Quantum Information IV

[Quantum • Information IV Edited by

T. Hida K. Saito

World Scientific

Quantum Information IV

Also published by World Scientific Quantum Information Proceedings of the First International Conference eds. T. Hida and K. Saito ISBN 981-02-3934-3 Quantum Information II Proceedings of the Second International Conference eds. T. Hida and K. Sait6 ISBN 981-02-4317-0 Quantum Information III Proceedings of the Third International Conference eds. T. Hida and K. Saito ISBN 981-02-4527-0

Proceedings of the Fourth International Conference

Quantum Information IV Meijo University, Japan

27 February-1 March 2001

Edited by

T. Hida & K. Saito Meijo University Japan

V f e World Scientific wb

Jersey * London • Singapore • Hong Kong New Jersey'London'Singapore*

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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QUANTUM INFORMATION IV Proceedings of the Fourth International Conference Copyright © 2002 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, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-020-5

Printed in Singapore by Uto-Print

V

PREFACE During the past four years we have had the most fruitful time for the research aiming at the development of quantum information theory. Here, in this volume, are reports on some of our main results that have been obtained so far by the members of the Academic Frontier Project at Meijo University. The papers in this volume are based on the lectures presented at the Fourth International Workshop on Quantum Information at Meijo University for the period February 27-March 1, 2001. The editors are pleased to accept all papers for publication in this volume at the suggestion of the referees. It is noted that there has been a great achievement of the theory in both white noise analysis and quantum probability theory starting from the seventies of the last century. It was our original intention to develop these two theories and further to establish unification of white noise analysis and quantum probability in order to explore new important directions including quantum computation and quantum information theory. We are very happy to see that this idea has been realized to a certain extent. In addition, a new impetus has come to probability theory from its strong interaction with quantum dynamics. Thus, we are encouraged to launch a new direction of our research. The Fourth Workshop was really a good opportunity to report the recent results on various hot topics and exchange new information in the fields of interest. At the same time we have been trying to proceed the revolution of the established areas. In this sense the workshop was quite exciting and has received good reactions from all the participants. As for the white noise theory, which is basically connected with the quantum information theory, is now about to meet its renaissance. Having good collaboration with quantum probability, as well as more intimate relationship with physics, we are sure that we are steadily approaching our goal. The quantum probability has now built a bridge between its results and quantum computer. We are most hopeful of success.

VI

Our Academic Frontier Project will be closed at the end of the 2001 academic year. The series of the publication "Quantum Information" shall also be closed. We promise to edit the last volume with a new idea so that the quantum information theory will develop towards new directions and the theory becomes more popular in science and technology. The organizers are grateful to the speakers at the workshop who contributed papers for this volume. December 30, 2001 Takeyuki Hida Kimiaki Saito Meijo University

CONTENTS

Preface

v

A Quarter Century of White Noise Theory H.-H. Kuo

1

Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials JV. Asai

39

Notions of Independence in Algebraic Probability Theory and Set Partition Statistics Y. Hashimoto

49

Some Thoughts on the Infinite Dimensional Harmonic Analysis T. Hida and Y. Hara-Mimachi

79

A Treatment of Quantum Baker's Map by Chaos Degree K. Inoue, M. Ohya and I. V. Volovich

87

Poisson Noise Analysis Based on the Levy Laplacian A. Ishikawa, K. Saito and A. H. Tsoi

103

A Hausdorff-Young Inequality for White Noise Analysis H.-H. Kuo, K. Saito and A. Stan

115

The Clark Formula of Generalized Wiener Functionals Y.-J. Lee and H.-H. Shih

127

Inverse S-Transform, Wick Product and Overcompleteness of Exponential Vectors N. Obata

147

Topics on Complex Gaussian Random Fields Si Si and Win Win Htay

Yll

Quantum Information in Space and Time and Theory of Stochastic Processes I. V. Volovich

187

Quantum Information IV (pp. 1-37) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

A Q U A R T E R C E N T U R Y OF W H I T E N O I S E T H E O R Y HUI-HSIUNG KUO Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA A B S T R A C T . T. Hida created the mathematical theory of white noise in his Carleton Mathematical Lecture Notes "Analysis of Brownian Functionals" (vol. 13, 1975). This theory has since been extensively developed and is now recognized as an important branch of stochastic analysis in the MSC 2000 subject classifications as "white noise theory" with the code number 60H40. We will review the shaping of white noise theory and the progress in various applications during the last quarter century.

1. Pre-theory period (white noise without a theory) White noise is a sound with equal intensity at all frequencies within a broad band. The sound in a large orchestra before the conductor raises his baton is an example of white noise. Mathematically it is informally defined as a stochastic process z(t) such that it is independent at different times and has identical distribution with mean 0 and infinite variance in the following sense: E(z(t)z{s))

= — f e^-^x

-ldx

= S0(t - s),

27T 7 R

where S0 is the Dirac function at 0. With this definition of white noise z(t) we can informally derive the following equality for a real-valued function / £ L2(a,b),

E(jj(t)z(t)dty=j*f(t)2dt. But what is the definition of the integral ja f(t)z(t) dt? Moreover, is it possible to define z(t) for each t? We can think of z(t) as the derivative z(t) = B(t) of a Brownian motion B(t). Then by regarding z(t) dt = dB(t), the integral /a6 f{t)z(t) dt can be defined as the Wiener integral /„ f(t) dB(t). However, since every Brownian path is nowhere differentiable, B(t) does not exist for any t. More importantly, consider a stochastic process /(£), is it possible to define the integral Ja f(t) dB(t) or even the integral /* f(t)B{t) dtl

2

It is well-known that the Ito integral fa fit) dB(t) is defined for all nonanticipating stochastic processes f(t) with almost all sample paths in L2(a, b). But in order to define the white noise integral j"f{t)B{t)dt we really need to give a rigorous definition of Bit) at least for almost all*. 2. Creation of white noise theory (1975) The mathematical theory of white noise was created by T. Hida in his 1975 Carleton Mathematical Lecture Notes [123]. In fact, five years earlier he already envisioned the white noise theory in his Princeton University Press book [116] by discussing Gaussian white noise, Poisson white noise, infinite dimensional rotation groups, etc. But it was in [123] that Hida proposed to use the family {B(t); t 6 T} as a coordinate system to analyze white noise functions. Recently he has advocated calling {B(t); t £ T } a system of idealized elemental random variables as a result of reductionism. To explain the original idea of Hida in [123], take the Schwartz space 5(E) of rapidly decreasing real-valued functions on E. Let 5'(R) be the dual space of 5 ( E ) . Since 5(E) C L 2 (E) and L2(R) can be identified with its dual space by the Riesz representation theorem, we get a Gel'fand triple 5(E) ^ L 2 (E) *-> 5'(E). By the Minlos theorem there exists a unique measure \i on 5'(E) such that /

ei<x^dli{x)=e-^2«,

( 6 5(1),

where | • | 0 is the £ 2 (E)-norm. The elements in 5'(E) can be regarded as B and for this reason the probability space (5'(R),/x) is called the white noise space. For simplicity, let (L2) denote the complex Hilbert space L 2 (5'(E), /*). By the Wiener-Ito theorem each

/n E L2ymJRn),


and the (L 2 )-norm of if is given by / oo

\l/2

IMIo= E«!|/-I2 V n=0

, /

where /„ is the multiple Wiener integral of order n.

3

Define a space (L2)+ of test functions and the corresponding space (L )~ of generalized functions by 2

{

°°

n±l

°°

1

£ J n(/n); /„ € Hs/mm(Rn) and £ «!|/n| 2 »±i, < °° , n=0 « 2 2 («") J 2 2 (L )- = { £ I B (F„); F n G F sym m (K») and £ n!|P n | ^ < oo [ H 2 (R I n=0 n=0 "' J s where H denotes the Sobolev space of order s. Then we have the triple n=0

T> + 1

(L 2 ) + «-* (L2) ^

(L 2 )-.

The space (L2)~ contains elements such as the white noise B(t) as well as the renormalizations : B(t)n: and : eB^ : for each fixed t e R . The renormalization of B(t)n is similar to Ito's idea of multiple Wiener integral, but the limit is taken in the generalized sense. Consider the case n = 2. Ito's idea is the following limit: :B{tf:=

f [ Jo Jo

ldB(u)dB(v)

= i l ^ E {B(ti) - B ^ - i ) ) (B(tj) = B(t)2-t,

Bit^))

limit in (L2).

On the other hand, here is Hida's idea to define the renormalization of B(t)2 as a generalized function in the space (L 2 )~: B{t + A)-B(t)\\_

\

z

1.

,rt+A , rt+A ,

JD/..,JD,..)

~ 2 hm Y, {B(U) - B ( ^ - i ) ) (B&) A - •-- . . . 1 (5(t + A ) - 5 ( t ) ) 2 - A A2

- B(t,._i)) (2.1)

The limit of Equation (2.1), as A ->• 0, does not exist in (L2). But if we take the limit in the space (L2)~ of generalized functions, then the limit, denoted by : B(t)2:, exists and defines a generalized function in

3. Kubo-Takenaka's construction (1980) In 1980 Kubo and Takenaka [260] constructed test and generalized functions on a general space. Let £ <-4 E <—• £' be a Gel'fand triple.

4

Here E is a real Hilbert space with norm | • | and £ is a nuclear space with norms {| • | p ; p > 1} satisfying the conditions: (a) There exists 0 < p < 1 such that | • | < p\ • \i < p2\ • | 2 < • • • < PV\-\P<---

for all p> 1.

(b) For any p > 1, there exists q > p such that the inclusion map iqiP : £q <—> £p is a Hilbert-Schmidt operator, where £p is the completion of £ with respect to the norm | • | p . Let p; be the probability measure on £' with characteristic function J£'

The probability space (£',fi) is an abstract white noise space. Each

p(*) = £ < : s 0 n : , / „ ) ,

/ne^§",

(3.1)

71=0

where : x®n: is a Wick tensor (see page 33 in the book [290]. Moreover, the (L 2 )-norm ||(^|| of tp is given by /

IMI=

\l/2

oo

E^l/nl \n=0

2

, /

where | • | is the norm on E®n induced by the norm on E. For

1, define oo

m\p =

\ 1/2

£"'l/n|p n=0

, /

where | • | p is the norm on £®n induced by the norm on £p. Let (£p) = {tp e {L2); \\ip\\p < oo}. The projective limit (£) of {(£p); p > 1} serves as a space of test functions on the abstract white noise space (£',//). The dual space {£)* of {£) serves as the corresponding space of generalized functions. Thus we have the Gel'fand triple {£) M . (L 2 ) M- (£)*. The space {£) of test functions is an infinite dimensional analogue of the Schwartz space »S(IRd) on the finite dimensional space M.d and has many similar properties as <S(lRd). For example, it is closed under pointwise multiplication of two test functions. The differential operators, translation operators, scaling operators, Fourier-Gauss transform, Gross Laplacian, and number operator are all continuous linear operators on (£). Hence their adjoint operators are continuous on the dual space (£)*. The integral kernel operators can be defined and are

5

continuous linear operators from (£) into (£)*. Moreover, the HitsudaSkorokhod integral can be defined as a random variable on the white noise space <S'(K).

4. Characterization theorems of Potthoff-Streit and Lee (1991) In 1991 two important theorems were obtained by Potthoff and Streit [420] (for generalized functions) and by Y.-J. Lee [328] (for test functions) . The Potthoff-Streit theorem characterizes generalized functions in the space {£)* in terms of their 5-transform S * ( 0 = «*,:*<••«:»,

(££c

where £c is the complexification of £. The theorem says that a complexvalued function F on £c is the 5-transform of a generalized function in (£)* if and only if it satisfies the following conditions: (a) For any £,»/" 6 £c, the function F(z£ + rj) is an entire function of zeC; (b) There exist constants K,a,p > 0 such that |F(0|
V£e£c.

The Lee theorem describes the space (£) of test functions directly in terms of analyticity and growth condition. It says that a function tp on £' belongs to (£) if and only if for any p > 1 the function ip is analytic on £' c (the complexification of £'p) and there exists a constant Kp > 0 such that "1, \ip{x)\ < Kpexp -\x\2 Vx&£'On the other hand, test functions in {£) can also be characterized in terms of their 5-transform [304]. A complex-valued function F on £c is the 5-transform of a test function in {£) if and only if it satisfies the following conditions: (a) For any £,77 G £c, the function F(z£ + 77) is an entire function of zeC; (b) For any a,p > 0, there exists a constant K > 0 such that

|F(0|<#exp[a|£| 2 _ p ],

V£G£c

6

5. More spaces of generalized functions (1992-2000) In 1992 Kondratiev and Streit introduced a family of Gel'fand triples [243] [244]. Let 0 < (3 < 1. For tp represented by Equation (3.1) and p > 1, define 1/2

1+/

M|p*=(f>!) UG) \n=0

/

Let (£p)p = {

1} and its dual space (£)p give a new Gel'fand triple (£)0 ^

(L 2 ) ^

(£);.

Kondratiev and Streit showed in [243] [244] that generalized functions in (£)p and test functions in {£)p can be characterized as follows. A complex-valued function F on £c is the S-transform of a generalized function in (£)% if and only if it satisfies the conditions: (a) For any £, fj G £c, the function F(z£ + TJ) is an entire function of 2GC; (b) There exist constants K, a, p > 0 such that \F(0\
V£ £ £C.

(5.1)

On the other hand, a complex-valued function F on £c is the 5transform of a test function in (£)p if and only if it satisfies the conditions: (a) For any £, r\ £ £c, the function F(z£ + TJ) is an entire function of zeC; (b) For any a,p > 0, there exists a constant K > 0 such that \F(t)\
V£ G £e.

(5.2)

Test functions in the space (£)p have also been described in [290] in the same spirit of Y.-J. Lee, namely, a function ip on £' belongs to {£)p if and only if for any p > 1 the function

0 such that \ip{x)\ <

Kpexp

oU + ^ N '

VXG£"!

The collection (£)%, 0 < /3 < 1, is an increasing family of generalized functions. Kondratiev and Streit [244] also constructed a space (£)~1 of generalized functions dealing with the case (3 = 1. But the corresponding space of test functions is not a nuclear space.

7

Between the union Uo 0 a(n)crn > 0 for some constant a > 1. (2) l i ^

M

( ^ ' " - O .

For ip represented by Equation (3.1) and p > 1, define

|MU=(£n!a(n)|/4)12. \n=0

/

2

Let [£p]a = {f £ {L ); \\ 1}- The dual space [£]* is a new space of generalized functions and we have the Gel'fand triple [£]a ^

(L2)^

[£]'a.

Important examples of {a(n)} given by the Bell numbers [82]. A characterization theorem for generalized functions in the space [£]* is given in [82] with the growth condition: there exist constants K,a,p > 0 such that

|F(0l<^G a ( a |£|3 1/2 ,

(5.3)

where Ga(r) = 2jJL0 ^ ^ r " - The corresponding characterization theorem for test functions in the space [£]a is given in [36] with the growth condition: for any a,p > 0, there exists a constant K > 0 such that \F(0\
(5.4)

where G 1 / a ( r ) = E ~ = o ^ r » . For the special case a(n) = (n!)^, the growth functions in Equations (5.3) and (5.4) are different from those in Equations (5.1) and (5.2), respectively. Moreover, for a given sequence {a(n)}, it is impossible in general to sum up Ga and G\ja in close forms. This is the motivation for Asai et al. [33] [35] [38] [39] to introduce CKS-space associated with a growth function. Let u be a continuous function on [0, oo) satisfying the following conditions: (1) l i m ^ ^ = oo. (2)limsup^00^
8

Define the Legendre transform £u of u by Un) y

'

= inf —n , r>o

n = 0,1,... ,

r

and the dual Legendre transform u* of u by e 2v^S

u*(r) = sup —-—-, s>o w(s)

r > 0.

For such a function u, define a sequence au(n) by

With this sequence we get a CKS-space denoted by

[£\u M. (i 2 )

M.

[f]:.

Characterization theorems for generalized functions in [£]* and test functions in [£}u are given in [39]. A complex-valued function F on £c is the S'-transform of a generalized function in [£]* if and only if it satisfies the conditions: (a) For any £, 77 £ £ c , the function F{z(t + 77) is an entire function of zeC; (b) There exist constants K, a, p > 0 such that \F(t)\ 0, there exists a constant if > 0 such that

\F(t)\
V£££c.

6. Applications of white noise theory There is a wide range of applications of white noise theory. mention just a few of them below with very brief comments. 1. Integral kernel operators

We

In 1992 Hida, Obata, and Saito introduced the integral kernel operators [193]. The generalization to operator-valued generalized functions has been done by Obata, Chung, and Ji, among others. These operators are important for applications in quantum probability.

9 2. Stochastic

integration

In 1981 Kubo and Takenaka related the white noise integral Ja d*ip(t) dt to the ltd integral /^ ip(t) dB(t). The integral / ' d*ip(t) dt turns out to be the same as the integral introduced by Hitsuda in 1972 [206] and by Skorokhod in 1975 [472]. In the paper [247] Kubo introduced the ltd formula for f(B(t)) with / being a generalized function. In 1990 Kuo and Potthoff [302] [303] studied anticipating stochastic integrals and stochastic differential equations. For more information, see the book [290]. Recent progress on white noise methods for stochastic integration has been done by Potthoff and Streit and their collaborators. 3. Infinite dimensional harmonic analysis In 1975 Hida [123] outlined several things about the infinite dimensional rotation group on the white noise space. Through the years the subgroups of this group, in particular the Levy group, have been analyzed very much by Hida, Obata, and Saito. The Levy Laplacian and related semigroups have been studied extensively by Saito and his collaborators. Recently Lee and Stan [344] obtained the infinite dimensional Heisenberg inequality. The finite dimensional Paley-Wiener theorem has been generalized to the white noise space by Stan in [473]. 4. Stochastic partial differential

equations

Since 1991 0ksendal and his collaborators have made significant progress using white noise theory to study stochastic partial differential equations. See the book [217]. Recent achievements of white noise methods to study SPDE's (in particular the Burgers equation) have been made by 0ksendal, Potthoff, Streit, and their collaborators [49] [50] [53] [87] [88] [90] [109] [212] [213]. 5. Feynman integral Feynman integral was one of the motivations for Hida to introduce white noise theory in 1975. Streit and his numerous collaborators have made revolutionary work to treat Feynman integrands as generalized functions in various spaces of generalized functions. In the paper [267] it is shown that with a rather general potential function the associated Feynman integrand is a generalized function in some big space of generalized functions. Recently, Asai et al. [17] [18] have shown that for the Albeverio-H0egh-Krohn and Laplace transform potentials the associated Feynman integrands are generalized functions in the space [£]* for some growth function u. 6. Random fields and stochastic variational calculus

10

Since 1988 Hida and Si Si have made significant progress for this application of white noise theory. For further information, see the new book by Hida [184]. 7. Intersection local times In 1991 H. Watanabe [500] used white noise theory to study the local time of self-intersections of Brownian motions. Recently Streit and his collaborators have obtained rather interesting results in [101]. 8. Infinite dimensional stochastic differential equations White noise theory can be used to study the laws of solutions of <S'(R.)-valued stochastic differential equations. In [311] it is shown that under certain conditions the laws of the solution of an <S'(R)-valued SDE induce generalized functions in the space (S)* of generalized functions. Moreover, these generalized functions satisfy an infinite dimensional partial differential equations. 9. Positive generalized functions Positive generalized functions in white noise theory was first studied by Potthoff [409] in 1987. These functions are associated with Hida measures on a white noise space. Interesting results have been obtained by Yokoi, Y.-J. Lee, Asai-Kubo-Kuo, on various spaces of generalized functions. For Hida measures on the Kondratiev-Streit space, see the

book [290]. 10. Dirichlet forms In 1988 Hida, Potthoff, and Streit [195] initiated the application of white noise theory to Dirichlet forms. Later they were joined by Albeverio and Rockner [23] [24] to achieve revolutionary work on the construction of Dirichlet forms for quantum field theory. 11. Quantum probability Recent development of quantum probability by Accardi and his collaborators (in particular, Lu, Obata, and Volovich) has shown a strong connection between white noise theory and quantum probability. See the HAS Report [16] and the Volterra Center Preprint [17] and the references therein. 7. References on white noise theory In the references below we have compiled a list of publications on white noise theory since 1975 with a few exceptions, which are listed for their influence on white noise theory. This list is by no means complete, but it gives some ideas on the development of white noise theory since 1975.

11

In addition, there have been several conference proceedings and special volumes which contain many papers related to white noise theory: (1) White Noise Analysis: Math, and Appls., T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit (eds.), World Scientific, Singapore, 1990 (2) Gaussian Random Fields, K. Ito and T. Hida (eds.), World Scientific, Singapore, 1991 (3) Stochastic Analysis on Infinite Dimensional Spaces, H. Kunita and H.-H. Kuo (eds.), Longman Science & Technical, Pitman Research Notes in Math. Series, vol. 310, 1994 (4) Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability, L. Accardi, N. Obata, K. Saito, Si Si, and L. Streit (eds.), Italian School of East Asian Studies, Natural and Mathematical Sciences Series 3, Intituto Italiano di Cultura, Kyoto, 2000 (5) Recent Developments in Infinite Dimensional Analysis and Quantum Probability, in honor of T. Hida's 70th birthday, L. Accardi, N. Obata, K. Saito, Si Si, and L. Streit (eds.), a special issue in Acta Applicandae Mathematicae, vol. 63, nos. 1-3, 2000 (6) Collected Papers of T. Hida, L. Accardi, N. Obata, K. Saito, Si Si, and L. Streit (eds.), World Scientific, 2001 A c k n o w l e d g e m e n t s . (1) This article was prepared during my visit to Hiroshima University, May 20-August 19, 2001. I would like to give my deepest appreciation to Professor I. Kubo for submitting a proposal to Monbu-Kagaku-Sho (Ministry of Education and Science) for my visit to him and Hiroshima University. I am most grateful for the warm hospitality of the mathematics department and the personnel office of the Graduate School of Science, Hiroshima University. I want to give my best thanks to Monbu-Kagaku-Sho for the financial support for my visit to Hiroshima University. (2) I would like to thank the Academic Frontier in Science of Meijo University for financial supports and to give my deep appreciation to Professors T. Hida and K. Saito for the warm hospitality during my many visits over the years 1998-2001.

REFERENCES [1] Aase, K., 0ksendal, B., Privault, N., and Ub0e, J.: A white noise generalization of the Clark-Haussmann-Ocone theorem, with application to mathematical finance; Finance and Stochastics 4 (2000) 465—496

12 [2] Aase, K., 0ksendal, B., and Ub0e, J.: Using the Donsker delta function to compute hedging strategies; Preprint University of Oslo 6/1998 and Potential Analysis (to appear) [3] Accardi, L.: Q u a n t u m stochastic calculus; Probability Theory and Mathematical Statistics 1 (1985) 1-21 [4] Accardi, L.: Yang-Mills equations and Levy-Laplacian; in: Dirichlet Forms and Stochastic Processes, de Gruyter (1995) 1—24 [5] Accardi, L. and Bogachev, V.: The Ornstein-Uhlenbeck process and the Dirichlet form associated to the Levy Laplacian; C. R. Acad. Sci. Paris Ser. I Math, bf 320 (1995) 597-602 [6] Accardi, L. and Bogachev, V.: T h e Ornstein-Uhlenbeck process associated with the Levy Laplacian and its Dirichlet form; Probab. Math. Statist. 17 (1997) 95-114 [7] Accardi, L. and Bogachev, V.: On stochastic analysis associated with t h e Levy Laplacian; Dokl. Akad. Nauk 358 (1998) 583-587 [8] Accardi, L., Boukas, A., and Kuo, H . H . : On t h e unitarity of stochastic evolutions driven by t h e square of white noise; Preprint (2000) [9] Accardi, L. and Bozejko, M.: Interacting Fock space and Gaussianization of probability measures; Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 (1998) 663-670 [10] Accardi, L., Gibilisco, P., and Volovich, I. V.: Yang-Mills gauge fields as harmonic functions for the Levy Laplacian; Russian J. Math. Phys. 2 (1994) 235-250 [11] Accardi, L., Hashimoto, Y., and Obata, N..: Notions of independence related to the free group; Infinite Dim. Anal, Quantum Prob. and Related Topics 1 (1998) 201-220 [12] Accardi, L., Hashimoto, Y., and Obata, N..: Singleton independence; Banach Center Publications 4 3 (1998) 9-24 [13] Accardi, L., Hashimoto, Y., and Obata, N..: A role of singletons in quantum central limit theorems; J. Korean Math. Soc. 35 (1998) 675-690 [14] Accardi, L., Hida, T., and Kuo, H.-H.: Ito table for the square of white noise; Preprint (2000) [15] Accardi, L., Hida, T., and Win Win Htay: Boson Pock representations of stationary processes (in Russian); Mathematical Notes 6 7 (2000) 3-14 [16] Accardi, L., Lu, Y. G., and Volovich, I.: T h e Hilbert module and interacting Fock spaces; HAS Reports 1 9 9 7 - 0 0 8 (1997) [17] Accardi, L., Lu, Y. G., and Volovich, I. V.: White noise approach to classical and quantum stochastic calculi; Centra Vito Volterra, Universita di Roma "Tor Vergata" 375 (1999) [18] Accardi, L., Lu, Y. G., and Volovich, I. V.: A white noise approach to stochastic calculus; Acta Appl. Math. 6 3 (2000) 3-25 [19] Accardi, L., Rozelli, P., and Smolyanov, O. G.: Brownian motion generated by the Levy Laplacian; Mat. Zametki 54 (1993) 144-148 [20] Albeverio, S., Daletzky, Y., Kondratiev, Yu. G., and Streit, L.: Non-Gaussian infinite dimensional analysis; J. Fund. Analysis 138 (1996) 311-350 [21] Albeverio, S., Fukushima, M., Karwowski, W., and Streit, L.: Capacity and quantum mechanical tunneling; Comm. math. Phys 81 (1981) 501

