stochastic msonance (SR) proved this assertion wrong: indeed, the appropriate amount of noise can facilitate signal detection in noisy environments. Due to its simplicity and robustness, SR has been implemented by mother nature on almost any scale, from celestial mechanics t o ion channels. The present minireview outlines the basic mechanism underlying SR phenomena, and discusses recent quantum optical scenarios, in the deep quantum regime.

Stochastic resonance was introducedl in 1981 as a possible mechanism to explain the apparently near-periodic occurence of ice ages (with a period of approx. 100000 years) in the earth climate, during the last 700000 years. In this particular model, noise from atmospheric dynamics cooperates with a weak periodic signal (which determines the above period) provided from the earths celestial mechanics. Whilst the latter alone is too weak to drive

457

45%

the earth’s climate (which is supposed to be bistable) from warm to cold periods (with a difference in the average temperature of approx. 10 K), almost periodic transitions from cold to warm and back are observed when the signal is assisted by the proper amount of noise. Given the not really satisfactory statistics of climatic changes (in the above example, only seven events define the periodicity), SR remains controversial as the real cause of ice ages (see, however, ref. 2), though the model of ref. 1 was soon carried over to simple electronic systems with bi~ta bi l i t y.Next ~ came ring laser^,^ and the field really started to blossom once the relevance of SR for biological systems was realized in the early nineties of the last century. Cray fish predator detection (see the beautiful article by Moss and Wie~ en feldand ,~ notably the impressive snap shot of the crayfish getting away from the big fish which is after him - a n example for all small fries in delicate circumstances), transport across ion channels, the auditory system, and crickets - all exploit SR such as to detect weak periodic signals in a noisy environment.6 Also medical applications of SR are under discussion, what promoted the subject twice to The Economist’s Science and Technology ~ e c t i o nWhat .~ characterizes SR? It is a nonlinear effect - most directly illustrated by the noise assisted dynamics of a periodically driven particle in a one dimensional double well potential, with the periodic force not strong enough to carry the particle across the barrier; due to the cooperativity of signal and noise, in the sense that the typical time scale (given by Kramers’ law8) on which the stochastic force drives the particle across the barrier matches half the period of the signal; expressed by an optimal signal enhancement a t a nonvanishing noise level, in the response of the bistable system to the periodic force.

To be a little more specific, consider the bistable potential in Fig. 1, with stable (unstable) equilibria at z = f c (z = 0) - representing, e.g., the metastable cold and warm periods of the earth’s climate - and a potential barrier of height UO. If we consider an overdamped particle moving in this potential, additionally subject to a periodic force of strength E and frequency W O , in the presence of a stochastic, delta-correlated force <(t) with intensity D, the equation of motion reads’

dU

t = -dx

+ €cos(wot) + a<@),

459

Figure 1. Typical setting t o observe Stochastic Resonance (SR): An overdamped particle moves in a bistable potential, under the combined influence of a weak periodic drive (smooth arrows), and of a stochastic force (ragged arrows). The latter defines transition rates W+,which are periodically modulated by the periodic signal, see eq. (2), provided the signal period is much larger than the typical relaxation time scale of the particle within the potential wells.

and the potential barrier seen by the particle is modulated with a n amplitude Ul = E C . Provided the modulation of the potential fulfils the adiabaticity condition wo << U”(fc), i.e. the time scale of the drive is much larger than the intrastate relaxation time of the particle, then Kramers’ law can be generalized to the following, modulated transition rates

between the metastable sites. This allows to deduce the spectral density of the response z ( t ) of the particle to the forcing, via a suitable Fourier transformg of the autocorrelation function C ( T ) = (z(t)z(t T ) ) of the particle. For sufficiently small driving amplitude E one obtains:

+

Inspection of the prefactor of the delta function - centered at the frequency of the periodic signal and therefore the signal part of the reponse - shows that the (positive) weight of the signal tends to zero for D + 0 as well as for D -+ m. Hence, besides the Lorentzian noisy backgroundg in the spectral density - the first term in (3) - the periodic response of the particle is most pronounced at a finite noise level, and this is precisely the signature of SR. This is qualitatively easily understood when we compare the Kramers activation rate and the period of the drive, see also Fig. 2: At weak noise level

460

(small D),the average residence time in either metastable state is much longer than the driving period W O . As the noise level is increased, the residence time eventually becomes comparable to half the modulation period, and transition events become most likely at that phase of the drive when the potential barrier is strongly suppressed. If the noise level is increased even further, too many transitions are activated by the noise during half a modulation period, and the cooperativity between signal and noise is lost again.

w w 0

Figure 2. The fundamental mechanism of stochastic resonance: in the presence of an optimal level of noise, the stochastically activated transitions between the two metastable states are most likely after one half-cycle of the injected periodic signal. Hence, the response is optimally synchronized with the external modulation of the double-well potential at nonvanishing noise strength.

Note that there are different definitions of SR floating around in the literature, and that they do not always come to the same conclusion whether a system displays SR or not.1° The most common are 0

0

0

the signal to noise ratio (SNR), this is the ratio of signal strength to noise level at w = W O ; the signal enhancement - the prefactor of the delta function discussed above; and the residence time distribution - the probability density function of the time intervals spent by the particle in a metastable site between entry and exit.

The reason for the fact that different conclusions may be drawn from the application of these different indicators of SR appears to be often the in-

461

trawell dynamics of the particle, which may hidell the actual noise ifiduced synchronization of the dynamics with the signal - the essence of SR. The current intuition of the authors suggests that the residence time distribution is the most reliable indicator, though possibly requires a larger amount of sampled data, in order to make the synchronization effect emerge from the noisy background. So far, our outline of SR is completely classical, with Kramers’ law and the adiabaticity condition as the essential ingredients. Since SR is so successful in the macroscopic world, from celestial scales to biological and medical applications, it appears somewhat natural to ask what happens when we enter the microcosm. This, however, is the realm of quantum mechanics, and we know that a quantum particle finds other ways to overcome the potential barrier in Fig. 1 than climbing the hill or being kicked over it by a stochastic force. Hence, which is the effect of generic quantum mechanical transition mechanisms such as tunneling or quantum fluctuations? This question was first addressed in the mid-nineties, for various model systems,12 and some modifications of the SR mechanism have been described. A first implementation of SR in a fundamental quantum system readily accessible in the laboratory13 was suggested in ref. 14, with a detailed elaboration of the theory in ref. 11. This fundamental quantum system is the micromaser, and we shall now briefly sketch how SR can serve as a means to synchronize the quantum jumps of a bistable maser field. Let us first recollect some essential ingredients: The m i c r o m a ~ e r ~ ~ > ~ ~ consists of a single mode resonator which is crossed by a beam of atoms, at such low flux r that at most one atom is present in the cavity at the same time. The atoms are initially prepared by a pump laser, possibly combined with a n additional microwave field, in a coherent superposition I$) = alu) bld) (“coherent pumping”) of high lying Rydberg levels Iu) and Id). Once in the cavity, the upper level Iu)is resonantly coupled to the lower-lying Id) by the field mode of frequency w . The atoms (obtained from a thermal atomic beam) are velocity selected, such that they arrive with a fixed velocity, and consequently interact with the radiation field inside the cavity for a precisely defined interaction time tint .13 Hence, they accumulate a well defined Rabi-angle q5 = Rti,t during the coherent interaction, with R the vacuum Rabi frequency. On exit from the cavity, the atoms are detected in) . 1 or Id) (or in a superposition of both, established by a second microwave field between cavity exit and detection region) by state selective static field ionization. For Jal>> lbl, and in particular if all atoms enter the cavity in lu) (“in-

+

462

coherent pumping”) , the atoms can possibly deposit a photon in the cavity, i.e. feed energy into the cavity mode. In contrast, between exit of one atom from the cavity and entrance of the following one, the population of the energy levels of the field mode evolves like a damped harmonic oscillator with damping constant y, due to the coupling to the cavity walls which are assumed to be in thermal equilibrium with the environment at temperature T . Given that y-l >> r-l >> tint, the complete time evolution of the cavity field density matrix is described by the master equation

P = Plat -tPlenv t >> tint ,

(4) where the first term describes the state evolution induced by the coherent atom field interaction, whilst the second describes the damping due to environment coupling. A steady state of the cavity field is reached when the gain due to the coherent atom-field interaction compensates for the losses through the walls. Fig. 3 shows a typical steady state density matrix of the (coherently pumped, a, b # 0) maser field in the Fock basis, with parameters chosen such as to induce a bistable photon distribution. In a single realization 1

0.14

Ill

Figure 3. Stationary state of the photon field density matrix (in the Fock state representation), with two metastable states centered around nl N 4 and ng N 21. The and atoms enter the cavity in a coherent superposition I+) = alu) bld), with a = b = - i m . T=0.5K,w=21.5GHz,~-’=0.06sec,r=40~,~=1.1.

+

of the maser dynamics, such a dichotomous density matrix leads to quantum jumps of the field,13 on rather long time scales, as depicted in Fig. 4 (for an example with incoherent pumping, a = 1). Remember that, due

463

0' 0

1000

500

I

1500

time [s] Figure 4. Quantum jumps of the incoherentlypumped maser field, for a = 1, T = 0.6K, w = 21.5 GHz, y-' = 0.06 sec, r = 407,4 = 1.1. The quantum jumps are monitored by the time dependence of the probability P ( t ) t o detect an atom in Id), on exit from the cavity. To smooth measurement noise, P ( t ) has been averaged over approx. 500 detection events. The typical residence time of the field in either one of the metastable states is of the order of 50 sec.

to the coherent atom-field interaction in the cavity, the atom and the field are entangled after exit of the atom from the cavity, and only the detection of the atom forces the field into a well defined (metastable) state.16 However, a single detection event cannot be predicted with certainty, leading to measurement noise because of the entanglement between atoms and field. In the bistable situation depicted in Fig. 4 (generated numerically with a quantum trajectory method), the detection signal is averaged over approx. 500 detection events, what smoothes the measurement noise, and only leaves the detectable consequence of the jumps of the maser field between the two metastable states which are manifest in the maxima in the occupation of the density matrix of Fig. 3. Indeed, when we observe the atom predominantly in its lower state, on exit from the cavity, it is likely to have deposited a photon in the mode, and the cavity field state is therefore likely t o be centered around 722 in Fig. 3. Vice versa, if we observe the exiting atoms predominantly in the upper state, then the state of the mode is centered around 711 in Fig. 3, due to the quantum mechanical projection postulate. However, whilst the jumps of the maser field occur on a typical time scale of tens of seconds in Fig. 4 (in agreement with the experiments reported in ref. 13) we cannot predict when a single jump will occur. This is precisely the general setting of SR: The maser field displays bistability, with environment- (vulgo noise-) induced transitions between the metastable states, on time scales much longer than any other characteristic time scale of the maser dynamics, what will allow us to satisfy the adiabaticity requirement of SR. The only difference with respect to the classical examples listed above is the dominantly quantum origin of the noise - measurement noise, on the one hand, and quantum fluctuations of the environment at

464

low temperatures (T N _ 0.5 K), i.e. small values of the thermal occupation nb = (exp(h/lcT) - 1)-l < 1 of the cavity mode, on the other. Consequently, SR provides a means to reduce our uncertainty about the residence time of the field in either of the metastable states, by synchronizing it with a weak, nondeterministic signal (in the sense that the bistability of the dynamics is not suppressed). Indeed, if we introduce the signal by a simple periodic modulation of the pump rate r of the cavity (experimentally easily achieved through a periodic modulation of the laser which excites the atoms from their ground to the Rydberg level"), we still observe random jumps of the cavity field, with a slight periodic modulation of the intrawell dynamics induced by the field, see Fig. 5a. Now we do nothing but to increase the noise level by increasing the environment temperature from T = 0.3 K to T = 0.6 K, and observe . . . a n enhanced periodicity of the quantum jumps of the field, at the period t m o d = 42 sec of the signal, see Fig. 5b. Noise and signal cooperate optimally, since, in fact, the average residence time of the field in either state is close to t m o d / 2 cu 20 sec at this temperature, in the absence of any modulation" (see also Fig. 2). If we further increase T , the noise is too strong, and the stochastic stimulus induces too many transitions per signal period - the signal drowns in the noisy background, see Fig. 5c. To quantify this observation, we extract the spectral density from the autocorrelation function (or, alternatively, from the master eq. (4)11) which is plotted in Fig. 6. Clearly, there is a pronounced signal peak at the frequency of the coherent drive, and, additionally, at its higher (even and odd) harmonics. The latter are just a n expression of the nonlinear response of the system to the injected signal, and even harmonics do not vanish due to the asymmetry of the transition rates between the metastable states of the cavity field.l1)l5 SR is finally born out when we plot the strength of the signal peak (and equally of the higher harmonics, see ref. 11) as a function of the environment temperature, see Fig. 7. Clearly, the signal is strongly enhanced, with a maximum at T = 0.6 K. Hence, SR gives us a handle to control the noise induced quantum jumps of the maser field, by tuning only the cavity temperature. The periodic signal is easily injected via slow modulation of the pump laser intensity, for the examplary case of incoherent pumping we have demonstrated here. Fig. 3, however, shows that also in the case of coherent pumping the maser field exhibits bistability and quantum jumps, which can now be detected in essentially arbitrary components of the atomic Bloch vector, e.g., when f Id))/&. In this case, we can inscribe the detecting the atoms in ) . (1

465

Figure 5. Time evolution of the probability P ( t ) t o detect an atom in Id), with periodically modulated atomic flux r ( t ) / y= 40 6.9COS(27Tt/tm,d), at different temperatures. Incoherent pumping, a = 1,with q5 = 1.033. Clearly, optimal synchronization is observed at T = 0.6 K.

+

20

10

0

.10

-20 0

1

2

3

4

Figure 6. Power spectral density of the detection signal p ( t ) on output from the maser cavity, for T = 0.6 K. Other parameters as in Fig. 5.

466

T“

Figure 7. Temperature dependenceof the signal strength in the power spectral density of the detection signal P ( t ) . Parameters as in Fig. 5 . Optimal synchronization a t T = 0.6 K is clearly born out by the maximum in the plot.

weak periodic signal directly into the atomic coherence, by modulating the amplitudes a ( t ) ,b(t). Again, one observes noise induced signal enhancement, now in the detected atomic coherence, upon exit from the cavity, at an optimal temperature T N 0.6 K.” Other scenarios for SR in quantum optical systems have been identified since. Given the simple double well picture which we have been using as a motivation here, and which is implicitely reduced to a two-state model once the dynamics is completely described in terms of transition rates W , such as in (2), one immediately thinks of the dark and bright periods observed in the resonance fluorescence of a single ion, due to “electron shelving”.17 In the simplest case, three electronic states of the ion are involved in the dynamics: The ground state 11) is strongly coupled by a resonant laser to 12), and weakly coupled to the metastable level 13). Depending on whether the metastable state is populated or not, one observes dark or bright periods in the resonance fluorescence signal which stems from the coupling between 11) and 12). The corresponding dynamics can be described by a two-state model, where the transitions between dark and bright periods occur at random, uncorrelated times, activated by projective measurement noise due to the detection of resonance fluorescence photons projecting the atom onto the ground state 11). However, since the strength of the quantum noise cannot be tuned easily, it is not immediately obvious how to relate this scenario to SR. (Note that there is no thermal bath the temperature of which can be varied.) For this reason, a slightly modified setup was proposed in ref. 18, where an

467

additional, tunable source of noise is provided, see Fig. 8. Here, the two

Figure 8. Four level system under coherent and intensity-modulated resonant driving with time-dependent Rabi frequency R giving rise t o dressed states ti-).The dashed (dotted) lines indicate the energies of the dressed states for the maximum (minimum) value of R. Transitions - H 3 and H 4 are driven by noisy broad band fields, with frequency distributions represented by the dashed curves. Depending on the value of R, the interaction is resonant with either the dressed level I+) or I-). After Ref. 18.

+

dressed states I&) = (11)+ 12))/fi of the fluorescence transitions are coupled incoherently, via noisy driving fields, to two additional states 13) and 14), respectively, with corresponding effective pump (i.e., transition) rates W3 and W,. The state 13) decays rapidly into the metastable state 14), with spontaneous rate r33. Under these conditions, the resonance fluorescence signal (i.e., the emission of photons due to spontaneous transitions from 12) to 11))will again exhibit quantum jumps between dark and bright periods. Here, the dark period corresponds to the atom resting in state 14),whereas the states I*) of the fluorescence transition are occupied during the bright period. (The population of 13) can be neglected, due to its rapid decay into 14).) In contrast to the original electron shelving scenario sketched above, the jumps are now triggered by the incoherent, noisy driving fields, whose intensities can be tuned easily. Furthermore, a periodic signal can be inscribed by modulating the amplitude R of the coherent laser driving the 11) ++ 12) transition, what shifts the energies of the dressed states I+) and I-) (separated by the Rabi splitting R). The modulation is chosen such that the minimum value of R tunes the incoherent transition I+) ++ (3) into resonance (inducing large W3),whereas at the same time I-) t) 14) is out of resonance (W4 small), and vice versa half a period later, for the maximum value of R (see Fig. 8). On the other hand, W4 defines the transition rate from the dark period (in which mainly the state 14) is populated) to the bright period, whereas the opposite rate is given by (W3 W4)/2, since I+) and I-) are equally populated in the stationary state of the resonance

+

468

fluorescence, i.e., the transition from bright to dark may occur either via I+) + 13) (from where the system rapidly decays to 14)),or via I-) + 14). Consequently, the modulation of R leads to a modulation of the transition rates between the dark and bright periods of the fluorescence signal. Thereby, we have again arrived at our familiar SR setting. Not surprisingly, a stochastic resonance maximum of the signal to noise ratio in the power spectrum of the emitted fluorescence intensity occurs at an optimal strength of the noisy fields, as was shown in ref. 18 by solving the quantum optical master equation governing the evolution of the atomic state: If the noisy driving is too weak, several modulation cycles 21rw;' have t o pass before population is incoherently transfered between I&) and 14). If the noise is too strong, the modulation of the transition rates between bright and dark periods is swamped - hence, optimal synchronization of dark and bright periods with the drive occurs at intermediate noise levels. Besides quantum optical systems such as the above - with a clear mapping on the double well or two-state model of SR - there are also other cases of noise induced signal enhancement, e.g., in dissipative optical lattices, l9 which, however, need a more refined analysis.20 An alternative scenario for SR is provided by systems described by the Bloch equations,21 which we briefly outline here as a last example. Indeed, some sort of SR is to be expected for a wide class of driven, dissipative two-level systems,21 in the framework of the well-known master equation for the density matrix of a coherently driven two-level system coupled to a Markovian reservoir (i.e., memory effects due to the environment coupling are neglected):22

H ( t ) accounts for the unitary part of the system evolution, generated by the Hamiltonian W T L S C T ~ of the isolated two-level system and the coherent interaction 252 cos(wot)a, between the two-level system and the (classical) driving field,

+

H ( t ) = WTLSCT, 2 0 COS(WO~)O,,

(6)

with ~ W T Lthe ~ energy splitting of the two-level system, and R the Rabi frequency proportional to the driving field amplitude. Under rather general conditions,22 the (positive-definite) 'relaxation matrix' { u k l } can be specified by three independent parameters which appear in the nonunitary part of Eq. ( 5 ) : ayz = a,, = 0, a,, = ayy = (2T1)-', asz = (Tz)-l - (2Tl)-l, uZy = a t , = i(&T1)-'seq, with T1,2 the longitudi-

469

nal ( T I )and transverse (T2) lifetimes, and seqthe steady state population difference between excited and ground state in the undriven case R = 0. In the Bloch state representation, in which p = (l-ts.a)/2, (a= (D,,uy,o,)), the master equation (5) reduces to the Bloch equations for the components s,, sy, s, of the Bloch vector s: s', = W T L S S ~- TT1sX

+

s j = - W T L S S ~ - T2-1 sY 2Rcos(wot)s, s', = - ~ R c o s ( w o ~ )-s ~T;'(S, - s q ) .