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15 Betounes, D. and Redfern, M.: Stochastic integrals for nonprevisible, multiparameter processes; J, of Appl. Math. Optim. 28 (1993) 197-223 Betounes, D. and Redfern, M.: T h e Stratonovich integral via the renormalization operator on Fock space; Stochastics and Stochastics Reports 56 (1996) 161-178 Bltimlinger, M. and Obata, N.: Permutations preserving Cesaro mean, densities of natural numbers and uniform distribution of sequences; Ann. Inst. Fourier Grenoble 41 (1991) 665-678 Carmona, R. and Yan, J. A.: A new space of white noise distributions and applications to SPDE's; in: Seminar on Stochastic Analysis, Random Fields and Applications (1995) 51-66 Chow, P. L.: Generalized solution of some parabolic equations with a random drift; J. Appl. Math. Optim. 2 0 (1989) 81-96 Chow, P. L.: Stationary solutions of some parabolic Ito equations; Pitman Research Notes in Math. Series 3 1 0 (1994) 42-51, Longman Scientific & Technical Chung, D. M. and Chung, T. S.: First order differential operators in white noise; Proc. Amer. Math. Soc. 126 (1998) 2367-2376 Chung, D. M., Chung, T. S., and Ji, U. C : A simple proof of analytic characterization theorem for operator symbols; Bull. Korean Math. Soc. 34 (1997) 421-436 Chung, D. M., Chung, T. S., and Ji, U. C : Products of white noise functionals and associated derivations; J. Korean Math. Soc. 35 (1998) 559-574 Chung, D. M., Chung, T. S., and Ji, U. C : A characterization theorem for operators on white noise functionals; J. of Math. Soc. Japan 51 (1999) 437-447 Chung, D. M. and Ji, U. C : Cauchy Problems for a Partial Differential Equation in White Noise Analysis; J. Korean Math. Soc. 33 (1996) 309-318 Chung, D. M. and Ji, U. C : Wick Derivations on White Noise Functionals; J. Korean Math. Soc. 33 (1996) 559-574 Chung, D. M. and Ji, U. C : Transforms on white noise functionals with their applications to Cauchy problems; Nagoya Math. J. 147 (1997) 1-23 Chung, D. M. and Ji, U. C : Transformation groups on white noise functionals and their applications; Appl. Math. Optim. 37 (1998) 205-223 Chung, D. M. and Ji, U. C : Some Cauchy problems in white noise analysis and associated semigroups of operators; Stoch. Analy. Appl. 17 (1999) 1-22 Chung, D. M. and Ji, U. C : Multi-parameter transformation groups on white noise functionals; J. Math. Anal, and Appl. 2 5 2 (2000) 729-749 Chung, D. M. and Ji, U. C : Poisson equations associated with differential second quantization operators in white noise analysis; Acta Applicadae Mathernaticae 63 (2000) 89-100 Chung, D. M., Ji, U. C, and Obata, N.: Transformations on white noise functions associated with second order differential operators of diagonal type; Nagoya Math. J. 1 4 9 (1998) 173-192 Chung, D. M., Ji, U. C, and Obata, N.: Higher powers of quantum white noises in terms of integral kernel operators; Infinite Dim. Anal., Quantum Prob. and Related Topics 1 (1998) 533-559

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17 de Faria, M., Oliveira, M. J., and Streit, L.: A generalized Clark-Ocone formula; Random Oper. Stock. Eqs. 8 (2000) 163-174 de Faria, M., Potthoff, J., and Streit, L.: T h e Feynman integrand as a Hida distribution; J. Math. Phys. 32 (1991) 2123-2127 de Faria, M. and Streit, L.: Some recent advances in white noise analysis; Pitman Research Notes in Math. Series 3 1 0 (1994) 52-59, Longman Scientific & Technical Doku, I.: On a class of infinite-dimensional Fourier type transforms in white noise calculus; in: Probability Theory and Mathematical Statistics, World Scientific (1995) 51-61 Doku, I.: On the Laplacian on a space of white noise functionals; Tsukuba J. Math. 19 (1995) 93-119 Doku, I., Kuo, H.-H., and Lee, Y.-J.: Fourier transform and heat equation in white noise analysis; Pitman Research Notes in Math. Series 3 1 0 (1994) 60-74, Longman Scientific & Technical Drumond, C , de Faria, M., and Streit, L.: The square of self-intersection local time of Brownian motion; in: Stochastic Processes, Physics and Geometry: New Interplays I, a volume in honor of Sergio Albeverio, F. Gesztesy et al. (eds.) - CMS Conf. Proc. 28, Amer. Math Soc. (2000) 115-122 Gannoun, R., Hachaichi, R., Ouerdiane, H., and Rezgui, A.: Un theoreme de dualite entre espaces de fonctions holomorphes a croissance exponentiele; J. Fund. Anal. 1 7 1 (2000) 1-14 Gielerak, R., Karwowski, W., and Streit, L.: Construction of a class of characteristic functionals; in: Feynman Path Integrals, S. Albeverio et al. (eds.), Lecture Notes in Physics 106 (1979) 182-188, Springer Gjerde, J., Holden, H., 0ksendal, B., Ub0e, J., and Zhang, T.: An equation modelling transport of a substance in a stochastic medium; in: Seminar on Stochastic Analysis, Random Fields and Applications, E. Bolthausen et al. (eds.), Progress in Probability, Vol.36, Birkhauser (1995) 123-134 Gjessing, H., Holden, H., Lindstr0m, T., Oksendal, B., Ub0e, J., and Zhang, T.: T h e Wick product; in: Frontiers in Pure and Applied Probability, Vol. I, H. Niemi et al. (eds.), T V P Science Publishers, (1993) 29-67 Grothaus, M., Khandekar, D. C , da Silva, J. L., and Streit, L.: T h e Feynman integral for time-dependent anharmonic oscillators; J. Math. Phys. 38 (1997) 3278-3299 Grothaus, M., Kondratiev, Yu. G., and Streit, L.: Complex Gaussian analysis and the Bargmann-Segal space; Methods of Funct. Anal. Topology 3 (1997) 46-64 Grothaus, M., Kondratiev, Yu. G., and Streit, L.: Regular generalized functions in Gaussian analysis; Infinite Dim. Anal. Quantum Prob. and Related Topics 2 (1999) 1-25 Grothaus, M., Kondratiev, Yu. G., and Streit, L.: Scaling limits for the solution of Wick type Burgers equation; Random Oper. Stochastic Equations 8 (2000) 1-26 Grothaus, M., Kondratiev, Yu. G., and Us, G. F.: Wick calculus for regular generalized stochastic functionals; Random Oper. Stochastic Equations 7 (1999) 263-290

Grothaus, M. and Streit, L.: Construction of relativistic quantum fields in the framework of white noise analysis; J. Math. Phys. 4 0 (1999) 5387-5405 Grothaus, M. and Streit, L.: Quadratic actions, semi-classical approximation, and delta sequences in Gaussian analysis; Rep. Math. Phys. 4 4 (1999) 3 8 1 405 Grothaus, M. and Streit, L.: On regular generalized functions in white noise analysis and their applications; Methods of Funct. Anal. Topology 6 (2000) 14-27 Grothaus, M., Streit, L., and Volovich, I.: Knots, Feynman diagrams and matrix models; Infinite Dim. Anal. Quantum Prob. and Related Topics 2 (1999) 359-380 Hashimoto, Y., Obata, N., and Tabei, N.: A quantum aspect of asymptotic spectral analysis of large Hamming graphs; in: Quantum Information HI, T. Hida and K. Saito (eds.) (2001) Hida, T.: Stationary Stochastic Processes. Princeton University Press, 1970 Hida, T.: Harmonic Analysis on the space of generalized functions; Teor. Verojatnost. i Primenen. (Theory of Probability and its applications) 15 (1970) 119-124 Hida, T.: Note on the infinite dimensional Laplacian operator; Nagoya Math. J. 38 (1970) 13-19 Hida, T.: Quadratic functional of Brownian motion; J. Multivariate Analysis 1 (1971) 58-69 Hida, T.: Complex white noise and infinite dimensional unitary group, Lecture Notes, Mathematics Dept., Nagoya Univ. no.3, 1971 Hida, T.: A role of Fourier transform in t h e theory of infinite dimensional unitary group; J. Math. Kyoto Univ. 1 3 (1973) 203-212. Hida, T.: Functionals of complex white noise; Proc. Symp. on Continuous Mechanics and related problems of Analysis 1 Tbilisi, USSR, 1973, 355-366 Hida, T.: Analysis of Brownian Functionals. Carleton Mathematical Lecture Notes 13, 1975 Hida, T.: White noise analysis and nonlinear filtering problems; Applied Mathematics and Optimization 2 (1975) 82-89 Hida, T.: Analysis of Brownian functionals; Mathematical Programming Study 5 (1976) 53-59 Hida, T.: Functionals of Brownian motion; Transaction of 7th Prague Conf. and 191J, European Meeting of Statisticians A (1977) 239-243 Hida, T.: Topics on nonlinear filtering theory; Multivariate analysis 4 (1977) 239-245 Hida, T.: Generalized multiple Wiener integrals; Proc. Japan Acad. 5 4 A (1978) 55-58 Hida, T.: Generalized Brownian functionals; Complex Analysis and its Applications, I. N. Vekua Volume (1978) 586-590 Hida, T.: White noise and Levy's functional analysis; Lecture Notes in Math. 6 9 5 (1978) 155-163 Hida, T.: White noise and Levy's functional analysis; Lecture Notes in Math. 6 9 5 (1978) 155-163 Hida, T.: Generalized multiple Wiener integrals; Proc. Japan Acad. 5 4 A (1978) 55-58

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37 Streit, L.: Representations of diffeomorphisms on compound Poisson spaces; in: Analysis on Infinite Dimensional Lie Algebras and Groups, H. Heyer and J. Marion (eds.), (1998) World Scientific Streit, L. and Hida T.: Generalized Brownian functionals and the Feynman integral; Stochastic Processes and Their Applications 16 (1983) 55—69 Streit, L. and Hida, T.: White noise analysis and its applications to Feynman integral; Proc. Conf. Measure Theory and its applications, Lecture Notes in Math. 1 0 3 3 (1983) 219-226 Streit, L. and Westerkamp, W.: A generalization of the characterization theorem for generalized functions of white noise; in: Dynamics of Complex and Irregular Systems P h . Blanchard et al. (eds.), World Scientific (1993) 174-187 Takenaka, S.: On projective invariance of multi-parameter Brownian motion; Nagoya Math. J. 6 7 (1977) 89-120 Takenaka, S.: Invitation to white noise calculus; Lecture Notes in Control and Inform. Sci. 4 9 (1983) 249-257 Takenaka, S.: On pathwise projective invariance of Brownian motion. I; Proc. Japan Acad. Ser. A Math. Sci. 64 (1988) 41-44 Takenaka, S.: On pathwise projective invariance of Brownian motion. II; Proc. Japan Acad. Ser. A Math. Sci. 6 4 (1988) 271-274 Takenaka, S.: On pathwise projective invariance of Brownian motion. Ill; Proc. Japan Acad. Ser. A Math. Sci. 66 (1990) 35-38 Timpel, M. and Benth, F. E.: Topological aspects of t h e characterization of Hida distributions - a remark; Stochastic Rep 51 (1994) 293-299. Watanabe, H.: T h e local time of self-intersections of Brownian motions as generalized Brownian functionals; Lett. Math. Phys. 23 (1991) 1-9 Watanabe, H.: Donsker's (5-function and its applications in the theory of white noise analysis; Stochastic Processes, a Festschrift in Honor of G. Kallianpur, S. Cambanis et al. (eds.) (1993) 337-339, Springer-Verlag Yan, J. A.: Sur la transformed de Fourier de H. H. Kuo; Lecture Notes in Math. 1372 (1989) 393-394, Springer-Verlag Yan, J. A.: Inequalities for products of white noise functionals; Stochastic Processes 349—358, Springer 1993 Yan, J. A.: Products and transforms of white-noise functionals (in general setting); Appl. Math. Optim. 31 (1995) 137-153 Yokoi, Y.: Positive generalized Brownian functionals; White Noise Analysis, Mathematics and Applications, World Scientific (1989) 407-422 Yokoi, Y.: Positive generalized white noise functionals; Hiroshima Math. J. 20 (1990) 137-157 Yokoi, Y.: Properties of Gel'fand triplet in white noise analysis and a characterization of Hida distributions; Proc. Preseminar for International Conference on Gaussian Random Fields, P a r t 2 (1991) 34-48 Yokoi, Y.: Simple setting for white noise calculus using Bargmann space and Gauss transform; Hiroshima Mathematical Journal 25 (1995) 97-121 Yokoi, Y.: On continuity of test functionals in infinite-dimensional Bargmann space; Memoirs, Faculty of General Education, Kumamoto University, Natural Sciences 31 (1996) 1-8 Zhang, Y.: The Levy Laplacian and Brownian particles in Hilbert spaces; J. Funct. Anal. 133 (1995) 425-441

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Quantum Information IV (pp. 39-48) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

INTEGRAL TRANSFORM AND SEGAL-BARGMANN REPRESENTATION ASSOCIATED TO Q-CHARLIER POLYNOMIALS

NOBUHIRO A S M International Institute for Advanced Studies Kizu, Kyoto 619-0225, Japan. [email protected], [email protected]

Let fip be the g-deformed Poisson measure in the sense of Saitoh-Yoshida 24 and vp be the measure given by Equation (3.6). In this short paper, we introduce the g-deformed analogue of the Segal-Bargmann transform associated with fip . We prove that our Segal-Bargmann transform is a unitary map of L2(ppq)) onto the g-deformed Hardy space 'H2[yq). Moreover, we give the Segal-Bargmann representation of the multiplication operator by x in L2(fip'), which is the sum of the g-creation, g-annihilation, g-number, and scalar operators.

1

Introduction

T h e classical Segal-Bargmann transform in Gaussian analysis yields a unitary m a p of L2 space of the Gaussian measure on M. onto the space of L2 holomorphic functions of the Gaussian measure on C, see papers7,8.is,16,19,25,26 Recently, Accardi-Bozejko 1 showed the existence of a unitary operator between a one-mode interacting Fock space and L2 space of a probability measure on R by making use of t h e basic properties of classical orthogonal polynomials and associated recurrence formulas 1 4 , 2 7 . Inspired by this work, the author 3 has recently extended t h e Segal-Bargmann transform to nonGaussian cases. T h e crucial point is to introduce a coherent s t a t e vector as a kernel function in such a way t h a t a transformed function, which is a holomorphic function on a certain domain in general, becomes a power series expression. Along this line, Asai-Kubo-Kuo 6 have considered the case of the Poisson measure compared with the case of t h e Gaussian measure. However, the case of L2 space of Wigner's semi-circle distributions in free probability theory is beyond their scope. On the other hand, van Leeuwen-Maassen 2 2 considered a transform associated with q deformation of the Gaussian measure 1 0 ' 1 1 ' 1 2 and showed t h a t for a given real number q £ [0,1) it is a unitary m a p of I? space of qr-defomed Gaussian measure onto the g-deformed Hardy space H2(iyq) where vq is given

40 in (3.6). Biane 9 examined the case of q = 0 (Free case). Roughly speaking, their methods do not give the relationship between Szegb-Jacobi parameters and kernel functions for their transforms. As observed in Section 3 and Appendix A, our approach clarifies the relationship between them. In this paper, we shall consider the (/-deformed version of the SegalBargmann transform S (,) associated with (/-deformed Poisson measure, denoted by /ip , in t h e sense of Saitoh-Yoshida 2 4 . As a main result, we shall provide Proposition 4.1, which claims t h a t 5* (,) is a unitary m a p of L2(/j,pq ) onto % 2 (i/ q ). Moreover, in Theorem 4.3 we shall give the representation in H2(vq) of t h e multiplication operator by x in L2(/j,p ), which is the sum of the (/-creation, (/-annihilation, (/-number, and scalar operators. We remark t h a t our representation is compatible with t h a t on the (/-Fock space by SaitohYoshida 2 3 and can be viewed as t h e (/-analogue of the H u d s o n - P a r t h a s a r a t h y 1 7 decomposition of the usual Poisson random variable on the s t a n d a r d Boson Fock space (q = 1). Ito-Kubo 1 8 also studied a similar decomposition in details from the point of white noise calculus 2 0 , 2 1 (For more recent formulation, see papers4'5'13). T h e present article serves a good example to papers by Accardi-Bozejko 1 and Asai 3 . The present paper is organized as follows. In Section 2, we recall the recurrence formula for q-Charlier polynomials. In Section 3, we introduce a (/-deformed coherent state vector and a Segal-Bargmann transform associated to a (/-deformed Poisson measure. In addition, we quickly define the Hardy space as the Segal-Bargmann representation space. In Section 4, our main results are given. In Appendix A, we give some remarks on known results 7 ' 9 ' 2 2 ' 2 5 ' 2 6 related to (/-Hermite polynomials. Notation. Let us recall s t a n d a r d notation from (/-analysis 2 . We put for n 6 No, [n], := 1 + q + • • • + qn~x

with [0], = 0.

Then (/-factorial is naturally defined as [ n ] , ! : = [ l ] , - - - [ n ] , w i t h [ 0 ] , ! = 0. T h e (/-exponential is given by

n=0

l

>q

whose radius of convergence is 1/(1 — q). In addition, another symbol used

41

is the q-analogue of the Pochhammer symbol, ra —1

oo

{a;q)n = J J ( l - a q J ) and {a-q)^ = J J ( l - a g ' ) j=o

3=0

with the convention (a;q)0 = 1. 2

q-deformed Charlier Polynomials

From now on, we always assume that q 6 [0,1) is fixed. Recently, SaitohYoshida24 calculated the explicit form of the g-deformed Poisson measure with a parameter (3 > 0 for q € [0,1). We denote it by /j,pq . The orthogonal polynomials associated to fj,p are the g-Charlier polynomials {C„ (a)} with the Szego-Jacobi parameters an = [n]q + /3,uin = (3[n]q, See papers 23,24 . We also refer the book 14 for the standard Charlier polynomials case (q = 1). Let A = {/3™Hq!}^Lo- The following relations hold for each n > 1: (x - \n]q-(3)C^(x)

= C i t ( x ) +P[n]qC%l1(x)

(2.1)

where CQ = 1, C\q = x — (3. Then, for any L2-convergent decompositions f{x) = J2anCn{x) and g(x) = Y, KCn {x), the inner product (•, •)L2{flM) is given by the form
(2-2)

n

Moreover, let H/H 2 ,,^,,, = E~=o / ^ K ' K I 2 3

Segal-Bargmann Transform and Hardy Space

Let us define the q-deformed coherent state vector for { C „ ( * ) } by E

i>>

*) = E

^

^

z E fl? := {z

€

C : N 2 < ^-q}.

(3.1)

It is easy to see that Ep [x, z) £ L2{/j,pq ) due to

Moreover, it can be shown that {E)9' (x, y)} is linearly independent and total in L2(np).

42

Now we are in a position to introduce our key tool in this paper. Let us sider the g-analogue g-analc consider of the Segal-Bar gmann transform S (,> associated to {Ciq\x)}

given by for any / G L2{^).

( S . w / K z ) = (B%l(x,z),f(x))L2(iit))

(3.3)

Lemma 3.1. Let f £ L2(fj,p ) . Then (S («)/)(z) converges absolutely for all

z e nf. Proof. For f(x) = ^2^L0 anCn

(x), it is quite easy to see oo

By the Schwartz inequality, we get the inequality 00

I 12

n

£K* l
n=0

This shows that the (5 u)/)(z) converges absolutely for all z £ Q^. MP

•

The completion of the space of holomorphic functions F, G on fi^ with respect to the inner product, (F,G)K

= JF\z)G{z)vq{dz),

is nothing but the g-deformed Hardy space 7i2(vq).

vq(dz) = (q;q)00Y,-r-VX?Mz)> j=0 V^'^/j

(3.5) Here vq means that

(3-6)

9 ^ [M), ^ = c j U - f V

where \P.(dz) is the Lebesgue measure on the circle of radious TJ. Lemma 3.2. {z n }£L 0 forms an orthogonal basis ofH2(i/q).

?

43 Proof. We adopt t h e same idea as in t h e proof 22 . (zn,zm)nl=

znzmvq[dz)

f

_

(g; g)oo

-

9J

V">

n+m

2n £j( 9 ;g)/>

f

*

pi(m-n)8M

/„

/3"(g;g)oo^g(ra+1)i

_ ,

= 5 n , m /3"[n] g !. Note t h a t we have used t h e q-Gamma function 2 ' 2 2 , (a- a)

^

0 ("+

1

)i

n Remark. It can be shown by Proposition 4.4 in the recent paper 6 t h a t vq is a unique measure satisfying ( z " , z m ) ^ 2 = SntTn/3n[n]q\. Hence, for any F = ^^L0anzn, product {-,-)-H* i s written as

G = £ ^ = 0 bnzn

E 7i2(uq),

t h e inner

00

(F,G)nl

= J^(3n[n}q\anbn

(3.7)

71 = 0

and t h e corresponding norm of F is 00

ll^ll«> = £ 0 > ] « ! l a » l 2 -

(3-8)

71 = 0

4

Main Results

P r o p o s i t i o n 4 . 1 . 5 (,) is a unitary

map of L2(nP

) onto 7i

{yq).

Proof. As we have seen in L e m m a 3.1, (S,q)cM)(z)

= z".

(4.1)

44 In addition, we derive by L e m m a 3.2, l|Ci' , Hl, ( | .«., ) = l l * - | | a K J = / ' - Therefore, we finish the proof.

•

Let us define operators Zq and Dq in 7i2(vq) ZqF{z)

satisfying

= zF{z).

(4.2)

Dq,pF(z):=m*){i-_FJ}qz)).(z?0)

(4.3)

and Dqif,F(z):=0(z

= 0),

Operators Zq and Dqp play the roles of t h e q-creation operator and qannihilation operator respectively and satisfy the g-deformed commutation relation Dq^Zq — qZqDqp = / . T h e q-number operator acting on H 2 (z/ g ) is defined by NqF{z) In addition, the operator a^

n > 0.

(4.4)

acting on K 2 ( f g ) is defined by

aNqF(z) Remark t h a t a^qF{z) calculation, we have L e m m a 4 . 2 . (1) S

= [n}qF{z),

= ([n], + 0)F{z),

n > 0.

(4.5)

= 0 for ^-Gaussian case, see Appendix A. By the direct U)l

UP

= 1

(2) D, i / 5 z» = (3[n]qz ra-1 (3) Zq n

=

zn+l

T h e transformation of t h e multiplication operator Qp satisfies the following relation. T h e o r e m 4 . 3 . S MQpq) = (Dq,p + Zq + SN)SM Mp

by x in L2(fip

)

Mp

Proof. By t h e recurrence formula (2.1), Equation (4.1) and L e m m a 4.2, we derive (S
= (E^hx^cH,

+ / 3 [ n ] , C < £ 1 + ([n], + / 3 ) C ^ ) i

n Moreover we obtain

45 C o r o l l a r y 4 . 4 . The operators Zq,Dq,Nq,a^ (1) SNq

= \ZqDq#

(2) Zq + Dqfi

+pi

=

have the following

•properties:

Nq+/3I

+ aNq = {-feZq + ytfl)

{-±Dq,p

+ y^J) -

Proof. T h e proof is done by Equations (4.2), (4.3), (4.4), (4.5), and L e m m a 4.2. • Therefore, by Theorem 4.3 and Corollary 4.4, the multiplication operator Q{pq) by x in L2(fj,p ) is represented as the sum of the q-creation, qannihilation, g-number, and scalar operators in the Hardy space %2(yq). In the g-deformed Gaussian case, due to Equation (A.4) in Appendix A, t h e multiplication operator Qg in L 2 (/ii 9 ) is represented as a linear combination of g-creation and q-annihilation operators in t h e "same" Hardy space "H2(vq). Remark, (1) T h e analogous results in this paper to q = 1 have been considered by Asai, et al. 6 . (2) In general, if variances of two given measures fi\ and fi2 are the same, t h e n the actions of creation and annihilation operators are the same. In addition, the transformed function spaces by 5 M l and S^ are also the same. However, if /Uj is non-symmetric and /X2 is symmetric, t h e n the representation of multiplication in L2(fi\) is different from t h a t in L2(fj,2). Appendix A

On q-Hermite Polynomials

First of all, we refer to the p a p e r s 1 2 ' 2 2 and references cited therein for the detailed description of q-Hermite polynomials. For 5 = 1 , see b o o k s 1 4 , 2 1 , 2 7 . Let fig b e the g-deformed Gaussian measure with mean zero and variance j3 > 0. It is well-known t h a t an associated orthogonal polynomial to fig is the g-Hermite polynomial {Hn (x)} with the Szego-Jacobi parameters an = 0,w„ = /3[n] q . In this case the following relations hold for each n > 1: xH^(x) = H^^+fSln^Ux) where H{0g) = 1 and H[q) g(x)

=

Y,KH(nq\x)

= x. Then for any f(x)

(A.l) = £ anH(nq) (x) and

in L 2 ( / 4 ? ) ) , the inner product (•, •) 2 . (,>. is given by the

form

(/>S)L3(/Iu>) = J ] / 3 n [ n ] g !a„b n .

46 Moreover, let H / H ^ , , , , = A q-deformed

Z7=oP>]A«n\2•

coherent state vector E^(x,z)

for {if„ (x)} is defined by

*S(*.*) = E ^ r * " . *enf. n=0

P

n

It can be shown t h a t the set {E^g(x,z) 2

q

and total in L (^ig ').

(A.2)

"

: z G £1^} is linearly independent

T h e g-analogue of the Segal-Bar g m a n n transform S (,)

9)

asso d a t e d to { i f i ( z ) } is given by (S^f)(z)

for any / G L 2 ( ^ ) .

= {E^g(x,z),f(x))L2{^)}

(A.3)

W i t h this transform, we can reproduce t h e same results as for q G [0,1) Theorem III.4 in Leeuwen-Maassen 2 2 and for q — 0 Proposition 1 in Biane 9 . T h a t is, S (,)-transform yields a unitary isomorphism between L 2 (/ig ) and U2{vq)

and S

^Q(9q)

= (D^

+

Z

i )

S

^

(A.4)

where Qg is the multiplication operator by x in L2(fj,g ) . Acknowledgments T h e author is grateful for a Postdoctral Fellowship of the International Instit u t e for Advanced Studies, Kyoto, J a p a n . He also thanks Prof. Bozejko for his comments. References 1. L. Accardi and M. Bozejko, Interacting Fock space and Gaussianization of probability measures. Infinite Dimensional Analysis, Q u a n t u m Probability a n d Related Topics, 1 (1998), 663-670. 2. G. E . Andrew, R. Askey and R. Roy, Special Functions. Cambridge, 1999. 3. N. Asai: Analytic characterization of one-mode interacting Fock space. Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics, 4 (2001), 409-415. 4. N. Asai, I. Kubo, and H.-H. Kuo, Roles of log-concavity, log-convexity, and growth order in white noise analysis. Infinite Dimensional Analysis,

47

5.

6.

7. 8. 9.

10. 11.

12.

13.

14. 15.

16.

17. 18. 19.

Q u a n t u m Probability and Related Topics, 4 (2001), 59-84. Universidade da Madeira CCM preprint 37 (1999) N. Asai, I. Kubo, and H.-H. Kuo, General characterization theorems and intrinsic topologies in white noise analysis. Hiroshima M a t h . J., 3 1 (2001), 299-330. LSU preprint (1999) N. Asai, I. Kubo, and H.-H. Kuo, Segal-Bargmann transforms of onemode interacting Fock spaces associated to Gaussian and Poisson measures, preprint (2001), to appear in Proc. Amer. Math. Soc. V. B a r g m a n n , On a Hilbert space of analytic functions and an associated integral transform, I. Comm. P u r e Appl. Math., 14 (1961), 187-214. V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, II. Comm. P u r e Appl. Math., 2 0 (1967), 1-101. P. Biane, Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal., 1 4 4 (1997), 232-286. M. Bozejko and R. Speicher, An example of a generalized Brownian motion I. Comm. M a t h . Phys., 1 3 7 (1991), 519-531. M. Bozejko and R. Speicher, An example of a generalized Brownian motion II. in Q u a n t u m Probability and Related Topics VII, L. Accardi (ed.) World Scientific, 1992, pp. 67-77. M. Bozejko, B. Kummerer and R. Speicher, q-Gaussian processes: Noncommutative and classical aspects. Comm. M a t h . Phys., 1 8 5 (1997), 129-154. W. G. Cochran, H.-H. Kuo, and A. Sengupta, A new class of white noise generalized functions. Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics, 1 (1998), 43-67. T. S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, 1978. T. Dwyer, Partial differential equations in Fischer-Fock spaces for Hilbert-Schmidt holomorphy type. Bull. Amer. M a t h . Soc. 7 7 (1971), 725-730 L. Gross P. and Malliavin, Hall's transform and the Segal-Bargmann map. in: Ito's Stochastic Calculus and Probability Theory, M. Fukushima et al. (eds.) Springer-Verlag, 1996, pp. 73-116. R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions. Comm. M a t h . Phys., 9 3 (1984), 301-323. Y. Ito and I. Kubo, Calculus on Gaussian and Poisson white noises. Nagoya M a t h . J., I l l (1988), 41-84. I. K u b o and H.-H. Kuo, Finite dimensional Hida distributions. J. Funct. A n a l , 1 2 8 (1995), 1-47.