(7)

(7) is derived in the weak coupling and/or high temperature limit,z3i24 which is appropriate in the range of microwave and optical driving, for fields with R << WTLS. Numerous experiments on magnetic and optical resonance were successfully described by these equations. Let us now turn towards a description of SR in systems governed by the Bloch equations. There is a coherent signal driving a two-state system, and there is (white) noise due to the coupling to the reservoir. Two types of noise-induced processes are to be distinguished: The first one, related to transitions between the energy levels of the two-level system, is described by the longitudinal relaxation time T I . The other one, related to pure d e phasing processes involving no energy exchange between the system and the bath, gives rise to the transverse relaxation time Tz. Since longitudinal and transverse relaxation times are induced by different elementary processes of the system-bath interaction, there is no general relation between these two times. However, there are systems in which only one single relmation time emerges. As shown in ref. 21 this condition is necessary for the observation of SR in such systems. To demonstrate that, let us consider the steady-state response of the two-level system to the coherent driving field. This response is expressed in terms of the components of the Bloch vector s ( t ) . In the context of SR, s, ( t )is considered as the output variable, since the periodic driving couples via ox,see Eq. (6). The spectral response S1 can be calculated from the asymptotic steady-state value s?'(t) = limt,oo s,(t), and is expected to exhibit a maximum at a finite noise level. In the following, we will use the quantity rl=

@ l

(8)

instead of 5'1 t o quantify the spectral response. (This leads to simpler equations] since r] is simply proportional to the length of the transverse

470

component of the Bloch vector.) For resonant driving (WO = WTLS), the rotating wave approximation can be applied to get rid of the rapidly oscillating terms in (7).z5Then, in the frame rotating at the external field's frequency, the equations of motion are time-independent, and the expression for the response is given by

It follows that, for given R, the spectral response r] is a monotonous function of T I ,Tz considered as independent parameters. However, if, additionally, T1z = TI = const Tz,r] reveals a local maximum for T12 fulfilling +

R2T1T2 = 0'- T& - 1. const The emergence of a maximum in the system's response as a function of the noise level is a distinct feature of SR in the two-level system. It arises due to the synchronization between the Rabi frequency and the time scale fixed by the decay rate TG'. This reminds us of the matching condition for the relevant time scales in the original two-state model of SR above. Note, however, that the relevant frequency scale is not given here by the signal frequency wo, but rather by its amplitude 0, compare Eq. (6), which is nothing but the effective, field-induced (tunneling) coupling matrix element between the two eigenstates of the isolated two-level system. It is clear from Eq. (9) that the requirement of a sing2e relaxation time is necessary for the synchronization to occur. The SR peak shifts towards shorter relaxation times for larger Rabi frequencies, and the magnitude of the peak does not depend on the driving field amplitude. Let us stress here that the occurrence of SR is a consequence of the resonance condition wo N W T ~ S (in contrast to the case of adiabatic driving considered in our previous examples of SR): the maximum of the response r] vanishes if this condition is not fulfilled.26 As noted in ref. 21, there is a wide class of systems described by the Bloch equations, where the present model of SR could be verified experimentally. In nuclear magnetic resonance (NMR), for example, the Bloch equations describe the time evolution of the macroscopic magnetization of an ensemble of spin-1/2 nuclei exposed to a magnetic field with a static component along the z-axis, and a radio-frequency component in the t r a n s verse plane. For a liquid spin 1/2 sample (e.g., water), the main source of r e l a x a t i ~ are n ~ fluctuations ~~~~ of the local dipolar field, induced by the motion of the 'H nuclei and usually implies Tz 5 2T1. In the steady state,

471

the magnetization vector precesses around the z-axis with the injected radiofrequency, and the response r] is the magnitude of the transverse magnetization, which can be measured experimentally. Such a n experiment on water was reported on in ref. 21. By addition of paramagnetic CuSO4, the relaxation times can be adjusted such that TI N Tz. Indeed, the experimental results reveal the above matching condition between Rabi frequency and relaxation rate: for a given value of T12, the normalized response r]/s,, exhibits a maximum for some optimal value of the driving field amplitude (i.e., of the Rabi frequency), and vice versa. In conclusion, the above examples suggest that SR has a wide range of applications in quantum optics, which only starts being explored. Note that SR tells us what can be done if we can only tune either the noise, tmod, or more realistic in an evolutionary setting, where the species has to adapt to the environment - the potential barrier UO and its modulation amplitude Ul in Figs. 1,2. Therefore, when we seek deterministic control over quantum states in low dimensional quantum systems, we will certainly try to minimize the noise and to implement coherent control schemes rather than using noise to optimize performance. However, as we try to control quantum systems of increasing dimension, which will be ever harder to isolate from their noisy environments, SR might turn into a robust alternative.

References 1. R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14,L453 (1981); Tellus 34, 10 (1982). 2. A. Ganopolski and S. Rahmstorf, Phys. Rev. Lett. 88,38501 (2002). 3. S. Fauve and F. Heslot, Phys. Lett. A 97,5 (1983). 4. G. Vemuri and R. Roy, Phys. Rev. A 39,4668 (1988). 5. F. Moss and K. Wiesenfeld, Scientific American (August 1995). 6. K. Wiesenfeld and F. Moss, Nature 373,33 (1995). 7. Science and Technology, Come Feel the Noise, The Economist, July 29th, 61 (1995); Stochastic Resonance: Feel the Noise, The Economist, September 19th, 99 (1998). 8. H. Kramers, Physica (Utrecht) 7,284 (1940). 9. B. McNamara and K. Wiesenfeld, Phys. Rev. A 39,4854 (1989). 10. A. Bulsara and L. Gammaitoni, Physics Today 49,39 (March 1996). 11. T. Wellens, Stochastische Resonanz im Mikromaser, diploma thesis, University of Munich (1998); T. Wellens, Entanglement and Control of Quantum States, PhD thesis, Ludwig-Maximilians-Universit at Miinchen, 2002, ht t p: //edoc/ub.uni-muenchen.de/archive/0000008 1; T. Wellens and A. Buchleitner, J. Phys. A 32,2895 (1999); Phys. Rev. Lett. 84,5118 (2000); Chem. Phys. 268,131 (2001).

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12. R. Lofstedt and S. Coppersmith, Phys. Rev. E 49, 4821 (1994); M. Grifoni and P. Hanggi, Phys. Rev. Lett. 76,1611 (1996). 13. 0. Benson, G. Raithel, and H. Walther, Phys. Rev. Lett. 72,3506 (1994). 14. A. Buchleitner and R. Mantegna, Phys. Rev. Lett. 80, 3932 (1998). 15. P. Filipowicz, J . Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986). 16. S. Haroche, in Fundamental Systems in Quantum Optics, Les Houches, Session LIII, ed. by J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, NorthHolland, Amsterdam 1992. 17. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797-2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. Rev. Lett. 57, 1696-1698 (1986). 18. S. F. Huelga and M. Plenio, Phys. Rev. A 62,52111 (2000). 19. L. Sanchez-Palencia, F. R. Carminati, M. Schiavoni, F. Renzoni, and G. Grynberg, Phys. Rev. Lett. 88, 133903 (2002). 20. T. Wellens, V. Shatokhin, and A. Buchleitner, Rep. Prog. Phys. 67, 45 (2004). 21. L. Viola, E. Fortunato, S. Lloyd, C.-H. Tseng, and D. Cory, Phys. Rev. Lett. 84, 5466 (2000). 22. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Springer, Berlin 1987. 23. C. Cohen-Tannoudji and J. Dupont-Roc and G. Gilbert, Atom-Photon Interactions, John Wiley & Sons, New York 1992. 24. A. Abragam, Principles of Nuclear Magnetism, Oxford University Press, Oxford (1961). 25. L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms, Dover Publications, Inc., New York (1987). 26. F. Shuang and C. Yang and H. Zhang and Y. Yan, Phys. Rev. E 61, 7192 (2000). 27. R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles ofNuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, (1987).

ON A CLASSICAL SCHEME IN A NONCOMMUTATIVE MULTIPARAMETER ERGODIC THEORY

A.G. SKALSKI Faculty of Mathematics University of Lddz' Banacha 22 90-238 Lbdz', Poland E-mail: adskal~imul.math.uni.lodz.pl In the first part of the paper we describe the natural scheme for proving noncommutative individual ergodic theorems, generalize it for multiple sequences of measurable operators affiliated with a semifinite von Neumann algebra M , and apply it to theorems concerning unrestricted convergence of multiaverages. In the second part we prove convergence of ergodic averages induced by several maps satisfying specific recurrence relations, including secalled Multi Free Group Partial Sums. This is the multiindexed version of results obtained earlier jointly with V.I.Chilin and S.Litvinov.

2000 Mathematics Subject Classification: Primay 46L51, Secondary 4 7A35

Many interesting cases considered in ergodic theory, both classical and quantum, can be cast in the common framework. One usually deals with a family of transformations (representing or evolution of a system, or averages of some quantities over periods of time), and asks questions about the convergence of these maps. Positive answers to these questions can be understood as the existence of some limit behaviour of a system, or of a mean value of a given quantity. The domain and range of these maps, the types of convergence, and the sense attributed to them, all depend on the specific problem. However, often one can assume that the evolution/averaging is linear, and that the domain of definition of our maps is some normed space B - clasically this might be the space of integrable functions over a probability space; in the quantum context it might be a C*-algebra, a von Neumann algebra or a noncommutative LP-space. In this paper we will

473

474

be especially interested in the multiparameter case, corresponding physically to the existence of several (not necessarily independent) evolutions of our system. We shall work in discrete time, and investigate behaviour at infinity. The aim of this paper is to present applications of the well-known classical scheme of proving individual ergodic theorems in the noncommutative context. After establishing some necessary notations in the introductory section, in Section 1 we describe how to extend this scheme to multisequences of maps acting in von Neumann algebras, as was done in [GL], [LM] and [CLS] for sequences indexed by one parameter. Section 2 collects and briefly summarizes known facts concerning unrestricted convergence of multiaverages and shows how to reprove them using the aforementioned scheme. Finally in Section 3 we present a few ergodic theorems on averages induced by several families of maps satisfying specific recurrence relations (of which the so-called Multi Free Group Actions are special examples). This is a multiparameter extension of results established using similar methods in [CLS], and derives from earlier work of A.Nevo, E.Stein and T.Walker.

Let d be a fixed positive integer. All multiindices will be underlined and will usually belong to &‘ or Nd, where & = N U (0). When Ic = {kl,. . ,kd} we will write mink = rnin(k1,. . .,kd}, maxh = max(k1,. . . ,kd}. Below we recall the notion of unrestricted convergence (convergence in Pringsheim’s sense) for a multisequence.

.

Definition 0.1. Let ( &)kENod be a multisequence of real numbers. We say that it converges to 7-C R in Pringsheim’s sense when for each E > 0 there exists n E N such that I& - 771 < E whenever E mink 2 n.

g,

Let M be a semifinite von Neumann algebra with a faithful normal semifinite trace T (in some places we will loosen or strengthen the assumptions on M and 7).Its positive part will be written as M + , its hermitian part as Msa and the lattice of all projections belonging to M as PM. We we shall denote the space of will also denote 1 - p by p’- for p E PJM.By all measurable operators affiliated with M , by L1(M) and L 2 ( M ) respectively the spaces of integrable and square-integrable operators (see [Ne]). The space % will be equipped with the topology of convergence in measure. Apart from the standard convergences (in norm or in measure) one can define in all these spaces various equivalents of the classical almost ev-

475 erywhere convergence. In our paper we will basically use two of them, the almost uniform convergence introduced by E.Lance in [L] and the bilateral almost uniform convergence introduced by F.J.Yeadon in [Y]:

-

Definition 0.2. A sequence ( x ~ ) : ?of~ operators of M is almost uniformly (a.u.) convergent to 2 E if for each E > 0 there exists p E P M ,+'-) < E and such that

z

A sequence (X~):=~

of operators of M is bilaterally almost uniformly (b.a.u.) convergent to 2 E if for each E > 0 there exists p E P M , T@'-) < E and such that

z

n+w

-+ 0.

IIP(Ga -4Pll,

We mention below several other counterparts of classical properties holding 8.e.. Let B be a linear space.

z

Definition 0.3. A map a : B + is bilaterally almost uniformly subadditive (b.a.u. subadditive) if for all E > 0 there exists p € PM,.@'-) < E and such that 11x42

5 IlW(2)Pll, + IlW(?/)Pll,.

+ 31)Pl,

A map a : B + is bilaterally almost uniformly homogenous (b.a.u. homogenous) if for all E > 0 there exists p E PM,.@I) < E and such that for all 2 E B , a E IR IlW(QZ)Pll,

= I4 IlW(4Pll,

*

The basic notion used in the following is that of kernel (also called

absolute contraction). Definition 0.4. A linear map positive contraction: VZEL'(M)"M

Q

: L1(M)

+ L 1 ( M )is a kernel if it is a

* 0 IQ ( X ) I I

0 Ix I I

and has the property V Z E L ' ( M ) , 2 2 0 .(Q(Z))

I (.)

476

Each kernel defines uniquely a o-weakly continuous map transforming M into M coinciding with the original map on M n L 1 ( M ) and usually denoted by the same letter. Moreover, for each kernel Q there exists a (unique) adjoint kernel a* such that for all u E L 1 ( M ) ,b E M

.r(a(a)b)= T ( U Q * ( b ) ) (the proof of all these facts can be found in

M).

We will also need a specific type of dense subsets of a Banach space.

Definition 0.5. Let ( B , 11 11, 2 ) be an ordered real Banach space with the closed convex cone B+, B = B+ - B+. A subset BOc B+ is said to be minorantly dense in B+ if for every z E B+ there is a sequence (2,) in Bo such that 2, 5 z for each n, and 112 - znJJ+ 0 as n + 00. For example, M+ n L 1 ( M )n L 2 ( M )is a minorantly dense subset of

1. A classical scheme for proving individual ergodic

theorems in the noncommutative context According to the description in the introduction we consider the following situation: let B be a Banach space, and let - be a family of linear maps acting from B to the algebra of all measurable operators affiliated with some semifinite von Neumann algebra M (in the classical case is replaced by the algebra & ( X , p ) of all measurable functions on a measure space ( X ,p ) ) . The typical way of proving pointwise (or b.a.u.) convergence of a family ( U ~ ( -Z ) ) for ~ ~ each ~ ; z E B consists of three parts:

z

A prove that we have the desired convergence for each z E BO- some dense subset of B B establish some estimate of the maximal type (separately for each x E B) C deduce the convergence for all z E B , using A, B and continuity properties of the maps uk We will now briefly describe each of these steps. In most cases (and commutative and noncommutative) B is the L1space, and BO is either the L2-space or the span of indicator functions/projections. The technique usually used in obtaining A is first to prove some kind of mean ergodic theorem (in the spirit of von Neumann’s

477

theorem on averages of contractions in a Hilbert space) and then use it to deduce the pointwise convergence - see section 3. As concerns B, basically all of the known maximal lemmas used in the noncommutative ergodic theory are based on the maximal lemma of F.Yeadon (PI).For our purposes it is convenient to formulate the version of this lemma proved in [PI,for a sequence of operators. In fact it was proved in a greater generality which we shall use in Remark 1.1.

Theorem 1.1. If QC : M + M is a kernel, ( x ~ ) % =is~ a sequence of operators belonging to L 1 ( M ) + and is a sequence of positive real numbers (estimation numbers), then there exists a projection p E PM such that

m= 1

For t.-e case of finite M, and operators Xm E M + n . r), one can replace the above estimates with the one-sided version (using the Kadison inequality). As we are interested here in the case of several kernels, we will need the following extension of the above maximal lemma:

.

Theorem 1.2. Let ( ~ 1 , .. ,(Yd : M + M be mutually commuting kernels. Then there exists a constant X d > 0 such that f o r every sequence of operators ( X ~ ) : = ~ belonging to L1(M)+ and (cm)E=i - a sequence of positive real numbers (estimation numbers), there exists p E P ( M ) such that

m= 1

Proof. The proof of the theorem stems from a fact proved by A.Brune1 in [B] for commuting contractions acting in the classical L1. Repeating his reasoning we can find for each k E F$ a nonnegative number a(&)such that

478

(9 C&N,d4b) = 1, (ii) the mapping U defined by U = the following inequality:

.

n-1

n-1

CkEN; - a(b)atl o . . . o afd

satisfies

nd-1

for any x E L l ( M ) , n E N, where x d > 0 depends only on d and n d E N depends only on d and n. As this is clear that U is also a kernel, the above version of Yeadon’s theorem ends the proof. 0

Remark 1.1. The above theorem remains true if M is any von Neumann algebra with a n.s.f. weight 4, ai : i = 1 , . . .d are positive linear maps acting in M such that for each x E M,O 5 x 5 I , we have ai(x)5 I , for 5 4(x),and the sequence (z,):=~ consists each x E M+ we have $(ai(x)) of operators in M + . Naturally, we have to replace everywhere r by $. The classical tool for C is the Banach Principle, established already in 1926. Its noncommutative generalization was proved by M.Goldstein and S.Litvinov in [GL] for the quasi uniform convergence, and then also by V. Chilin, S. Litvinov and the author ([CLS]) for the b.a.u. convergence. Here we need the extension of this result for multisequences (due in the classical case to F.Moricz [MI). Because of some technical subtleties we need t o work with minorantly dense subsets of a Banach space. Theorem 1.3. Let d E N, let B be a n ordered real Banach space with the closed convex cone B+, B+ - B+ = B , and for each Ic := (kl,. . . ,kd) E let ak- : B -+ %f be a continuous positive linear map. Asszlme that the following conditions are satisfied: (i) for each b E B+ and 6 > 0 there exists y E M + , 0 # y 5 I and K E N such that

and r ( I - y) 5 6, (ii) there exists Bo,a minorantly dense subset of B+ such that for each - a,(b) b.a.u. converge t o 0 as k,m -+ 00, b E BOthe operators ak(b) in Pringsheim’s sense. is b.a.u. convergent t o some element of %f as T h e n for each b E B, ak(b) k + 00 in Pringsheim’s sense.

479

Proof. The method of proof is typical; it is based on the Baire Category Theorem, and is reminiscent of that adopted in [GL] and [CLS]. Whenever we say that some multisequence of real numbers converges to a real number, it is to be understood in Pringsheim’s sense. Let’s fix E > 0. For each j , L , K E N we put = €12j+~,

Ej

Bj,L,K

=

{

B+ ’

3 g € k f + , y S 1 , r ( l - - y ) < a jv&Nod,mink>K

~ ~ u k ( b ) ’ Y5~ ~ L} O0

*

(1)

Fix now j E N. It is easy to see that

u 03

B+ =

Bj,L,K.

L,K=1

Moreover, using a-weak compactness of the unit ball of M one can show (exactly as was done in [CLS] for the sets X L , ~that ) each of the sets B ~ , L , K is closed. Once this has been done, the Baire Theorem allows us to infer that there exist L j , Kj E N,bj E B+ and Sj > 0 such that B ~ , L ~contains ,K~ the ball with centre in bj and radius Sj. This means that for any b E B+, Ilb - bjll 5 S j there exists yb,j E M+ satisfying conditions mentioned in (1). Let

1

1

Yb,j

=

be the spectral decomposition of

yb,j.

xdEb,j(A)

Define

We have

for all

and using

min

Whenever

we get for all

putting

min

480

< -

IIgbj ,jak(bj)gbj2j I

I00

IIgbj -4jL;~,jak(b j

- 4jLjC7 j)gbj- 4 j L ;

c,j

I

100

4L3j 1

57.

(5)

3

Now let b be a fixed element of B+. There exists a sequence ( c j ) p l of elements of B such that for each j E N

We choose a sequence (pj)gl of projections from M such that: for each jEN T(Pf)

<Ej,

ll(a@j(b+

Cj)

k

-

m+co

4- Cj))PjII, -’---)

0.

Put Q=

The definition of

03

00

j=1

j=1

A Pj A A fb+cj,j-

(pj)gl, together with (2) and (4), gives 00

00

j=1

j=1

It remains to prove that Jlq(ak(b)-am(b))qJJWtends to 0 as &,m+ 00. Fix S > 0, and let j E N be such that S > 3j-l. Then llq(ak(b> - am(b))qIlca

II P ~ (ak(b+cj 1-am(b+cj 1) ~llmj +IIf ci ,jak(cj)f c , j lloo +I lfcj ,jam(cj f c j ,j I loo < S for all lc,m E I++,‘ such that minb 2 K and m i n z 2 K , moreover K j

depends only on b and j (so actually only on b and 6). As the algebra is complete with respect to the topology of b.a.u. convergence (Theorem 2.3 of [CLS]), this ends the proof of the desired convergence for any b E B+. The general case follows immediately. 0 The above theorem remains true when one replaces throughout (both in the assumptions and in the hypothesis) convergence in Pringsheim’s sense

48 1

by the so-called ’maximal’ con\rergence. Obviously one also has to reformulate properly the condition on the b.a.u. boundedness of the maps considered. Moreover, when one replaces b.a.u. convergence by quasi uniform convergence, one may prove the theorem considering a dense subset of a Banach space (instead of a minorantly dense subset of an ordered Banach space). A careful reader would also notice that actually it is enough to k :B + is positive, b.a.u. homogeneous and assume that each map a subadditive (see definitions (0.3)) instead of assuming that the U k ’ S are linear. In our context the technical complication (using additionally the order structure in a given Banach space) will not be a serious obstacle. Working with complex Banach spaces of operators (say L1 ( M ) )we can first concentrate on the selfadjoint parts of them, and then use the existence of the convenient decomposition of a given operator into its real and imaginary part to conclude the convergence of an investigated sequence. This kind of reasoning will be further used without any comments.

2.

Unrestricted convergence of multiparameter averages

The theorem below appeared first in [GG] and then was also mentioned in [JX]. Here we show how to deduce it immediately from the maximal lemma established in the second-mentioned paper and the older result of D.Petz.

+

Theorem 2.1. Let d E N, p E (1,00), a; : L 1 ( M ) L 1 ( M ) (i = 1 , . . .,d ) be kernels. For each y E LP(M) denote by i@i(y) the n o r m limit

(a

00

of the sequence C:zi a : ( y ) ) n=l (which exists by the reflexivity of LPspace). T h e n f o r each x E LP(m) the multisequence ( S ~ ( -X ) ) where ~ ~ ~ ~ ,

b.a.u. converges to the operator sheim’s sense.

@I(.

. . ( @ d ( X ) ) . . .)

as Ic

+ 00

in Pring-

482

Proof. In [JX] it was proved that for each y E Lp(M)+, and each kernel /3 : M + M there exists an operator 5 such that for all n E N

.

n-1

In our context this immediately implies that there exists Z such that s&)

52.

This is sufficient for part B of our scheme. Putting BO = M+ n L1(M) n L 2 ( M ) and using theorem 4 of [PI we obtain part A. Theorem 1.3 shows that the multisequence considered is b.a.u. convergent, and standard reasoning allows us to conclude that the b.a.u. limit is equal to the norm limia

Remark 2.1. In [GG] a formula analogous to formula ( 6 ) was obtained for some 5 E Lp-'(M) (for any given sufficiently small E > 0). This clearly also suffices to conclude the proof in the same way as was done above. We would also like to briefly describe the situation concerning norm convergence. Here nothing depends on the number of kernels considered, the only important factor is the finiteness of the trace. This is illustrated by the following basic example

Example 2.1. Let M = Lw(R), with trace given by the Lebesgue integral, and let cr be the standard shift operator,

M f ) ) ( t )= f(t - 1) for all f E L1(R), t E R It is clear that cr is a kernel, and

IISk(X(0,1))II1

for all k E N (by interval (0,l)).

x(0,l)

=1

we understand the characteristic function of the

The situation described above cannot happen when M = L"(X, p ) and (X,p ) is a finite measure space. On the algebraic level this corresponds to the finiteness of the trace on the algebra M .

483 Theorem 2.2. Let M be a von Neumann algebra with a faithful normal finite trace r. Let d E N,cri : L 1 ( M ) + L 1 ( M ) (i = 1 , . . . , d ) be kernels and let x E L 1 ( M ) . Then the multisequence sk(x) converges in L1-norm as k + 00 in the Pringsheim’s sense. Proof. The proof follows by induction with respect to the number of kernels. For each r E ( 1 , . . ., d } and x E L 1 ( M )if only the sequence (S:(Z))~=~ is convergent in &-norm its limit will be denoted again by ar(x) (in the case of one kernel we shall write simply @(x)). Let Q : L 1 ( M ) + L 1 ( M ) be a kernel. For each n E N we define the set

A , = {y E L’(M) : y* = y, -nI 5 y One can see that if (&)& in PM,and y E A, then

I nl}.

is a sequence of mutually orthogonal projections

I I 03

Ir(yl)k)l

5 I T ( ’ W k ) l 5 72

T ( c P i ) i=k

7

so the expression on the left side of the above inequality tends to 0, as k tends to cm, uniformly with respect to y. The theorem 11.2 of [A] allows us to infer that A, is weakly relatively compact. As A, is a convex and norm-closed subset of L 1 ( M ) , Mazur’s theorem shows that it is actually weakly compact. Now we can apply Theorem 2.1.1 of [K] (notice that cr : A , + A,) to conclude that for each x E A, there exists @(x) E A, k+w such that S k ( Z ) + @(x), Q(@(z))= @(x). As the set A, is normdense in the hermitian part of L 1 ( M ) ,and each operator in L 1 ( M )can be expressed as a sum of two hermitian operators, the proof of the theorem for the case d = 1 is finished. Assume now that we know that the theorem holds for d - 1. Then we can write the following inequalities:

u,”==,

11@1(.*

I 11@1(.

* *

(@d-.l(@db))).

(@d(Z))*

*

0

)

*

.) - Sk(4lll

- Sm(@d(4)lll + IISfIl(@d(2)

- QP(4)Ill

I ll@l(. ( @ d - l ( @ d @ ) ) ) .) - Sm(@d(.))lIl + ll(@d(.) - Q;d(x)lllr where m E @-’, m = {kl, . . .,k d - l } , and we used the fact that each kernel * *

*

-

is a contraction. It is easily seen that the first part of the above expression tends to 0, as m tends to infinity in Pringsheim’s sense (by the induction assumption), and the same can be said about the second part as kd co. 0

484

Remark 2.2. A version of the above theorem for one kernel is due to C.Radin ([R]). However since our assumptions are slightly different, we do not need to introduce the abstract notion of a unit in a predual of a von Neumann algebra. Moreover, we give a more detailed proof. As was mentioned in Theorem 2.1, for p E (1,co)the norm convergence of (mu1ti)averages follows immediately from the reflexivity of the space in question. The same is true in the non-tracial situation.

3.

Ergodic theorems for multiparameter Free Group Actions

For the sake of clarity we shall restrict ourselves to the case of d = 2 throughout this section - all the results hold for general d E N. We begin with a general fact concerning the strong convergence of averages, formulated in the spirit of von Neumann’s ergodic theorem. Let for any p E ( 0 , l )

D, = { z E C : [dz+ 4 p - 4p2 + dz - 4 p - 4 9 1 5 2&,

The next theorem is an easy consequence of the following lemma from [WI:

(fn)r=l

Lemma 3.1. Assume that p E (0,1] and is the sequence of functions o n the complex plane such that f o ( z ) = 1, f I ( z ) = z and z f n ( z ) = P f n - l ( z ) -t(1 -P)fn+l(z) f o r 72 >_ 1 (2 E men fk(z) converges pointwise i f fz E D,. I t converges to zero on D, \ { 1). Moreover f k ( z ) ( 5 1. there exists N E N such that for all n 2 N and z E D, : 1

i~ i z i

zizi

Theorem 3.1. Let H be a Hilbert space, xo,1,x1,o - commuting normal operators in B ( H ) whose spectra are respectively subsets of D,, and D,, f o r some. p 1 , n E (0,1]. Let XO,O = I and xm,n (for m n 2 2), operators satisfying the relations x1,OXm1n = P1xm+l,n (1 - Pl)Xm-l,n ( 1 - n ) Z m , n - I . Then the multisequence and xO,1xm,, = p2xm,,+1 k1-1 &a-1 converges in Pringsheim’s sense to the pro~ m = o C n = o ~ m , nkEN2 )

(A

+

jection P onto the set { q E H : 2177 = 2277 = 77).

+

+

485

Proof. Let to the pair X O J ,

denote the spectral measure on D,, x Dp2corresponding It is easy to see that

x1,o.

kl-lk2-1

k1-lk2-1

+,

(fp’)r=l, (f?’)r=l

where by we understand the sequences introduced in lemma 3.1 with respectively p = pl and p = p z . Therefore

.

_.

ki-lka-1

kl-lka-1

and the desired strong convergence follows from standard properties of spectral integrals. 0 The following notations will be used: let A be a von Neumann algebra with a faithful normal semifinite weight 4. Further let N4 = { A E A : $(A*A) < CO},

do = N i nN+,

and let 3cg be the Hilbert space completion of (with respect to the scalar product ( A , B ) @= d(B*A)).We will write A4 for the canonical injection of in 34,and 7r4 : A + B(3t) for the faithful normal representation such that for all A E A, B E do

r + ( A ) ( b ( B )= ) AdAB) (left regular representation). We will also occasionally use the standard language of Hilbert algebras, as in [TI. Let us describe the averages we will consider. Fix real numbers p l , p 2 E (0,1]. Let o1,0,a 0 , l be normal, completely positive, unital, &invariant and commuting maps acting on the algebra A. Moreover let gm,n (for m.n 6 NO,m n 2 2) be positive maps acting on A defined recursively by the following relations:

+

gl,Ogm,n

+

= ~ l g m + l , n (1 - p l ) g m - l , n ,

where 00,o = Idd. Clearly for all m, n E &

4866

and each of the sequences (Om,o)z=o and ( O O , ~ )satisfies ~ = ~ the independent one-parameter recurrence relations of the type similar to ones described It is therefore easy to observe above (respectively for constants p l and n), that each of the maps om,,, is normal, completely positive, unital and $invariant, as the initial maps 01,o and OOJ had these properties. We will write for each k , k l , kz E N, kl-lka-1

Example 3.1. Let us now describe the basic example of maps satisfying the above conditions. Let { ai}i&, { bi}zlbe respectively sets of generators of Frl and Fr2 (the free groups on T I and r2 generators; if T I = r2 we consider isomorphic copies of the same group) and let {ai}i&, {/3i}Izl be sets of $-invariant fiautomorphisms of the algebra A, such that aj o pi = /3i 0 aj for i E (1,. . , T I } , j E (1,. , r 2 } . Assume that we have group homomorphisms @ I : Fq + Aut(d) and @ 2 : F'2 + Aut(d) defined on the basis elements by @1(ai)= ai,i E (1,. .,TI}, @ 2 ( b j ) = P j , j E (1,. . ., r 2 } . Let for each n E N w2)(respectively wp))denote a set of reduced words belonging to FTl (Fra) of length n. Further let Iw?)~ (respectively lwi2'I) denote the cardinality of this set (e.g. Iwil)I = 27-1). The following elements are double-indexed equivalents of objects introduced in [NS] and will be called the Multi Free Group Actions and the Square f i e Group Partial Sums:

.

..

.

The respective recurrence relations follow from properties of the free group, and = 1- each word of length n in with parameters pl = 1 a free group on r generators multiplied from the left by one of the words of length 1 yields either the word of length n 1 (in 2r - 1 cases), or the word of length n - 1 (in 1 case).

&

+

Note that in the situation described in the beginning we can define operators 61,o and 50,1 acting on A@(&) by

Gl,O(A@(B))= A & J l , O ( B ) ) ,B

.A0

(similarly 6 0 ~ ) The . complete positivity, unitality and $-invariance of the initial maps imply that 51,0,&,I are contractive, and as such can be continuously extended to the whole I?@ (the extension will be denoted by the

487

same symbols). Using recurrence relations we define in a natural manner &,or a m , , (for m + n 2 2), etc.. The important fact concerning maps 81,o and &,I defined in this way is that they are commuting normal operators, so they satisfy all assumptions of theorem 3.1, except possibly the spectra conditions.

A. If the spectra of &,I and SI,O(as operators in B ( H 4 ) ) are respectively contained in D,, and in D,, then the rnultiseqzlence (Sm,n(A))z,,=Iconverges strongly in Pringsheim’s sense to E do. Moreover zf P E B ( H 4 ) is a projection onto {q E H+ : a&iq = a;,oq} then A,j,(A) = PAb(A). Theorem 3.2. Let A E

a

Proof. Theorem 3.1 shows that ,!?& for any ~ 1 , 7 1 2E .Ah, A E do

(7rb(Sk(A))WI7?2) =

k+w -+ P

strongly. This implies that

(~b(Sk(A))I7l27l!) = (~k(~b(A))lWl!)

k+w

-+ (P(~dA))Irlzr)!).

If II, E d* is defined via $ ( B ) = (7rb(B)qlIq2),B E A,then the absolute value of the right-hand side of the above expression can be estimated by 11$~11. \lAllw (all Sk are contractive). Using the fact that the set of the above forms is dense in A, we may conclude that the functional *4,31c)+ lim $(&(A) k+w

EC

is well defined and continuous. Therefore there exists all $ E A,

a E .A0 such that for

llAllw 5 llAllw. Consider again any ql ,q2 E 4.We have

It is clear that

( 4 ) mIrl2) = (pAb(A)Irlzrl;)= (KL(rl1 )PA&)

Irl2)

,

where 7rL denotes the right regular representation. As q2 was arbitrary, we obtain ~4

( ~ ) v= I mk (qi)PLj( A ) .

As all maps Sk are *-preserving, we can easily prove that This applied to (7) gives ~ b ( J ) * q= l

(qi)PAb(A*).

(7)

(a*)= (a)*. (8)

488

Using Proposition 10.4 of [SZ], we infer from (7) and (8), and the fact that the Hilbert algebra A is full, that PA,(A) E A, PA,(A) = A,(a). Now the required strong convergence can be obtained almost immediately, again taking any rl E 4 :

4 S k ( A ) ) r l = n;(rl)AdSk(A))= +rl)%(A,(A)) k+w

-+ T;(rl)PA,(A) = %4477

and using the fact that ~6 is normal.

0

We need the following lemma, which is a straightforward generalisation of Lemma 1 of [NS].

Lemma 3.2. There exist Cp,,Cp2 > 0 such that, f o r all m,n E A E A+,

.