48 20. I. K u b o and S. Takenaka, Calculus on Gaussian white noise I, II, III, IV; Proc. Japan Acad. 5 6 A (1980) 376-380, 5 6 A (1980) 411-416, 5 7 A (1981) 433-437, 5 8 A (1982) 186-189. 21. H.-H. Kuo, White Noise Distribution Theory. CRC Press, 1996. 22. H. van Leeuwen and H. Maassen, A q deformation of the Gauss distribution. J. M a t h . Phys., 36 (1995), 4743-4756. 23. N. Saitoh and H. Yoshida, The q-deformed Poisson random variables on the q-Fock space. J. M a t h . Phys., 4 1 (2000), 5767-5772. 24. N. Saitoh and H. Yoshida, The q-deformed Poisson distribution based on orthogonal polynomials. J. Phys. A: Gen., 3 3 (2000), 1435-1444. 25. I. E. Segal, Mathematical characterization of the physical vacuum for a linear Bose-Einstein field. Illinois J. Math., 6 (1962), 500-523. 26. I. E. Segal, The complex wave representation of the free Boson field, in: Essays Dedicated to M. G. Krein on t h e Occassion of His 70th Birthday, Advances in Math.: Supplementary Studies Vol.3, I. Goldberg and M. Kac (eds.) Academic, 1978, pp. 321-344. 27. M. Szego, Orthogonal Polynomials. Coll. Publ. 2 3 , Amer. Math. S o c , 1975. 28. D. Voiculescu, K. J. Dykema and A. Nica, Free Rondom Variables. C R M Monograph Ser. 1, Amer. M a t h . S o c , 1992.

Quantum Information IV (pp. 49-78) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

N O T I O N S OF I N D E P E N D E N C E IN ALGEBRAIC PROBABILITY THEORY A N D SET PARTITION STATISTICS

YUKIHIRO HASHIMOTO * Graduate

School of Mathematics, Nagoya University Chikusa-ku, Nagoya 464-8602, Japan

A non-commutative analogy of the combinatorial probability theory is studied. We investigate roles of singletons in the set partition statistics which appear in the non-commutative probability theory, and introduce general notions of statistical independence. To clarify structures of the set partition statistics, we introduce an idea of non-commutative decomposition of discrete random walks, which is a sort of discrete time analogy of Quantum Ito theory.

1

Singleton condition and set partition statistics

An algebraic probability space (.A, ip) is a pair of a *-algebra A and a state

C . Let us take another *-algebra B. A *-homomorphism J : B -4 A is called an algebraic random variable associated with a sample algebra A and a state algebra B. Here and then we restrict ourselves in the category of unital *-algebras and t h e n t h e algebraic r a n d o m variables are unital *homomorphism. A stochastic process on an algebraic probability space is a one-parameter family {Jt} of algebraic r a n d o m variables. Throughout this paper, we consider a stochastic process {Jn} with discrete parameters n G N . As long as we fix a set of generators {6^''}jgN C B it is sufficient to consider the set {a„ := Jn(b^) | n, j £ N } instead of {Jn}. Thus we may identify {(a„ )^Lj | j = 1, 2, .. .} with t h e discrete process. D e f i n i t i o n 1 2-3 (1) A finite or countably infinite set of sequences {(a„ )£Lj \ j = 1,2,...} of elements in A is said to satisfy the singleton condition with respect to ip if t h e factorization ¥>(«&>••• a<£>) = ?{<#?) occurs for any choice of j \ , . . . , j

m

•

• • • a £ > • • • a < £ > ) E N (TO > 1), and m,...,nm

(1) £ N

with an index ns different from all other ones. Here &«/ stands for the omission of o„ s . *JSPS RESEARCH FELLOW

50 (2) A stochastic process {Jn} is said t o satisfy t h e singleton respect to ip if t h e factorization

condition

with


• V (Jni(b(1))

• • • Jn,(b{3))

• • • Jnm(b{Tn)))

(2)

holds for any choice of b^\ ... , 6 * m ' £ B, and n\, • • • , n m £ N with an index ns different from all other ones. (3) We say sequences (a„ )^L 1 , (a„ ) , . . . of elements of A satisfy t h e condition of boundedness of the mixed momenta if for each m E N and ji, • • • ,jm there exists a positive constant Bm(ji,... ,jm) > 0 such t h a t

K a ^---^rOI- B m ( i l '---' i m )

(3)

for any choice of n j , . . . , nTO £ N . Note t h a t by setting a„ :— Jn{b^'), (1), while t h e converse is false.

the singleton condition for Jn implies

For a sequence aS*> = (a„ )^Lj C A, we p u t

S„(a«>) = £ > « > .

(4)

n=l

We shall discuss t h e partition statistics associated with {Sjv( a scribing t h e mixed momenta of S J V ( O " ' ) ' S
N<*

'''

J\[a

J'

)}• For de-

( >

we introduce t h e following notations. N o t a t i o n 2 For m £ N , we define a set of equivalent classes P(m)

= {(W1,...,Wm)\l<Wi<

m}/6m

where (W1,W2,..., Wm) ~ (W[,W'2,..., W'm) if and only if there exists a p e r m u t a t i o n 9 E G m satisfying 6(Wi) = W'{. \W\ ... Wm] stands for an equivalent class. For T = \W\ • • • Wm\ £ P(m), c(T) denotes t h e number of alphabets in T different from each other, t h a t is, c(T) \— #{Wi,..., WVn}Then we may take a m a p p : { l , . . . , m } —» { 1 , . . . , c(T)} characterized by p(j) = P{k) if and only if Wj = Wk, which give rise to a partition J J ^ l j Sj of t h e set { l , . . . , m } by putting Sj := {fc | p(k) — j}. Since there is an ambiguity on t h e suffix of Sj, we choose t h e m a p p uniquely determined by

51 t h e property mm Sj < minS*. if and only if j < k. PP{2m) denotes the set of equivalent classes with length 2m which consist of m-pairs, t h a t is, the set of all pair-partitions ]J"L1 Sj of the set { 1 , . . . , 2 m } , #Sj = 2 for j = 1 , . . . , m . For an m-tuple (ni,..., nm) £ N m , we construct a m a p p' : { 1 , . . . , m} —> { 1 , . . . , m } defined by p'(j) = min{fc | n*. = rij}, which has a property p'{j) = p'(fc) if and only if rij = n*.. Then we associate an equivalent class T = [p'(l) • • • p'(m)] to each m-tuple ( n i , . . . , n m ) , denoting T = [ ( n 1 ; . . . ,n TO )] for short. We put for T £ P(m), TN = {(n1,...,nm)&{l,...,N}m 2

\

[(nu...,nm)]=T}.

L a w of large n u m b e r s , c e n t r a l l i m i t t h e o r e m s a n d e n t a n g l e d ergodicity

One can deduce some fundamental results from the singleton condition. Here we just mention the results and omit the proofs. See Accardi-HashimotoO b a t a 2 ' 3 for details. L e m m a 3 2 ' 3 Let (<x„ ) ^ i , (a„ ) ^ L l 5 . . • be sequences of elements of A satisfying the condition of boundedness of the mixed momenta. Then, for any 0
lim -J—

fU^)---a{^))

Y

=0

(6)

(ni,...,nm)6TN

holds provided that c(T) <

am.

2,s

Lemma 4 Let {(ajj ) " = 1 } be sequences of elements A with mean 1/2 or if a = 1/2 and m is odd, we have lim

ip

—

•••

—

(ii) In the case of a = 1/2 and m = 2p, h m ip '

JV-+00 NP TePP(2p)

(n 1 ,...,nj,)eT J V

0.

(7)

52 holds in the sense that one limit exists if and only if the other does and the limits coincide. Moreover, the following Gaussian bound takes place: lim sup

5jv(a(il))

SV(a(i2p))

< ^ B

2 p

.

(9)

JV->oo

D e f i n i t i o n 5 5 A discrete stochastic process Jn : B —} A is said to be weakly stationary with respect to tp if the distribution tp = tpoJn is determined independent of n. T h e following is a non-commutative version of t h e law of large numbers. T h e o r e m 6 5 Let (A, (p) be an algebraic probability space and Jn : B —» A be a stochastic process. Suppose that {Jn} *s weakly stationary and satisfies the singleton condition (2) with respect to 0 such that \^(Jni(b^)---Jnm(b^))\
,™,(P(M^1 where a „

n1,...,nmGN

non-commutative

(10)

polynomial P

M p ) ) = w < . > , , . . . , « (11)

= «/„(&"') and tp ~ ip o Jn.

In t h e next, we observe t h e n a t u r a l role of t h e non-crossing partitions in the proof of the existence of the limit (8) and show how this naturally leads to t h e idea of entangled ergodicity. D e f i n i t i o n 7 Let U = 1 Sj b e a partition of { 1 , . . . , m) a n d p u t Sj = min{s £ Sj}, Then U j = i Sj other,

ls

called non-crossing

Ui, si) n 5 j = 0

lj = m a x { s £ Sj}. if for any i, j £ { l , . . . , m } distinct each

or (SJ, SJ) n 5i = 0

(12)

holds. Here (a, b) stands for the set {a, a + 1 , . . . , b}. T h e block Sj is called a non-crossing block of a partition if (12) holds for any S{, i ^ j . T h e partition I I j = i Sj is called totally crossing if there is no non-crossing block.

53 Suppose t h a t the process {J„} satisfies the singleton condition. Then, for non-crossing pair partition T = ]]^=i{hk, Ik} 6 PP(2p), the factorization property (2) shows t h a t t h e limit

JT^

E


(13)

(n1,...,n,p)eTN

is reduced to t h e limit of usual ergodic average:

,&, * (h t J-^(fcl,6(,i))) •••*{*£

J hp)b{lp))

^

) •

However, due to the non-commutativity we need the existence of the limit for entangled partitions. D e f i n i t i o n 8 2 ' 3 A process { J n } is said to have t h e property of entangled ergodicity with respect to tp if for any m € N , any choice of fe'1',..., b^m' G B and any totally crossing partition T £ P{m), c(T) > 2, t h e limit

,£LNm


T, («l

(H)

nm)GTJV

exists.

T h e o r e m 9 ' Suppose that a process {J n } satisfies the singleton condition. Let (b^) be a sequence ofB with ip{Jn(bU))) = 0 for any n and j , satisfying uniformly bounded condition (10). Put a„ := J „ ( 6 ' J ' ) . Suppose that the mean

covariance lim 1 N—Kx> i v

exists for any i,j.

Then the

f>(Jn(6W6<>)))

(15)

z — ' n=l

limit

b,f«...«)

(16)

exists if and only if {Jn} satisfies the entangled ergodicity. C o r o l l a r y 1 0 2 ' 3 Suppose that a process {Jn} satisfies the singleton condition. Let (b^>) be a sequence of B with )) = 0 for any n and j , satisfying uniformly bounded condition (10). Put a„ := Jn(b^'). The central limit theorem holds if any one of the following conditions is satisfied: (i) (q-commutation

relations) for each i,j 6 N , i ^ j , there exists a complex

number qij such that a\^'a„

= qijOn'am

for

any m,n

£ N;

54 (ii)

(symmetry)

v(«{j)-aLJi:,) = *'(«gi)-«fei,). holds for any injection

(17)

6 : N —» N ,

(Hi) (pair partition freeness)
Set partition statistics on discrete groups

Let G be a discrete group generated by £ : = { B(Z 2 (G)) b e the left regular representation. Let AQ b e t h e *-algebra generated by 7r(G) with *-operation (an(g) + bTr(h))* = a7r(g _ 1 ) + b-K(h~l) (a, b £ C , g, h £ G). In this section we take a vacuum state <j>(g) :— (0|7r(p)|0) on AQ where |0) € l2(G) stands for t h e characteristic function 5e of t h e unit e £ G and consider the pair (AG,4>) a s a n algebraic probability space. T h e *-linear extension of a group homomorphism Jn : Z / 2 Z 3 a i-> <x„ £ G is taken for an algebraic r a n d o m variable associated with a sample algebra AQ and state algebra B :— (Z/2Z) < r . a j K e j 3 r a . An algebraic central limit theorem describes a limit behavior of rescaled sum of algebraic r a n d o m variables with mean 0 and variance 1, SN = -j=(ir(cri)

+ ...+

7r(CTjv)).

Note t h a t a family of subsets £ W := {
= lim 4>(S%)

55

lim

—•==-—

>

d>(o-i,ai

Kz,

= lim

*

im
#TN.

V

n->oo C\//V1TO

^

We call { M m } combinatorial moments. Let A& be a fc x fc-matrix given by (Afc)ij = Mi+j, 0
<

(18)

then there exists a unique distribution p.G on real line of which moments coincide with {Mm}. For a word (cr^, • • • ,cr i m ) e S TO , we define a m a p p : { l , . . . , m } —> { 1 , . . . , m } by p(j) = min{fc | i*. = ij}. Note t h a t p ( j ) = p{k) if and only if ij = ifc. Then we associate an equivalent class T = [p(l) • • • p(m)] to each word ( u i j , . . . , <J;m). We use an abbreviation T = [cr^ • • • o~im]. If a product o"ij • • • <7;m is reduced to the unit e, we say the word (ff^,..., .

then X satisfies the singleton

condition

PROOF. If a product a^---aik includes a singleton u ^ , we have °»i ' ' ' °~ik T^ e - Otherwise we have o~is = o~itl • • • er^ • N . If the generators are symmetric t h e n one has PG(N)(m) = PQ(ITI) for N > m and the combinatorial moments {.MTO} are given by M m = lim

^

_ *

T&Pa(m)

\V1S

# T w

(19)

>

T h e o r e m 1 4 7 Let G be a discrete group generated by symmetric minimal generators {o~i}. Then all combinatorial moments Mm exist and are given by M2m

= #PPG(2m)

and M2m+1

= 0.

(20)

56 P R O O F . Take a closed word ( t r ; , , . . . , o-im), t h e n 0 ( 0 ^ • • • a ; m ) = 1 holds. Since S is minimal, it follows from L e m m a 4 and the symmetry assumption t h a t for odd m we have M m = 0, a n d for even m = 2p t h e right h a n d side of (19) equals to

J™,

n->oo

E

±W» = #PPa{2P).

•<—' A^P T€i>P 0 (2p)

Here are two typical results in algebraic probability theory. • Let A be a direct sum of Z / 2 Z with minimal symmetric generators: A = © t ^ 1 Z / 2 Z e i . It is easy to see t h a t PPA(2m) = PP(2m), and hence (2m)' M 2 m = ^— -^ 2mm!

and M 2 m + 1 = 0.

Then the central limit distribution for (A, {e;}) under the vacuum state

=

1 _i J .—e 2 1 dx. \j2ir

• Let F be a free product of Z / 2 Z , F = * ^ = 1 Z / 2 Z / i . It is shown t h a t P P i r ( 2 m ) is a set of non-crossing pair partitions of { l , . . . , 2 m } , and hence M 2 m = #PPF(2m)

=

^ } and M 2 m + 1 = 0. (771 + l)!m!

(See e.g. 1 4 .) Then t h e central limit distribution for (F,{f{}) under the vacuum state equals to the normalized semi-circle distribution 2 = — „ -X[-2,2]V4^a: X [ - 2 , 2 ] V 4 - a ; 2cd x

dfiF(x)

We shall discuss these examples again in section 8.1 from the viewpoint of distance regular graphs. Here we only note t h a t these examples are extremal in the discrete group category. T h e o r e m 1 5 7 ' 4 Let G be a discrete group generated by symmetric minimal generators {<X;}. Then the moments of limit distribution satisfy

(2m)!

<M ! m ( G ,K},<S' ~ 2 m\

(m + l)!m! ~ Therefore condition.

the sequence

{Mm(G,

v

L

J

{&{})} satisfies

m

the Carleman's

<»U uniqueness

57 P R O O F . By t h e universal property of free groups, there exists a canonical surjection IT : (F, {/<})-* (G, S ) with n(fi) = a{. T h e n P P F ( 2 m ) C PPG(2m) holds for all m , hence ( 2 m ) ! / ( m + l)!m! = # P P F ( 2 m ) < # P P G ( 2 m ) = A^2m(G). On t h e other hand, P P o ( 2 m ) is a subset of P P ( 2 m ) a n d one sees # P P ( 2 m ) = ( 2 m ) ! / 2 m m ! . Hence t h e inequality follows. Since t h e moments { ( 2 m ) ! / 2 m ! m ! } fulfill t h e condition (18), t h e rest of t h e assertion follows. | Remarks. (1) T h e results in this section are valid with slight modification in case where all generators are of order more t h a n 2 (cf. Hashimoto 7 ). In t h a t case, one needs to consider { S V } a s

SN =

7ilv ( 7 r ( C T l ) + 7r(ffl"1) + "'+ *^ x

+ 7r(a 1))

^ -

1

Then a 'pair' in a product means t h a t a\ • • • a^" is given by ( c i p , o^*) with ip = iq and ep = — eq. T h e free abelian group A = © ^ Z and t h e free group F = *°°Z play t h e same extremal role as stated in T h e o r e m 15. (2) T h e concept of minimal generator is extended t o t h e category of monoids, see Accardi-Hashimoto-Obata 4 for details. (3) Lu 1 5 ' 1 6 a n d M u r a k i 1 7 independently introduced a new notion of independence in algebraic probability theory, called chronological or monotone independence, where t h e symmetric condition on 'generators' breaks down. T h e n we fail to apply our result to a sequence of monotone independent random variables. Indeed, the limit distribution associated with a sum of monotone independent random variables with respect to a vacuum state coincides with t h e arcsine distribution, of which combinatorial moments are determined by a certain set of pair partitions which have much smaller cardinality t h a n t h e set of non-crossing pair partitions. (4) In general, we face h a r d combinatorial problems to determine combinatorial moments associated with (G, {c;}). If we construct a suitable space on which G acts, there is a possibility to determine combinatorial moments by observing t h e action of G. We illustrate such a m e t h o d in the following example. We forget t h e order 2 assumption on generators here. For a fixed k G N , let A/"'fc' be the normal subgroup of infinite free group F = *ZtZfi generated by {(fhfl2 • • • A i ) f e i + 1 | i , i i G N } . We p u t (fc) (fc) 6 = F/jV a n d n = fiNik). We note t h a t t h e infinite p e r m u t a t i o n group Soo is a factor group of 6 ^ .

58 P r o p o s i t i o n 16

7

The generators

{T^} are symmetric

PPeW{2m)

=

Then, the central limit distribution with the normalized

semi-circle

and minimal,

and

PPF{2m).

associated

with ( © ' * ' , {i\})

coincide

law.

P R O O F . Let X b e a discrete space {0} U Z x {1, 2 , . . . , fc}. Define the action of Tj's on X by Ti(0) = (i,l),

Ti{i,k)=0,

Ti(i,a)

= (i,a + 1), for 1 < a < k — 1,

Ti{j,a)

= (j,a),

forj^j,

and T^~ 's on X inverse way. We see t h a t (T^TY, • • • Til)kl+1 acts on X identically for every / G N . Then t h e action of ©(*' on X is well defined. Suppose r / 1 • • • rf" = e and t h a t 7V is a singleton. Then r ? 1 • • • T?"'1 and T*"* 1 •••?£* leave (* p ,a) fixed, while T*" changes (i p , a ) . This is a contradiction. Hence the minimality of {TY} follows. By Theorem 14 we focus on a closed word (r,? 1 , • • •, T'*™ ) belongs to a pair partition. Choose a pair (TY , r t - s ) (p < g), i.e., r : = i p = iq and e q = —e p , such t h a t Tip+1,..., Ti , are different each other. We assume eq = 1 without losing generality. Then we have T"' • • • Tiq(r,k) — (iq-i, 1) and it follows t h a t

whenever q ^ p+1. By t h e assumption r ? 1 • • • Tpm = e, we need q = p + 1 . Repeat this argument, then we see t h a t the closed word ( T ? 1 , • • • ,T t £2m ) forms a non-crossing pair partition, and hence the assertion follows. | In the argument above, the action of S^ fc ' on X is used to exclude all kinds of pair partitions except non-crossing ones. This method is useful to determine t h e moments. 4

Singleton independence

Motivated by the study of central limit theorem for the Haagerup states on free groups 8 , we introduce an analytic generalization of the singleton condition as follows.

59 D e f i n i t i o n 1 7 2<3 Let A b e a *-algebra and let (a„ )™=1 b e sequences of A. Assume t h a t we are given a family of states y>7, 7 > 0, on A. T h e sequences (a„ ) is called singleton independent with respect to 0 such t h a t

l^(^ ) ---^r ) )l<^|^(^; , )|-|^(^ 1 l) ---^; ) ---^r ) )|

(22)

holds whenever n s is a singleton for {n\,... ,nm). An algebraic r a n d o m process Jn : B —t A is called singleton independent with respect to ip1 if for any m £ N and any choice of b^n\ .. . ,b^Jm' £ B, there exists a constant Cm := Cm(b^l\..., 6 ( j - ' ) > 0 such t h a t (22) holds for a„ : = J n ( & ^ ) whenever n s is a singleton for ( n x , . . . ,nm). We assume t h a t there exist constant numbers (3 > 0, n > 0 and A > 0 such t h a t sup \Vy(a^)\

< / •

(23)

j'.nSN

Then in the case of 7 = 0, the condition (22) is reduced to the singleton condition. (Here we put 0° = 1.) Our goal is to study the partition statistics in t h e limit associated with mixed m o m e n t a

= &,Wz;

E TeP(m)

where j(N)

W)(< ) -"-£ ) )'

E

is a function of N satisfying lira -y(N)NK

= X.

(25)

L e m m a 18 Let s(T) denote the number of singletons P{rn). It holds that for any partition T £ P(rn),

in the partition

m + s(T)>2c{T),

For a partition T = L I ^ Sj £ P(m),

T £ (26)

and the equality holds if and only if T consists of pairs and PROOF.

(24)

(n1,...,Tim)€TJV

singletons.

we have

c(T)

™ = E *si = S(T) + E 3=1

*Si ^ 2«T) -
#Sj>2

T h e equality holds if and only if # 5 j = 2 for all j .

I

60 L e m m a 19 Suppose that sequences (an ) satisfy the condition of boundedness of mixed momenta (3) uniformly on 7, and are singleton independent. If a partition T £ P(m) satisfies (27)

am + (3Ks(T) > c(T) then we have the

limit 1

tLN^ PROOF. inequality

T h e boundedness (3) and repeated application of (22) yields an

where Dm : = Bm_s^CmCm^1

N°

*7<*)(°ii , -«& ) )=°-

£

(nlt...,nm)eTN

• • • C m „ s ( T ) + 1 . Hence we have

E (ni

^(^•••W)

< Dr

nm)€Tjv

\^T\c{T))\( N<*m+pKs(T)

N \c(T)J'

T h e assertion then follows from the condition (27). From Lemmas 18 and 19, we see t h a t a partition T £ P(m), tributes to the limit (24) if the inequality am

+ (3KS(T)

< c(T)

< -(m

+

m > 0, con-

s(T))

holds. ,U) ) satisfies the boundedness condition (3) Suppose t h a t t h e sequence (a„ uniformly on 7 and
(ni,...,nm)€TAr |c(si) . . .

< C

m

— o«

• • • (7m_s_|_i -

a+l|c (

c

(s»)

J\[am. s

i ) . . . >

AT

•7(<sc(r)!^(r)J|^(^(T')| A^c(T)! ( * \ I JSJam+pKs \ C ( T )

y { N ) { r )

\

(28)

61

Here we put

+ (3KS{T)

(29)

holds. Only the partition that holds the equality in (29) contribute to the limit. Suppose that m = 2p > 0 and take a partition T £ PP(2p). Then c(T) = p and s(T) - 0. It follows from (29) that a > 1/2. Suppose that a partition T e P(m) holds the equality in (29). It follows from Lemma 18 and a > 1/2 that 0 < ( i - a)m + ( 1 -

< (^ - /3«) a (T).

/3K)S(T)

(30)

Suppose that a > 1/2, then the partition T contains a singleton, that is, s(T) ^ 0, otherwise the inequality (30) breaks down. Thus we have /3K < 1/2. Since s(T) < m, we have 0 < ( - - a)m

+ ( - - (3K)S(T)

< (1 - a - (3K)TTI,

hence a+/3K
(31)

Take a partition T G P{m) with c(T) = s(T) = m. It follows from (29) and (31) that a + /3«= 1. Suppose that a = 1/2. Again take a partition T £ P(m) with c(T) = s(T) = m. From (29) we have j3n > 1/2. If j3n > 1/2, s(T) - 0 holds from (30), and then we see from Lemma 18 that only pair partitions may survive in the limit. Let us consider the converse. Suppose that a > 1/2 and a + (3K = 1. We see that for T € P(m), am + PKs(T) - V1+AH

=

(Q _ ^

m +

( / 3 K - i ) s{T)

>(a + /3K- l)m = 0 since s(T) < m and /3K, < 1/2. Then Lemma 18 shows the inequality (29). The equality in (29) holds if and only if c(T) — s(T) = m. Suppose that a = 1/2 and (3K, > 1/2. Combining with Lemma 18, we have the inequality (29). The equality holds only for the partitions T e P(m) with s(T) = 0 and m — 2c(T), that is, the pair partitions. Finally, suppose that a = (3K = 1/2, Lemma 18 shows (29) and its equality holds only when the partition T consists of pairs and singletons. In summary, we come to the following limit theorem.

62 T h e o r e m 2 0 Assume that sequences (a„ ) C A are singleton independent and satisfy the boundedness condition (3) uniformly, with respect to a family of states ipj, 7 > 0. Let j(N) = \/NK and suppose that for each j there exists C-i' G R such that y> 7 (a„ ) = c^'yP for any n. Then for any m, the mixed moment (24) has non-zero finite limit if one of the followings holds: (i) a+Pn = 1 and a > 1/2. In this case the partition T G P(m) may survive in the limit only if c(T) — s(T) = m, that is, T consists of singletons only. Then the mixed moment (24) of order m is given by

(nu...,nm)ETN

where T E P(m) with c(T) = s(T) = m, that is, the partition singletons only. The moment is bounded from above by

consists of

C1--Cm\c^---c^\X'3m. (ii) a = 1/2 and /3K > 1/2. In this case the partition T G P{m) w a y survive in the limit only if T is a pair partition. The mixed moment (24) of order 2m is given by

^

^

E

E

TePP{2m)

¥W«4i>•••«)•

(ni,...,n, m )eT J V

( m ) a = (3K = 1/2. In this case the partition T G P{in) "rnay survive in the limit only if the all blocks Sj contained in T satisfy # 5 j < 2, that is, the partition consists of pairs and singletons only. The mixed moment (24) of order m is given by

Ji*]^E

E

3=0 TePP[m;s)

E

^,(a£-a£)),

(n 1 ,...,n m )6T A r

where PP(m; s) C P(m) stands for the set of partitions and s singletons. 5

consists of pairs

A limit t h e o r e m on t h e Haagerup states

T h e notion of singleton independence is found in investigation of t h e Haagerup s t a t e on a free group, T h e Haagerup s t a t e is a non-trivial instance t h a t fails to satisfy t h e singleton condition. Here we give a summary of t h e studies 2 ' 3 on combinatorial aspects of t h e Haagerup state.

63 Let F be the free group on countably infinite generators £ = {gn \ n £ Z x } , where gr_n stands for g~*. Denote by AF t h e group *-algebra of F and for each n £ N by An the *-subalgebra generated by {g±n}- For 0 < 7 < 1 we denote by y>7 the Haagerup state defined by
jef,

(32)

where | • | denotes t h e reduced length function, t h a t is, for g = g%- • • • g^™ with | n j | 7^ \n2\ 7^ • • • 7^ | n m | , its length |g| is given by | e i | + • • • + |em|- We set [e| = 0 for the unit e 6 F and 0° = 1. {gn} satisfy t h e singleton condition with respect to tp1 if and only if 7 = 0. In particular, the algebras An are not free independent with respect to <^7, 7 7^ 0. However we have the following. P r o p o s i t i o n 2 1 The sequence {gn | n g Z x } are singleton independent with respect to the Haagerup states tpy, 0 < 7 < 1. In view of t h e reduction gn • g_n = e, we attach a partition T £ P(m) to each index tuple ( o n , . . . , Q m ) in a slightly different way from Notation 2: for an m-tuple ( o ^ , . . . , am) £ ( Z x ) m , the partition T = [|Q 1 ; . . . , am\] £ P(m) is defined by

t=i

j=\

where a3i 7^ — a*, for any 1 < k < m and Sj's are given by the form Sj — {*i,. . . M I | a t l | = • • • = | a t | | and 3p,q such t h a t atp =

-atq}.

T h e factor aSi is called a signed singleton. A signed singleton aSi is said to b e inner if there is a block Sj such t h a t s < Si < SJ (see Definition 7), and is said to be outer if there is no such a block. i(T) and b(T) denote t h e number of inner singletons and the number of blocks which are not signed singletons, respectively. An m-tuple (QJ , . . . , am) is said to be separable at k if there is no block Sj such t h a t s < k < s~j. Note t h a t t h e existence of a n outer singleton Q 3 implies t h e separability at s — 1 and s. D e f i n i t i o n 22 Let NCPI{rn; s) C P(m) be the set of partitions which consist of non-crossing pair partitions and s inner singletons, t h a t is, an element T £ NCPI(ms) is given by

T = ]l{st}]l{p3,qj}, »=i

3=1

where s;'s are inner singletons and U j = 1 { P j , 9 j } forms a non-crossing pair partition.

64 T h e o r e m 23

2 3

'

For T G P(m)

with s inner singletons,

we have

( a 1 , . . . , a m ) € T J V VV^Vj

-A)s,

if

TeNCPI(m;s), (33)

otherwise. Here we put ga = ga — JHence for aN : = (<7±i + • • • + <J±N)/VN, the mixed momenta fx/VN^N '' ' aiV*) w^h £ = ( e i , . .. , e m ) , £{ = ± , in i/ie /imti is given by

YJ(-^Y-*NCPI{m-s]e). Here NCPI(m;s;e) tuple ( a 1 ; . . . , am)

denotes the subset of NCPI(m; s), that consists where the sign of each a^ equals to £{.

of m-

In view of Theorem 23, we construct a representation of the limit process for the Haagerup states on a GNS space. Let oo

oo

8

r(c)„ = c © 0 c " ( = 0 c ) , n=l

V = L,R

n—0

denote two copies of t h e full Fock spaces on C with free creations a+ and free annihilation av. Let ri = © „ n = 0 ^ m , n D e the free product T(C)i * T(C)R, t h a t is, its (m, n)-particle space rim,n is the complex linear span of the vectors {|i/i,.. .,vm+n) I #{j I Vj = L} = TO, # { j I Vj = R} — n} equipped with a scalar product given by {v\,

• • • ,VkWi,

• • • ivi)u

= <5(t/i,...,i/ fc ),(i/(,...,i/ 1 ')-

T h e creation operators L

: = aL

* 1 : rlm,n

~^ rlm+l,n

i

R

: =

1 * aR

'• iim,n

~^ " m , n + l

are given respectively by £ + | " i , •••,"*) = |L,^i,...,^fc),

R+\v!,...,vk)

= li?,!/!,...,^),

and the annihilation operators are given by t h e duals L — (L+)* and R = ( i ? + ) * . Let P : H —> M be t h e orthogonal projection onto H^Q, the subspace orthogonal to Tiofi = C|0), where |0) denotes the vacuum vector. P u t A~ = L+ + R - XP, where A > 0 is a constant.

A\

= L + R+ - XP,

65 T h e o r e m 2 4 2 ' 3 The limit process for (a^,aN, fx/\/N') That is, all its correlations in the limit are given by lirn^

Vv

*s represented on %.

^ ( a * > . . . » « " ) = <*, A ? • • • A'^)u

.

R e m a r k . We have seen t h a t t h e field operator a~^ + a~^ converges in the sense of correlations to A+ + A~ = (L+ + R+) + (L~ + R-) - 2XP. This means t h a t t h e field operator is decomposed into a s u m of 'creation' L+ +R+, 'annihilation' L~ +R~ a n d a function of t h e number operator, —\P. This decomposition leads us naturally to an idea of ' q u a n t u m decomposition' introduced in t h e next section. 6

Q u a n t u m decomposition in a discrete group

We here introduce a new idea of ' q u a n t u m ' decomposition of a graph Laplacian on a Cayley graph associated with a discrete group. To make t h e point clear, we focus on discrete groups satisfying condition (A2) below. More general situations are discussed in section 8.3 a n d 8.4. Let G b e a discrete group equipped with a length function | • | : G —t No : = N U {0} a n d a set of generators S : = {ga \ a 6 Z x } . T h e length of t h e unit e £ G is 0. Throughout this section we adopt t h e convention {g~a = 5 „ J } and assume g-a ^ ga for simplicity. W e say t h a t S is compatible with respect to t h e length function | • | if ( A l ) \ga\ = 1 for any ga € S , (A2) \ga -g\ = \g\±l

for any j

a

e S a n d j e G.

T h e property (A2) leads us t o t h e following idea: we take t h e left regular representation 7r : G -> B(Z 2 (G)). Let Tr(g^) € B(Z 2 (G)) be bounded operators denned by H9a)0g

|

Q ]

otherwise,

*\9a Pg

10>

9\ = \g\-h Qther otherwise .

Here 6g denotes t h e characteristic function of a singlet {g}. Then we have a ' q u a n t u m ' decomposition of ga; T ( S C ) = *(
a n d \\v{g£)f\\P

(34) < | | / | | | i for / € / 2 ( G ) .

66 A family of subsets S ( J V ) : = {ga \ \a\ = 1 , . . ., N} C £ gives a filtration of G: GW C G<2> C • • • C G, where G<w> is the subgroup generated by tfN\ For each g G G ^ put " f ' ( s ) ••= {(ft,,*) e E<"> x G<"> | * ( , £ ) * . = J 9 } , WL

W)

( 5 ) == {fa,*)

S £

W

x G<"> | n(g-)Sx

= 5g}.

We note t h a t n(g£)6x = Sg implies | 5 | = |as| ± 1 and #J_f\g) + #J_N)'(g) = 2N by assumption (A2). We also note t h a t for t h e unit e 6 G we have w^_ '(e) = 0 and w_ '(e) = E ' ^ ' by definition. W i t h these notations we assume (A3) for each n G N , there exist u „ 6 N and Cn > 0 such t h a t # { 5 € G<"> | H = n and # w f >(<,) /

w„}

< Cn(2iV)"-\

(A4) for each n G N , sup sup #wf](g) JveNgeGt^i.iff^n

=: Wn < oo,

and l i m s u p , , ^ , ^ W v " < oo. (A5) if |p| = n and 5 G G ( J V ) , there exists an n-tuple ( 5 Q I , . . . , p Q „ ) G ( £ W ) n such t h a t <59 = 7 r ( f i ' a 1 ) - - - T ( s f a „ ) ^ -

Canonical examples associated with a free group and a free abelian group are again observed in section 8.1. 7

Interacting Fock spaces associated w i t h a discrete group

For each subgroup G^N\ we construct a one-mode interacting Fock space as follows. See Hashimoto 1 0 for proofs. T h e vacuum vector is defined by |0) := 5e. A vector v G l2(G^N^) is called n-homogeneous if it has a form of v=

Yl

v(g)w(g)\0),

v(g) € C .

(35)

g€GW,\g\=n

D e f i n i t i o n 2 5 (1) Let T„ b e a one-dimensional complex vector space equipped with a pre-scalar product (z\w)W

:= zw{*(n)\*(n))W,

z,weC,

67 where ^ ( n ) } ' ^ ' stands for a number vector defined by

«»»"" := TMT. (v2JV)"

E |g| = n*WW.

ggG(i»),

with (w n )! = WiU}2 • • • U!n. T h e completion of the orthogonal sum ©n{r»i , ( - | ) n } is denoted by T(G^ '), which we call the interacting Fock space associated with G^N\ (2) T h e operators a+N := (n{g±„)

+ ••• + T T ^ ) + n(g+) + •••+

a^ := HgZN)

+ --- + Tr(gZ1)+A9i)

are called asymptotic

creation a n d annihilation

7r(g+N))/
+ --- +

H9N))/^N

operators

respectively.

(3) For A 6 C we define An

I^))W:=E7^1TI*H)(JV,I whenever (£(A)|£(A)) = £ ~ = o | A | 2 " / ( w n ) ! converges. We call | £ ( A ) ) W an asymptotic coherent vector. L e m m a 26 (1) For any

10

n>0,

a^|$(n+l))(JV»=a;„+1|n)W+Un(-i), where vn(Nm) 0{Nm).

is an n-homogeneous

vector

and

such that ||v„(A^ m )||;2 —

(2) For each n > 0 and iV > 1 it holds that

<*(n)|*(n))W = K ) ! + 0 ( l ) . (3) For A £ C m a neighbourhood

a~ |$(0)> = 0,

of 0 me have

1 a ^ | £ ( A ) > W = A | f ( A ) > W + W (l^ ) AT

68 (4) Let v G l3(GW) be an n-homogeneous vector. are respectively (n + 1)- and (n — 1)-homogeneous estimates: \\a+Nv\\2p < 2NWn\\v\\2P,

\\a-v\\2,

Then, a^v and aNv vectors with the norm

<

2NWn\\v\\22.

By virtue of lemma 26 we associate a one-mode interacting Fock space r ( G W ) with parameters {A0 := 1, A n := (w„)!} for each G^ . (See AccardiBozejko 1 for definitions.) Next, we construct an interacting Fock space for the limit of a~^ and a^ as N —> oo. Let T(G) denote the one-mode interacting Fock space with parameters {A0 : = l , A n := (w n )!}. By definition T(G) is the completion of the orthogonal sum of one-dimensional space Tn := C | $ ( n ) ) equipped with a pre-scalar product (z\w)n

:= zw($(n)\$(n)),

z,w £ C ,

where | $ ( n ) ) stands for a number vector with norm \ / A ^ = ^/(u,,)!. creation a+ and annihilation a~ are uniquely determined by a+|$(n)) = |*(n+l)>,

a~\9{n

+ 1)) = w B + i | * ( n ) ) ,

a

The

-|*(0))=0.

For A S C , we define a coherent vector

whenever (£ (A)|£(A)) = ^ ^ _ 0 l ^ | 2 " / ( w n ) ' converges. By definition, it holds t h a t a-\£{\)) = A|£(A)). For u = £ u n | $ ( n ) ) 6 T ( G ) , we put

*<">:= 5>„|*(n))< w )er(G< w >).

(37)

T h e o r e m 27 10Let m > 1 and e j , . . . , e m e { ± } . 77ien, / o r any u 6 T(G) and n 6 No we /iaue w

8

8.1

limJuW|a^...a^|*(n)>W(JV)) = (uK'.-af»|i(n))r(G).

Applications t o limit t h e o r e m s in algebraic probability theory Central limit theorems

on the vacuum

state

Since Theorem 27 is nothing b u t a general form of limit theorems in algebraic probability theory, it particularly leads to the algebraic central limit theorem

69 with respect to t h e vacuum s t a t e (0| • |0):

^ J ° K ^ ^ r | 0 C < " > ) = (^(0)l(«+ +^n*(0))r(G)-

(38)

Let A (resp. F) be a free abelian group (resp. a free group) generated by S : = {ga}, where g~l = g-a according to our convention. Each element g G A (resp. g £ F) has a canonical expression in terms of S: 9 = 9a\

•••Sa™.

where 0 < oti < a 2 < • • • < am, ej 6 Z (resp. \ai\ ^ \a2\ # • • • ^ |a TO |, £i € Z), and a reduced length function is defined by \g\ = |ei| H

h km|-

Conditions (Al)(A2)(A5) are obvious. As for (A3)(A4), we see for g G A (resp. gf G F) with |g| = n > 1, # « V % )
\g\ = n,#J_?\g)

< n} < ( 2 J V ) - 1 ,

2AT - n < #wL W ) (ff) < 2iV - 1, ( resp. # 0 , ^ ( 3 ) = L # « ^ ( f l ) = 2iV - 1). Hence we have w n = n (resp. w n = 1), which implies t h a t t h e interacting Fock space in the limit is the one-mode boson (resp. free) Fock space. 8.2

Limit theorems for Haagerup

states

In this paragraph, we again observe the limit process associated with the Haagerup s t a t e (32). Contrast to Theorem 24, we here give a fully q u a n t u m representation of the limit process. T h e result in Hashimoto 8 was thought as an analogy of a central limit theorem for a vacuum state, though t h e limit distribution h a d peculiar properties. In fact, t h e limit distribution coincides with the free Poisson law 2 3 u p to translation. It is noticeable t h a t limit behaviors of t h e field operators under a vacuum s t a t e are completely described by pair-partition statistics, as is seen in L e m m a 4, while the Poisson laws are induced from all set partition statistics 1 3 ' 2 1 ' 2 0 . Proposition 28 implies t h a t the Haagerup states give rise to a transform of set partition statistics through a coherent state expression of the Haagerup functions. See Hashimoto 1 0 for proofs. We note t h a t for any A G C with |A| < 1 a n d g G F^N\ we have

V^.(s) = (£(A)K(fl)|*(0))S^(N)).

70 It follows from Theorem 27 t h a t the moments of the field operator in the limit are given by

^

((^^) m ) = {£ma+

^

+--n*(0)>r(-)-

(39)

Note t h a t ( $ ( n ) | a + a - | $ ( n ) ) = 1 = ( $ ( 0 ) | P 0 | $ ( 0 ) ) holds for n > 1, where P 0 denotes t h e vacuum projection. P u t t i n g 4>o{A) = ( $ ( 0 ) | J 4 | $ ( 0 ) ) , we have a recurrence formula (S(\)\(a+

+ a - ) m | $ ( 0 ) ) = 0 O ((o+ + a~)m)

(40)

L(m-1)/2J

+

£

0o((a++a-)2i)0o(APo)-(f(A)|(a+ + a-)m-2(-1|$(O)),

1=0

while the expansion of (a+ + a~ + APo) m gives ({a+ +a~ + XP0)m)

= (j>{{a+ + a")" 1 )

[(m-l)/2j

+

J2

H(a+ + a~)2l)^Po){(a+ + a~ +

xp

o)m~21'1)-

1=0

By induction we then obtain the following. P r o p o s i t i o n 28 10In the free Fock space {T(F),{Xn

= 1}}, the

identity

m

<£(A)|(o+ + a - n $ ( 0 ) ) r ( F ) = (*(0)|(a+ + a " + A P 0 ) | * ( 0 ) ) r ( F ) holds, where Po is the vacuum Remarks.

projection.

(1) Since in t h e free Fock space the vacuum projection is given by Po = / — a+a~, the r a n d o m variable o + + a~ + APo is nothing b u t the free Poisson one 2 1 , t h a t is,

Q++a-+APo=A+I-(^--L)(v

/

Aa---L).

(2) Proposition 28 describes a Gaussian-Poisson transform in terms of coherent vectors, while Oravecz 1 9 recently introduced a Gaussian-Poisson transform called the T-transform for an arbitrary symmetric measure from t h e viewpoint of orthogonal polynomials. (3) W i t h t h e recurrence formula (40) one obtains a functional identity for the moment generating functions. P u t F(t) = 1 + J^^Li Fmtm with Fm = (£(A)|(a+ + a - ) r a | $ ( 0 ) ) and C(t) = (1 - \/l - 4t2)/2t2, which is

71 the moment generating function of t h e normalized semi-circle law. Then we have t h e identity F(t) = C(t) + XtC(t)F(t), from which the limit distribution is computed easily:

8.3

Multiplication

of free

elements

Proposition 28 implies t h a t under the Haagerup state, SN/V2N = n(gi) + ^ ( s r 1 ) " ! " ' •+ 7r (5'Jv)+ 7r (3Jv 1 ) *s decomposed into a creation a+, an annihilation a~ and conservation a~*~a~ in the limit. In this subsection we present a central limit theorem associated with a multiplication of free elements, and again we obtain the same decomposition as Proposition 28. Let F be a free product of Z / 2 Z , F = *£fLjZ/2Zaj, where cri's are generator of order 2. Note t h a t a^s are free from each other with respect to the vacuum state (0| • |0) in t h e sense of Voiculescu 2 3 . Let us consider a set of products of free elements S 2 : = {wij :— (TiO-j (i ^ j)}, which are not free from each other, and a subgroup F2 C F generated by £2- We use a new length function | • | 2 : F2 —> N 0 with respect to £ 2 different from the one introduced in Section 8.1. Since any product u>i Ul • • • wimjm has a reduced expression in terms of cr;'s, w

where k\ ?k;

i #

••• #

31 • • '

ii

h,

s e e 1for

i

m

• • •

any g =--crkl • • • °~km

Kjsh

j ,„

=

crfcl

• • • CTfc,,

its length is defined by \u>ii i i

Then we occur:

W

-I

w

i

(41)

„iJ2:=l/2£ F2 a n d

wij,

the following three cases

M2 + I, if J ^ h, M 2 - I , if (i,j) = (^2 , * i ) , \g\2,

if j = fci and i ^

(42) k2.

It follows from \wij\2 = 1 and (42) t h a t \g\2 £ N 0 for any
(43)

72 where g* are defined by (34) and g°a is given by *U<*>°9

=

/ <W if I

1\ 0,

otherwise. otl

Let us return to the example ( ^ , £ 2 ) . We P u * ^2 j < N}.

F2

denotes the subgroup generated by E 2 w

define J±\g)

F(2N),

by (35) where we take G< > for u < w , ( 5 ) := {(wijt

x) e S f

I 1 < * 7^

. For g £ F2 N)

and wi (g)

we

by

x F 2 W I * ( < £ ) * . = <5J.

By definition, we have u+ '(e) = u>0 '(e) = 0 and u>l unit e S Fi- It is easy to see t h a t for g 6 F2 N, #J+N)(g)

'•— iw'j

(e) = £ 2

(44) for the

with J£jr|2 = n > 1 and for large

= 1, #o,LW)(<7) = (W - I ) 2 and # W < w ) ( f l ) = AT - 2.

Also note t h a t # ^ ( 5 ) + #J_N)(g) + #u(0N\g) = #S< J V ) = N(N According to Section 7, we define a number vector

\y#S

2

7

- 1).

9 eF(^(2),| 9 | 2 =„

and define canonical operators corresponding to the decomposition (43),

where e = ± , o. Then by similar arguments in Section 7, we have a+|$(n))W = |*(„ + l ) > W + t , B + 1 ( l ) , a-N\(n + l))W a°N\*{n))W

=

= \*(n))(»)

\$(n))W+vn(±), + «„(!),

«JV|0)(JV)=a^|0)(JV)=0,

for n > 1, (45)

where vn is a homogeneous vector denned in Section 7, and t h e n we construct an asymptotic one-mode Fock space r(-F 2 ) associated with F2 . Let T(F) b e t h e one-mode free Fock space, a+, a~ and 0° : = PQ be the creation, the annihilation and the vacuum projection respectively, used in the previous subsection. It follows from (45) t h a t we have a ' q u a n t u m ' central limit theorem associated with (F2 , X 2 ' ) as follows.

73 T h e o r e m 2 9 Let m > 1 and e i , . . . , eTO 6 { ± , o } . Then for any u G and n S N o , we /iat>e

lim ( „ ^ ) |

-±L=

-..

- ^ =

^

T(F)

|*(n))<">. =•<">*

"-" ' V v ^ / v^/#sF ^"/'^(F^', = (u|a £ l • • • a £ m | $ ( n ) ) r ( F ) , where we use the notation theorem,

(37). Particularly

l

lim (01 ( ,

Y

we obtain a classical central

-K(Wij)Y\Q)(N\ a°)m\0)T{F).

= (0\(a+ + a~ + 8.4

Anti-commutation

of semi-circular

limit

elements

Throughout this paragraph, we use t h e notations in Section 8.3. By taking subsets of T,2 , we obtain various kinds of ' q u a n t u m ' decomposition. Here we observe one of the decompositions, which leads us an analysis of anticommutations of semi-circular elements. Let us take a subset £3,7 C ^2 for 0 < 7 < 1, S

V?

:

= iwH

e £ 2 * ' I 1 < * < m a x { l , jN}

<j
or 1 < j < m a x j l , yN} F\ ' denote the subgroup of F2 again obtain a decomposition

generated by Y,\

< i < N}.

. Since (43) holds, we

w

ii = wij + wij + wij-

(46)

According to the decomposition, we define operators

a£ :

(47)

" =-b E ^K)-

where e = ± , o and v :— # S ^ ' ~ 2^(1—f)N2. T h e purpose of this p a r a g r a p h is to establish a limit theorem on t h e operators (47) for a constant 7. General cases are discussed in Hashimoto 9 . It is easy to see t h a t for any g £ F2 , we have 2(7AT - 1)((1 - 7)N

- 1) < #u(N)(g)

< 27(1 -

7 )JV

2

.

(48)

74 However, to determine #w^_ (g) and #Wo (g), we need a delicate argument as follows. Let / := [1, yN] fl N and J := (jN, N] n N . For each n > 0, let us decompose the set V„ of elements g £ F 2 ' with \g\2 = n into 2 2 " disjoint subsets,

V„ := { 5 S F 2 ( ^ | \g\2 = n} =

]J

A ( m , . . ., r,2n)

m.—.»»n€{/,./}

where A ( r 7 i , . . . , »?2n)'s are given by A ( r h , . . . , r ? 2 „ ) : = {°"ii •••o-i 2 „ e V„ | ii e T j i , . . . , i 2 „ e % „ } . W i t h t h e notations above, we see #J^\g)

= 1, for g E A{I,J,m,

#wV V) (fl) = 0, for (1 - l)N 7

5

€ A(I,I,r)3,

N)

- 1 < M (g)

Ar-l<#wi

JV)

• • • ,Vn) U A(J,I,r,3,.. ...,r,n)U

.,%„),

A(J, •/,%,.. . , % „ ) ,

< (1 - 7 ) ^ , for 3 e A ( / , 7}2, . . . . r ^ ) ,

( 5 ) < 7 7 V , f o r g e A(J,Th,...,rjan).

(49)

According to the decomposition of V n , the number vector

Kn)>(JV):=

(s)|0> (^)"5> g€V

\ Jxi I

n

has an orthogonal decomposition into 2 2 n homogeneous vectors

|#(n))W=

|m,...,»»-> (W) ,

£ '71,--,';3ne{/,J}

where | r j ! , . . . ,?7 2 „) (JV) := Z) g € A( m ,...,^ 2 „) ^ ( f l ) ! 0 ) / ^ (

space r ( F 2 ^ ) := ®m,...,mne{i,j}\{vi, (49), we have

• • • ,V2n)

{N)

T h e n we

define a Fock

• It follows from (48) and

a+N(\m, • • • ,V2n)iN)) = \i, J, m, • • •, mn)(N) +1 J, i, m, • • •, mn){N) +vn

aN\I,I,V3,---,V2n){N)

=a-\J,J,r]3,...,Ti2n){N)

=0,

I

l ,- I ,

75

a%\I, / , % , . . . , V2n)(N)

= J2(11

a%\JJ,m,...,r)2n)^

=

J J » J ^ 3 , • • • , mn)(N)

+ Vn (-j=)

,

^l^\IJ,r,3,...,mn)^+vn^-l=y

a^J, J, ,*,,..., %„)<"> = JlEl\I,J:rl3,...,Tl2n)W+vn

(-±=\

. (50)

In view of the relations (50); we prepare a two-mode free Fock space, T 2 := ©~= 0 r 2 (n), where T 2 (n) := C-linear span{|jji,...,Tj2n) | Vi G {I, J}}, equipped with a canonical inner product (m,---,mkWi,---,V2l}

=

5

(m,-,V2k),(v[,v'2l)^

and operators o + , o~ a n d a° specified by 0+

(hl)- •

-JTOH))

= \I,J,Vl:---

J ^ n ) + \J, I, Vl, • • • , »?2n),

a~\I,J,m,---,r)2n)

= a~\J,I,r]3,...,r)2„)

= -\q3, . .. ,T)2n),

a~\I,I,r]3,...,ri2n)

= a~\J,J,ri3,...,r)2n)

= 0,

a°\I, J, %,-••, »?2n) = J oil-

J,7',7'r?3'---'T?2")'

o ° | / , / , r j 3 , . . . ,ry 2 „) = W

J J , J,T?3, . . . ,7?2n),

,

aC|^,-f,»73,---,'?2n) = W - ^

a°\J,J,r]3,...,r]2n)

= J——17,

\I,I,V3,--,mn),

J,rj 3 ,...,fj2n)-

(51)

Summing up the arguments above, we have established a central limit theorem associated with the decomposition (46) as follows. Theorem 30 Let m > 1 and e1,...,em £ {±,o}. Then for any constant 0 < 7 < 1, u G r 2 and TJI, . . . ,r)n £ {I, J}, we have

SS&m\{%)

-($)'•"••• ••*%?,-H*"" '-I- " • > " •

where we use the notation (37). 77ie operators a+, a" and a° are specified by {51). In particular, combinatorial moments with respect to the vacuum state

76 are given by

lim(0|(-L

Y,

- M ) > C < « >

=<0|(a++a-+a°n0)r2.

(52)

Remarks. (1) By a combinatorial argument, we see t h a t the combinatorial moments (52) are given as the number of closed walks on a induced subgraph of a weighted binary tree. (The weights A = l / \ / 2 are given in the figure below.) Indeed rath combinatorial moment Fm is given by the number of m-step walks which leave o and return to o. Let fm be the number of m-step walks which leave z and return to z without arriving at o. By the self-similarity of the graph, we see for m > 2,

Fk

-fc-2, /™ = 5 > * +V24)/' J

Fm — 2_^

k=0

fkFm-k-2

fc=0

where / 0 = FQ = 1. Then the moment generating function F(t) ZmFmtm and f(t) = £ m fmtm satisfy

f{t)-l

= t2(f(t)

+

^-)f(t),

F{t)-l

=

t2F{t)2,

=

hence t2F{t)3 + t2F(t)2 - 2F(t) + 2 = 0. T h e Cauchy transform G(t) of the distribution associated with t h e combinatorial moments (52) is given as a solution of tG(t)3 + G{tf - 2tG(t) + 2 = 0. (2) T h e limit distribution associated with (52) coincides with the one in Examples 1.5 (1.16) and (1.17) of Nica-Speicher 1 8 , u p to t h e variance, where the anti-commutation ab + 6a of semi-circular elements a, b which are free from each other is observed. Indeed we see t h a t

4= y - , e ^

(Ti++ 7r(wt])^((?i

h
--^ )

V

h <*N J

^(l-j)N

77

\^ v/(i- 7 )AT

; v

vw

)'

which is nothing but t h e anti-commutation of semi-circular elements in the limit N —t oo.