N and

3m-13n-1

The following consequence is needed below.

Lemma 3.3. For any A E A and m E @ the multisequence IlSk(S,(A) A)llw tends t o 0 as maxk tends to 00 (and so also in Pringsheina’s sense).

Proof. We begin with the following observations: &(A) - A = Cjz0 ml-1 C z i l (&(gj,l(A) - A ) ) , so it is enough to prove convergence

for expressions such as llSk(oj,~(A) - A)llw. In turn we can reduce this to proving that for each j , Z E N llSk(q& O ~ : , ~ -A)llw ( A ) tends to 0 as maxk tends to 00. However this can be obtained with the help of the previous 0 lemma by considering standard Cesaro averages. We will use the existence of a convenient decomposition of a selfadjoint operator in A, proved by D.Petz in [PI:

Lemma 3.4. Suppose that B E A, B = B*. T h e n there exist C E A, C = C*, D , EE A+ such that B = C D - E , llCllw I 4(B2);, 4(D) L 4(B2) 4,+(E)5 4(B2)4 and IICIIw, IPIIm, IIEIIw IIIBIIw-

+

489

Now we can formulate the first of the two main results of this section.

Theorem 3.3. Let A E Jto. If the spectra of &,J and ii1,o (as operators in B ( H 4 ) ) are respectively contained in D,, and in D, then the sequence (Sn(A))Sp=l is b.a.u. convergent to E A.

a

Proof. Theorem 3.2 implies that if k E N and tk

= gk(A#(A))- PA+(A)

a

tends to 0 as k tends to cx). As is invariant under 01,o and mhl, we have s k ( A ) = for all k E N. Moreover & E A$(&) and if &. = A4(Bk), B k E A ,We have then

a

Decomposing A into its real and imaginary part we can assume that Bk = B i . Let us fix e > 0 and choose a subsequence such that 4(3k2,)4L: n-12-n-1 E . For each n E N we can decompose Bk, according to Lemma 3.4, B k n = c k , 4-Dk, - Ek, Without loss of generality we assume that say Ek, = 0. Now we apply Lemma 1.2 (or rather actually its version in Remark 1.1) for maps O O J , 01,o and a sequence (Dk,)z=l, with estimation numbers respectively equal to as a result finding a projection p E PA such that

(Icn)z=l

-

i, W

n=l

and an application of Lemma 3.3 ends the proof.

0

The scheme described in section 2 allows us to deduce immediately the second important result.

490

Theorem 3.4. Let M be a von N e u m a n n algebra with a normal semifinite faithful trace r and let x E L1(M). If ( u k ) k E ~ is 2 the sequence of maps acting o n M and satisfying the conditions described before Theorem 3.2 t h e n the sequence (Sn(x))F!.l converges b.a.u. to some B E L'(M).

Proof. Assume that x 1 0. As it is clear that OOJ and u1,o are commuting kernels, we can (as was done above) use Lemma 1.2 and Lemma 3.2 to deduce that for every E > 0 there exists p E PM such that r(pL)< E and for all n E N llPSn(Z)Plla, 5 E-lCp, cp,x211x111. Obviously each Sn treated as a map from L1(M)sa to is positive and continuous. Theorem 3.3 implies that for any x E M+nL2(M) the sequence (Sn(x))r=l is b.a.u. convergent. As M+ n L2(M) is a minorantly dense subset of L1(M)sarwe are in position to apply the noncommutative Banach principle (Theorem 1.3) to end the proof.

As a special case, putting p1 = . . . = Pd = 1 we obtain the noncommutative generalization of the classical result of A.Brune1: Corollary 3.1. Assume that (~1,.. . ,ad are commuting, n o m a l , completely positive, unital, r-invariant maps acting o n M . T h e n f o r each x E L1(M) the sequence ( S ~ ( X ) ) ~ = ~ ,

,Y

il=O

id=O

is b.a.u. convergent. All the results remain true if instead of considering the averages over squares we deal with so-called sequences of indices tending to infinity but remaining in a sector of Nd. This means that we consider averaging over sets of the type { 1,. . ,kl (n)}x . . . x { 1, . .. ,kd(n)}, for which there exists C > 0 such that < C for all i , j E (1,...,cl}, rz E N.

.

Acknowledgments During the preparation of this work the author was partially supported by the KBN Research Grant 2P03A 030 24 and by the European Comission HPRN-CT-2002-00279, RTN QP-Applications. The author would like to express his gratitude to Vladimir Chilin and Semyon Litvinov, whose remarks essentially improved the final form of this paper.

49 1

References [A].

C.Akemann, The dual space for an operator algebra, Trans. Amer. Math. SOC.126 (1967), 286-302 [B]. A.Brune1, Theordmb ergodique ponctuel pour un semigroupe commutatif finiment engendrd de contractions de L1,AIHP B 9 (1973), 327-343 [CLS]. V.I.Chilin, S.Litvinov and ASkalski, A few new results i n noncommutative ergodic theory, to appear in Journal of Operator Theory [GG]. M.S.Goldstein and G.Y.Grabarnik, Almost sure convergence theorems i n von Neumann algebras (some new results), Special classes of linear operators and other topics (Bucharest, 1986), 101-120, Oper. Theory Adv. Appl., 28, Birkhuser, Basel, 1988 [GL]. MSGoldstein and S.Litvinov, Banach principle i n the space of T - measurable operators, Studia Math. 143 (1) 2000, 33-41 [J]. R.Jajte, Strong limit theorems i n noncommutative L2 spaces, Lecture Notes in Math. 1477, Springer, Berlin-Heidelberg-New York 1991. [JX]. M.Junge and &.Xu, Thoremes ergodiques maximaux dans les espaces L, non wmmutatifs, C. R. Math. Acad. Sci. Paris 3 34 (2002), no. 9, 773778 [K]. U.Krenge1, Ergodic theorems, Walter de Gruyter, Berlin-New York 1985 [L]. E.C.Lance, Ergodic theorems for convex sets and operator algebras, Invent. Math. 37 (1976), 201-211 [LM]. S.Litvinov and F.Mukhamedov, On individual subsequential ergodic theorem in von Neumann algebra, Studia Math. 145 (1) (2001), 55-62 [MI. F.M6ricz, Extension of Banach’s principle for multiple sequences of operators, Acta Sci. Math. 45 (1983), 333-345 [Ne]. E.Nelson, Notes on the noncomutatawe integration, Journal of Functional Analysis 15 (1974), 103-116 ”1. A.Nevo, Harmonic analysis and pointwise ergodic theorems for noncommutang transformations, Journal of the AMS 7 (1994), 875-902 [NS]. A.Nevo and EStein, A generalization of Birkhofl’s pointwise ergodic theorem, Acta Math. 173 (1994), 135-154 [PI. D.Petz, Ergodic theorems i n won Neurnann algebras, Acta Sci. Math. 46 (1983), 329-343 [R]. C.Radin, A noncommutative L1-mean ergodic theorem, Advances in Mathematics 21 (1976), 110-111 [SZ]. S. Stratila and L.Zsido, Lectures on von Neumann algebras, Abacus Press, 1979 [TI. M.Takesaki, Theory of operator algebras. 11, Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003 ywl. T.Walker, Ergodic theorems for free group actions on von Neumann algebras, Journal of Functional Analysis 150 (1997), 27-47 M. F.J.Yeadon, Ergodic theorems for semifinite von Neumann algebras - I, J. London Math. SOC.16 (1977), 326-332

LEVY PROCESSES AND TENSOR PRODUCT SYSTEMS OF HILBERT MODULES

MICHAEL SKEIDE CP-semigroups on C*-algebras lead to product systems of Hilbert modules and quantum LBvy processes have an associated CP-semigroup. In these notes we review our latest results about the relation of the Arveson system of the LBvy process and the product system of its associated CP-semigroup and discuss some examples how the generators and product systems of classical Markov semigroups are related to that of brownian motion which is both a Markov process and a LBvy process. All results were obtained in collaboration with one or more of the following people: Luigi Accardi, Franc0 Fagnola, Uwe Franz, Volkmar Liebscher, Michael Schiirmann.

(1) The Arveson system of a Lkvy process and the product system of its CP-semigroup (joint with U. Franz and M. Schiirmann) (2) The product system of a classical Markov semigroup (joint with L. Accardi) (3) The product system of the brownian semigroup and of the OrnsteinUhlenbeck semigroup (joint with F. Fagnola and V. Liebscher) (4) Hunt’s formula for generators of CP-semigroups?

492

493

1. The Arveson system of a LQvy process and the product system of its CP-semigroup

In this section we discuss a subclass of quantum LQvy processes (LQvy process, for short), namely, those defined on a C*-algebra mapping into some IB(H), and on % ( H ) there is a time shift acting with respect to which the Ldvy process behaves covariantly. Then we have a look at the relation to the product system of Hilbert modules associated with the CP-semigroup of the Levy process. The results are joint work with U. F’ranz and M. Schurmann.

Ldvy prozesses with Arveson system Let B be a C*-bialgebm (B,A,d). By this we mean that 13 is a unital C*-algebra that contains a dense *-subalgebra BO such that (230,A t B0,S t Bo)is a *-bialgebra. This happens, for instance, if B is a compact quontum group [Wor98]. Consider a quadruple ( j , H , 6, R) of a family j = ( j t ) t l o of unital representations of l3 on a Hilbert space H , an Eo-semigroup (time shift) 19 (that is a semigroup of normal unitd endomorphisms) on % ( H ) and a unit vector R E H . The Arueson system (see Arveson [Arv89]) fy@ = ( f y t ) t l o of 6 consists of the Hilbert spaces

At = {hi E IB(H): 6t(a)ht = hta with inner product

(a,

0)

(U E

IB(H))}

defined by (hi,h:)l = hi hi. We find

fyt@H = H

via

h i @ h = hth

98 @ A t = As+t

via

ha @ ht = h8ht.

(The identifications are clearly isometric. Surjectivity follows by standard arguments. See, for instance, [Arv89,Ske03b,Ske03c].)We see that

6t(u)(ht €3 h) = 6t(u)hth = htuh = (id3, @ u ) ( h t @h). 1.1 Definition. The quadruple ( j , H , 6 , 0) is a stationary LCvy process with Arveson system, if (1) po = OR* is increasing for 6, i.e. R = 6 t ( R R * ) R . (2) j is adapted, i.e. j t ( l 3 ) C 6t(!B(H))’ = !B(At)@ idH so that j t = gt @I idH for a (unique) representation j t : B + %(At).

(3)

j8+t

=

*&

( := ( j , @ i t ) 0 A ).

494

(4) Setting cp(u) = (R, an) and

cpt

= cp o j t we have lim cpt(b) = 6(b) = t+O

cpO(b).

Observe that cpt are states on 23, while cp is a vector state on IB(H). Some immediate consequences are: 0

0

0

0

By 1. there is a unique vector Rt E fit such that R = &(RR*)R = (id3, @RR*)R = Rt @ R. One checks that R, @ Rt = R,+t so that 0" = (Rt),20 is a unit for fie. By 2. and the preceding conclusion the increments (j,,t)OSsit with j 8 , t = 29, o j t - 8 to disjoint intervals factorize in the state cp. That is the increments are (tensor) independent in the state cp. By 3. and the preceding two conclusions we find q 8 + t = cps*cpt ( := ( 9 8 @qt) o A ), i.e. the cpt form a convolution semigroup of states. By 4. this convolution semigroup is pointwise continuous.

All these properties mean that the restrictions of j to BO form a Lkvy process on Bo with values in 'B(H) with the state cp in the sense of Schiirmann [Sch93]. Stationary Lkvy processes are characterized (up to stochastic equivalence) by their infinitesimal generator $ defined as the pointwise derivative of cpt at t = 0. By an application of the fundamental theorem of coalgebras (see Abe [Abe80]) this generator exists, at least, on Bo. By [Sch93] the possible generators are exactly the (closures of) the hermitian linear functionals on ,130 that are normalized ($(1) = 0) and conditionally positive (i.e. positive on ,130 n kerd). By [Sch93] every such generator (on a general *-bialgebra Bo) has a realization as Lkvy process by operators on an invariant dense domain on some symmetric Fock space I'(L2(&, K ) ) where K is some Hilbert space. Whenever the j t map all hermitian elements in ,130 to operators which are essentially self-adjoint on a common invariant domain, then by spectral calculus we may pass to algebras of bounded functions of these elements. The operators j t ( b ) are then affiliated to the von Neumann algebra generated in !B(H) by all spectral projections of all hermitian elements and defining (sufficiently continuous) mappings amounts to define mappings on the algebra generated by continuous functions of self-adjoint elements in ,130.

+

1.2 Remark. Whether finding such an invariant domain is possible, can be seen from the generator. Indeed, a certain dense subspace KO of K carries a representation of the *-bialgebra for which it is invariant. It is sufficient that this representation maps all hermitian elements to operators

495

which are essentially self-adjoint on KO.

1.3 Example. The case of a classical LQvy process on the real line is a particularly useful example. We set

f3

= eb(q

“f>l(w>

=

f(z+?/)

cptm =

/WPt

(recall that S = 90). Here B is the algebra of continuous bounded functions of a centered gauBian random variable and pt N ( 0 ,t ) . It is essential that the algebra does not depend on the variance t . Brownian motion is now represented by the operators a * ( H p t ]+) a ( H p t ] )on the Fock space r ( L 2 ( R + ) ) . Passing to bounded functions eiwz (which span a nice *-subbialgebra of Q(R)) we obtain Weyl operators. N

1.4 Remark. Every generator has a minimal realization as a LQvyprocess ( H is generated from R by the j t ( b ) ) and we know it is on a Fock space (with a K constructed from the generator of the process by a GNS-type construction). All realizations we know are Fock and, therefore, have type I Arveson systems. Every LQvy process with Arveson system has at least the unit R@ so that its Arveson system is, therefore, type I or type 11. We might ask: Do there exist LCvy processes with type I1 Arveson systems?

Product systems from LCvy processes with Arveson system By Bhat and Skeide [BSOO] for every CP-semigroup on a C*-algebra 23 (i.e. a semigroup @ = (Gt),,, of, usually unital, completely positive mappings @ t on B) there is a product system EO = (Et)t,O of Hilbert B-B-modules = Et (i.e. E , @ Et = Eg+t in an associative wayywith a unit (i.e. It E Et and I , @ It = <,+t) fulfilling (&, b&) = at@).If the product system is generated by the unit, then it is determined uniquely by @. With every LQvyprocess there is associated a unital CP-semigroup = (@dt>O This semigroup is strongly continuous. It is continuous, if and only if the generator is bounded. Let us see, whether we can find directly a product system with unit as described in the preceding paragraph. We put

Et = B B f i t

and

It

= 1got,

where 23 @I f i t , as a right Hilbert module, is the exterior tensor product with inner product ( b 8 ht, b’ @ hi) = b*b’(ht, hi). Defining on Et the left

496

multiplication

b(c @ ht) = (ids*&)(b)(c@ ht), we find ipt = (&, a&). It is an illuminating exercise to check that

( b @ h8)

@

(I @ g t )

* b(1 @ g t ) @

h8

(*I

defines a family of isomorphisms E8OEt = E8+t which iterates associatively. (Mind the flip in (*).) For every unit u@ for B@by 1 @ tit we obtain a unit for EO.In particular, the & form a unit. 1.5 Theorem. E0 is the minimal product system of ip, i.e.

if and only if j is the minimal L h y process. 1.6 Remark. E0 is continuous in the sense of Skeide [SkeOSa]. However, Theorem 7.7 of [SkeO3a] tells us that, if the generator is unbounded, then E0 may not contain a single continuous unit. In other words, for unbounded generator E0 is type III. 2. The product system of a classical Markov semigroup

Like every CP-semigroup, in particular, the CP-semigroups of classical Markov processes have a product system. In this section we describe its construction directly from the data provided by Daniell-Kolmogorov construction. This is joint work with L. Accardi. See also Skeide [SkeOP]. So let Pt be a semigroup of transition functions on R and fix an initial distribution pa. The choice of the initial distibution is of a rather technical nature. What we, actually, need is the measure type of po or, in other words, the structure of the von Neumann algebra 23 = L"($po), and we require that (almost) all measures Pt(x,0 ) on R are absolutely continuous with respect t o po. This ensures that [ipt(f)](y) = J Pt(y, dx)f (x)defines a completely positive mapping on B so that @ = ( i p t ) t,O is a CP-semigroup. The actual initial distribution may vary from application to application. For instance, if the process is also a LCvy process, then the initial distribution would be do (which does not satisfy the hypothesis). Denote by X = ( X t )t20 the Markov process associated with P and po in the representation where X t is the t-coordinate function on nt,,+R

497

and the probability measure p on this space is that obtained by DaniellR,p) and observe that Kolmogorov construction. We put A = LO3 this von Neumann algebra is generated by the process X . By dt(X8) = Xs+t we define a time shift Eo-semigroup on A. By p = E(oIXo) we denote the conditional expectation onto the unital subalgebra Z? of A. A carries a natural filtration. For I C It.+.set A1 = Loo(Ute,R,p ~ ) and At = A p t ] . Observe that all A1 are can be viewed as semi-Hilbert &modules by defining the inner product (2,2’) = p(x*z’). Set

(n,,,,

-

E = A/N = 1+N

Et = At/&
<

where the N indicate the corresponding nullspaces of (.,a). Then E is a Hilbert &module, Et are Hilbert B-&modules (left action via X t , i.e. f ( X o ) acts on xt E Et by multiplication with f ( X t ) = d t ( f ( X O ) ) ! )such that

EOEt = E < @
=

<

E , O E t = E,+t

&(a) = a O i d ~ ,

<8+t

T t ( f ) = (
<8

@
=

(
via

F O Gt = 6t(F)Gt

F, O Gt = 8t(F,)Gt.

(rt)

= is a unit. In particular, E0 = ( E t ) is a product system and At is spanned by f n ( X t , ) . . . f i ( X t l ) f o ( X o ) ,t = t , > ... > t~ > 0. Therefore,

Et = s p a n { f , < t n - t n - l 0.. * @ f 1 t t 1 f 0 (n E N, fi E a, t = t , > . . . > tl > O ) } = span { fn 0, t , . . . tl = t )} .