5.5

Commutative

association

scheme and its quantum

decomposition

It is quite n a t u r a l to apply our approach of the q u a n t u m decomposition to isotropic random walks on homogeneous graphs. In fact, it will be seen t h a t any adjacency operator of a certain graph is decomposed into a sum of a 'creation', an 'annihilation' and a 'conservation' operators. Such a decomposition is motivated by the s t a n d a r d theory of stochastic evolutions 1 3 . This idea is useful for q u a n t u m limit theorems associated with r a n d o m walks on graphs, particularly, on distance regular graphs. Indeed, Tabei 2 2 and HashimotoO b a t a - T a b e i 1 1 carried out our idea on the Hamming scheme. A q u a n t u m decomposition of adjacency m a t r i x of a H a m m i n g graph is defined with the help of Euler's unicursal theorem, and a q u a n t u m central limit theorem is proved. As a result, we reproduce one of Hora's results 1 2 on t h e asymptotic distribution of eigen values of t h e adjacency matrix associated with a Hamming graph.

References 1. L. Accardi, M. Bozejko, Interacting Fock spaces and Gaussianization of probability measures, Infinite Dimen. Anal. Q u a n t u m P r o b . 1 (1998), 663-670. 2. L. Accardi, Y. Hashimoto and N. O b a t a , Notions of independence related to the free group, Infinite Dimen. Anal. Q u a n t u m P r o b . 1 (1998), 201220. 3. L. Accardi, Y. Hashimoto and N. O b a t a , Singleton independence, Banach Center Publications 4 3 (1998), 9-24. 4. L. Accardi, Y. Hashimoto and N. O b a t a , A role of singletons in quantum central limit theorems, J. Korean M a t h . Soc. 3 5 (1998), 675-690. 5. L. Accardi and N. O b a t a , Introduction to algebraic probability theory, Nagoya Mathematical Lectures 2 (1999). (in Japanese) 6. K. Fichitner, W. Freudenberg and V. Liebscher, Time evolution and invariance of boson systems beam splittings, Infinite Dimen. Anal. Quant u m P r o b . 1 (1998), 511-531.

78 7. Y. Hashimoto, A combinatorial approach to limit distributions of random walks: on discrete groups, preprint, (1996). 8. Y. Hashimoto, Deformations of the semi-circle law derived from random walks on free groups, P r o b . M a t h . Stat. 18 (1998), 399-410. 9. Y. Hashimoto, Samples of algebraic central limit theorems based on Z / 2 Z , In Infinite dimensional harmonic analysis, Transaction of a JapaneseGerman Symposium, Kyoto,1999, 115-126. 10. Y. Hashimoto, Quantum decomposition in discrete groups and interacting Fock spaces, Infinite Dimen. Anal. Q u a n t u m P r o b . (2001) (to appear). 11. Y. Hashimoto, N. O b a t a and N. Tabei, A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in " Q u a n t u m information IV (T. Hida and K. Saito Eds.)," (2001). 12. A. Hora, Central limit theorems and asymptotic spectral analysis on large graphs, Infinite Dimen. Anal. Q u a n t u m P r o b . 1 (1998), 221-246. 13. R. Hudson a n d K. P a r t h a s a r a t h y , Quantum Ito's formula and stochastic evolutions, Commun. M a t h . Phys. 9 3 (1984),301-323. 14. P. Hilton and J. Pederson, Catalan Numbers, Their Generalization and Their Uses, M a t h . Intelligencer 1 3 (1991), 64-75. 15. Y. Lu: On the interacting free Fock space and the deformed Wigner law, Nagoya M a t h . J. 1 4 5 (1997), 1-28. 16. Y. Lu: Interacting Fock spaces related to the Anderson model, Infinite Dimensional Analysis and Q u a n t u m Probability 1 No.2 (1998), 247-283. 17. N. Muraki, A new example of noncommutative "de Moivre-Laplace theorem", in "Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium (S. W a t a n a b e , Ed.)," Tokyo, 1995, 353-362, World Scientific (1996). 18. A. Nica and R. Speicher, Commutators of free random variables, Duke M a t h . J. 92 No.3 (1998), 553-592. 19. F . Oravecz, Deformed Poisson-laws as certain transform of deformed Gaussian-laws, preprint (2000). 20. N. Saitoh and H. Yoshida, Graphical representations of the q-creation and the q-annihilation operators and set partition statistics, preprint (2000). 21. R. Speicher, A new example of 'Independence' and 'White Noise', P r o b . Th. Relat. Fields 8 4 (1990), 141-159. 22. N. Tabei, Quantum decomposition on Hamming graphs and central limit theorem, Master Thesis, Nagoya University (2001). (in Japanese) 23. D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, C R M Monograph Ser. Amer. M a t h . Soc. (1992),

Quantum Information IV (pp. 79-86) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

SOME T H O U G H T S ON T H E INFINITE D I M E N S I O N A L H A R M O N I C ANALYSIS T A K E Y U K I HIDA Faculty

of Science and Technology, Meijo Nagoya 468-8502, Japan

University

YUKO HARA-MIMACHI Department

of Information Sciences, Meijo Nagoya 468-8502, Japan

University

White noise analysis has extensively developed during the past quarter century, and it is now able to think of the significant characteristics of the theory. Among others we should like to emphasize that the white noise analysis enjoys an aspect of an infinite dimensional harmonic analysis. In this article we shall discuss the role of the infinite dimensional rotation group in the study of white noise theory and propose some of future directions as well as its frontiers that shall be discussed in this line.

It is important for him who wants to discover not to confine himself to one chapter of science, but to keep in touch with others. - J. Hadamard Mathematics Subject Classification (2000): 60H40

1 1.1

Background Rotation group

Let E cL2(Rd)

C E*

be a Gel'fand triple. The white noise measure /i is introduced on E*. Let O(E) be the group of linear isomorphisms g which are orthogonal. It is called the infinite dimensional rotation group. Let 0*(E*) be the collection of the adjoints g* with g £ O(E). The set 0*(E*) also forms a group which is isomorphic to 0(E) under the correspondence There is an important property stated by the following Proposition 1. The white noise measure fi is 0*(.E*)-invariant.

80 This assertion can b e proved by the invariance of the characteristic functional of (i under the action of g, and the functional provides a link between the infinite dimensional rotation group and white noise analysis; indeed we can carry on the harmonic analysis arising from the group O(E). 1.2

Fock space

We form a complex Hilbert space (L 2 ) = L2(E*,fj,). A member of (L 2 ) is a white noise functional. T h e following assertion is known. P r o p o s i t i o n 2. T h e space (L2) admits a direct sum decomposition: oo

71 = 0

This is called the Fock space. A member of Hn is a homogeneous chaos of degree n. 1.3

Subgroups of

O(E)

T h e group 0(E) is topologized by the compact-open topology. It is neither compact nor locally compact. In addition, as we shall see below, there are various subgroups of 0(E), each of which enjoys its own interesting property from the viewpoint of probability theory a n d / o r analysis. I. Finite dimensional subgroups of rotations. Fix a complete orthonormal system {£ n } of L2(Rd). Then, one can easily find a subgroup Gn of O(E) such t h a t Gn is isomorphic to SO(n). Let G^ be t h e inductive limit of the Gn. Associated with a member g of 0(E) is a unitary operator Ug acting on (L 2 ) which is defined by (Ug
It is known t h a t

P r o p o s i t i o n 3. 1) T h e unitary representation {Ug, Hn} is irreducible. 2) T h e subgroup G^ characterizes t h e infinite dimensional Laplace-Beltrami operator Aoo which is given by

A

d2

d

- = S^-(*,f»>^}-

81 As a consequence t h e subspace Hn is t h e eigenspace of the infinite dimensional Laplace-Beltrami operator Aoo. In fact, t h e eigenvalue is — n. II. T h e Levy group Q. Still a complete orthonormal system {£„} is fixed. Let 7r b e a p e r m u t a t i o n of the natural numbers. Then, a change of the coordinate vectors is defined: 9n '•

?n

^ S7r(n)-

Then, gn extends to a linear transformation on E. Now assume t h a t the density d(7r) of n is zero:

d(7r) = lim sup — # { n < N; n(n) > N} = 0. T h e collection G = {gieGO(E);d{1r) forms a group which is called the Levy T h e Levy Laplacian defined by

= 0}

group.

has a close connection with the Levy group. It is noted the following assertion. T r i v i a l i t y T h e Levy group has a continuous representation on the space of generalized white noise functionals. III. Whiskers. A whisker we mean t h a t a one-parameter subgroup {gt} of O(E) t h a t comes from a one-parameter family of diffeomorphisms of the parameter space Rd. It is expressed in the form

gt •• &*)-^

t(Mu)W\AM«))\,

where J means t h e Jacobian. There is assumed t h a t ipf^>

— 4>t+s-

Example. T h e shift -{St}- It is defined by

Stt(u) = f(« - *)•

82 Let Tt be t h e adjoint of St. Then {Tt,t £ R} is t h e flow of Brownian motion after S. Kakutani [4]. His pioneering work has stimulated our study. Note t h a t t h e shift describes t h e propagation of chaos as time t goes by. T h e subgroups in II and III are, as it were, essentially infinite dimensional, although they have different flavour from each other. As for t h e subgroup III, we can see this property through t h e their spectrum; in general, they have count ably Lebesgue spectrum.

2

Roles of whiskers

T h e plan of this report is to emphasize the important roles of whiskers, in particular conformal group, in the study of white noise analysis. In particular, special roles can b e seen in t h e study of r a n d o m fields which are indexed by a manifold and are formed by a white noise functional. Let {gt} be a whisker as is given in the last section. Then, as is known, it is conjugate to the shift. More exactly, the function ipt{u) defining the gt can b e expressed in the form

iM«) = / [ / - 1 («) + *]. where / is a strictly monotone, continuous function on R. For important examples of a whisker with which we shall be concerned, t h e function / can be a function t h a t is smooth enough. (See, e.g. [2, C h a p t 5].) We now give an interpretation why a whisker is essentially infinite dimensional. Remind a notion of t h e average power r(x,g) of a rotation g. T h e definition is (see [1])

r(x,g)

1 " =limsup-y](a:,£fc fc=i

g£k)2.

Note t h a t this definition depends on the choice of a complete orthonormal system. If the average power is positive, then g is considered to be infinite dimensional. P r o p o s i t i o n 4. For the shift {St}, dimensional.

essentially

any St, with t ^ 0 is essentially infinite

83 Proof. Take a n St. Choose a complete orthonormal system {£*,}. We have a n equality (x,St£)

=

(x,exp[i\t]i),

from which we can easily prove t h e assertion. T h e o r e m 1. All t h e nontrivial whiskers are essentially infinite dimensional. Proof. From Proposition 4 and t h e fact t h a t any whisker is conjugate to t h e shift except trivial cases, we can prove t h e theorem. Remark. There are also many other members in t h e Levy group which are essentially infinite dimensional. Coming back to whisker, it is convenient to have its infinitesimal generator a, which can b e expressed in t h e form i \ d + -a 1 //(u), \ a = a[u)— du

I

with a suitable function a{u). W i t h t h e help of t h e generators we can discover complex relationship between whiskers, and even think of their roles in white noise analysis. 3

Conformal group

Starting from t h e shift, which describes t h e flow of Brownian motion (and hence t h e most i m p o r t a n t whisker), we can find some other whiskers which have good commutation relations and have significant probabilistic meaning. Actually, we are given several interesting whiskers and p u t t i n g t h e m together it is proved t h a t they form a conformal group, denoted by C(d). T h e parameter space of white noise is now taken to be R . W i t h some generalization of what has been discussed in [2], we claim t h a t those whiskers in question can be listed as follows: 1) Shift, 2) isotropic dilations, 3) subgroup isomorphic to

SO(d),

84

4) special conformal transformations. It is known that the subgroup C(d) of 0(E) is isomorphic to the classical group SO(d + 1 , 1 ) . Its Lee group structure help our study of white noise. One of the interesting and in fact significant applications is that a unitary representation of the group C(d) contributes to the innovation theory of a random field. Let X(C) be a random field parameterized by a closed convex manifold C. Assume, in particular, that the parameter C runs through a class C of ovaloids in Rd. We are interested in the case where X(C) is expressed as a causal (generalized) functional of white noise x(u),u G (C), where (C) denites a domain enclosed by C. Generally speaking, geometric structure of the manifold C can describe the information theoretical behavior of the random field X(C). The C, unlike the time parameter t, enjoys complex structure, so that when C deforms X(C) carries more information than X(t). In particular, the variation SX(C) by using conformal transformations is useful for our study. The basic idea is illustrated below. The infinitesimal deformation of C is represented by a family of functions {5(s),s G C}. We apply the conformal transformations that leave the C invariant. Then, we are given various family of functions 6(s) which is dense in the L2-class of functions over C. This comes from the irreducible unitary reprersentation theory of the conformal group. This claims that the {x(s), s G C} can be obtained if X(C) is Gaussian. The system is the innovation for X(C). We assume that C is taken as before and and the representation of X[C) is causal in x(u). Then, we can prove Theorem 2. For a Gaussian random fields X(C), the innovation is obtained by the conformal transformations acting on the basic parameter space Rd. Further we can introduce the so-called stochastic variational equation which determines a random field. See [8].

4

Concluding remarks

There are three remarks.

85 I. Reversibility. As soon as we come to t h e study of a r a n d o m field X(C), we have t o worry about how to define t h e reversibility of t h e random phenomena. For X(t) only one direction and reverse can b e considered when we think of propagation of the r a n d o m phenomena. As for the X(C) the direction of propagation is not unique, but should not vaguely defined. Deformations of the parameter C are recommended to be taken within the conformal group. Here is a useful information. In the two dimensional parameter case, we prefer to use notations in the complex plane. A conformal mapping from unit circle onto itself is written as z —a w - 1T, az — 1 where j —y | = 1. II. Applications to physics. Stochastic variational equation determines a Euclidean field. We hope this can be extended to relativistic q u a n t u m fields. We have hope to establish a p a t h integral for a r a n d o m field indexed by C. This would be a good approach to quantization of a q u a n t u m field. III. T h e white noise theory has so far much related (not based on) to t h e socalled L 2 -nonlinearity. It should be refeminded t h a t t h e original idea of the white noise theory came from, as it were, the reductionism; nemely, we start with a system of basic and atomic random variables, like white noise, and analyse the functionals of those variable in the system. T h e assumption to restrict functionals in L2 -class is somewhat too restrictive. There are many other interesting classes of r a n d o m functions which have not variances, so t h a t the analysis requires suitable topology which is not the mean square, and hence it needs new tools from analysis, like entropy. Anyhow, we should now think of some new frontier of white noise analysis. T h e authors have already started a study in this direction and given lectures at the present Academic Frontier Project Seminar. Some part of our study will also be printed as t h e literature [3]. Further development will b e reported in the Q u a n t u m Information Series.

86 References [1] L. Accardi et al ed. Selected papers of Takeyuki Hida. World Scientific P u b . Co. 2001. [2] T. Hida, Brownian motion. 1975 (in Japanese); English Translation 1980 Springer-Verlag. [3] T. Hida, W h i t e noise analysis: A new frontier, Univ. Center P u b . N. 499, 2002.

Roma, Volterra

[4] S. Kakutani, Determination of the spectrum of the flow of Brownian motion. Proc. National Acad. Sci., 36 (1950), 319-323. [5] H.-H. Kuo, W h i t e noise distribution theory, 1996, CRC Press. [6] P. Levy, Problemes concrets d'analyse fonctionnelle. 1951 Gauthier-Villars. [7] P. Levy, Processus stochastiques et mouvement brownien. 1948: 2eme ed. 1965, Gauthier-Villars. [8] Si Si, Innovation for some r a n d o m fields. J. Korean Math. Soc. 35, no.3, 1998, 793-802. [9] Si Si, Gaussian processes and Gaussian random fields. Q u a n t u m Information II2000, World Scientific P u b . Co., p p 195-204. [10] Si Si and Win W i n Htay, Topics on Complex Gaussian r a n d o m fields, 2001. Q u a n t u m Information IV, World Scientific P u b . Co., (in this volume).

Quantum Information IV (pp. 87-102) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co. A T R E A T M E N T OF Q U A N T U M BAKER'S M A P B Y CHAOS DEGREE

KEIINOUE, MASANORI OHYA Department of Information Sciences Science University of Tokyo Noda City, Chiba 278-8510 Japan IGOR V. VOLOVICH Steklov Mathematical Institute, Russian Academy of Science Gubkin St. 42, Moscow, GSP1, 117966, Russia We study the chaotic behaviour and the quantum-classical correspondence for the baker's map. Correspondence between quantum and classical expectation values is investigated and it is shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.

1

Introduction

T h e study of chaotic behaviour in classical dynamical systems is dating back to Lobachevsky and H a d a m a r d who have been studied t h e exponential instability property of geodesies on manifolds of negative curvature and to Poincare, who initiated the inquiry into the stability of t h e solar system. One believes now t h a t the main features of chaotic behaviour in the classical dynamical systems are rather well understood, see for example [1, 27]. However t h e status of " q u a n t u m chaos" is much less clear although the significant progress has been made on this front. Sometimes one says t h a t an approach to q u a n t u m chaos, which a t t e m p t s to generalize the classical notion of sensitivity to initial conditions, fails for two reasons: first there is no q u a n t u m analogue of the classical phase space trajectories and, second, the unitarity of linear Schrodinger equation precludes sensitivity to initial conditions in the q u a n t u m dynamics of s t a t e vector. Let us remind, however, t h a t in fact there exists a q u a n t u m analogue of the classical phase space trajectories. It is q u a n t u m evolution of expectation values of appropriate observables in suitable states. Also let us remind t h a t the dynamics of a classical system can be described either by t h e Hamilton equations or by the liner Liouville equations. In q u a n t u m theory the linear

88 Schrodinger equation is the counterpart of the Liouville equation while the q u a n t u m counterpart of the classical Hamilton's equation is the Heisenberg equation. Therefore t h e study of q u a n t u m expectation values should reveal the chaotic behaviour of q u a n t u m systems. In this paper we demonstrate this fact for the q u a n t u m baker's m a p . If one has the classical Hamilton's equations dq/dt = p,

dp/dt = -V

(q),

then the corresponding q u a n t u m Heisenberg equations have the same form dqh/dt-ph,

dph/dt

= -V'

(qh),

where qh and ph are q u a n t u m canonical operators of position and m o m e n t u m . For the expectation values one gets t h e Ehrenfest equations d

/dt =< ph>,

d

/dt = - < V (qh) >

Note t h a t t h e Ehrenfest equations are classical equations b u t for non nonlinear V (qh) they are neither Hamiltonian's equations nor even differential equations because one can not write < V' (qh) > as a function of < qh > and < Ph > • However these equations are very convenient for t h e consideration of the semiclassical properties of q u a n t u m system. T h e expectation values < qh > and < ph > are functions of time and initial data. They also depend on the q u a n t u m states. One of important problems is to study the dependence of expectation values from t h e initial data. In this paper we will study this problem for the q u a n t u m baker's m a p . T h e m a i n objective of " q u a n t u m chaos" is to study t h e correspondence between classical chaotic systems and their q u a n t u m counterparts in t h e semiclassical limit [8,7]. The quantum-classical correspondence for dynamical systems has been studied for many years, see for example [3,10,30] and reference therein. A significant progress in understanding of this correspondence has been achieved in the W K B approach when one considers the Planck constant h as a small variable parameter. T h e n it is well known t h a t in the limit h —> 0 q u a n t u m theory is reduced to the classical one. However in physics the Planck constant is a fixed constant although it is very small. Therefore it is import a n t to study the relation between classical and q u a n t u m evolutions when the Planck constant is fixed. There is a conjecture [29,5] t h a t a characteristic timescale r appears in t h e quantal evolution of chaotic dynamical systems. For time less then r there is a correspondence between q u a n t u m and classical expectation values, while for times greater t h a t T the predictions of the classical and q u a n t u m dynamics no longer coincide. T h e important problem is

89 t o estimate t h e dependence r on the Planck constant h. Probably a universal formula expressing r in terms of h does not exist and every model should b e studied case by case. It is expected t h a t certain q u a n t u m and classical expectation values diverge on a timescale inversely proportional to some power of h [4]. Other authors suggest t h a t a breakdown may b e anticipated on a much smaller logarithmic timescale [20,9,22,14,17,24,25,18]. T h e characteristic time T associated with the hyperbolic fixed points of the classical motion is expected to be of the logarithmic form r — j In ^ where A is the Lyapunov exponent and C is a constant which can be taken to be the classical action. Such the logarithmic timescale has been found in the numerical simulations of some dynamical models [30]. It was shown also t h a t t h e discrepancy between q u a n t u m and classical evolutions is decreased by even a small coupling with the environment, which in t h e q u a n t u m case leads to decoherence [30]. T h e chaotic behaviour of the classical dynamical systems is often investigated by computing the Lyapunov exponents. An alternative quantity measuring chaos in dynamical systems which is called the chaos degree has been suggested in [19] in the general framework of information dynamics [12]. T h e chaos degree was applied to various models in [13]. An advantage of the chaos degree is t h a t it can be applied not only to classical systems but also to q u a n t u m systems as well. In this work we study the chaotic behaviour and the quantum-classical correspondence for the baker's m a p [4,21]. T h e q u a n t u m baker's m a p is a simple model invented for t h e theoretical study of q u a n t u m chaos. Its m a t h ematical properties have been studied in numerical works. In particular its semiclassical properties have been considered [20,9,22,14,17,24,25,18], quant u m computing and optical realizations have been proposed [11,23,6], various quantization procedures have been discussed [16,15,22,26], a symbolic dynamics representation has been given [26]. It is well known t h a t for the consideration of the semiclassical limit in q u a n t u m mechanics it is very useful to use coherent states. We define an analogue of t h e coherent states for t h e q u a n t u m baker's m a p . We study the q u a n t u m baker's m a p by using the correlation functions of the special form which corresponds to the expectation values of Weyl operators, translated in time by the unitary evolution operator and taken in the coherent states. To explain our formalism we first discuss the classical limit for correlation functions in ordinary q u a n t u m mechanics. Correspondence between q u a n t u m and classical expectation values for the baker's m a p is investigated and it is shown t h a t it is lost at the logarithmic timescale. T h e chaos degree for the q u a n t u m baker's m a p is computed and it is demonstrated t h a t it describes the chaotic features of the model. T h e dependence of t h e chaos degree on

90 the Planck constant is studied and t h e correspondence between classical and q u a n t u m chaos degrees is established.. 2

Q u a n t u m vs. Classical D y n a m i c s

In this section we discuss an approach to the semiclassical limit in q u a n t u m mechanics by using the coherent states, see [10]. Then in the next section an extension of this approach to the q u a n t u m baker's m a p will be given. Consider the canonical system with the Hamiltonian function

H=£

+ V(x)

(1)

in the plane (p, x) G R 2 . We assume t h a t the canonical equations

x(t)=p(t),

p(t) = -V'(x{t))

(2)

have a unique solution (x (i) ,p (t)) for times \t\ < T with the initial d a t a

p ( 0 ) = «o

SB(0) = ;CO,

(3)

This is equivalent to the solution of the Newton equation x(t)

= -V'(x(t))

(4)

with the initial d a t a

x(0) = x0,

x(0)=v0

(5)

We denote

a = —=. (x0 + iv0) T h e q u a n t u m Hamiltonian operator has the form Hh = ^

+

V(qh)

where p/, and qh satisfy the commutation relations [Ph,qh] =

-ih

(6)

91 T h e Heisenberg evolution of the canonical variables is defined as ph (t) = U {t)phU

(i)* ,

Qh (t) = U (t) qhU ( 0 *

where U (t) = exp

(-itHh/h)

For the consideration of the classical limit we take the following representation ph = -ih1/2d/dx,

qh =

h1/2x

acting t o functions of t h e variable x £ R . We also set

a=

^&{qh

+ iPh) =

a =

-k{X + £)'

^{qh-iph)=T2(:x-i)'

*

then \a,a*} = 1. T h e coherent state \a) is denned as \a) = W(a)\0)

(7)

where a is a complex number, W (a) — exp (aa* — act*) and |0) is t h e vacuum vector, a |0) = 0. T h e vacuum vector is the solution of the equation

{qh + iph) |0> = 0

(8)

In the x - representation one has |0) =exp(-x2/2)

/V2^.

(9)

T h e operator W (a) one can write also in t h e form W (a) = CeiqhVa/hl'2e-lphXa'h}12

(10)

where C — exp (—vQxQ/2h). T h e mean value of t h e position operator with respect to the coherent vectors is the real valued function

92

q (t, a, h) = (h-l/2a\

qh {t) | f c ~ 1 / 2 a )

(11)

Now one can present the following basic formula describing t h e semiclassical limit \hnq(t,a,h)

= x{t,a)

(12)

h-yO

Here x (t, a ) is the solution of (4) with the initial d a t a (5) and a is given by (6). Let us notice t h a t for time t = 0 the q u a n t u m expectation value q (t, a, h) is equal t o the classical one: q(0,a,h)

= x{0,a)

= x0

(13)

for any h. We are going t o compare the time dependence of two real functions q(t,a,h) and x(t,a). For small t these functions are approximately equal. T h e i m p o r t a n t problem is t o estimate for which t the large difference between t h e m will appear. It is expected t h a t certain q u a n t u m and classical expectation values diverge on a timescale inversely proportional to some power of h [10]. Other authors suggest t h a t a breakdown may b e anticipated on a much smaller logarithmic timescale [20,9,22,14,17,24,25,18]. One of very interesting examples of classical systems with chaotic behaviour is described by t h e hamiltonian function

H = £ + £ + \x\x22

l 2 2 2 T h e consideration of this classical and q u a n t u m model within t h e described framework will be presented in another publication.

3

C o h e r e n t S t a t e s for t h e Q u a n t u m B a k e r ' s M a p

T h e classical baker's transformation maps the unit square 0 < q,p < 1 onto itself according t o

r„^_J( 2 9'P/ 2 )' l 9

'

P

if

^ \ ( 2 g - l ) ( p + l)/2),if

°<9
9

<1

This corresponds t o compressing the unit square in the p direction and stretching it in t h e q direction, while preserving t h e area, t h e n cutting it vertically and stacking the right part on top of t h e left part.

93 To quantize t h e unite square one defines the unitary displacement operators U and V in D - dimensional Hilbert space, which produce displacements in the m o m e n t u m and position directions, respectively, and which obey the commutation relation UV =

eVU,

where e = e x p ( 2 7 r i / D ) . We choose D = 2N. T h e "Planck constant" h = 1/D = 2~N. T h e operators U and V can be written as U = e2"4,

V = e2™?

T h e position and m o m e n t u m operators q and p b o t h have eigenvalues j/D, j = 0,..., D — 1. If {|<7j)} are the eigenvectors of the position operator q then t h e eigenvectors of t h e m o m e n t u m operator {|pj)} are obtained by using the discrete Fourier transform i*V :

\Pi) = FN\*i) =

-J5iie2*ikilD\
T h e q u a n t u m baker's m a p is written as the following matrix

We define the coherent states by

\a) = Ce2^ve-2KiPx

|i/>0)

(15)

Here a = x + iv, x and v are integers, C is t h e normalization constant and \tp0) is the vacuum vector. This definition should be compared with 10. T h e vacuum vector can be defined as the solution of the equation {qh + iph) \i>o) = 0 (compare with 8). We will use the simpler definition which in the position representation is (< 7i |Vo) = C e x p ( - 9 2 / 2 ) (compare with 9). Here C is a normalization constant.

94 4

Chaos Degree

Let us review the entropic chaos degree defined in [19]. This entropic chaos degree is given by the probability distribution ip of a orbit generated by a dynamics (channel) A* sending a s t a t e to a state; tp = ^iPfc^fc, where 5k 1 (k = j) - ,, . .{ . Then the entropic chaos

{

degree is defined as D(^A*)

= YtPkS(A*Sk)

(16)

k

with the von N e u m a n n entropy S, equivalently to t h e Shannon entropy because the probability distribution ip is a classical object. T h e above channel A* produces a dynamics F of t h e orbit, so t h a t let xn be the orbit or a certain function of the orbit at time n and F be a m a p from xn to xn+i, For a m a p F on / = [a,6] c R N with xn+± — F (xn) (a difference equation), let / = \Jk Bk be a finite partition with Bi n B}: = 0 (i ^ j). T h e state tpW of the orbit determined by the difference equation is defined by the probability distribution (p\ J , t h a t is, ipW = p ' " ' = J2iPi "i> w n e r e for an initial value x £ / and the characteristic function 1 A m+n m+n

1

-ri £ i* (rk*) •

Pi

m+ _

W h e n the initial value x is distributed K due to a measure v on / , t h e above (n) .

p\

.

is given as -.

« m+n fc=n

T h e joint distribution ( p | " , n

) between the time n and n + 1 is defined by

or

^n'"+1) = ~ r / £ ^ (Ffc») in, (rfc+1*) */. Then the channel Aj^ at n is determined by

95

and the chaos degree is given by

This classical chaos degree was applied t o several dynamical maps such logistic map, Baker's transformation and Tinkerbel m a p , and it could explain their chaotic characters [19,13]. Our chaos degree has several merits compared with usual measures such as Lyapunov exponent. We can judge whether t h e dynamics causes a chaos or not by t h e value of D as D > 0 <=> chaotic, D = 0 «=> stable. 5

C h a o s D e g r e e for t h e Q u a n t u m B a k e r ' s m a p

In this section, we show a general representation of the mean value of the position operator q for the time evolution, which is constructed by the q u a n t u m baker's map. T h e n we give t h e algorithm to compute the chaos degree for the q u a n t u m baker's m a p . Recently a whole class of q u a n t u m baker's maps has been defined in [26]. It is a q u a n t u m analogy of t h e symbolic dynamics [2] for the classical baker's map. For any n, 1
Tn | 6 6 • • • £ n « £ n + l £ n + 2

-

" - 6v) = | 6 6 ' ' • £n+l«£n+26> + 3 ' ' ' 6v) ,

where the vector | ^ £ 2 • • • 61.61+161+2 • • • 6 v ) 1 6 6 • • • U.U+ltn+2

is

g

iven

(18)

W

• • • 6 v ) = |£»+1> ® ' ' • ® I6v> J*0*""*1)

®

v/I72 {|0) + exp [2ixi (0.& 1)] | 1 » ® y/l/2{\0)

+ exp [2ni ( 0 - 6 6 1 ) ] |1>} ® • • • ®

\ / l 7 2 { | 0 ) + e x p [2iri (0.&, • • • Cil)] | 1 »

96 for n, 1
TN-I 166 • • -6v-i-6v) = | 6 6 • • -6v) •

(19)

For n = 0, the m a p is (20)

To 1-66 •••£*> = 1 6 - 6 6 • • • £ * ) .

To study the time evolution and the classical limit ft. —> 0 which corresponds to N —t oo of the q u a n t u m baker's m a p To, we introduce the following the mean value of the position operator q for time n £ N with respect to a single basis |£):

r»£ ' = <£|r o »gT 0 -»|0.

(21)

2 — 1

where |f) = | 6 6 " - 6 v ) and g = S J = 0 9; |j) ( j | with eigen values qj = * ± J £ , j = 0 , 1 , . . . , 2N - 1, j = E f = 1 jk2N-k and j k = 0 , 1 . T h e simple formula of the matrix elements of To with respect to two different bases is given by

<e°| To IO = V I I

5

(^ -ri-i)«p ( ^ Itf - &

(22)

*i=2

where|£°) = | t f # - - - & ) and | ^ ) = ! # £ • • • & ) [28]. From (22), the following formula of t h e matrix elements of T£ for any n G N is easily obtained.

'mn(uL7HC+k-ek)) = <

(nr=i^?ci,_.+1) I] {Am)e

e

*n
^=1

B

AT

(¥) fc=inu B w

if n = miV,

where|£°) = | t f $ • • • ?N) , If 1 ) = ^ ' ' " & > . ^ is t h e 2 X 2 matrix with the element ^4i l X 2 = exp (-|i |«i — x2\) for x\,xz = 0 , 1 , p = 1, • • •, iV — 1 and m e N.

97 Using these formula, t h e following theorem are proved. T H E O R E M 5.1 r £ } = ( S~^N~n

2_/fc=l

t

9 -fc _|_ 2 " 2»+l

ifn
Sra+fcZ

2

1

V

2

W

- 1 j + 1/2

/r

n (^r 't +i

2 N-p

/,m+l\ N-p+k

r

0l

if n = mN,

^ k 3k

fc=l

if n = mAf + p

(24) w/iere | f ° ) = |£°£° • • • & ) , | £ : ) = \£l£ • • • £}N) , A is the 2 x 2 matrix with the element AXlX2 — exp (ji \x^ — x2\) for xi,x2 = 0 , 1 , p = 1, • • •, N — 1 and m £ N. By diagonalizing the matrix A, we obtain t h e following formula of the absolute square of the matrix elements of An for any n £ N . L E M M A 5.2 For- any n £ N , we have l A >*i

-|

2

"sin

2

= J ( ^ ) t/fe

Combining t h e above theorem a n d lemma, we obtain t h e following two theorems with respect to t h e mean value rN of t h e position operator. T H E O R E M 5.3 For the case n = mN +p, p = 1, 2 , . . . AT - 1 and m £ N , we have

»

Yjk=i ?p+k 2 + 2i7+T ,JV Efc=iv^p+i Vk-(N-P)2'k + 2 27f+Pi+1 = < 1 ^ 2^fc=i Vp+k* -*-+- -JFHT: -2" + l . Z fc=AT-p+l ?*:-( AT-p)' - f c + 2£ N l J

-p+1 s « - ( J V - p j "

i

2 +

if m = 0 [mod 4) ifm = l (mod 4) ifm = 2 (mod 4)

(25)

if m = 3 (mod 4 ) .

where r\\. = £*. + 1 (mod 2 ) , A; = 1, • • •, N. T H E O R E M 5.4 For the case N - mN,m £ N , we have (n)

E £ = i 6 2 - f c + ^ r i / m = 0 (mod ^) i if m = 1 , 3 (raoii ^) r

2

!

i m

Efc=i ?fc ~* + 2 ^ f

= 2

(

mod

(26)

4) •

Using these formulas (24), (25) a n d (26) , t h e probability distribution [Pi

) °f t n e orbit of mean value r£

of the position operator q for t h e time

evolution, which is constructed by t h e q u a n t u m baker's m a p , is given by

98 m+n

(n) —

for an initial value rN distribution ( p | ?

,n

!

V^ 1

(

W\

€ [0,1] and the characteristic function 1 ^ . T h e joint ) between the time n and n + 1 is given by m+n

i-ir'^sTix:'-.^") 1 *^")Thus the chaos degree for the q u a n t u m baker's m a p is calculated by

(n)

(

,

^(P - ;A;)=E*"

+1)

f I). ^I^

( 27 )

whose numerical value is shown in the next section.

6

N u m e r i c a l S i m u l a t i o n of t h e C h a o s D e g r e e a n d Classical-Quantum Correspondence

We compare the dynamics of the mean value rN of position operator q with t h a t of the classical value g*™' in the g direction. We take an initial value of the mean value as

where & is a pseudo-random number valued with 0 or 1. At t h e time zero we assume t h a t t h e classical value g' 0 ' in the g direction takes the same value as the mean value rjy of position operator g. T h e distribution of r^ for the case N = 500 is shown in Fig.l u p to t h e time n = 1000. T h e distribution of the classical value q^ for the case N = 500 in the q direction is shown in Fig. 2 up to t h e time n = 1000.

99

» ' N

• *

•

•

* • • * ; • • •

.8

* w v%" * V: *

.6

v?-V\-; "•*•**»/• " • ' • * •

*

•

• • • • . .

';•

;

•••'

.4 1 .2 • •

I

•

. . « • • • •

i T i l '

J * |

v

• • « . *

* *

t

» »•

»

n

0

200

400

600

1000

800

Fig.l. The distribution of ^ for the case Af=500 .(»)

XT W^

-TV

0.7 ••••*•>

•.••

•••

* n / \ ; «

0.5

-.

0.2

-*-*?

».*

1

._•» . - ^

-+-*

«.*•:••• *

0.1 0

200

V*.

»-«—V^

• »•

400

•

600

1000

800

n

)

Fig.2. The distribution of the classical value q"' for the case N=5Q0

Fig.3 presents the change of the chaos degree for the case AT = 500 u p to the time n = 1000.

100

D 0.6 0.5 0.4 0.3 0.2 0.1 0

-0.1

Fig.3. The change of the chaos degree for the case N=500 up to time n=1000

T h e correspondence between the chaos degree Dq for the q u a n t u m baker's m a p and t h e chaos degree Dc for the classical baker's m a p for a fixed N (500, here) is shown for t h e time less t h a n T = log 2 \ — log 2 2N = N, and it is lost at the logarithtic time scale T. T h e difference of the chaos degrees between the chaos degree Dq for the q u a n t u m baker's m a p and the chaos degree Dc for the classical baker's m a p for a fixed time n (1000, here) is displayed w.r.t. N in Fig.4.

101

N 700

900

1000

Fig.4. The difference of the chaos degree between quantum and classical for the case n=1000 T h u s we conclude t h a t t h e dynamics of the mean value r^ reduces the classical dynamics q(n) in t h e q direction in the classical limit N —> oo (h —t 0). References 1. D.V.Anosov and V.I.Arnold (eds.), Dynamical Systems, VINITI, Moscow, 1996. 2. V.M. Alekseev and M.N.Yakobson, Symbolic dynamics and hyperbolic dynamic systems, P h y s . Reports, 7 5 , 287-325, 1981. 3. I.Y. Arefeva, P.B. Medvedev, O.A. Rytchkov a n d I.V. Volovich, Chaos in m(atrix) theory, Chaos, Solitons & Fractals, 1 0 , No.2-3, 213-223, 1999. 4. N.L.Balazs and A.Voros, T h e quantized baker's transformation, Ann. Phys., 190, 1-31, 1989. 5. M.V.Berry, Some quantum-to classical asymptotics, Les Houches Summer School "chaos and q u a n t u m physics", Edits. Giannoni, M.J. Voros, A. and Justi, Zinn, North-Holland, Amsterdam, 1991. 6. T . B r u n a n d R.Schack, Realizing t h e q u a n t u m baker's m a p on an N M R q u a n t u m computer, Phys. Rev. A, 5 9 , 2649-2658 , 1999. 7. G.Casati and B.V.Chirikov (eds.), Quantum Chaos: between Order and Disorder, Cambridge Univ. Press, Cambridge, 1995. 8. M.C.Gutzwiller, Chaos in classical and Quantum Mechanics, Springer, Berlin, 1990. 9. F.M.Dittes, E.Doron and U.Smilansky, Long-time behavior of the semi-

102

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27. 28. 29. 30.

classical baker's m a p , P h y s . Rev. E, 4 9 , R963- R966,1994. K.Hepp, T h e classical limit for q u a n t u m mechanical correlation functions, C o m m u n . M a t h . P h y s . , 3 5 , 265-277, 1974. J.H.Hannay, J.P.Keating and A.M.Ozorio de Almeida, Optical realization of t h e baker's transformation, Nonlinearity, 7, 1327-1342, 1994. R.S.Ingarden, A.Kossakowski and M.Ohya, Information Dynamics and Open Systems, Kluwer Academic Publishers, 1997. K.Inoue, M.Ohya and K.Sato, Application of chaos degree to some dynamical systems, Chaos, Soliton and Fractals, 1 1 , 1377-1385, 2000. L.Kaplan and E.J.Heller, Overcoming the wall in the semiclassical baker's m a p , Phys. Rev. Lett., 76, 1453-1456, 1996. A.Lakshminarayan, On the q u a n t u m baker's m a p and its unusual traces, Ann. Phys., 2 3 9 , 272-295,1995. A.Lakshminarayan and N.L.Balazs, T h e classical and q u a n t u m mechanics of Lazy baker maps, Ann. Phys., 2 2 6 , 350-373, 1993. M.G.E. da Luz and A.M.Ozorio de Almeida, P a t h integral for the quant u m baker's m a p , Nonlinearity, 8, 43-64, 1995. P.W.O'Connor and S.Tomsovic, T h e unusual n a t u r e of the q u a n t u m baker's transformation, Ann. Phys., 207, 218-264, 1991. M.Ohya, Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics, 37, N o . l , 495-505, 1998. A.M.Ozorio de Almeida and M.Saraceno, Periodic orbit theory for the quantized baker's map, Ann. Phys., 2 1 0 , 1-15, 1991. M.Saraceno, Classical structures in the quantized baker transformation, Ann. Phys., 1 9 9 , 37-60, 1990. M.Saraceno and A.Voros, Towards a semiclassical theory of t h e q u a n t u m baker's m a p . Physica D, 7 9 , 206-268, 1994. R.Schack, Using a q u a n t u m computer to investigate q u a n t u m chaos, Phys. Rev. A, 5 7 , 1634-1635 ,1998. R.Schack and C M . C a v e s , Hypersensitivity to perturbations in the quant u m baker's m a p , Phys. Rev. Lett., 7 1 , 525-528, 1993. R.Schack and C M . C a v e s , Information-theoretic characterization of quant u m chaos, Phys. Rev. E., 5 3 , 3257-3270, 1996. R.Schack and C M . C a v e s , Shifts on a finite qubit string: A class of quant u m baker's m a p s , Applicable Algebra in Engineering, Communication and Computing, A A E C C 10, 305-310, 2000. Ya.G.Sinai, Introduction to Ergodic Theory, Fasis, Moscow, 1996. A. N. Soklakov and R. Schack, Classical limit in terms of symbolic dynamics for the q u a n t u m baker's m a p , e-print qunatu-ph/9908040. G.M.Zaslavskii, Stochasticity of Dynamical Systems, Nauka, Moscow, 1984. W.H.Zurek, Pointer basis of q u a n t u m a p p a r a t u s : Into what mixture does the wave packet collapse?, Phys. Rev. D, 2 4 , 1516-1527, 1981.

Quantum Information IV (pp. 103-114) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

POISSON NOISE ANALYSIS BASED ON THE LEVY LAPLACIAN

ATSUSHIISHIKAWA Graduate

School of Mathematics Meijo University Nagoya J, 68- 8502, Japan KIMIAKI SAITO

Department

of Information Sciences Meijo University Nagoya 4 68- 8502, Japan A L L A N U S H. T S O I

Department of Mathematics 202 Mathematical Sciences Building University of Missouri Columbia, MO 65211, USA In this paper we discuss the Levy Laplacian acting on the space of Poisson noise functionals. In particular we introduce a semigroup generated by the Laplacian and give a stochastic expression of the semigroup.

Mathematics 1

Subject Classification

(2000):

60H40

Introduction

T h e Levy Laplacian was introduced by P. Levy 12 and was applied to the theory of generalized white noise functionals initiated by T. Hida 2 . This Laplacian has been studied by several authors within the framework of Gaussian white noise analysis (e.g. Refs. 1, 9, 10, 11, 13, 14-16). On the other hand the Poisson noise analysis has been also discussed in Refs. 3-8, 17, 18 and others. In t h e previous paper 1 7 , we introduced a domain of t h e Levy Laplacian to get the self-adjointness of the Laplacian on the domain. In this paper we introduce a domain consisting of Poisson noise functionals which are eigenfunctions of the Laplacian. The domain is larger t h a n the one given in Ref.17. We define an equi-continuous semigroup of class (C 0 ) generated by the Laplacian on this domain and give also a stochastic expression of the semigroup acting on t h e domain.

104 T h e paper is organized as follows. In Section 2 we summarize some basic definitions and results in the Poisson noise analysis. In Section 3 we give the definition of the Levy Laplacian acting on Poisson noise functionals and also give eigenfunctions of t h e Laplacian. In Section 4 we introduce Hilbert spaces EJV I P depending on the parameters N G N and p G R related to weights of norms. T h e spaces consist of eigenfunctions in Section 3 and so these are suitable to act as domains of the Laplacian. For p € R , the following inclusion relations hold: • • • C EN+itP

c Ejv, P C • • • C E i i P c E 0 | P C • • • C E _ J V I P c E_jv-i,p C • • •

T h e Levy Laplacian becomes a self-adjoint operator densely defined on EJV, P for each J V e N and p 6 R , and an equi-continuous semigroup of class (Co) generated by the Laplacian is defined on the inductive limit space E_oo i P of Ejv, P , N G N , p G R . In the last section we give stochastic expressions of the semigroup generated by t h e Laplacian on the image space i / j E ^ p ] of the projective limit space Eoo ]P of EJV I P , N G N , p G R , by t h e ZY-transform and the semigroup generated by t h e Laplacian on t h e space E ^ p , respectively. 2

Preliminaries

In this section we summarize the basic definitions and results in Poisson noise calculus following Refs. 3 and 5 ( see also Refs. 4 and 6. ) We start with the following Gel'fand triple E = 5 ( R ) C L 2 ( R ) c E* =

S'(K),

where <S(R) is the space of rapidly decreasing functions defined on R , L2(R) is the Hilbert space consisting of square-integrable functions on R with norm | • |o, and <S'(R) is the dual space of <S(R), i.e. the space of tempered distributions. Let A — —(d/du)2 + u2 + 1. Then A is a densely defined self-adjoint operator on L 2 ( R ) and there exists an orthonormal basis {e^; k — 0 , 1 , 2 , . . . } C E for L 2 ( R ) satisfying Aek = 2(k + l)ek, k = 0 , 1 , 2 , . . . . For each p G R , we define |£| p = I^P^IO and let Ep = {£ G L 2 ( R ) ; |f|p < oo} for each p > 0 and let Ep be the completion of L 2 ( R ) with norm | • | p for each p < 0. We introduce the projective limit topology and the inductive limit topology of spaces Ep,p G R to E and E*, respectively. T h e Bochner-Minlos Theorem admits the existence of a probability measure fj, on E* whose characteristic functional is given by J

exp{i(x,

0)MX)

= exp [ f

( e l « ( u ) - l ) du

UE,

105 where (•, •) is the canonical bilinear form between E* and E. Let (L2) = L2(E*,(J,) be a Hilbert space consisting of complex-valued square-integrable functionals defined on a Poisson noise probability space (E*,fi). Take a sequence {r]n}™=1 c E converging to l( 0i t] in L 2 ( R ) for each t > 0 and to — l(t,o] i n L2(R) for each t < 0. T h e n the sequence { ( - , ^ T I ) } ^ L I converges to a Poisson noise functional N(t) = N.(t) in (L 2 ) and N(t) becomes a s t a n d a r d Poisson process defined on (E*,fi). T h e W-transform is defined by U
f ei^^tp(x)dfi(x), J E*

£ G E,

for each tp G ( L 2 ) , where C ( f ) = exp [/ R ( e ^ ^ - l ) du] , £ G £ . If we introduce an inner product (U)u = (y,V')(L'),

for

£{L2),

t o U = W[(L 2 )], then U is a reproducing kernel Hilbert space with kernel

T h e ^/-transform is an isomorphism from (L2) onto U . P u t Q(t) = N(t) — t. Then Q(t) is the compensated Poisson process and the space (L 2 ) has the Wiener-Ito decomposition: oo

(L2) = ®tfn, where Hn is a closed subspace of (L2) given by Hn = | l n ( / ) = J

/ ( u ) d Q ( m ) • • • dQ(u„); / G L 2 C ( R ) ® " | ,

where u = ( u i , . . . ,un) and L 2 -,(R)® n is the n-fold symmetric tensor product of the complexification L 2 -,(R) of L 2 ( R ) . T h e (L 2 )-norm \\
IMIo = I 53 n ! l^lii(R) s " 1 • T h e W-transform Utp of

n

M o = E / /n(u)n(«<£K)-i)^-

106 T h e second quantization operator T(A)

of A is densely defined on (L 2 )

by oo

r(A)

for f = £ ~ = 0 I „ ( / » ) . For pEB., be the domain of T(A)p. If p < respect to the norm || • || p . T h e n || • ||p. It is easy t o see t h a t for p (E)_p. Moreover, for any p £ R ,

let | M | p = | | r ( A ) ^ | | 0 . If p > 0, let (E)p 0, let {E)p b e the completion of (L 2 ) with (E)p,p G R , is a Hilbert space with t h norm > 0, the dual space (E)* of (E)p is given by we have the decomposition oo

71 = 0

where Hn is the completion of { I „ ( / ) ; / £ E^n} with respect to || • || p . Here n E^ is t h e n-fold symmetric tensor product of the complexification Ec of E. We also have Hnp) = { I „ ( / ) ; / £ E®^p} for any p e R , where E®Tp is also the n-fold symmetric tensor product of the complexification EQIP of Ep. T h e norm ||v?|| p of

IMIp=(E n! W)

- /«e^,

where the norm of EQ is denoted also by | • | p . T h e projective limit space (E) of spaces (E)p:p £ R is a nuclear space. T h e inductive limit space (E)* of spaces (E)p,p £ R is nothing but the dual space of (E). T h e space (E)* is called t h e space of generalized Poisson noise Junctionals. We denote by <§;•,• ^> the canonical bilinear form on (E)* x (E). Then we have «:$)¥p»=^n!(Fn,/„) 71 = 0

for any $ = £ ~ = 0 I „ ( F „ ) € (E)* and

{(-) = C ( £ ) - 1 exp{i(-,f)} £ (E), t h e U-transform can be extended to a continuous linear operator on (E)* by

w$(0=«<Mc»,

UE.

We have the following characterization theorem of the W-transform.

107 T h e o r e m 2 . 1 . Let F be a complex-valued function defined on EQ. Then F is the U-transform of some generalized functional if and only if there exists a complex-valued function G such that 1) for any £ and r\ in Ec, z G C, 2) there exist nonnegative

the function

constants

G(z£ + rf) is an entire function

of

K, a, and p such that

\G(£)\
V£ G

Ec,

3) F{Cl = G(e* - 1) for all £ G E. 3

T h e Levy Laplacian acting on Poisson noise functionals

Let $ be in (E)*. Then for the W-transform W$, there exists a complex-valued function G satisfying conditions 1), 2), 3) in Theorem 2.1. In particular, for any £ G E, the second variation of W$ is given by ^ " ( O f o - O = G"(e l « - l N x r ^ t C e * ) + G'(e l « - l ) ( i C e * ) ,

r,,(e£.

Fix a finite interval T of R . Take an orthonormal basis {Cnj^Lo C E for L2(T) satisfying the equally dense and uniform boundedness condition ( see Refs. 9, 12, 13). Let VL denote t h e set of all $ G {E)* such t h a t t h e limit ALU^)

= lim

-^X>*)"(0(Cn,C„) n=0

exists for any £ G Ec and is in U[(E)*}. Then we consider the Levy A i defined by

for $ G £>L- We denote a set of all functionals <& EVL for all r) G £ with supp(jy) C T c by 2?J.

such t h a t U$(i]) = 0

For any / G J5c™ with s u p p ( / ) C Tn, a Poisson noise functional y? = /

/(ui,...,u„)dAT(ui)---diV(u„)

is in D j and is equal to ^2 7= 0

-if

1 -v /

JTn

/(u)d<3(ui)"''

dQ(un-j)dun

Laplacian

108 Hence, the W-transform Utp of tp is given by

U
/(u)e if(ui) • • • e ^ - W

We p u t

D

" = { / - f^dNi-Ul^'''

diV

K); / e Ecn, suPP(/) c TA

for each n G N . Then for each n G N , D n is a linear subspace of (E) and therefore for —(p)

p G R , we can define a space D n

by the completion of D „ in (E)p with respect

to || • ||p. Define —(P) D 0 by the complex field C . T h e n for each n G N U {0} and —(p)

p G R, D n

becomes a Hilbert space with t h e inner product of (E)p. T h e Levy —(p)

Laplacian Af, becomes a linear operator defined on D n and p G R .

for each n G N U {0}

For each n G N U {0},p G R and tp G D ^ p ) , tfie

T h e o r e m 3.1.[Ref.l7] e<jiua£icm

holds.

4

A semi-group generated by the Levy Laplacian

If $ e ( £ ) * is expressed in t h e form £ ^ 1 0 $ „ , $ « e ^ p ) , n - 0 , 1 , . . . , then t h e expression is uniquely determined in {E)*. Then we can define the following spaces. if

Let aN(n)

- Ylt=* (|f|)

{

OO

for each N

G N U {0} and n G N U {0}. Set

OO

"\

^ ^ n ; ^ a ^ ( n ) | | ^ „ | | p < oo, tpn G D„ P , n = 0 , 1 , 2 , . . . S n=0

7i=0

J

for each JV G N and p G R . Then for any J V 6 N , EJV, P is in (-E) p and is a Hilbert space with norm ||| • |||JV ) P given by

(

oo

\

^QAf(n)||¥>n||M n=0

/

1

/

2

oo

,

(p=^2ipne.ENtP. n=0

109 P u t E Q Q ^ = fljvLi ^ff,p with the projective limit topology. For any TV 6 N U {0} and p G R , we define E_AT I P by the completion of Eoo, p with respect to norm ||| • | | | - w l P given by

(

oo

\ V2

^a A r(n)- 1 ||(^ n ||M ii=0

oo

, if = ] T ipn G E ^ p .