+ +

The first line emphasizes end points of time intervals [tn-l, t,] and is more useful in the following section, while the second line emphasizes the length of such intervals and is suited better to see that E 0 is exactly the product system from [BSOO] in our special case. 3. The product system of the brownian semigroup and of the Ornstein-Uhlenbeck semigroup

It is well known that Ornstein-Uhlenbeck processes can be thought of as modifications of brownian motion. In this section we illustrate different possibilites t o identify the product systems of Ornstein-Uhlenbeck semigroups

498

and show how they are related to the product system of the brownian semigroup. These results are joint work with F. Fagnola and V. Liebscher. The brownian semigroup on 23 = ea(R) is given by

[W)l(4=

JdY e - G ( Y

+

while the Ornstein-Uhlenbeck semigroup is given by

with ~ ( t=)d p . The member EF of the product system of the brownian semigroup as constructed in the previous section is generated by functions F t ( z n ,...,x1,x) (t = ( t = t, > ...tl > 0)), where xi and x are to be thought of as the values of Xti and of X O ,respectively. The inner product of two such functions is I

--

-

(GtFt)( yn

+ ...y

1 +a:,

yn-1+ . . . y 1 + Z

7

* * .

Y1

7

+x, x)

that the yi can be thought of as the values of the increments Xti - Xti-l. (Different tuples ti, t 2 must be made equal by adding suitable intermediate time points to each of the two tuples obtaining a tuple t. The involved functions, then, simply do not depend on those xi that correspond to added time points.) Passing to functions Ft defined by SO

-

Ft(Yn,. . . ,Y1,Z) = Ft( yn+. . . y1 +x

, yn-l+. . .y1 +x ,

0

.

.

, y 1 + 2 , x),

we see that E: is generated by functions Ft with inner product

and left multiplication

corresponds to the function

499

and in

to

The tensor product is (F5@

Gt)(Ym,.. . ,Yl,Zn,. . .,Zl,Z)

=

. .,91,Z n + . . . + ZI + z)Gt(Zn, . .

F 5 (ym,.

21

x).

For the Ornstein-Uhlenbeck semigroup we find Etw 'Y EF as right modules, however, left action of f on Ft is multiplication by

fn&n-tn-l

0.. . 0fl(tt, fo now corresponds to the function

...

The tensor product is

+Z

n - l ~n-1~ e - ( t - t n - l )

+ . . . + zl@e-(t-tl)

+ ze-t)Gt(zn,. .. ,zl,z).

E m is spanned by its strongly continuous units but contains no continuous unit. It contains no elements which commute with B . In particular, E m is not a time ordered Fock module nor does contain one.

500

3.1 Remark. There is a third way to describe these product systems based on the description of both processes as Markov processes with continuous paths worked out by Liebscher. This description works rather in the set-up of von Neumann modules (strong closures). The statement that neither E B nor Em contain central elements can be seen most easily in this picture and shows, additionally, that this effect (unlike other simple examples for type I11 product systems of Hilbert modules) does not disappear under strong closure. 3.2 Remark. There are so-called quantum extensions of the CPto CP-semigroups on 'B(L2(R)). semigroups 'B and cR1on Cb(R) c 'B(L2(R)) It is well-known that the product systems of these extensions are timeordered Fock modules of !B(L2 (R))-modules which contain the original product systems. We see that the type of the product system of a CPsemigroup may differ from the type of the product system of one of its extensions. This shows that the theory of extensions of CP-maps as initiated in Gohm [GohOS] is an important task and needs to be generalized to CP-semigroups. 4. Hunt's formula for generators of CP-semigroup?

In this section we discuss a Hilbert B-B-module, introduced and called the tangent bimodule in Sauvageot [Sau89], constructed from the generator L of a CP-semigroup a. We illustrate with some considerations, in how far this bimodule could help to identify the product system of the CP-semigroup. We comment on a suggestion for a notion of locality from [Sau89]. Let L be the generator of a unital strongly continuous (CO)CPsemigroup 9 on a C*-algebra B with a dense *-subalgebra Bo of B in its domain. Define a sesquilinear mapping on 230 €3 B by

(bo @J b, bb €3 b') = b*L(b:bb)b'.

L is conditionally complete19 positve what means exactly that positive on the 230-B-submodule

(0,o)

is

span{ (bo €3 1 - 1 €3 bo)b}

of BO €3 B. The completion (modulo length zero-elements) F is a Hilbert Bo-B-module. Moreover, approximating C by the CCP-maps +(at - idB), we see that the canonical homomorphism from BO into !Ba(F) is a contraction. Therefore, F is even a Hilbert B-B-module.

501 4.1 Example [BBLSOO].If L is bounded, then the product system of @ [BSOO]is a strongly dense subsystem of the strong closure of the time ordered Fock module over F . 4.2 Example. For the Ornstein-Uhlenbeck and the brownian semigroup we find (modulo length-zero elements and completion)

F = { F E e'(R2): F ( z , z ) = 0)

(F,G)($) = &F(z,z)&G(z,z).

Obviously, f F = Ff for all f E 23. The same is true for the time ordered Fock module over F . We see that CP-semigroups with bounded generators (which, therefore, cannot have a local part) can be described conveniently on time ordered Fock modules over the tangent bimodule, while (generally) the tangent bimodule of classical Markov semigroups with local generators has the trivial bimodule structure so that its time ordered Fock module has nothing in common with the product system of the CP-semigroup. Sauvageot [Sau89] called a bimodule triuialbable, if it can be embedded as a bimodule into a free bimodule, and suggested to call a generator of a CP-semigroup local, if it is trivializable (what is more or less equivalent to say that the bimodule is centered in the sense of Skeide [Ske98]). We reply to this suggestion by remarking that in the case of l3 = !B(H) all (sufficiently closed) bimodules have the form fj @ B ( H ) (fj some Hilbert space) and, therefore, are free bimodules and a fortiori trivializable. The suggested notion of locality would, thus, imply that all generators, also bounded ones, of (normal) CP-semigroups on B ( H ) are local generators, what seems unsatisfactory. We would like t o contrast this by the suggestion to say, a generator has no local part, if (the strong closure of) its product system is the time ordered Fock module constructed from (the strong closure of) its tangent bimodule. This definition would be satisfactory, if we could proof a theorem which asserts that every generator decomposes into a certain nonlocal part and another part from which no nonlocal part can be separated. For classical LBvy processes this separation is exactly the separation given by Hunt's formula [Hun561 where the first part is the Poisson part and the second part is the gaujlian part of the generator. For classical Markov processes one might hope that the separation corresponds to a separation into jump processes and processes with continuous paths. It is, however, clear that such results depend on concrete technical versions of the definitions and it is likely that it holds only for a large subclass of Markov processes.

502

If the programme has success in the quantum case, then it should provide us with a satisfactory notion of locality. However, so far, the problem of finding the analogue of Hunt’s formula is solved not even for (quantum) L6vy processes. Only special case like, for instance, Woronowicz’s SUq(2) wor87] could be treated [SS98,Ske99] and confirm completely the outlined programme. References E. Abe, Hopf algebras, Cambridge University Press, 1980. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC.,no. 409, American Mathematical Society, 1989. BBLSOO. S.D. Barreto, B.V.R. Bhat, V. Liebscher, and M. Skeide, Type Iproduct systems of Hilbert modules, Preprint, Cottbus, 2000, To appear in J. F’unct. Anal. 2003. BSOO. B.V.R. Bhat and M. Skeide, Tensor product systems of Halbert modules and dilations of completely positive semigroups, I n h . Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519-575. GohO3. R. Gohm, Elements of a spatial t h e o y f o r non-commutative stationary processes with discrete time index, Habilitationsschrift, Greifswald, 2003, To appear in Lect. Notes Math. Hun56. G.A. Hunt, Semigroups of measures o n Lie groups, Trans. Amer. Math. SOC.81 (1956), 264-293. Sau89. J.-L. Sauvageot, Tangent bimodule and locality f o r dissipative operators o n C*-algebras, Quantum Probability and Applications IV (L. Accardi and W. von Waldenfels, eds.), Lect. Notes Math., no. 1396, Springer, 1989, pp. 320-338. M. Schiirmann, White noise o n bialgebras, Lect. Notes Math., no. 1544, Sch93. Springer, 1993. Ske98. M. Skeide, Halbert modules in quantum electro dynamics and quantum probability, Commun. Math. Phys. 192 (1998), 569-604. -, Hunt’s formula f o r SUq(2); a unified view, Information DySke99. namics & Open Systems 6 (1999), 1-27. SkeO2. -, Independence and product systems, Preprint, Cottbus, 2002, To appear in Proceedings of the “First Sino-German Meeting on Stochastic Analysis” , Beijing, 2002. SkeO3a. -, Dilation theory and continuous tensor product systems of Halbert modules, QP-PQ: Quantum Probability and White Noise Analysis XV (W. Reudenberg, ed.), World Scientific, 2003. Intertwiners, duals of quasi orthonormal bases and represenSkeO3b. -, tations, Preprint, Bangalore and Iowa, 2003. Three ways t o representations of !Ea(E), Contribution to the SkeO3c. -, MEETING ON OPERATOR ALGEBRAS, CHENNAI, DEC 15-16, 2003 and the AMS-INDIA MEETING, BANGALORE, DEC 17-20, 2003. Available at Abe80. Arv89.

~~

~

503 2003. M. Schiirmann and M. Skeide, Infinitesimal generators on the quantum group SUq(2), Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 573-598. S.L. Woronowicz, Twisted SU(2) group. A n example of a noncommutative diferentaal calcpllus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181. -, Compact quantum groups, Symetries quantiques (A. Connes, K. Gawedzki, and J. Zinn-Justin, eds.), Elsevier Science B.V., 1998. http://uvv.math.tu-cottbus.de/INSTITUT/lsvas/_skeide.html,

SS98. Wor87.

Wor98.

THREE WAYS TO REPRESENTATIONS OF B A ( E )

MICHAEL SKEIDE We describe three methods to determine the structure of (sufficiently continuous) of all adjointable operators on a Hilbert representations of the algebra 'Ba(E) B-module E by operators on a Hilbert C-module. While the last and latest proof is simple and direct and new even for normal representations of ' B ( H ) ( H some Hilbert space), the other ones are direct generalizations of the representation theory of ' B ( H ) (based on Arveson's and on Bhat's approaches to product systems of Hilbert spaces) and depend on technical conditions (for instance, existence of a unit vector or restriction to von Neumann algebras and von Neumann modules). We explain why for certain problems the more specific information available in the older approaches is more useful for the solution of the problem.

1. Introduction

A normal unital representation 6: B ( H )

B ( K ) of the algebra B ( H ) of all adjointable (and, therefore, bounded and linear) operators on a Hilbert space H by operators on a Hilbert space K factors K into the tensor product of H and another Hilbert space A, such that elements a in B ( H ) act on this tensor product in the natural way by ampli(-fic-)ation, i.e. K = fj@H

or

6 ( a ) = idfi@a

or

K = H8.A

with

6 (u ) = a m i d ~ .

It is the goal of these notes to report three different proofs of the following analogue result for Hilbert modules and dicuss their interrelations. 1.1 Theorem (Muhly, Skeide and Sole1 [MSS03a]). Let E be a Halbert module over a C*-algebra B and let F be a Hilbert module over a C*-algebra C . If 6 : IBa(E)+ !Ba(F) is a unital strict homomorphism, then there exists a Hilbert Bil-module F,q and a unitay u:E 0 F8 + F such that 6(a) = u(a 0 idFs)u*. The same result is true, if we replace C*-algebras by W*-algebras, Hilbert modules by W*-modules (and their tensor products) and 6 by a normal homomorphism.

504

505

Here, the strict topology of 'Ba(E) is the strict topology inherited by considering B a ( E )as multplier algebra of the C*-algebra X ( E )of compact opemtors, which is the norm completion of the *-algebra 3(E)of finite rank opemtors spanned by the rank-one opemtors xy* : z -+ x(y,z ) . A linear mapping is strict (and, therefore, bounded), if it is strictly continuous on bounded subsets of 'Ba(E). The proof from [MSSOSa], being both the simplest available and the most general, is based on the observation that the tensor product E 0E* of the X(E)-B-module E and the dual B-X(E)-module E* (with inner product (x*,y*) = xy* E X(E) and module operations bx*a = (a*xb*)*) may be identified with X(E). (The canonical identification is x O y 9 xy'.) Now F is a Hilbert 'B"(E)<-module (left action via 6)and, therefore, also a Hilbert X(E)X-module. Since 6 is strict and since X ( E ) has a bounded , have approximate unit (converging strictly to idE E ' B a ( E ) )we

F = X ( E ) @ F = ( E @ E * ) Q F= E O ( E * O F ) = E O F 8 where we set F8 := E* 0F . The canonical identification is

E O F $ = E O ( E * @ F )3 a:O(y*Oz)

t--)

~ ( x Y * ) zE F.

Clearly, 8(a) = a 0idF*. 1.2 Remark. A more detailed version can be found in [MSS03a]. The mechanism of the proof can be summarized by observing that, if E is full (i.e. if the range of the inner product of E generates Z? as a C*-algebra), then E may be viewed as Morita equivalence from X(E) to B. (If E is not full, then replace 23 by the closed ideal BE in 23 generated by the inner product.) Then X ( E ) = E 0E* and B = E* 0E serve as identities under tensor product of bimodules. The identifications of the bimodule F8 and of EOF8 with F are highly unique. For instance, we may establish the equality F = E 0E* @ F by showing that F furnished with the embedding i: E x E* x F + F , i(x,y, z ) = 6(xy*)z has the uniuersai proper& of the threefold tensor product E O E *O F . By these and similar considerations one may see that all identifications are essentially unique by canonical isomorphisms. We investigate these and other more categorical problems in [MSS03a]. Among the applications of Theorem 1.1there is the answer to the question when 6 is a (bistrict) isomorphism, namely, if and only if F8 is a Morita equivalence. We will investigate consequences of this insight in Muhly, Skeide and Sole1 [MSS03b].

506

Even in the case of normal representations of %(I?) on another Hilbert space the preceding proof (or, more acurately, its modification to normal mappings) seems t o be new. In the remainder, we discuss two known ways of treating the representation theory of B ( H ) (Section 2). Then we describe modifications t o adapt them to Hilbert modules, at least, under certain additional conditions (Sections 3 and 4). The two approaches correspond to the two basic constructions of product systems of Hilbert spaces from Eo-semigroups on 'B(H), the original one by Arveson [Arv89] based on intertwiner spaces and an alternative one by Bhat [Bha96] based on rankone operators. In the generalization to Hilbert modules it turns out that the two product systems constructed by Arveson and by Bhat are well distinguished. The product system constructed by Arveson is, actually, a product system of Hilbert C'W-modules where C' is the commutant of C when represented in the only possible (non-trivial) way by operators on the Hilbert space C. In terms of the cornmutant of Hilbert bimodules (as introduced in Skeide [SkeO3a] and also, independently, in Muhly and Sole1 [MS03]) the Arveson system of an Eo-emigroup is the commutant of its Bhat system and the Bhat system is that which corresponds to the representation theory (applied t o endomorphisms of 'B(H)) in Theorem 1.1. All three proofs of Theorem 1.1 lead to the construction of product systems of Hilbert (bi-) modules when applied to the endomorphisms of Eo-semigroups. We compare the three possibilities. In particular, we emphazise those aspects where the more concrete identifications in Sections 3 and 4 help solving problems which are more difficult in the above approach. A detailed discussion with complete proofs and specifications about how to distinguish identifications via canonical isomorphism from identifications just via isomorphism can be found in Skeide [SkeO3c].

2. Respresentations of

B(W)

In this section we repeat two different ways to look at the representation theory of ' B ( H ) . The goal of this repetition is two-fold. Firstly, it prepairs the terrain for the more subtle arguments in the Hilbert module case. Secondly, we use this opportunity to point at the crucial differences between the two proofs already in the case of Hilbert spaces. We hope that the present section will help the reader to understand why these two approaches, whose results may easilly be confused and mixed up in the case of Hilbert spaces, later on, lead to well distinguished directions in the case

507 of von Neumann modules. Let H denote a Hilbert space and let 19 be a normal unital representation of B ( H ) on another Hilbert space K . There are many ways to prove the well-known representation theorem which asserts that there is a Hilbert space Ej such that

K

E

EjBH E H R E j

and

s(a) = idB@u = a@idrj

in the respective identifications. Which order, Ej@H or H@Ej,is the natural one depends heavily on the proof, and the apparent equality of Ej €3 H or H €3 Ej is, sometimes, able to cause a certain confusion about the choice. Following what Arveson [Arv89] did for endomorphisms of B ( H ) , we introduce the space of intertwiners EjA = ( 2 E

B ( H , K ) :19(a)z= 20 (a E B ( H ) ) } .

One easily checks that x*y is an element in Cl, the commutant of B ( H ) , so that (z,y)l = x*y defines an inner product. (Observe that there are well distinguished commutants of C in B ( H ) and of C in C.) Of course, being obviously complete, EjA is a Hilbert space. A well-known result (for instance, [MSOPILemma 2.10) asserts that intertwiner spaces of normal representations act totally, if one of the representations is faithful:

m B A H = K.

(2.1)

From (z @ h, 2‘ €3 h’) = (h,(2,z’)h’) = (zh,z’h’)

it follows that z €3 h t)xh is a unitary EjA €3 H -+ K and that O(a)(zh)= $(ah) is the image of x €3 ah = ( i d A A €3u)(z€3 h). 2.1 Remark. The reader might find it strange that in the middle term we write (h,(2,z’)h’) instead of (2,z’)(h,h’). However, recalling that (z,z’), acutally is a n operator in B(H)’ c !B(H), the way we wrote it appears, indeed, more natural. Additionally, in the module case only this way of writing remains meaningful and we must dispense with the attitude to put the “scalars” outside of the inner product.