/

n=0

P u t E _ o o p = Ujv=i E_jv,p with the inductive limit topology. T h e n for p G R , we have the following inclusion relations: Eoo.p C • • • C E J V + I ) P C Ejy iP C • • • C E i ] P c E 0 i P C - C E

CE

C •••

C E — oo, P -

(p)

T h e space E ^ ^ includes D n for any n G N U {0} and p G R . T h e Laplacian A i can be extended to a continuous linear operator denned on E - ^ p into E_oo,p, denoted by the same notation A ^ , for each p G R . For each t > 0 and p G R , we consider an operator Gt on E_oo i P defined by

Gtip=J2e

t]%]


71=0

for ip = ^ ^ = o fn S E - o o ^ . Then we have the following: T h e o r e m 4 . 1 . [cf. Ref. 16] Let p G R. Then the family {Gt;t > 0} is an equi-continuous semigroup of class (Co) generated by AL as a continuous linear operator defined on E-oo p. Proof: Let p G R. For any $ G E - o ^ p , there exists AT G N such t h a t $ G E-N,P- Then, for any t > 0 t h e norm | | | G t $ | | | _ i v , P for $ = Y,7=o $ « e E - o o , P is estimated as follows:

2 P 71=00 OO oo

2 P n ==00

= 111$

2 -N,p-

110

Hence the family {Gt; t > 0} is equi-continuous in t. It is easily checked that G 0 = / , GtGs = Gt+s for each t, s > 0. We can also estimate that oo

| G t $ - G t o $ | | | 2 _ •N,p =

^ " ^ W

1e

-e

*\T\

'°m

\*r,

n=0 oo

<4^aJV(n)-1[|$n| 4|||$|

-N,p

< oo

for each t, t0 > 0. Therefore, by the Lebesgue convergence theorem, we get that lim G t $ — G ( o $ in

t—>t0

E-^p

for each to > 0 and $ £ E-oo :P . Thus the family {Gt;t > 0} is an equicontinuous semigroup of class (C 0 ). We next prove that the infinitesimal generator of the semigroup is given by A^,. For any <£ = Yln=o ^« ^ E_oo]P, there exists A f e N such that |||$|||_Ar,P < oo. We note that Gt$-$ t

= ^ajv+1(n) *

Ar$

-(JV+l),p

r

e

2

m -1

t

n=0

_n

"~ W\

By the mean value theorem, for any t > 0 there exists a constant 6 6 (0,1) such that \T\

<

\T\

Therefore we can estimate the following term:

*N +i W

1

e cm - 1

t

_n_ n

~ W\

= aN+i(n)

1

e

m -1

<4aw(n)-1||$„||2. By

n

iri

l*»l

111 and the Lebesgue convergence theorem, we obtain lim

G,$-$

2

= 0.

-A,,4

t-5-0

-(JV + l ) , p

Thus the proof is completed. 5

•

Stochastic expressions of t h e semigroup generated by the Levy Laplacian

Let {Xt\t given by

> 0} be a Cauchy process with the characteristic function of Xt

Take a smooth function rjT € E with TJT = TJTT on T. Let p £ R and set Gt = UGtU~x

on ^[Eoop] with the topology induced from E ^ p by the U-

transform. Then by Theorem 4.1, {Gt; t > 0} is an equi-continuous semigroup of class (Co) generated by the operator A ^ . Let { X ( ; t > 0} be an i5-valued stochastic process given by X t = £ + XtVT, £ & E. Then we have the following a stochastic expression of the operator Gt. T h e o r e m 5.1.

Let p be a real number.

Then we have

G > ( £ ) = E [ F ( X t ) | X 0 = f] for o l l F £ W [ E m i P ] . Proof: Using the similar m e t h o d in the proof of Theorem 7 in Ref. 16, we can obtain this theorem. • By Theorem 2.1 we can prove t h a t for $ and \& in (E)*, there exists a unique generalized Poisson noise functional whose W-transform is given by (U^)(U^). We call this generalized functional in (E)* the Wick product of $ and * in {E)*, denoted by $ o * , i.e. W ( $ o * ) = (W*)(W¥). Since for a; £ E* and rj £ E, t h e product e^a: is given by elT)x = x + (eIT? — l ) x , we can define a continuous linear operator Me,n on E ^ p characterized by M e i ,[o"=i(-, /i>](*) = ^ = i ( e 1 ^ , />> n £ N U {0}.

112 Here o? = 1 (-, fj) means (•, / j ) o • • • o (•, / „ ) and we note t h a t it is equal to /

/ i ( w i ) • • • fn(un)dNx(ui)

•••

dNx{un)

with s u p p ( / j ) c T , j = 1, 2 , . . . , n. T h e n we have the following. T h e o r e m 5.2. the equality

Let p be a real number. Gtip =

Then for any t > 0 and ip £ Eoo i P ,

E[MeixtnTip]

holds. Proof: P u t ip = JTn f\(ui)-• • fn{un)dN{ui)-•-dN(un) s u p p ( / ; ) G T, j — 1,2,.. . ,n. T h e n we have E[Me,xtr,Tlp(x)}

in E ^ p

=

E[*?=1{eiX,VT*,fi)}

=

E[e*ThX
= e~l^ip(x)

=

with

Gtip{x).

Let tp = ^ n = 0 tpn £ Eo^p. Then for any n £ N U {0}, ipn is expressed in the following form: Nc


Y,

<

fc„o"=i

(•'/!%)'

i

E,oo,p,

kl,...,fc„ = l

where (a^,

)fcXl...,*;„/ is a sequence of complex numbers and (/^ k)k,t is a

fc

sequence of functions in E^n.

Hence for any J V g N U {0}, we have

J2n\\\Me,Xt.T
M

Yvtnt-

lim

V aM fc1,...,fcn = l

J ] Ilk" l l l T V . p n=0

I

AXtr]Tl

Nt

oo

= V

.n

< OO.

o"

/• f[e] " N,p

M

hK

N,p

113 By the Schwarz inequality, ^ ^ L o E[|||M e .x e „ T (^ r l |||.| V | p] < oo for all N £ N U {0}. Therefore by the continuity of Gt we get t h a t oo

E[Me,xtVTip]

= ^E[Me.x„T^>„] 71=0 OO

= ^

Gt
n=0

= Gttp. Thus the proof is completed.

D

Acknowledgments This work was written based on the previous works K.S. and A.H.T. 17 and also K.S. 15 1 6 . This work was supported in p a r t by the Joint Research Project " Q u a n t u m Information Theoretical Approach to Life Science " for the Academic Frontier in Science and was also supported in part by J S P S - P A N Joint Research Project "Infinite Dimensional Harmonic Analysis" promoted by the Ministry of Education in J a p a n . T h e authors are grateful for the support.

References 1. D.M. Chung, U.C. Ji and K. Saito: Cauchy problems associated with the Levy Laplacian in white noise analysis, to appear in Journal of Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics, Vol. 2, No. 1 (1999). 2. T. Hida; Analysis of Brownian Functionals, Carleton M a t h . Lecture Notes, No.13, Carleton University, O t t a w a (1975). 3. T. Hida a n d N. Ikeda; Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, in: Proc. 5th Berkeley Symp. on Math. Stat, and P r o b . Vol. II, P a r t 1 (1967) 117 -143. 4. K. Ito; Spectral type of the shift transformation of differential processes with stationary increments, Trans. Amer. M a t h . Soc. 81 (1956) 253-263. 5. Y. Ito; Generalized Poisson Functionals, P r o b a b . Th. Rel. Fields 77 (1988) 1-28. 6. Y.Ito and I. Kubo; Calculus on Gaussian and Poisson white noises, Nagoya M a t h . J. , Vol. I l l (1988) 41-84.

114 7. Y.G. Kondratiev, J.L. Da Silva, L. Streit and G.H. Us; Analysis on Poisson and G a m m a spaces, Infinite Dim. Anal. Q u a n t u m P r o b a b . Rel. Topics 1 (1998) 91-117. 8. Y.G. Kondratiev, L. Streit, W. Westerkamp and J.-A. Yan; Generalized functions in infinite dimensional analysis, HAS Report, No. 1995-002 (1995). 9. H.-H. Kuo; W h i t e noise distribution theory, CRC Press, Boca R a t o n (1996). 10. H.-H. Kuo, N. O b a t a and K. Saito; Levy Laplacian of generalized functions on a nuclear space, J. Funct. Anal. 94 (1990) 74-92. 11. H.-H. Kuo, N. O b a t a and K. Saito; Diagonalization of the Levy Laplacian and Related Stable Processes, submitted to Infinite Dim. Anal. Q u a n t u m P r o b a b . Rel. Topics (2001). 12. P. Levy; Lecons d'analyse fonctionnelle, Gauthier-Villars, Paris (1922). 13. N. Obata; W h i t e Noise Calculus and Fock Space, Lecture Notes in Mathematics 1577, Springer-Verlag (1994). 14. K. Saito; A C 0 -group generated by the Levy Laplacian II, Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics Vol. 1, No. 3 (1998) 425-437. 15. K. Saito; A stochastic process generated by t h e Levy Laplacian, Acta Applicandae Mathematicae 6 3 (2000) 363-373. 16. K. Saito; T h e Levy Laplacian and stable processes, Chaos, Solitons and Fractals 12 (2001) 2865-2872. 17. K. Saito, K., A. H. Tsoi; T h e Levy Laplacian acting on Poisson noise functionals, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 2 (1999), 503-510. 18. A.H. Tsoi; L-transform, Normal Functionals and Levy Laplacian in Poisson Noise Analysis, Preprint (2000).

Quantum Information IV (pp. 115-126) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

A H A U S D O R F F - Y O U N G INEQUALITY FOR W H I T E NOISE ANALYSIS HUI-HSIUNG KUO Department

of Mathematics, Louisiana State Baton Rouge, LA 70803, U.S.A.

University

KIMIAKI SAITO Department

of Information Sciences, Meijo Nagoya 468-8502, Japan

Department

of Mathematics, Northwestern Evanston, IL 60208-2730, U.S.A.

University

A U R E L STAN University

A Hausdorff-Young inequality and Young inequality are proven for White Noise Analysis. The Fourier transform from the classical case is replaced by the 5transform or the Segal-Bargmann transform.

1

INTRODUCTION

The classical Hausdorff-Young inequality [10] says the following: Theorem 1 Let n G N. / / 1 < p < 2 and 2 < q < oo such that ± + ± = 1, then the Fourier transform is a bounded linear map from L P (K") into L 9 (R"). This theorem is related to the following Young inequality [10]: Theorem 2 Let n £ N. Let 1 < p, q, r < oo such that 1 + 1 = 1 + 1. / / / e Lp(Rn) and g £ Lq(Rn), then their convolution product f *g is in Lr(Rn) and the following inequality holds:

ii/*siir
BACKGROUND

We present below the basic background from White Noise Analysis. more details see [7] and [9]. (a)Gel'fand triples

For

116 Let £ b e a real separable Hilbert space. We denote the norm of E by | • |o- Let A b e a densely defined self-adjoint operator on E, having a countable spectrum and whose eigenvalues {A„}„>i satisfy the following conditions: • 1 < Aj < A2 < A3 < • • •.

• ir =1 A„- 2 0, let £p = {/ e E | | A P / | 0 < oo}. For any / e £ p , we define | / | p := | A p / | o . We observe t h a t £p is a Hilbert space with t h e norm | • | p . T h e first condition implies t h a t £q C £p, for any q > p. T h e second condition on the eigenvalues above is equivalent to t h e fact t h a t A is a Hilbert-Schmidt operator from E into itself. Thus, for any p > 0, the inclusion m a p i : £p+\ —> £p is a Hilbert-Schmidt operator. Let £ = p L x ) £p- T h e space £ equipped with the topology given by the family {| • | P } P >o of norms is a nuclear space. We denote by £' the dual of the nuclear space £ and by £' t h e dual of the Hilbert space £p, p > 0. For any p > 0, the space £' is isomorphic to the space £-p, which is the completion of t h e initial Hilbert space E with respect to the norm | • | _ p defined in the following way: if / S E, t h e n | / | _ p := | A _ p / | o . In this way the dual of EQ = E is isomorphic to EQ = E. Therefore we have identified the dual of the initial Hilbert space E with E by Riesz representation theorem. If 0 < p < q, then £_p C £-q- It turns out t h a t £' = (J P >o ^-p- T h e space £' is equipped with the inductive limit topology of the topologies of the increasing family of Hilbert spaces £ _ p . T h e duality between £' and £ is denoted by (,). If x £ E and £ 6 £, then (a;, f) is t h e inner product of the elements x and £ in E. If 0 < p < q, then we have the following continuous inclusions: £ C £q C £p C E c £'p C £'q C £'. T h e following sequence of inclusions £CEC£' is called a Gel'fand triple. By Minlos' theorem there exists a unique probability measure /i on the Borel subsets o f f such t h a t for all £ 6 £, the random variable (•, £) is normally distributed with mean 0 and variance |£|g. T h e characteristic function of fi is given by / ei<>x'S)dn{x) = e-*W°, it'

Vfe£.

(1)

117 Because £ is dense in E, we m a y define for any / £ E a r a n d o m variable (•, / ) on £'. This r a n d o m variable is normally distributed with mean 0 a n d variance | / | 2 . We call t h e probability space (£', n) a white noise space. We denote by (L 2 ) t h e space of all functions ip : £' —I C t h a t are measurable a n d square integrable with respect to t h e measure /J,, t h a t means J£l \ip(x)\2dfj.(x) < oo. Ec denotes t h e complexification of E, while Efn t h e space of all symmetric n-tensors in t h e space Efn. Throughout this paper a h a t above a tensor will denote t h e symmetrization of t h a t tensor. For example if u, v E Ec, then u®v = ( l / 2 ) ( u (g> v + v ® u). For any x 6 £' and n G N U {0} we define the Wick tensor : x®n : 6 £'®" by t h e formula: [»/2]

/

x

= Z(i)(>»-»)«(-Dk ®{n-2k)X®k x

fc=0

x

'

where r is t h e so-called trace operator, which is t h e unique linear bounded operator from £c ® £c into C such t h a t {T,Z<8>T)) = (t,T)),

Vf.fjGfc

2

For any ip G (L ) there exists a unique sequence {/„} n >o such t h a t Vn > 0, / „ 6 Ef",

and

n=0 2

T h e above equality is in t h e (L )-sense a n d from it we can easily calculate the (L 2 )-norm of t h e function ^! i/. n

VII5

n=0 n

where | • |o denotes t h e £ ® - n o r m induced from t h e norm | • | 0 on E. We define the second quantization operator of A, denoted by r ( A ) , as a densely defined self-adjoint operator on (L2) in the following way: If*> = E r = o ( : - ® n : > / n ) , t h e n OO

r(AV=^(:-®":,^® B /«>. n=0

118 T h e operator T(A) has also a countable spectrum. Its eigenvalues satisfy the same conditions as those of the operator A except t h a t the smallest eigenvalue of T(A) is not strictly greater t h a n 1, but equal to 1. For any p > 0, we define (£p) = {

0(£p). T h e space (£) equipped with the family {|| • || P } P >o of semi-norms (In fact norms) is a nuclear space. For any p > 0, the dual of the Hilbert space (£ p ) is isomorphic to t h e Hilbert space (£-P), which is t h e completion of the space (L 2 ) with respect to t h e norm || ip | | - p : = | | T(A)~pip || 0 . If we denote by (£)* the dual of the nuclear space (£), then (£)* — U p > o ( ^ - p ) - T h e elements in {£) are called test functions on £', while t h e elements in (£)* are called generalized functions on £'. T h e bilinear pairing between (£)* and (£) is denoted by ((•,•))• If

/ „ G £fn. Moreover, OO

iM£ = X>!i/»ip<°°>

v

^°-

n=0

In the same way, any element

= E r = o ( : '®" : > F ")> Fn G (^c)®"- K 0 G (£)* has the above representation, then there exists p > 0 such t h a t oo

= $ > ! | F„ |2_p< oo.

H\\-P

n=0

If = E r = o ( ; •*" ^Fn) following relation holds:

e (£)* ^ d ^ = £ ~ = 0 < : •«» : , / „ ) G (£), then t h e oo

<(*,¥>» = 2 > ! < F „ , / n > . n=0

(7>J T/ie Exponential

Functions

and The

S-Transform

For any x G ££, w e define the renormalized exponential function in the following way: oo

{ x)

:e ' :=^2^(-.-®n:,x®n). n=0

:e^',x':

119 A simple calculation shows t h a t , for any p £ R , \\:e^:\\p

= e ^ .

(2)

Hence for any x G £'c, we have : e ' ' , z ' : G (£)*• Moreover, it follows from (2) t h a t -.e^'^-.e (L2) if a n d only if x G Ec, and :e< ,,ae) : G (£) if a n d only if x G felt is also easy to check t h a t if x G £ c and £ G £ c , then ( ( : e < - ' * > : , : e « > : ) ) = e<*'«>.

For any £ G Ec, we have: : e <*,«>._ e <>:,0-(i/2)<«,0_

T h e renormalized exponential functions {:e(''&: \ £ G £c} are linearly independent and span a dense subspace of (£). For any $ 6 (£)*, the S-transform of $ is the function S ( $ ) : £ c -> C denned by

S($)(0 = «*,:e<--«>:»,

£ G £c.

Since t h e renormalized exponential functions span a dense subspace of (£), t h e S-transform is injective. T h a t means, if $ , $ G (£)* such t h a t S ( $ ) = S(tf), then $ = * . For $ G ( £ 2 ) , t h e S-transform of $ is also called the Segal-Bargmann transform of $ . (c) The Wick

product

T h e Wick product of two generalized functions $ and ty in (£)*, denoted by $ o $ , is the unique generalized function in (£)* such t h a t :

S ( $ o * ) = (S$)(S#). T h e mapping (, t/>) H-» 0 o i/> is continuous from (£) x (£) into (£). It is also continuous from (£)* x (£)* into (£)*. If y = E ^ 0 < : -®n : , / „ ) G (£) and V- = £ r = o < : -®" :,fln> G (£), then

n=:0

p+q=n

120 3

HAUSDORFF-YOUNG INEQUALITY FOR WHITE NOISE ANALYSIS

A starting point in formulating a Hausdorff-Young inequality is t h e following theorem established by t h e contribution of Kondratiev [3], Kree [4, 5, 6], a n d Lee [8]. T h e o r e m 3 The S-transform is a unitary operator from (L2) onto the Hilbert space rlL2(Ec,m) consisting of all holomorphic functions f : Ec —> C such that || / | | 2 = supljp, \f(z)\2dmp(z)} < oo, with the supremum being taken over all finite dimensional subspaces F of Ec. If F = Cn, then dmp(z) = (•7r)_n exp[— |z| 2 ]
real number q, we define the space:

oo

! //

V =

oo

G {£)* I £

£"=„(=

n=0B

-®

:./»>

e

(L»),

^

(f) n " ! l/«lo < °° •

n=0

iAen we de/ine

||

J

p

12

ll(q)._

E~=o(f)"»!|/-l8D e f i n i t i o n 5 For am/ rea/ number q > 1, we define the space HLq(Ec,m) consisting of all holomorphic functions f : Ec —> C such that || / ||?i = sup{f_ \f[z)\qdmF(z)y < oo, with the supremum being taken over all finite dimensional subspaces F of Ec. If F = C™, then dmp(z) = (7r) _ ™e~' z ' dz, where dz = dxdy, for z — x + iy G C " , x, y G R". O b s e r v a t i o n 6 f o r any <7i, 92 G R s w c ^ ^ ffl * ° < 9i < 92, ( £ , : ! ) C ( £ ? 1 ) - / / ¥?G (L« 2 ), then || v? || ( q i ) < ||
II ^

II',] =

SU

e

is a bounded

( L ")-

\S(p{z)\qdmF(z) J-

P\

= sup < / \S(ip o ip o • • • o V

k times

ip)(z)\2dm,F{z)

W e

have:

121 By Theorem 3 we have: sup < / \S(<po<po- •• oip)(z)\2dmF{z) JF

'

v k times

\

> = || ipoipo-

'

•• o ip ||Q .

v k times

'

'

T h u s , we obtain: II Sip \\q[q] = || ipoipo

•••op

\\l

k times

= £n! n=0

E »i+*aH

^En! n=0

/
E Lti+t2H

l/ii®/i,®"'®/n Hfc=n

Since the symmetrization operator is a projection operator, we have: l/ii<8>/*a<8» • - • <8>/«fc|o < | / n ® / « , ® " - < 8 > / d o Hence we get:

S^"9

[9]

<£n!

.»i+«j-|

oo

= En=0 oo

=£

i/ii ® /<

E

n—0

•••®/u|o

hifc = n T 2

!

E .«l+ijH

l/«i|o|/ij|o ••• | / u | o htfc=n

E

^l/iilo|/i,k--|/Jo

.«i+»H

E

Nk=n

E

V • r T

n = 0 Lti+t 2 H

• ,V/^i|/nloV/^|/i2|o---\Afc!|/iJo

hik=n

Applying the Cauchy-Bunyakovskii-Schwarz inequality we get:

E

E

71 — 0 .11+12-I

hu =™

ii!i 2 !---ife!

V ^ l / i i lo V^-lfi,

lo • • • V ^ l / u |o

OO

<E n=0 ii+tj-l

E

E hu=n

»i+»j-l

*i!|/lJo*2!|/i2lo---^!|/ulohu=n

122 Hence we obtain:

II SV ||f„ oo

sE

E

n=0t,+i2H

%i

h«k=n

2o

'

= y:(i+i+---+i)"

= Efen n=0 oo

E ii+iH

n=0 ii+t2H fc

•A «i+iH

E >i+«2H

fc times

Hk=Ti

E

• • • Zl>

'

HViAW.\fiAl---ik\\fik\l h»k=n h»k=n

Mii/^isi2!i/ijg ••• ikVi«Jo k Mk=n

»i!iAlis*a!iA,is ••• w j s

htfc=Ti

oo

i = l iJ=0 oo

£(i)'«i/.e = IMI?.) • Thus we obtain: Sp | | H < || ip ||(,) .

(3)

This proves t h a t t h e linear operator S : (L 9 ) —» ~HLq{Ec,m) is continuous and its operatorial norm || 5 || satisfies the inequality || S ||< 1. To see t h a t || S ||= 1, let's consider the function ip = e^''v', where r\ S E, |r/|o = 1. Let F b e the one dimensional complex vector subspace of Ec spanned by r). We have: oo

Hif„ = E (!)"-•

-Edrf lH5

= e ^

123

Hence || ip \\(q) — e*. We have: II S || • || v || ( „ > || Stp | | w V?

-\nfF lS^)l"<W(0

1/?

= \lj |e<"-«|'d^(0 1/9

= f~ /" |e<"'z'»>|«e-lz,'l2dz =

[\e*\<>e-^dz1/9

\i

y|9P-^2-y2da.dj/

1/9

L ^ ./1R ./R

l

= \ ff'

1/9

•ixe-x2-y2dxdy

L71" J R ./M

eVe-V+9^-^

LVWK

dx-= —

// 'e - " dy

1/9

1/9

7-A

-(-iYdx

3

= imi»Hence || S ||> 1 and so || S ||= 1. Therefore we have proved the theorem for any q 6 {2, 4, 6, • • •}. • If B : E —> E is a bounded linear operator, then we may define the second quantization operator of B, denoted by T(B), as a densely-defined operator on (L 2 ), in the following way: if

= 53(:.®-: > B®"/n>Observation 8 If I denotes the identity operator on E, then for any q > 2 and ip G {Lq), we have

rU/f/]v

(4)

124 Thus the last theorem can be rephrased in the following natural number q and for any if £ (Lq), S

¥> \\[q] <

way: for any

r\Jii)v>

even

(5)

T h e following theorem is a n analog of t h e classical Young inequality. T h e o r e m 9 Let p, q, and r be strictly positive numbers such that - + - = -. If ip £ (Lp) and tp € (L9), then

£ (Lr) and the following inequality holds: ll(r) < || V|l(p)l| V> ||(g) •

(6)

Proof. Let

€ ( i p ) and t/> = £ ~ = 0 < : -®" =.S») G (L»). Then y> o V = £ r = o < : -®" :,/»„>, where hn = E u + „ = „ /«®
II V o V» ||?r)

n—0 oo

-£(£)"*

53 /«®5*

n=0 oo

^5Z©"n! ra=0 oo

- S(D =

1 3 l/«®0»lo L"U+f=n

n!

£ l/«®S»lo

n=0 oo

|_u+i>=n T

71=0

/u-f-u=n

£(D"n! £

= £(£)"

IUIU n!

_ii+v=n

/2\

u!u! y p

u

/2

x t

'

(f)^|A|o/(ff^k|o

Applying t h e Cauchy-Bunyakovskii-Schwarz inequality we obtain:

125

Thus, we have:

^G)rE^G)"a)'i:(i)>/.B i H* =f:(i)'(; + j)"E(f)'-i/.B(!r-w -t(i)"(;)"E(i)'-l/.B(§)--W5 oo

= E E (?)"««(§)"i« = E(|)"«n«E(l)"*.K u=0

t>=0

= IMI( P )IIV>|l( g )D References 1. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, P a r t I; Communications of Pure and Applied Mathematics 2 4 (1961) 187-214. 2. Gross, L. and Malliavin, P.: Hall's transform and the Segal-Bargmann map; in "Ito's Stochastic Calculus and Probability Theory", N. Ikeda, S. W a t a n a b e , M. Fukushima, and H. Kunita (eds.), 73-116, Springer-Verlag, 1996. 3. Kondratiev, Yu. G.: Nuclear spaces of entire functions in problems of infinite-dimensional analysis; Soviet Math. Dokl. 22 (1980) 588-592. 4. Kree, P.: Solutions faibles d'equations aux d'erivees functionnelles, I; Lecture Notes in Math. 4 1 0 (1974) 142-181, Springer-Verlag. 5. Kree, P.: Solutions faibles d'equations aux d'erivees functionnelles, II; Lecture Notes in Math. 4 7 4 (1975) 16-47, Springer-Verlag. 6. Kree, P.: Calcul d'integrales et de derivees en dimension infinie; J. Fund. Anal. 3 1 (1979) 150-186.

126 7. Kuo, H. H.: White Noise Distribution Theory, Probability a n d Stochastic Series, C R C Press, Inc. 1996. 8. Lee, Y.-J.: A characterization of generalized functions on infinitedimensional spaces and Bargmann-Segal analytic functions; in "Gaussian r a n d o m Fields", K. It6 and T. Hida (eds.) (1991) 272-284, World Scientific. 9. O b a t a , N.: White Noise Calculus on Fock Space, Lecture Notes in M a t h . 1 5 7 7 , Springer-Verlag (1994). 10. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness, Academic Press, 1975. 11. Segal, I. E.: Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 8 1 (1956) 106-134. 12. Segal, I. E.: Mathematical characterization of t h e physical vacuum for a linear Bose-Einstein field, Illinois J. Math. 6 (1962) 500-523. 13. Segal, I. E.: T h e complex wave representation of the free Boson field; in "Topics in functional analysis: essays dedicated to M.G. Krein on t h e occasion of his 70th birthday", Advances in Mathematics: Supplementary Studies (I. Gohberg and M. Kac, eds.), vol.3, Academic Press, New York, 1978, p p . 321-344.

Quantum Information IV (pp. 127-145) Eds. T . Hida and K. Saito © 2002 World Scientific Publishing Co. THE CLARK F O R M U L A OF GENERALIZED

WIENER

FUNCTIONALS

YUH-JIA LEE* Department of Applied Mathematics National University of Kaohsiung Kaohsiung, TAIWAN 811 H S I N - H U N G SHIH f Center of General Education Rung Shan University of Technology Tainan, TAIWAN 710 The Clark formula is reformulated on the classical Wiener space for a large class of generalized Wiener functional via the idea of white noise analysis. The underlying spcae is taken to be the Gel'fand triple E C C C E* associated with the space C of Cameron-Martin functions. Then we adapt the space £Q of analytic function of exponential growth as the space of test functionals in the construction of the space £Q of generalized Wiener functionals. It is shown that the S-transform SF of a generalized Wiener functionals F satifies the following formula SF(V)

= <(F, 1 ) ) ^ + / iJo at

SF(Pt(r,))dt,

which follows easily from the fundamental theorem of calculus, where n £ Ec, the complexification of E, and Pt{n)(s) = J7(min{s, (}). The the Clark formula follows immediately.