Although the only von Neumann algebra involved is ’B(H),the preceding proof uses elements from the theory of general von Neumann algebras like the commutant of all operators a @ e(a) in B ( H @ K ) and the fact that bijective algebraic homomorphisms are isomorphisms. Most other proofs make more or less direct use of the fact that a normal mapping on a von

508

Neumann algebra is known, when it is known on the finite-rank operators 3 ( H ) (i.e. the subalgebra of ’B(H)spanned by the rank-one operators hl ha : h I+ hl (hz,h ) ) . One of the most elegant ways to do this we borrow from Bhat [Bha96]. Choosing a reference unit vector w E H , we denote by f j B the subspace 6(ww*)Kof K . From

(6(hw*)z, 6(h’w*)d)= (z, 6(w(h,h‘)w*)z’) = ( h 8 z, h‘ 8 2‘)

(2.2)

we see that h 8 x I+ 6(hw*)xdefines an isometry H 8 f j B + K . To see surjectivity we have to make use of an approximate unit for 3 ( H ) which converges strongly to 1 (cf. the proof of Theorem 1.1 in Section 1). Also here we see that 6(a)6(hw*)x= 6((ah)w*)zis the image of ah I8 z = ( a I8 i d p ) ( h8 z). 2.2 Remark. By the uniqueness results mentioned after Theorem 1.1 the Hilbert space fj such that K = H 8 fj and 6 ( a ) = a 8 idrj is unique up to (unique) canonical isomorphism. For instance, the construction of f j B depends on the choice of w , but, if w’ is another unit vector, then S(w’w*) 1 f j B defines the unique unitary onto the space $ I B constructed from w’. Moreover, if fj = H* @ K is the Hilbert space according to Theorem 1.1 (with inner product (h: 0 k ~ , h a0 k2) = (k1,6(hlha)k2)), then h* 0 k I+ 6(wh*)kis the unique unitary fj + $ j B . We see that f j and ~ fj are very similar. Indeed, we may say that the construction of fj is just freeing the construction of f j B from the obligation to choose a unit vector.

2.3 Remark. Of course, 4’ ?% fjA, but this is an accidental artifact of the fact that C’ = C and that f j A 8 H Z H @.AA, canonically. Considering fjA as the C‘-C’-module it is, both the expressions H 8 f j and f j 8 H do not even make sense without additional effort. Indeed, as indicated in Observation 4.3, to deal with such expressions we have to introduce the tensor product of a Hilbert module over 0 and a Hilbert module over the commutant 0‘of 0. 2.4 Remark. Suppose that 61 and 92 are unital normal representations of ’ B ( H )on K and of B ( K ) on L , respectively, and denote by 6 = 792 o d1 their composition. Then fjA = f j f 8 fjf while f j B = fjf 8 fjf . There is no possibility to discuss this away as, for instance, by arguments like 6, o 6t = 9,+t = 6t 0 6, when (dt)t,R+ is an Eo-semigroup. The corresponding isomorphism f j f 8 fjf fjf 8 fjf (or, similarly, for 3 ): would not be

509

the canonical one. A clear manifestation is Tsirelson’s result [TsiOO] that a product system of Hilbert spaces need not be isomorphic to its anti product system. (For Hilbert modules we may not even formulate what an anti product system is.) 3. Generalizations of Bhat’s approach

Under the hyposthesis of Theorem 1.1 (both for strict representations and for the W*-version) Bhat’s approach generalizes easily, as shown in Skeide [SkeO2], if E has a unit vector t ,i.e. if (t,t)= 1 (what, of course, includes that B is unital). As in the proof for Hilbert spaces we define a Hilbert submodule Ft = 6 ( t t * ) Fof F . As additional ingredient (as compared with Hilbert spaces) we define a left action of B on Ft by setting by = d((b(*)y. With these definitions one checks that zoy

-

6(zt*)y

defines an isometry E 0 Ft -+ F . Like in the in proof of Theorem 1.1 surjectivity follows from existence of an approximate unit for X ( E ) whose image under 6 converges (strictly or a-weakly) to idF. Of course, 8(u) = U

0 idFB.

In this section we describe a construction from Skeide [SkeOSc] which frees the preceding construction from the requirement of having a unit vector, at least, for the case of W*-modules. Then, as in Remark 2.2, we compare the construction with that one from Section 1. Finally, we point out why the construction here, although not canonical (in the sense that it depends on the choice of a complete quasi orthonomal system for E * ) ,can have advantages over the intrinsic construction from Section 1. By making B possibly smaller, we may always assure that E is full and proofs of Theorem 1.1which work for full E work for arbitrary E. Existence of a unit vector is, however, a serious requirement. Our standard example is the W*-module E = 8 g g C M3 which is a Hilbert module over (c 0 0 ) B = ($ ,&) C M3 (with structures inherited from the embedding into M 3 ) which is full but does not admit a unit vector. (Actually, E is a bimodule and as such E is a Morita equivalence, because W ( E )= K ( E ) = B.) There are several equivalent possibilities to characterize W*-modules. A W*-module is always a Hilbert module E over a W*-algebra B fulfilling a further condition. We can require that E be self-dual or that it has a predual Banach space. In the following section we consider von Neumann modules as introduced in Skeide [SkeOO] as strongly closed operator spaces.

510

A W*-module over a von Neumann algebra B c B(G) is a von Neumann module and every von Neumann module is a W*-module. Many results on W*-modules have particularly simple and elementary proofs, when we transform them into von Neumann modules by choosing a faithful representation of B on a Hilbert space G. Here we need the facts that B a ( E )is a W*-algebra and that every W*-module admits a complete quasi orthonormal system, i.e. a family ( e p , p p ) p E B of pairs ( e p , p p ) consisting of an element ep E E and a projection pp E I3 such that

where the sum is a a-weak limit over the increasing net of finite subsets of B. There is also a tensor product of W*-modules denoted by 0". So let us start with the assumptions of the W*-version of Theorem 1.1. As explained before, we may assume that E is full (which for W*-modules means the a-weakly closed ideal in B generated by the range of the inner product of E is B). It follows that the dual B-Ba(E)-module E* of E is a Morita equivalence, in particular, that B a ( E * )= B. Now choose a family ( e p ) p E B of elements in E such that (e;, epei;)gEB is a complete quasi orthonormal system for E*. It follows that p p := (ep,ep) are projections in B fulfilling CpEB p p = 1. 3.1 Remark. If B consists of a single element ,f3, then ep is a unit vector. The following construction shows that the family ( e p ) p E B , indeed, plays the role of the unit vector in the construction explained in the beginning of this section.

Now we define the W*-submodules Fp = 29(epe;)F of F and set FB = Fp. (The direct sum is that of W*-modules. Observe that the submodules Fp of F need not be orthogonal in F so that FB is not a submodule of F.) On FB we define a left action of b E B by setting byp = $p,EBt9(ep~be;)yp ( y p E Fp). (This defines, indeed, a *-algebra representation of B by adjointable operators on the algebraic direct sum, so the representing operators are bounded and, therefore, extend also to the a-weak closure.) For x E E and Y B E FB set xp = x p p and y p = p p y ~ . Then the mapping

51 1

defines a unitary E 0" FB + F and O(a) = a O idFe. The proof of isometry and surjectivity is exactly like in the version with a unit vector, except that now there is one index, p, more. See [SkeOSc] for details. 3.2 Comparison. In the case when there are unit vectors the comparison of Ft, 41and F8 works as in Remark 2.2 (( and (' being possibily different 2 0 y t) unit vectors). S(('(*) defines an isomorphism Fc + 41and ' O((z*)y defines an isomorphism Ffi + Ft. For the family ( e g ) p E Bthe mapping

x* 09 =

Cpp*0 3 LJEB

e, @ O(epz*)y BEB

defines an isomorphism F8 + FB. The identification of F ' and FBI (B' indicating the dual of some different complete quasi orthonormal system for E') follows by iterating the preceding formula with the inverse of its analogue for the other basis. The resulting formula is slightly complicated and does not give any new insight, so we do not write it down. 3.3 Advantages. If 19 = (&),+ is an Eo-semigroup on P ( E ) (i.e. a semigroup of unital strict or normal endomorphisms of 'Ba(E)), then the Et := Efit = E* Ot E form a product system in the sense of Bhat and Skeide [BSOO]. Indeed, if we identify E, 0Et with E,+t via (2: 0 s Ys)

0 (2;Ot Yt)

c-) 2: Os+t

&(Ysz;)Yt,

then

( E , O E,) O Et = E, O ( E , O Et) and ( E O E,) O Et = E O ( E , O Et) Also the identifications via a unit vector 4 or a family ( e p ) p E Brespect these associativity conditions. So far, the two constructions can be used interchangeably. This changes, however, when we wish to include also technical conditions on product systems. A product system of Hilbert spaces in the sense of Arveson [Arv89] is supposed to be derived from an Eo-semigroup on B ( H ) that is pointwise a-weakly continuous (in time). The product system has, therefore, the structure of a Banach bundle, more precisely, the structure of a trivial Banach bundle. In Skeide [SkeOSb] we have investigated the Hilbert module version in presence of a unit vector. Requiring the product system to be isomorphic to a trivial Banach bundle seems too much. (We do not even know, whether all members Et (t > 0) of a product system are isomorphic as right modules.)

512

However, our product systems have the structure of a subbundle of a trivial Banach bundle. This can be derived easily from the observation that in presence of a unit vector all Et can be identified with submodules &(<<*)E of E. Since 6 is sufficiently continuous, the corresponding subbundle of the trivial Banach bundle [0, 00) x E is a Banach bundle (there are enough continuous sections). A o-weak version in presence of a unit vector does not seem t o present a difficulty. Now, if we have a family (ea)s,B instead of a unit vector, FB need no longer be a submodule of E. However, each Fs = 6(eper;l)F is a submodule. It follows that FB = Fs is a submodule of F. Therefore, it seems reasonable to expect that the product system is a a-weak subbundle of the trivial o-weak bundle [0,00) x E. This requires a convenient definition of a-weak bundle and is work in progress. In both cases the construction according to Section 1 does not seem to help to identify a good candidate for the trivial bundle of which the product system is a subbundle.

eBEB

eBEB eBEB

4. The generalization of Arveson's approach

For this section we need a longer preparation. If E is a B-B-module, then the B-center of E is the space

Ca(E)=

{X E E : b~

= xb ( b E a)}.

In what follows it is essential that von Neumann algebras and von Neumann modules (or, more generally, Hilbert modules over von Neumann algebras) always come along with an identification as concrete subspaces of operators on or between Hilbert spaces. A von Neumann algebra B is given as a concrete subalgebra of !B(G)acting (always nondegenerately) on a Hilbert space G . Every Hilbert B-module E may, then, be identified as a B-submodule of B(G,H ) for a suitable Hilbert space H in the following way. Set H = E 0 G . Then, an element x E E defines an operator L,: g c) x 0 g in !B(G,H). Clearly, ( q y ) = L i L , and Lxb = L,b. Moreover, E acts nondegenerately on G in the sense that LEG is total in H and the pair H , q : x + L, is determined by these properties up to (unique) canonical isomorphism. We, therefore, identify E as a subset of 'B(G,H)by identifying x with L,. Following Skeide [SkeOO] E is a uon Neumann module, if it is strongly closed in B(G, H ) . One may show (see [SkeOO,Ske03d]) that a Hilbert B-module E over a von Neumann algebra B c 'B(G)is self-dual, if and only if E is a von Neumann module.

513

On H we have a normal unital representation p’ of B’,the cornmutant lifting, defined by p’(b’) = idE Ob’. The space CBI(B(G, H ) ) is a von Neumann B-module containing E as a submodule with zero-complement . Since Ca, (!B(G, H ) ) is self-dual, it follows that E = Cat (!B(G, H ) ) , if and only if E is a von Neumann module. Observe that in this case $(a’)’ is exactly ’BO(E). The identification of C ~(B(G, I H ) ) as the unique minimal self-dual extension of E (in the sense of Paschke [Pas73]) was already known to Rieffel [Ftie74]. The definition of von Neumann modules seems to be due to [SkeOO]. In Skeide [Ske03d] we show directly (without self-duality of von Neumann modules) that E = Cot(’B(G,H)), if E is a von Neumann module, and then give a different proof of Rieffel’s result that Cat (B(G, H ) ) is self-dual. Muhly and Solel [MS02] show that, conversely, every normal unital representation p’ of B’ on a Hilbert space gives rise to a von Neumann B-module Cal (!B(G, H ) ) c B(G, H ) acting nondegenerately on G. Summarizing, we have a one-to-one correspondence (up to canonical isomorphisms)

B(G,H) 2 E

t--)

(p’,H)

between von Neumann B-modules and normal representations of B‘. If E is a von Neumann A-B-module (that is, A C B ( K ) is another von +B(H) Neumann algebra and the canonical homomorphism A + !B”(E) defines a normal unital representation p of A on H ) , then we have a pair of representations p and p‘ with mutually commuting ranges. p’ gives back the right module E as intertwiner space CBf(B(G, H ) ) and p gives back the correct left action. This works also if we start with a triple ( p , p ’ , H ) . For the standard representation of B so that B’ S B o p and p’ may be viewed as representation of B o p , we are in the framework of Connes and others where von Neumann bimodules and pairs of representations of A and B o p are interchangeable pictures of the same thing. The more general setting where 23 is not necessarily given in standard representation seems not to have been observed before Skeide [SkeO3a] and Muhly and Solel [MS03]. Going only slightly further, by exchanging the roles of A and B in the triple ( p , p’, H ) , we find the following one-to-one correpondence

where E’ = Cd(’B(K,H))is a von Neumann 8‘-A’-module.

We refer to

514

E’ as the cornmutant of E (and conversely), because when E is the von Neumann B-B-module 23, then E‘ = 23’. Also this correspondence was observed in Skeide [SkeO3a]and, later, in Muhly and Sole1 [MS03]. See also Gohm and Skeide [GS03] for another application of the commutant. 4.1 Observation. It is crucuial for what follows to notice that the preceding correpondences between (bi-)modules and (pairs of) representations enables us to identify von Neumann (bi-)modules by, first, identifying Hilbert spaces and, then, showing that representations on them coincide.

4.2 Observation. E = Cal(B(G,H))and E’ = CA(B(K,H) act nondegenerately on G and K , respectively. Therefore, spanEG = H = span E’K. Since we canonically identify H = E 0 G and H = E’ 0 K (by setting xg = x 0g and x’k = x’ 0 k), we have E 0G = E’ 0 K . Writing down this identity is an invitation to the reader to take an element x 0 g in E 0G and write it as a sum of elements x’ 0 k in E’ 0K . There is no canonical way how to do it, like there is no canonical way how to express a general element in a tensor product by a sum over elementary tensors. We just know that it is possible and that how ever we do it our conclusions do ot depend on the choice. For instance, it is important to keep in mind how the representations p and p’ act in these pictures. We have p(a)(x 0 g) = a2 0 g, while p(a)(z’ 0 k) = x’ 0 ak and, conversely, p’(b’)(x 0 g) = x 0 b’g, while p’(b’)(x’ 0 k) = b’x’ 0k.

Now we come to sketch the third proof of Theorem 1.1 where we, x u tally, first construct the commutant of FG. We assume the hypothesis for the W*-version of Theorem 1.1. As in Section 3 we assume that E is full. Furthermore, we assume that B c B(G) and C c B ( L ) so that E and F are von Neumann modules. We make up the following dictionary.

H = E@G p’(b’) = idEOb‘ p(a) = a O i d ~

K = FOL LT’(c’)= idpac’ .(a) = d(a) O i d ~

(b’ E B ’ , d E C’) ( a E ’B”(E))

It makes, therefore, sense to define the intertwiner space Fi = CB~(E)(B(H,K)) which is the subspace of B(G,K) of all mappings intertwining the actions of Ba(E) via LT and p = idBB”(E)where by definition B a ( E )is a von Neumann algebra on H via the identity representation p .

515

Recall that the commutant of ’BQ(E)is p’(B’). Therefore, in the above correpondence between von Neumann bimodules and pairs of representations, we may consider F i as the von Neumann C’-p’(B’)-module determined by the triple (o’,o,K) with inner product yi’y; E p’(B’), and left and right multiplication given simply by composition with d ( d ) from the left and with p’(b’) from the right, respectively. Now, since E is full, p’ is faithful so that $(a’)E 23’. Therefore, we may, finally, and will consider FA as von Neumann C’-B’-module where (yi,y;)

:= p’-l(y;*&)

and

c’y’b’ := d(c’)y’p’(b’).

We may now construct F$ = Cp @(L,F i O G ) ) where (see Observation 4.2) F; o G = F i 0 L is another Hilbert space which we do not name with an own letter. The following identification

F O L = K = Fi@H = FiOEOG = EOFiOG = EOFiOL identifies the Hilbert spaces F 0 L and E 0 F$ 0 L of the von Neumann modules F and E 0F;. 4.3 Observation. The “tricky” identification is F i 0E O G = E O F i O G. One easily checks that there is a canonical identification of these spaces simply by flipping the first two factors in elementary tensors. This also shows that operators on E and on F i , respectively, act directly on the factor where they belong. In [SkeOSc] we investigate systematically the tensor product E b” E’ CY E’ b” E of a von Neumann 23-module and a von Neumann B’-module which is a von Neumann (Bn 23’)‘-module. This tensor product may be viewed as a generalization of the exterior tensor product with which it has many properties in common.

An investigation how the relevant algebras act on these Hilbert spaces show that the von Neumann C-modules F and E bsF$’coincide (in the sense of Observation 4.1) and that O(a) = aOidF; what concludes the third proof. 4.4 Applications. Taking the commutant of von Neumann bimodules is anti-multiplicative under tensor product; see [SkeO3c] for details. Taking into account that in Section 2, clearly, AA = (AB)’,we see that the Arveson system of an Eo-semigroup on B ( H ) is the opposite of its Bhat system. Also the product systems of B-B-modules in [BSOO] and of B’-8’-modules in [MSO2], both constructed from the same CP-semigroup on B, are commutants of each other. We explain this in [SkeO3a].

516

Also other applications are related to endomorphisms of !Ba( E ) .While every bimodule FS comes from a representation of ’Ba(E) on E 0Fs, the question, whether a bimodule comes from an endomorphism (i.e. whether there exists a (full) E such that E 0 FS E E ) is nontrivial. It is equivalent to the question whether Fi has an isometric fully coisometric covariant representation on a Hilbert space. In the semigroup version this means that the question, whether a product system stems from an Eo-semigroup on some ’Ba(E), is equivalent to the question, whether the commutant system allows for such a covariant representation. We investigate these and other questions in Muhly, Skeide and Solel [MSS03c]. 4.5 Comparison. How is F$ related to FS = E* 0” F from Section l? Of course, we know that they are canonically isomorphic, but we want to see the identification in the sense of Observation 4.1. In fact, we are able to identify (E’OF)’ = F’OE*’ and FA,but after the sketchy discussion earlier in this section it is not possible to present the subtle arguments (flipping continuously between the isomorphic von Neumann algebras 23’ and ~ ’ ( 2 3 ’ ) ) in a coherent way. (In fact, many readers will feel uncomfortable with our continuously used canonical identifications of spaces which a priori are different, and doing this consistently requires a skillful preparation.) Once more, we refer the reader to [SkeOSc] for a detailed discussion. Acknowledgements. The results of Section 1 are joint work with Paul Muhly and Baruch Solel and most of these and other results have been worked out during the author’s stays at IS1 Bangalore and University of Iowa in 2003. The author wishes to express his gratitude for hospitality during two fantastic stays to B.V.Rajarama Bhat (ISI) and Paul S. Muhly (University of Iowa). References Arv89. Bha96. BSOO.

GS03.

W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC.,no. 409, American Mathematical Society, 1989. B.V.R. Bhat, An index theory for quantum dynamical semigroups, ‘llans. Amer. Math. SOC. 348 (1996), 561-583. B.V.R. Bhat and M. Skeide, Tensor product systems of Halbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (ZOOO), 51S575. R. Gohm and M. Skeide, N o n a l CP-maps admit weak tensor dilations, Preprint, ArXiv:math.OA/03 11110, 2003.

517

MS02. MS03. MSS03a. MSSO3b. MSS03c. Pas73. Ftie74. SkeOO.

SkeOZ. SkeO3a.

SkeO3b. SkeO3c. Ske03d. TsiOO.

P.S. Muhly and B. Solel, Quantum Markov processes (correspondences and dilations), Int. J. Math. 51 (2002), 863-906. -, Hardy algebras, W*-correspondences and interpolation theory, Preprint , ArXiv:math.OA/0308088, 2003. P.S. Muhly, M. Skeide, and B. Solel, Representations of 'Ba(E), Preprint, Iowa, 2003. _ _ , (Tentative title) Endomorphisms, commutants and Morita equivalence, Preprint, Campobasso, in preparation, 2003. -, (Tentative title) O n product systems of W*-modules and their commutants, Preprint, Campobasso, in preparation, 2003. W.L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. SOC.182 (1973), 443-468. M.A. RiefFel, Morita equivalence for C"-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. M. Skeide, Generalized matrix C*-algebras and representations of Hilbert modules, Mathematical Proceedings of the Royal Irish Academy lOOA (2000), 11-38. ___ , Dilations, product systems and weak dilations, Math. Notes 71 (2002), 914-923. ___ , Comrnutants of von Neumann modules, representations of 'Ba(E)and other topics related to product systems of Hilbert modules, Advances in quantum dynamics (G.L. Price, B .M. Baker, P.E.T. Jorgensen, and P.S. Muhly, eds.), Contemporary Mathematics, no. 335, American Mathematical Society, 2003, pp. 253-262. -, Dilation theory and continuous tensor product systems of Hilbert modules, QP-PQ: Quantum Probability and White Noise Analysis XV (W. F'reudenberg, ed.), World Scientific, 2003. -, Intertwiners, duals of quasi orthononnal bases and representations, Preprint, Bangalore and Iowa, 2003. -, Von Neumann modules, intertwiners and self-duality, Preprint, Bangalore, 2003, To appear in J. Operator Theory. B. Tsirelson, h m random sets to continuous tensor products: answers to three questions of W. Aweson, Preprint, ArXiv:math.FA/0001070, 2000.

THE HAMILTONIAN OF A SIMPLE PURE NUMBER PROCESS

WILHELM VON WALDENFELS Institut f i r Angewandte Mathematik, Universitat Heidelberg Private Address: Poststrasse 1'7 16'798 Himmelpfort, Germany E-mail: Wilhelm. WaldenfelsOT-OnPne.de We consider the strongly continuous unitary one parameter group on L2(R)given bY utf(z) = j ( s - t) exp (i~xto). The Hamiltonian is given by the singular operator

H = -ici

+ Klb)(bI

on a Sobolev space. The subspace where H is non-singuly is the domain of the Hamiltonian H and the Hamiltonian coincides there with H. The spectrum of H is the real line. We calculate the generalized eigenvectors.

We consider in this paper the strongly continuous unitary one parameter group on L2(R)given by Utf(2)

= f(z - t) exp (iXX&)c)).

where

xt,(z) =

{

1 fors<x
By Stone's theorem there exists a selfadjoint operator H in L2(R), called the Hamiltonian such that

Ut = exp (-iHt). We shall define a singular operator H with domain D, such that the subspace Do C D, where H has no singularities is the domain of H and H is the restriction of H to DO.

518

519

Define the translation operator q : r t f ( z ) = f(z - t ) and the multiplication operator V: = exp (iXxt),then

Ut = rt

v,t.

V: is the restriction of the quantum stochastic process W,"defined on the Fock space to the one-particle space L2(R).The process Wj is a pure number process, its quantum stochastic differential equation is given in its Ito-form by &Wj = (eix - 1)dAtW; and in its Stratonovich form8 by

dWj/dt = -inafatWj in usual notation and n is given by ,ix

= 1 - in/2

1

+ in/2*

We treat here a very special case of a more general problem, to determine an explicite form of the Hamiltonian related to a quantum stochastic differential equation with constant coefficients. The problem was posed by Accardil in 1989, solved as a boundary value problem partially by Chebotarev4 in 1996 and finally solved by Gregoratti in 2000 in his thesis.l* The author presented in 2002 the singular operator for the problem of the damped oscillator and in 2003 the singular operator for the quantum stochastic differential equation containing no number term, cf. Refs. 13, 14 In order to formulate the Hamiltonian, we use symmetric differentiation and the symmetric Dirac function, which we want roughly to describe. Symmetric differentiation 8 is defined for functions, which are differentiable except a finite number of jumps and the symmetric Dirac function 8 is defined for functions continuous except a finite number number of jumps. Take as example the Heaviside function

Y ( t )=

1 for t 0 for t

>0 <0

and not defined for t = 0.

/

+

8(t)Y(t)dt= (1/2)(Y(O+) Y(0-) = 112

+

8Y(t)= ( Y ( t 0 ) - Y ( t - O))&t)= &t)

520 The notion of a symmetric Dirac function was introduced by by 3. G0ugh7 and by Accardi, Lu and Volovich.2 The symmetric differentiation was introduced in Ref. 13 and Ref. 14. We define the Sobolev space D = H 1(R \ ( 0 ) ) of all L2-functions, such that their Schwartz derivatives restricted to IR \ (0) are L2 as well. These functions are absolutely continuous in IR \ {0}, they admit right and left limits in 0 and if both limits of a function coincide, its Schwartz derivative is L2 on the whole of R Define DOC D as the subset of those functions f such that f(O+) = eixf (0-). Chebotarev showed, that Do is an essential domain of the Hamiltonian and the Hamiltonian is given there by f r-) -if', where f' is the usual derivative outside the point 0. In this article I want to apply the methods developped in Ref. 13 and Ref. 14. The elements of D admit the application of the symmetric 6 function

@If)= (1/2)(f (O+) +

m-))

and of the symmetric derivative yielding

Sf = (f(O+)

- f ( 0 - ) ) 8 + f'.

Denote by D' the set of all semilinear mappings from D to C, and define the operator H mapping D into D' given by

H = - is where

+ nl8)(8I,

18) is the semilinear functional on D given by

and IE is given by equation(1). The subspace DO is exactly the subspace of all those f , where the contributions of 18) vanish. We will show that DO is not only the essential domain, but the whole domain of the Hamiltonian, and that fi coincides with H on DO.By proposition 111.2 of Ref. 14, we see, that fi is symmetric on D and hence on DO.Of course, outside the point 0, Hf = -if'.

Proposition. The subspace DO is the subspace of all f E D , such that Hf E L2(IR) and DO is the domain of H and H coincides there with H .

521

Proof. Denote C(t) =

and Z=

1

C ( t ) q d t=

l*

e-t.rtdt.

One has

Zf =C*f, where * denotes the usual convolution. Especially 2 6 = C*S =

c.

The Schwartz derivative satisfies

a(

= 6 - C,

hence

azf = (a() * f = f - C * f = f - Zf. So if f E L2,then Zf and the derivative of Z f is in L2 as well. One obtains

f ED

9 f = Z(c6

+ g)

(2)

part results from what has just been for some c E C and g E L2. The said, the part results from the Schwartz derivative

af = c 6 + h with c = f(O+)

- f(0-)

and h E L2 and

f = Z f + a Z f = Z f + Z(af) = Zf + Z(c6+ h). The symmetric derivative is

Sf = c 8 +

f’=cB+g-Zg,

where f’ is the usual derivative outside 0. Furthermore

(4f) = (1/2)(f(O+)

4-

f(O-))

= 4 2 + (Zg)(O)-

We calculate

Bf = -icB - ig + izg + nB(B(f).

522

so f

E Do

*ic =

(3)

K(8lf).

= eiAf(O-) mentioned

This condition is equivalent to the condition f(O+) above. We write

f(z - t ) ( l + lo o ( U t f ) z )= f ( z - t)(I + It
{

'

and define the resolvent of Ut

-i

soweiztUtdt for 9 z > 0

R(z) = i ,J !

eiztUtdt for

$2

<0

and the resolvent of rt

After a straight forward calculation

with

a ( z ) = sign%. As

z = i&(i)

(5)

we have by the resolvent equality & ( z ) = (1

+ (z - i)&(z))Z/i

and for that reason we have R ( z ) f E D for f E L2 .Using = -8 (m.dz)S) =

-ih&(z)S

+ zRo(2)G

-4w

we obtain

&R(z)f = -f

+zR(z)f.

As the singular parts containing 8 vanish, we have R ( z ) f E DO and H coincides with H on R ( z ) f . The resolvent R(z) maps L2 onto the domain

523

DH of H one-to-one, so D H c DO.We have still to prove,that D H 3 DO. Go back to equation (3), then f = Z(c6 + g) is in Do iff ic = K(BI f ) = KC12

+ n(Zg)(O)

or

and

Recalling (4)and (5) we obtain

f

= iR(i)g

and f E DH.This finishes the proof. Using the theory of Ref. 13, we obtain that the spectrum of H is the real line and that the generalized eigenvectors are & with

These eigenvectors are orthonormal and complete, i.e.

References 1. L. Accardi, Noise and dissipation in quantum theory. Reviews in Math. Physics 2 (1990) 127-176. 2. L. Accardi, Y.-G. Lu, I.V. Volovich, White noise approach to classical and quantum stochastic calculus. Preprint 375, Centro Vito Volterra, Universita Roma 2, (1999). 3. L. Accardi, Y.-G. Lu, 1.V. Volovich, Quantum theory and its stochastic limit, (Springer-Verlag, Berlin, 2002). 4. A.M. Chebotarev, ”Quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrodinger equation”, Mathematical Notes, 1996, 60, No.5-6, 544-561. 5. A.M. Chebotarev, ”The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrodinger equation”, Mathematical Notes, 1997, 61,No.3-4, 510-518. 6. A.M. Chebotarev, Quantum stochastic differential equation is unitary equivalent to a symmetric boundary problem in Fock space. Inf. Dim. Anal. Quantum Prob. 1 (1998), 175-199.

524

7. J. Gough, Causal structures of quantum stochastic integrators.Theoret. and Math. Physics 111 (1997), 563-575. 8. J. Gough, Noncommutative Ito and Stratonovich noise and stochastic evolution, Theor. and Math. Phys. 113,N2, (1997) 276-284. 9. J. Gough, A new approach to non-commutative white noise analysis, C. R. Acad. Sci. Paris Ser. I Math. 326,1998, no. 8,981-985. 10. M. Gregoratti, The hamiltonian associated to some quantum stochastic differential equations. Thesis. Milano 2000. 11. K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhaeuser, Basel, Boston, Berlin 1992). 12. L. Schwartz, Thkorie des distributions I (Herrmann, Paris 1951). 13. W. von Waldenfels, Description of the damped oscillator by a singular F’riedrichs kernel. In Quantum prob. and rel. fields 2003. 14. W. von Waldenfels, Symmetric differentiation and hamiltonian of a quantum stochastic process. Preprint 24/2003 Greifswald, Germany.

ON TOPOLOGICAL ENTROPY OF QUOTIENTS AND EXTENSIONS

JOACHIM ZACHARTAS School of Mathematical Scaences University of Nottingham, University Park, Nottangham, NG7 2RD, UK E-mail: [email protected] Let 0 + J 4 A 4 B --t 0 be an exact sequence of exact C*-algebras. Let o E Aut(A) leaving J invariant and ir E Aut(B) induced by a. We study the ques), ht denotes the tion when ht(&) 5 ht(a) and h t ( a ) = max(ht(&), h t ( a ( ~ )where Voiculescu-Brown topological entropy. There is a close connection to the existence of equivariant completely positive lifts. We introduce local equivariant lifts and prove that they always exist for a stabilized action. It follows that ht(ir) 5 ht(a) and ht(a) = max(ht(d),ht(alJ)) at least for such stabilized actions.

1. Introduction Entropy - as a conjugacy invariant of measure preserving transformations of a probability space - has been invented by A.N. Kolmogorov in the late 1950’s. His definition is based on the orbit behaviour of finite measurable partitions. During the 1960’s it was extended to the setting of a homeomorphism T of a compact space X using finite open covers of X instead of partitions. The resulting topological entropy has been of central importance in topological dynamics and ergodic theory (c.f. [14]). There have been several attemps to extend this definition to the noncommutative setting. During the 1990’s Voiculescu ([13]) discovered an approach to topological entropy using finite rank completely positive approximations which works for automorphism (or other self maps) of nuclear C*-algebras and can be extended to exact C*-algebras ([4]), the largest class of C*-algebras where finite rank cp-approximations exist in some sense. Roughly, Voiculescu reformulates the classical definition in the commutative case using the fact that a partition of unity in C(X)(subordinated to a finite open cover) defines a finite rank completely positive approximation

525

526

of idc(x) : C(X) C(X).The resulting definition does not use commutativity any more but only the existence completely positive approximations. Many nice and powerful results about the Brown-Voiculescu entropy including its determination in numerous concrete examples have been obtained recently ([4], [5], [6], [3], [12] to name only a few references). But there are still some basic open questions. For instance it is not known under which circumstances the formula h t ( a 8 8 ) = ht(a) ht(8) holds. (a and 8 denote automorphisms of two different exact C*-algebras.) It is easy to see that the entropy can not increase when passing to invariant subalgebras. The question whether the same is true when passing to quotients by invariant ideals is still open ([4], Question 2.18). In the classical commutative setting the latter means that the entropy of a homeomorphism dominates the entropy of its restriction to an invariant closed subspace Y C X, an assertion which follows immediately from the definition using open covers. More generally, given an extension

+

O+J4AA,B+O and a E Aut(A) leaving J invariant, we can define ci E Aut(B) by &(a J) = a(.) J. Do we always have ht(a) = max(ht(ci),ht(alJ)), where ht denotes the Voiculescu-Brown topological entropy? Again this can be verified in the commutative case using the classical definition. As we will point out in the sequel the quotient problem is closely related to the existence of equivariant completely positive lifts for the quotient map A + B and to solve the extension problem we also need ’covariant’approximate units in the ideal. In this note we will show that ’local’ equivariant lifts and ’covariant’ approximate units always exist if we tensor the original sequence by the compact operators and replace the action by a tensor product action a 8 T . Here T acting on Ic has entropy 0. This enables us to show that the above two problems have at least affirmative answers for these stabilized actions.

+

+

2. Preliminaries 2.1. Nuclear and Ezact C*-Algebms We use the abreviations cpc for ’completely positive contractive’ and upc for ’unital completely positive’. Given C*-algebras A and B denote by CPA(A,B ) the set of all triples (cp, $, D),where D is a finite dimensional C*-algebra and cp : A + D and $ : D + B are cpc maps. Maps of the form $ o cp, where (cp, $, D) E CPA(A,B ) are called factorable.

527 A C*-algebra A is called nuclear if A amin B = A @maz B for all C*algebras B. Today it is well-known that A is nuclear iff idA is C*-nuclear i.e. idA lies in the point-norm closure of the factorable maps from A to A. Nuclearity passes to ideals, quotients and extensions but not to subalgebras in general. A C*-algebra A is called exact if the functor A @min - preserves short exact sequences of C*-algebras (c.f. [15]for an excellent survey of the theory of exact C*-algebras). A deep result of Kirchberg and Wassermann shows that a C*-algebra is exact iff it is nuclearly embeddable i.e. there is an embedding into another C*-algebra which can be approximated by factorable maps into the larger algebra in the point-norm topology. The foregoing means that A is exact if and only if there is a faithful representation x : A + B ( H ) and an approximating net (pi,&,Di)iE1 C CPA(A,B ( H ) ) such that Ilqi o cpi(a)- x(u)II + 0 for all a E A. Given another faithful representation 0 : A + B ( K )there are completely positive Arveson extensions .ir : B ( K ) + B ( H ) of x and 6 : B ( H ) + B ( K ) of 0 which allow us to pass to the approximating net (cpi,6 o $i, Di)C CPA(A,B ( K ) )for 0 with the same converges properties i.e. ll$i o cpi(a) - x(u)II = 116 o $i o cpi(a) - o(a)ll for all a E A. Thus for approximating nets and their convergence properties the choice of the faithful representation is unimportant. Given an approximating net (cpi,$i,Di) for A and a (C*)-subalgebra B E A the family (cpilB,$i,Di) forms an approximating net for B so exactness passes to subalgebras. A much deeper fact shown by Kirchberg is that also quotients of exact C*-algebras are exact. The proof uses his subquotient characterization of exactness (c.f. [15]).

2.2. Noncommutative Entropy

Given x : A + B ( H ) faithful, A exact (not necessarily unital) and (cp, $, D )E CPA(A,B ( H ) ) we let rk(D) denote the dimension of a maximal abelian subalgebra of D. Following [13] and [4] define for w C A finite and d > 0 the d-rank of w by

{

~ , ~ (wx, 6), = inf rk(D) I (cp, $, D )E CPA(A,B ( H ) ) :

As explained above rCp(7r, w , 6) does not depend on the choice of the faithful representation hence in the sequel we will suppress x from the notation and

528

define for a E Aut(A) (or any other selfmap of A)

ht(a,w,6) = lim supn-l log(r,,(w u a(w)u . .. u a”-l (w), 6)) n-+m

ht(a,w) = lim ht(a,w,6) 6-0

ht(a) =

sup

ht(a,w).

wcA,finite

ht(a) is the Voiculescu-Brown entropy of a. It is convenient to use the orbit notation orbE(w) = w U a ( w ) U . . . u a n - l ( w ) . Then ht(a,w,b) = limsupn-’ log(r,(orbZ(w),d)). n-+m

The full orbit is denoted by orb,(w) = UnEzan(w). If A is unital then we may require all triples (cp, $J, D ) in the definition of ht(a)to be in the set CPAl(A, B ( H ) ) (i.e. cpi and $Ji are unital). Moreover if a” is the canonical extension of a E Aut(A) to an automorphism of the unitization A” then ht(a)= ht(a”) ([4]).We mention one important property of the entropy: Theorem 2.1. (Kolmogorov-Sinai property, c.f. [13], [.I) If ( W A ) A ~ Ais a net consisting of finite subsets of A partially ordered by inclusion such that

U

A = clospa(

an(wA))

A€h,nEZ

then ht(a) = sup{ht(a,wA) I X E A}. 2.3. LijXngs

If J C A is a (closed twosided) ideal in a C*-algebra A we have the exact sequence

o + J A ~4 B + 0. is a lift of q : A + B if q o p = idB.

A cpc map p : B + A If A is nuclear then by a result of Choi and Effros there exists a lift for q (c.f. [l]for a short proof). Lifts do not generally exist if A is an exact C*-algebra. In this case local lifts play an important role. Recall that an operator system X is a closed selfadjoint subspace of a unital C*-algebra C containing the unit. Since X = span(X+), where X+ = {z E X 1 x 2 0) we can define complete positivity for maps between operator systems ([7]). A very important technical key result is Arveson’s extension Theorem. It says that any ucp map cp : X + B ( H ) has a ucp extension @ : C + B ( H ) (c.f. [lo]).

529 Now assume that A and hence B are unital. (No loss of generality; adjoin a unit and omit it again if necessary.) q : A -+ B is said to be locally liftable if for each finite dimensional operator system X 2 B there is a ucp map p : X + A such that q o p = idx. Kirchberg has shown that if A is exact (and unital) then q is locally liftable in the above sequence ([9]). Using local lifts and following ideas from [6] we will give an explicit procedure how to combine approximating nets for B and J to obtain an approximating net for A in the next Proposition. This will be used later. First we have to indicate some technical preliminaries. Let A be a unital exact C*-algebra and 0 + J 4 A 4 B + 0 a short exact sequence. Suppose 7r : A + B ( H ) in faithful. Consider the 7r(A)invariant subspace x ( J ) H =: H I # H. XI := n l and ~ 7r2 ~ := T I H ~ : = H : are representations of A such that 7r2 (J)= { 0). So 7r2 passes to a representation ir : A/ J = B + B(H2) which we may and shall assume to be faithful. (This holds for instance if 7r is a universal representation of A.) Thus 7r maps actually into B(H1) @ B ( H 2 ) . Let P = P H ~Q, = PH, be the orthogonal projection onto H I and H2. For exact A we have the following diagram were we have indicated generic approximating triples.

Q

0 - J - A - B - 0

* * 1.

DIJP1 0-

1 1 .

B(H1)-

D

B (

D@

lir

& H ) F B(H2)-

0

q=Q*Q

Note that by Arveson’s extension theorem local lifts p extend to ucp-maps p^: B(H2) -+ B ( H ) . Finally recall that an approximate unit (ex)xEAC J is called quasicentral if [ex,a] + 0 for all a E A. By ([l] Thm.1) quasicentral approximate units always exist. We will assume 0 5 ex 5 1~ for all X E A for such approximate units. A kind of decomposition of an element in A can be achieved using quasicentral approximate units: given a E A it is easy to see that U A := e:/2ae:/2 (1 - eA)1/2a(l- e#I2 -+ a as X + 00

+

([W Proposition 2.1. Let w C A be finite containing 1 , let E > 0 and (ex)xEA C_ J a quasicentral approximate unit. For X E A choose p E A so that Ilei/2e,, - e:/’Il < e(max{llall 1 a E w})-’. Given (p1,?,b1,Dl) E CPA(J,B(H1)) such that

530

and given

(92, $2,

Dz)E CPAl ( B ,B(H2)) szlch that

for all a E w define 9 :

A

+ D1 @ D z b y

Then ( ‘ P , $ J , @ D ~D z ) E CPA(A,B(H)) and a E w provided X is suficiently large.

ll$ o cp(a) - a / [ < 6.5 for all

Proof. Clearly $ and cp are cpc-maps. For all a E w we may assume Ilax-all < e and by the choice of p we have lle:j2e,ael,e:/2-e:/2ae:/211 < 2e Thus Il$l 0 cpl(e,ae,) - el,ae,ll < E implies IJelh’2$1(cpl(e,ae,))e:’2

- e1’2ae:/2 x 11 < 3e

for a E w . Moreover, j ( l - e x ) + 0 and ~~(l-e~)1~2j(l-e~)1~2-j(l-e~)~~ 5 11(1- e x ) 1 / 2 j- j ( 1 - eA)1/211+ o for j E J. Since p(q(a)) - a E J we may therefore also assume

for a E w . Putting everything together yields 1111, o cp(a) - ax11 ll$ 0 cp(a) - all < 6&for all a E w.

< 5~ hence 0

Notice that we could easily modify $ and cp to be unital. The forgoing Proposition also provides an explicit proof of the fact that locally liftable extensions of exact C*-algebras by exact C*-algebras are exact. (This has been mentioned by Kirchberg with an indirect proof.) Note however, that extensions of exact algebras by exact algebras are not always exact unless the quotient map is locally liftable ([8]).

531 2.4. Equivariant Lifting8

In the sequel we need equivariant refinements of local lifts which take into account a *-automorphism a E Aut(A) such that a ( J ) = J . In this case &(a + J ) := a ( a ) J defines a *-automorphism of B = A / J such that q o a = 6 o q. Thus we have the equivariant exact sequence

+

O + ( J , C Y4 ) (A,(Y)4 p , b ) -+o

(*)

A lift p : B -+ A is equivariant if a o p = p o 6. Even if A is nuclear and hence q liftable it does not always follow that there exists a n equivariant lift ([2]). However, in this case one can use Takai’s duality Theorem in order to show that

O + ( J @ K , ~ @ T4) ( A @ K , ~ @ T4) ( B @ K , ~ @ +TO ) has equivariant lifts. Here T = Ad(T) and T E B(12(Z))is the bilateral shift T ( e i ) = ei+l for i E Z. This is proved in [2] in the very general setting of duality for Hopf C*-algebras. For the reader’s convenience let us sketch the argument in our situation briefly. By taking crossed products we obtain from (*) the exact sequence of crossed products

0 + J x ,Z

+ A x ,Z 4B x

~ +Z0

consisting of nuclear C*-algebras. Hence 4 has a lift fi. Moreover, this sequence is equivariant for the dual actions & and & of 11‘. Replacing fi by ( 2 7 ~ ) - ~ & &o, fi o &dz we may assume that fi is also T-equivariant. Then a calculation shows that fi induces a cpc-map : B x b Z x T +

a

f

P

Ax,% x& Twhich is equivariant for the double dual automorphisms d and &. Now Takai’s duality Theorem ([ll]section 7.9) says that (A x, Z x &T, &) and ( B x b Z X & a,&)are covariantly isomorphic to (A @ K,a @ T ) and ( B @J K ,ti @ 7)respectively. This implies the result. In the setting of exact C*-algebras (*) is in general only locally liftable. But even then the stabilized sequence has better equivariant lifting properties. We will see below that, roughly speaking, we can lift approximately and approximately equivariantly on the full orbit of certain finite dimensional subspaces of B @ Ic. Before proceeding we need two facts about approximate units and local lifts which almost invariant. They follow immediately by averaging. Proposition 2.2. Suppose that A is unital and exact in the above sequence (*). Let E > 0 , w C B finite containing lg and N E N. Then

532 (1) if 2 = span{&"@) I In1 5 N , b E w U w ' } there is a local lift p : 2 + A such that IlcP o p o &-"(b) - p(b)II < E whenever 1121 5 N and b E w ; (2) there is a quasicentral approximate unit ( e A ) A E A C J such that JJa"(eA) - eA)J< E whenever In1 5 N . 3. Locally Equivariant LiRs and Entropy For the next Lemma we collect some notation. As before we consider an equivariant exact sequence

0 + (J,a) 4 (A,cr) 4 ( B , & )-+ 0, where we now assume that A is unital and exact. If w C B is a finite subset containing l g and r < s are integers denote by M[,,,](w) (resp. M~,,,](U)) all matrices [xij]E Moo(B)C B @ K: with entries xij in w and such that xij # 0 + i , j E [r,s] = { r , ~ 1 , . . .,s} (respectively xij # 0 + i , j E ]r,s]= { r + I , . . .,s}). Notice that ( c i @ ~ ) j ( M [ , , , ] = ( ~M ) )[ r + j , s + j ~ ( & ( ~ ~ ) ) . Given a fixed k E N define for K E N

+

XK

= spa{(& €4 ~

u

) j ( ~ [ - k , k ] (w~' ) )

I Ijl I K } .

X K is an 'operator system' with 1 E h f [ - K - k , K + k ] playing the role of the unit. Let X = clo UKENX K ) . The full orbit orbkB7(M[-k,k](w))= UnEZ(&8 ~ ) " ( M [ - ~ , , l ( wof) )M [ - k , k ] ( w ) is contained in X . Finally we will usually denote the stabilized quotient map q €4 idx also by q.

(

Lemma 3.1. Let w c B finite containing 1~ and let k E N. Given E there exists a linear contraction p : X -+ A €4 K: such that (1) p l X ~is completely positive for all K ; (2) llq 0 p ( x ) - 211 < E and IIp 0 (b €4 T ) ~ ( z )( a @ ~ all x E ~ r b ~ ~ ~ ( M [ - ~ ,and ~ ] (allwj) E) Z.

) 0 jp(z)ll

>0

< E for

Proof. Let 6 > 0. For N E N, where N >> k consider w ( N ) := U g - N & j ( w ) C B and M I ~ , ~ ~ ( C ~B ( N@ )K). By Prop 2.2 there is a local lift p : 2 + A such that w ( N ) c 2 and llan o p o &-"(b) - bll < 6 for 172.1 5 N and b E w ( N ) . Let's denote the inflation of p t o a map from M]O,NI to(MIO,NI(A), ~) [ ~ j ++ ] [ ~ ( z i j )also ] by 1.1. Let PI,,,]be the projection in K ( t 2 ( ( z ) )onto spa{e,+l,. . . ,e,} and let p~ = 1 @ P I I N , ( I + ~ ) N I . Then s = @ (a,, IEZ

0

p

0

K")

(pl

. PI)

533 is defined on X and so are also (a€3 T ) o~s o (d!€3 T ) - ~for all i E Z; in fact we have (a! €3 T ) o s~ o (ci €3 T ) - = ~ s at least on X. Finally let 1

p=-

c

N-l

(a!

€3 T ) i 0 s 0 (d!c3 T)?

i=o

For 6 > 0 small enough and N large enough p is as required.

0

Theorem 3.1. Let A be a unital and exact C’-algebra. Let a E Aut(A) leaving the ideal J 2 A invariant and ci be the induced automorphism of the quotient B = A / J . Then ht(&€3 T ) 5 ht(a €3 T). Proof. With Lemma 3.1 at hand the proof is a modification of an argument in [4] and will be sketched briefly. We have ht(iY) =

sup wcB@K:finite

sup lim sup n-l log(rcp(orb& ( w ) ,6)) 6>0

~ + C C

and

ht(a!)=

sup

SUP

w‘CA@K: finite

6>0

limsup n-l log(rcp(orbz@,,(w’),6)). n+03

Using the Kolmogorov-Sinai property it suffices to show that given w finite containing l ~k E , N and 6 > 0 there is w’ A finite such that

c

cB

for all n E N, where wk = M I - ~ , ~ ~ (wW i )=, M[-k,k](w’). By Lemma 3.1 there is p : X -+ A €3 Ic defined on the full orbit of wk such that IIqop(x)-zII < 6 and ( I p o ( d ! € 3 ~ ) j ( z ) - ( a ! ~ ~ ) j o p ( 2< ) 16I for z E Orb&@T(Wk) and j E Z. Let w’ = p(w). p maps orb&,,, ( w k ) approximateiy to orb:@,, ( w i ) up to a perturbation of norm less than 6 and q(orb&(wi)) is orb;@,,(wk) up to a perturbation of norm less than 6. Let (cp, $, D) E CPA(A 18 Ic,B ( H €3 l’))such that

II$J o cp(a) - all < S for a E orbE,,,wi and rk(D) minimal for 6 > 0. The following diagram tells us how to find ( q , p ,D) E CPA(B €3 Ic,B ( H 2 €3 l’)) such that IIp o q(b) - blJ < 36 for b E orb: (wk).

534

Since D is finite dimensional cp o p : X + D extends to a cpc map q : B €3 K + D. Letting p : D + B(H2 €3 t’) be the composite (4 €3 id) o $J we obtain the required triple ( q , p , D ) E C P A ( B , B ( H 2 €3 C’)). Thus the minimal possible 36-rank for an approximation in B €3 Ic on orb&,,(w6) is at most rk(D) which proves the claim for w’ = p(w). 0 Finally we apply Prop 2.1 to the extension question.

Theorem 3.2. Let 0 + ( J , a ) 4 ( A , a ) 3 ( B , & )+ 0 be an equivariant exact sequence, where A is a unital and exact C*-algebra. Then ht(a€3T)= mm(ht((a€3 T ) I J @ I c ) , ht(h €3 T I ) . Proof. Clearly ht((a€3 T ) I J @ K )5 ht(a €3 T ) and by Thm 3.1 we also have ht(&€3 T ) 5 ht(a €3 T ) . In order to show that max(ht(&€3 T ) , h t ( ( a €3 T ) ~ J @ K )2 ) ht(a €3 T ) it suffices to show the following. Given w C A finite J and w” C B finite such containing 1, k E N and 6 > 0 we can find w’ that

66) 5 rcp(orb&,,@JJL),6/31 rcp(Orb;@&JJk),

+ Gp(orb:@,,(~:),

6/31.

for all n (wk = M [ - k , k ~ ( wetc. ) as before). Let (ex) C J be a quasicentral approximate unit as in Prop 2.2.(2), specifically assume that Ilai(ex)aa%x) - exaexll

<

6

for ) i ) , ) j5) k and all a E w. (ex,l) defined by eA,l = ~ ~ = - , a ’ ( e x€3 ) err forms a quasicentral approximate unit of J €3 Ic as X + 00 and 1 + 00. Now observe that ex,k+n (orb&(wk))ex,k+n is approximately equal to orb&7(ex,kwkex,k) which is in turn approximately equal to orb:@,, ((exweA)k). (Both up to perturbations of norm less than 6/3.) Let w’ = exwex and w” = q(w) and let p be as in the proof of Thm 3.1. Then for a possibly larger X Prop 2.1 implies .cp(Orb:@T(Wk), 66) L rcp(ex,le+n(orb:@,(wk)) eA,k+nr 6)

+ Tcp(Q(orb:@,,(Wk)),6) < ~cp(Orb~@,,(~L), 6/3) + rcp(orb&(w:),

6/31

535

which shows the result.

0

In Thm 3.1 and 3.2 one may drop the assumption that A is unital by adjoining units. Also it is not hard to see that r acting on K: has entropy 0. If the formula ht(a 8 p) = ht(a) + ht@) were true in general then our result would show that the quotient and extension problem have affirmative answers in general. However counterexamples to the formula are known to exist and this route does not seem to lead to a solution of the quotient and extension problem in general. We leave this to future investigations. References 1. W. Arveson, Notes o n extensions of C*-algebras, Duke Math. J. 44 110.2 (1977) 329-355 2. S. Baaj, G. Skandalis, C*-algkbre de Hopf et the'orie de Kasparou e'quiuariante, K-Theory 2 (1989) 683-721 3. F. Boca, P. Goldstein, Topological entropy for the canonical endomorphism of Cuntz-Krieger algebras, Bull. London Math. SOC.32 no.3 (2000) 345-352 4. N.Brown, Topological entropy in exact C*-algebras, Math. Ann. 314 110.2 (1999) 347-367 5. N. Brown, N. Choda, Approximation entropies in crossed products with a n application t o free shifts, Pacific J. Math. 198 110.2 (2001) 331-346 6. N. Brown, K. Dykema and D. Shlyakhtenko, Topological entropy of free product automorphisms, Acta Math. 189 no.1 (2002) 1-35 7. E. Effros, Z-J. Ruan, Operator spaces, London Mathematical Society Monographs 23 Oxford University Press (2000) 8. E. Kirchberg, O n nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math. 112 no.3 (1993) 449-489 9. E. Kirchberg, Commutants of unitaraes in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994) 39-77 10. V. Paulsen, Completely bounded maps and dilations, Pitman Research Notes 146 Longman (1986) 11. G. Pedersen, C* -Algebras and their automorphism groups, Academic Press (1979) 12. C. Pinzari, Y. Watatani, K. Yonetani, K M S states, entropy and the uariational principle in f u l l C*-dynamical systems, Comm. Math. Phys. 213 110.2 (2000) 331-37 13. D. Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys. 170 no.2 (1995) 249-281 14. P. Walters, An introduction t o ergodic theory, Graduate Texts in Mathematics 79 Springer-Verlag (1982) 15. S. Wassermann, Exact C*-algebras and related topics, Lecture Notes Series 19,Seoul National University Research Institute of Mathematics (1994)