1

Introduction

T h e representation of functionals of Brownian motion by stochastic integral with respect to Brownian motion, known as t h e Clark formula, was first studied in [l] a n d later in [20, 5]. I t was generalized t o weakly differentiable functionals on t h e classical Wiener space C by Ocone [20] via Malliavin calculus. Later, it was generalized t o Malliavin distributions in [6, 22]. T h e first reformulation of t h e Clark formula using H i d a calculus (or w h i t e noise analysis) has been done in [2]. In this p a p e r we are devoted to t h e reformulation of t h e Clark formula on t h e classical Wiener space for a class of generalized Wiener functional via "^RESEARCH SUPPORTED BY THE NATIONAL SCIENCE COUNCIL OF TAIWAN

128 the idea of white noise analysis. T h e underlying spcae is taken to be the Gel'fand triple E C C C E* with a chain of Hilbert spaces Ep with the norm \x\p = \K~p/2x\o, where C is the Cameron-Martin space, Kx(t)

=

mm{s,t}x(s)ds,

x£L2[0,l],

Jo Ep is t h e domain of K~p'2 for any p > 0, and E = n p > 0 Ep is t h e projective limit of Ep's. T h e dual space of Ep is denoted by E_p. To construct t h e generalized Wiener functional, for p > 1, let £p be the space of functionals / defined on £ L p such t h a t (1) / has a analytic extension / to the complexification i ? - p , c of

E-p,

(2) There exist constants c,c' such t h a t |/(.z)| < c e x p { c ' | z | _ p } for z G

E-PtC.

Denote by £Q the corresponding space defined on C similarly as above . Set Eva = flp^j^p. £oo is t h e collection of analytic version of members of the Yan-Meyer space and we have t h e follwing relations £ „ C £P C £0 C L2[C, fi] C £; C £*p C

C

where /J, is t h e Wiener measure on C and £,*>* denote t h e dual space of £oo . Members of £* and £^ are called generalized Wiener functionals. In this paper we show t h a t the S'-transform SF of a generalized Wiener functionals F G £Q satisfies t h e following formula

SF(r,) = ((F, 1))^ + J j t SF(Pt(r,)) dt, which follows easily from the fundamental theorem of calculus, where 77 £ Ec, the complexification of E and Pt(r])(s) = r7(min{s, t}). T h e Clark formula of generalized Wiener functionals in ££ follows immediately. In particular, for ip 6 £1, we obtain ip(x) = E [ < ^ ] + /

H[D(p(x) l\t,i] I Tt ] dB(t; x)

fx-almost all x on C,

Jo where the integral is the Wiener-Ito stochastic integral. T h e following notations will be used frequently in this paper. Notations. 1. For a real linear space V, Vc denotes the complexification of V.

129 2. For each continuous function x on [0,1], x(t), x(t), x^p'(t) are respectively the first, second, and p-th order derivatives of x with respect to t £ [0,1], if they exist. 3. For a n-linear operator T on a complex normed space X, T(z,... , z) (n copies) for each z £ X.

Tzn

:•=

4. For any Hilbert space H, \\ • \\HSn^H^ denotes the n-linear Hilbert-Schmidt operator norm and || • ||,c»(#) t h e n-linear operator norm over H. 2

A G e l ' f a n d t r i p l e a s s o c i a t e d w i t h t h e classical W i e n e r s p a c e

Let C b e the collection of all real-valued continuous functions x on [0,1] vanishing at zero and the subclass C' of C consist of all absolutely continuous functions x with x E L2[0,1]. Then C is a Banach space with the sup-norm | • |oo and C is a Hilbert space with norm | • |o = \ / ( - , -)o and the inner product (-,-)o defined by (x, y)Q — L x(t)y(t)dt. T h e space C is usually called the Cameron-Martin space and it is well known t h a t (C',C) forms an abstract Wiener space (AWS, for abbreviation). T h e dual space C* of C, may be identified as a dense subspace of C given as follows Fact 2 . 1 . 2 3 ' 1 7 C* = J x £ C : x is right continuous and of bounded variation with x{\) = 0 By this identification, whenever x £ C and y £ C*,

(x,y)=

- f

x(t)dy(t),

Jo where (•, •) is the C-C* pairing. In particular, if x £ C , (a;, y) = (x, Let K be a bounded linear operator on L 2 [0,1] defined by Kx(t)=

/

(sAt)x(s)ds,

y)0.

i£L2[0,l],

Jo where s At denotes the minimum of s and t. T h e n K is a positive self-adjoint compact operator having a complete orthonormal basis (CONS, for abbreviation) consisting of functions {en(t) = \ / 2 sin(n — l/2)-7rt : n = 1 , 2 , . . . } with the corresponding eigenvalues given by { l / ( ( n — l/2)7r) 2 : n — 1, 2 , . . . }. For n £ N, let /„(*)=

y/Ken(t),

t£[0,l].

(2.1)

Observe t h a t , for each n £ N and for t £ [0,1], e „ ( l - t) = ( - l ) n + 1 fn(t), then, by Fact 2.1, {/ n : n £ N} C C*. Moreover, {/„ : n £ N} is also a CONS

130 of L 2 [0,1]. Thus, {/„ : n G N} is a CONS of C. Let A b e t h e inverse operator of \/K on C. Then A is self-adjoint densely defined on C and A fn = A n / „ for each n g N , where A n = (n — 1/2) 7r For any p > 0, let i? p b e the domain of Ap. Then Ep is a real Hilbert space with the norm \x\p = \Apx\0 (C' = Eo) and {(1/A P ) / „ : n G N} forms a CONS of Ep. The increasing family {| • | p : p > 0} of norms are compatible and comparable; and the embedding from Ep+a into Ep is of Hilbert-Schmidt type whenever a > 1/2. Next, let E_p b e t h e completion of C with respect to the norm \x\-p = |^4 _p a;|o- Then E-p is a Hilbert spaces with a CONS {A£ f„:ne N } . Identify z G £ £ with £ ~ = i (a, / „ ) / „ G £ _ p , where (•, •) is the E*-Ep pairing. £ L p becomes the dual space of Ep. Set E = n p > 0 Ep as the projective limit of .E p 's. Then E is a nuclear space with the dual E* = U p > 0 £ - p and E C C C E* forms a Gel'fand triple . Observe t h a t , for x £ C a n d p > 1,

n=l

n=l

Jo

V^O

x ( t ) 2 dt < \x I 2

/

•

I co-

l t follows t h a t C C E-p for all p > 1. Furthermore, by t h e denseness of C in C and in E-p, we have the following chain of continuous inclusion: EcEpcEqcE1=LZcC

cC

cCcL2

= E-iCE-qC

E-p

C E*,

2

w h e r e p > q > 1 and L2 denotes t h e space L [0,1]. For notational convenience, we will use the notation (-, •) to stand for all the dual pairings of E*-E, E-p-Ep (p > 1), and C-C*. Let »o and i _ p be respectively t h e embeddings from C into C and E-p with p > 1. Then ( C ' , B _ p , i _ p ) is a AWS. Let \i and H-p, p > 1, b e the abstract Wiener measures of C and E-p respectively. Then t h e measurable support of ^t_ p is contained in C and, for any integrable complex-valued function

ip(x) fj,_p(dx)

JE-„

—

/ tp(x)

n(dx).

Jc

P r o p o s i t i o n 2.2. Forp € N, Ep is the class consisting of all functions x € C' with the property: (i) x, x,... , a;' p ' are absolutely continuous with x ^ p + 1 ' S L 2 [0,1] and (ii) z(2fc>(0) = z ( 2 f c + 1 ' ( l ) = 0 for k = 0 , 1 , . . . , [ p / 2 ] . Moreover, (x, y) — -

/

Jo

x(t)dy(t)

for x £ L2 and y G Ep.

(2.2)

131 Proof. Denote by U the class of functions satisfying the conditions (i) and (ii). We shall show t h a t U = Ep. Let y be in Ep. Then y = E ^ L i (l/> fn)o fn in Ep(c defined as in (2.1). Since en = A„ en for each n G N,

C), where / n ' s are

oo

y(t)=

£

(1/A„)(y,/„>oCn(0

(in L2)

n=l

=

£

A„(y, / „ ) 0 /

e„(s)ds

(in L 2 ) ,

£ G [0,1].

Let yx = 2^°=i A n ( y , fu)oen. Then y, G L 2 and y(t) Jt yl(s)ds for all t G [0,1]. T h u s y is absolutely continuous with y ( l ) — 0 and y=yl. As p > 2, let t/2 = E ^ L i A« (^; /n)o e n - T h e n this series converges to y2 in L 2 and y(t)=

!/.(*)=

/" y 2 ( ^ ) ^ , Jo

ie[0,l].

It implies t h a t j / is absolutely continuous with y(0) = 0 and j / 3 ' = y2 £ i j . Continuing in this way, it is easy to see t h a t Ep C U. To show t h a t U C Ep, let j / G *7 be arbitrarily given. Then y ( p + 1 ) G 2 L [0,1] and hence oo

E

I

/

!/

(f+1)

\

1 _ (-l)P

/•!

W ^ cos(

-

J

2

—ir-(n-l/2)'irt)dt

<+oo.

n = l | Jo

Applying integration by parts formula, we see t h a t E ^ L i tf? (y> /n)o < + ° o . T h a t is, y G i? p . This proves t h a t U = £ p . Finally, from Fact 2.1 it follows t h a t for x G C and y G -Ep,

{x,y)=

f x(t)dy(t) = -f Jo

x(t)y(t)dt. Jo

Extending x to the whole space L 2 by continuity, we obtain the formula (2.2). •

C o r o l l a r y 2 . 3 . The nuclear space E consists of all real-valued almost everywhere differ•entiable functions x defined on [0,1] suchn that x'-2 > (0) = 2fc 1 + )(l) = 0 for each k G N U {0}. a;( It follows from the definition of the | • | p - n o r m t h a t we have

132 Corollary 2.4.

(i) ForpeN

and x G Ep, \x\2p = J* |a;(P +1 )(<)| 2 dt

(ii) For x E L2 and p G N, 2

laslip-i =

/ / •••/ / / J0 I Jap Jts JO Jti

x{s)dsdt1dt2---dtp-1

where ap — tp and bp = 1 if p is odd; ap = 0 and bp — tp if p is even. 3

T h e spaces of test and generalized functions

If V is a real normed space with the | • |y-norm, then the | • Iv^-norm of Vc is given by \x + iy\vc = sup{||e , e (a; + i3/)||v e : 0 € [0, 2n]} for x,y 6 V, where ||ac + iy\\\r = \x\\r + \y\y. In general, || • \\vc is a quasi-norm and for any i , i / £ V, we have t h e inequality: ||a: + *2/||ve < \x + iy\vc

<

IF +

iy\\vc-

Note t h a t when V is a Hilbert space, the quasi-norm || • Hv^ coincides with the | • 114-norm. In this section, we construct test functionals on t h e AWS (C',C) and the AWS ( C , -E_p) for p > 1. For notational simplicity, we use t h e symbols | • l ^ and | • | _ p to stand also for | • \cc and \-\E^ C, respectively. For a fixed Banach space B which is either C or £-p, let £(B) be the class of those functions ip defined on C so t h a t ip has an analytic extension ip(z) to Bc and satisfies the exponential growth condition: |<£>(;z)| < c e x p { c ' | z | s e } for some constants c,c' > 0. It is clear t h a t £(B) C L2(C,fi) by the Fernique theorem (see [7]). For m G N and ip G £(B), define |M|Cm(B)=

sup{\lp(z)\e-m^^

: z G Bc}.

Let £m(B) = { p = £m(Ep), £ m i 0 = £m{C), £p = £{E-p), and £0 = £(C). T h e n we have the following chain of continuous inclusions: £<x = n p > i £p C £p C £q C £o C L2[C, n]

as p > q > 1,

where E^ is topologized as the projective limit of {£p}. T h u s a sequence { 1, ipn conveges to tp in £p as n —>• oo. £oo will serve as the space of test functions for the generalized functions in our investigation.

133 For / 6 L2[C,[i], the Wiener-Ito decomposition theorem (see [8]) assures t h a t / can be decomposed into an orthogonal direct sum of multiple Wiener integrals / „ ( / ) of order n, n £ N. Let fj. * f b e the convolution of / and \i defined on C, i.e., /i * f(h) = Jc f(h + y) fi(dy) for h £ C. Then /i * / is infinitely Frechet-differentiable in t h e directions of C' and In{f) can be represented by / „ ( / ) = L 2 [C,/i]-lim

i

Jc Dnfx*f(0)(Pk(-)

+

iPk(y)r^(dy)

(see [10, 12]), where Z?" denotes the n - t h Frechet derivative in the directions = Yl)=i {z, fj) fj

of C and Pk(z)

for z

°°

f

G E*. Moreover, 1

/ |/(x)|V(^)= E -||^V*/(o)||2H5»(c.)Next, we briefly describe t h e space (E) of test functionals, which was introduced by Meyer and Yan (see [16, 19]). For m £ N and p > 1, define the II • ||m,p-norm on L 2 [C,/i] by oo

1

ll/ll^,P= E ^IU>V*/(o)|lW_,)71 = 0

''

(3-i)

l

Denote by {E)m>p the class of functions / in L2[C,fi] so t h a t | | / | | m , p < + o o . Then {[E)m

! (E)p, equipped with the projective limit topology. Then the following chain of continuous inclusions hold: (E) C (E)p C (E)q C L2[C,n]

whenever p > q > 1.

Obviously, each member / of (E) satisfies oo

E n=0

l

-T||0>*/(O)|&s»(i5_J>)<+oo n

forallp>l.

-

According to the work by Lee [13, 15], there exists a unique analytic function / defined on E* such t h a t f — f (J.-almost everywhere on Cc and

/(*)=

E i / ^ * / > + «'y)VW.

Z

^K,

n=o n\ Jc where the series converges absolutely and uniformly on bounded subsets of E*. f £ £00 and the space E^ is exactly the collection of analytic version of members of (E). We reformulate t h e related results and growth estimates as follows.

134 Theorem 3.1.

16

(i) Let f be in (E). For any p>l,

let mp £ N so that f £ (E)m

iP.

Then

ll/IUn.,., < Cm, \\f\\smp,p,

(3.2)

m y

where cm — Jc e \ \°° n(dy) for any m £ N. (ii) Let f be in EQO. For any p > 1, let mp £ N so that

In2/(2In7r — 2 I n 2 ) , and choose fhp be the least integer greater than Jmp e. Then

I M | m „ p - < 0mptP\\f\\€mp,p, where f3m (iii)

p

is a constant

depending

(3.3)

only on p and mp.

The mapping if —> /j, * tp is a homeomorphism let if be assumed as above. Then

from E^ onto E^. In fact,

(!/<*) I M I ^ , . , < |M *
(3.4)

mp

e .

(iv) Sao C (E) and 8^ = (E) as vector spaces, where (E) = {/ : / £

(E)}.

Proof. T h e statements (i), (ii), a n d (iv) follow from [16: Theorem 4.1 and Theorem 4.4]. For t h e statement (iii), it is obvious t h a t \\fj, * 1, let m p £ N so t h a t i/) £ £m ,p- Then, by t h e Cauchy integral formula, \Dnyj(Q)zn\ < em"n \\ip\\emp,p \z\% far each n £ N. Thus, for any z £ E-P:C,

l/(*)l<

E n=o

- / n:

|Z?nV(0)(z + iyr|/x(d2/)

7c

< I M U „ P . P Jc e x p { e m - ( | z | _ p + | y | _ p ) } /i(dy). Dividing b o t h sides of t h e above estimation by e fc l z l-», we obtain (3.4).

D

135

Example 3.2. (1) F o r / i , / 2 , . . . ,/„ 6 C, define : fi---fn

• {x)= / f[{x + iy,

fj)n{dy).

Jc j=i Then : j \ • • • fn :

G L 2 (C, /i). It is worth to note t h a t M / i ® " - ® fn)

=

•h---Jn--

Moreover, for r ? i , . . . ,r]n G Ec, we have • rji • • rj„ : G £oo(2)

T h e exponential vector functional e(r]) associated with r\ £ Ec which is given by

e(»j)= e x p | ( - , » , ) - | jT 77(t) 2 d*j. Then 6(77) G £ooFor more examples, we refer the reader to [17]. Remark 3.3. (a)

Identify (E) with £00, then Theorem 3.1 implies t h a t two families of norms {|| • ||m )P } and {|| • \\e } are equivalent. In other words, t h e space £x is equivalent to t h e Yan-Meyer space (E).

(b)

Let / j / s be functions as given in (2.1). T h e n for any ip G £oo, t h e series

E -

E

D«» *
fu ••(*)},

converges to y> in £00 and, for any p > 1, the series also converges absolutely and uniformly on each bounded set of E-pc. As a consequence, the linear space V spanned by all cylinder polynomials of the form 0((-,77i),... , (-,»7n)) for any n G N U {0} is dense in £00 and in £p for p > 1, where 1), and £00 respectively by £ Q , £*, and £^ which are topolozied by the weak*-topologies. T h e n we have t h e following chain of continuous inclusions: £oo C £p C £q C £Q C L2[C, H] C £0* C £* C £1 C ££,,

whenever p > g > 1.

136 Members of £^0 will b e referred as the generalized Wiener functionals . Next, let ((•, -)) 0 , ((•, -)) p , and ((•, -^oo stand for the dual pairing of £Q-£Q, £p-£p, and E^-Soo, respectively. E x a m p l e 3.4. Denote by Llxp[C,/j] / denning on (C, B(C)) such t h a t J f(x)emW°°

the space of all measurable functions

n(dx)

for all m 6 N.

By the Fernique theorem, L* xp [C,ju] can be regarded as a subspace of £Q by identifying each / £ L\ [C,fi] with the functional Gf defined by {(Gf,
J

f{x)
for if £ £Q. Members of Ll [C,JJ\ will b e called regular generalized Wiener functionals. For more examples we refer the reader to [17]. Denote the dual space of (E) by (E)* which is endowed with the weak*topology and let ((•, •)) denote the dual pairing of (E)* and (E). D e f i n i t i o n 3.5. The S-transform valued functional on Ec by SF{r1)=

SF

of F £ £^

((F, eir,)))^

is defined as a

complex-

r, £ Ec.

2

We remark that as F £ L (C,/x), SF = y, * F. Since (E) is dense in (E)p for any p > 1, (E) becomes the reduced topological projective limit of {{E)p}. Then, by [4: Theorem 6, p p 290], (E)* = U p >i (E)* on which t h e inductive limit topology is endowed. This implies t h a t if F £ (•£)*> t n e n there exists p > 1 so t h a t F £ (E)*, t h e dual of (E)p. Since ( £ ) * = n ^ ^ E ) ^ , we have the following P r o p o s i t i o n 3.6. any m £ N,

16

Let F he in (E)*.

Then there exists p > 1 so that for

So (^ll D " S F WH a *5-(- F )<+°°-

(3-5)

According to Theorem 3.1, t h e embedding j : £<*> —> (E) is continuous. So t h a t , for any G £ ( £ ) * , G o j £ £^. T h u s (E)* can be identified as a subspace of £^0. Conversely, for a fixed F £ £^, define a functional F on (E) by

((F, /)) := «F, /»«,,

/ e (£),

137 where / is t h e analytic version of / on E*. It follows from Theorem 3.1 t h a t F£{E)*. Recall t h a t a functional G on Ec is analytic if it satisfies t h e following two conditions: (A-l) for all 77, (/> S EC1 the one complex variable mapping C 9 A \-¥ G(r] + A 1 such 77 G Ec and \r)\p < c} < + 0 0 .

that

for

any

c

>

0,

sup{|Gfa)|

:

T h e following theorem characterizes the generalized functionals in £^ terms of its S'-transforms.

in

T h e o r e m 3.7. Let F E £ ^ be fixed. Then the S-transform analytic function on Ec such that the number 00

"m,-P(F)~

i:

m

SF of F is an

2 n

j-^\\DnSF(0)\\2Hsn{Ep).

(3.6)

n=0 V J

is finite. Conversely, suppose that G is an analytic function defined on Ec and satifies the condition (A-2) for some p > 1. Let q be sufficiently large so that e2 • E i = i \ < 1- Then there exists a unique F £ £ ^ such that nm^q(F) < + 0 0 and SF = G, where Xj = (j - (1/2))TT for j £ N. Proof. T h e first assertion follows from Theorem 3.1 (or Remark 3.3), Proposition 3.6, and the following identities: 00

SF(r,) = E

1

r

-

n=o "•

00

E

_

fa,

/ * ) • • • fa, fjJ

i(F, : fn

_

")

• • • />„ : )>oo •

lii,...,j„=i

J

(3.7) and £)"5 J P(0)(r ? 1 , . . . , , , „ ) =

E

fai,

/ * ) • • • fan, / ; „ ) ((F, : fh

• • • fJn : » „ . (3.8)

To prove the second assertion. Let G is an analytic function on Ec. For any m G N, since G is locally bounded, there is a positive number M m so t h a t |Gfa)| < Mm provided t h a t |»7|p < m, 77 £ Ec. By the Cauchy integral

138 formula, \\DnG(0)\\Cn{Ep)

< Mrnnn/mn

for each n G N. Then

x\D-G(0)Vh,..., oo

< Mlc*

„2n

f

oo

feaf

n=0 V n+

/

li,)\'\ „,

, \

n

A-2"-rtV<+oo,

j= l

/

2

where c = s u p { n ^ / ' / ( e ™ n!) : n e N} < +00 by the Stirling formula (see [8]). Next, for

1

(

«*»>«,= £ - {

00

£

^V^(o)(/ J - 1) ...,/ > jz?»G(o)(/ >1) ...,/,-j

n=o n- L >i,... , i „ = i

By Theorem 3.1, F £ C i and it is clear t h a t SF = G. • R e m a r k 3 . 8 . Let TX{EP) for p € R b e t h e class of analytic functions on EPyC such t h a t 00

\n

ll/im*,) = E -rllc/WH^^) < +00, n=0

"••

called t h e space of Bargmann-Segal analytic functions on EPiC with parameter A > 0. T h e n Lee [13, 15] has shown t h a t ^OO,A = n p > ! TX(E.P)

c T\E-P)

C Tx{E-q)

C J- A (£J,) C T\EP)

c L2(C,M) C Up>x ^ A ( £ p ) =

5(^)A),

P>9>1, s

where -Aoo,A i t h e projective limit of { . ^ A : P > 1} and «4 P ,A'S are Banach space of analytic functions / on E* with the || • \\_& A -norm given by H/H.4,,,, = s u p { | / ( z ) | e - ^ / ( 2 A ) ) l z | - » : z £ £ _ p , c } < + 0 0 ; and A^iX is the dual of A^x with t h e weak*-topology. Comparing with t h e result in Theorem 3.7, we have t h e following relations:

1. foe C A^x

C L2{C,ti) C A^x

C C

2. £ ( £ « , ) = £«, C yloo,A = S^Ax,,*) C U p >! F A ( £ p ) = S ( A ^ A ) C . 4 ( £ c )

= 5(C), where A(EC)

is t h e space of all analytic functions on Ec.

139 P r o p o s i t i o n 3.9. (i) SF = 0 iff F = 0 for F G ££,. Equivalently, a total subset of £x. (it) foo is a dense subspace of

the class {e(rj) : 77 G E} is

S^.

Proof. T h e first assertion (i) follows from Theorem 3.7. For the second assertion (ii), let F G ££,. By Theorem 3.7, there exists p > 1 so t h a t nmi-p(F) is finite for each m G N. Let Fk{z)=

^ ~. f DnSF{0)(Pk{z)+iPk{y))nfjL{dy), n=0 nl Jc

z G EPiC,

(3.9)

for any k G N. Then F^s are all in SQQ . Take an arbitrary tp G £00 • Then V G £m{E~p-T) for some m G N; and by Theorem 3.1, ||v||m,p < + ° ° , where r and m are defined as in (3.3). Then, by Theorem 3.7 and using t h e CauchySchwarz inequality,

\((Fk - F, ^))oo| - \((Fh - F, p»| < |M|™,P • n s ,_ p (F fc - F) -> 0. T h u s , ,Ffc —> _F in f^, as fc —> 00. It implies t h a t £<*, is dense in f^. 4

•

T h e C l a r k f o r m u l a for g e n e r a l i z e d W i e n e r f u n c t i o n a l s

Let B = {B(t) : t G [0,1]} b e the standard Brownian motion on (C, B(C), (J.), where B(t; x) = x(t) for x G C and £ G [0,1]; and let Tu t G [0,1], t h e er-field generated by B(u), Q < u < t. In [3], Lee and Huang show t h a t the conditional expectation E[<^| J-t\ for tp £ £0 and t G [0,1] admits an integral representation

n
J v(Pt{z) + Qt{y))n{dy),

zeCc,

(4.1)

where Pt(z)(s) = z(t A s), s G [0,1]; and Qt{z) — z — Pt{z). From such a representation, one see t h a t E J ^ I ^ ] G £0 and t h a t t h e mapping T : ip -) E[y>|JT t ] is continuous on £0. Since T extends to a self-adjoint operator on L 2 [C,/i]. It is n a t u r a l to define t h e conditional expectation for a generalized function F in £Q relative t o Tt, t G [0,1], by

«E[F|^t], ^ ) ) o : = «F,

nv\rt]))0.

Consequently, E [ F | . F t ] is in £ Q ( C £^)- In this section, we shall prove t h e Clark formula for generalized functions F G £g .

140 Note t h a t even for r\ £ Ec, Pt{r)) lies in C* b u t not necessarily lies in Ec. Therefore, in this section, we restrict our consideration only to the generalized Wiener functionals in £0*ForFG£0*

(=nm€Nf^(C)),let

| | F | | m = sup{|((F, ip))0\ :

< if

|z + i y | o c e 2 m " | z + i y U M ( d y ) | |fj(s)|ds

(by Fact 2.1)

< J f e(2mn+1)dzl~+l3'l~)M(dy)| r

|fj(«)|da.

It implies t h a t UPu(v))-
+ l[c)<

c

- ,

+ 1

/

Ws)lds>

(4-2)

,, is defined as in Theorem 3.1 (1); and thus

\SF(Pu(r,))-SF(Pt(r,))\ < c^^WFW-^-!

£

\ij(s)\dS

for 0 < t < u < 1. From (4.3) it follows t h a t t H-> SF(Pt(r})) continuous.

(4.3)

is absolutely •

R e m a r k 4 . 2 . Along the line of the proof of Proposition 4.1 with J0 \rj(s)\ds being replaced by JQ |d?7(s)|, it is easy t o see t h a t Proposition 4.2 remains valid.

141 For any 77 £ Ec, Proposition 4.2 implies t h a t (d/dt) SF(Pt(r])) exists almost everywhere in (0,1) and (d/dt) SF(Pt(r])) is Lebesgue integrable so that

SF(Pb(r1))-SF(Pa(r1))

= J

j t SF(P^))

dt,

0 < a < b < 1.

In particular,

SF(V) = ((F, 1)U + 1

Jt SF(Pt(r,)) dt.

(4.4)

For any t G [0,1], let Tp(t, •) b e a complex-valued mapping on Ec given by TF(t,r,)=

jtSF(Pt(V)):

V£EC.

P r o p o s i t i o n 4 . 3 . For F e £ 0 * and t G [0,1], Tp(t, •) is analytic

on

Ec.

Proof. First of all, we observe t h a t for a sufficiently small e > 0, it follows from (4.3) t h a t SF(Pt+£(r,))-SF(Pt(r,))

^•||F||_2m <

C

„ • 11*11-

ft+e . • /

\H(s)\ds W«)l' (4.5)

2m - 1 • \V\

Let 77, 0 be arbitrarily given in Ec. By (4.5) and the dominated convergence theorem, we obtain t h a t

f TF(t,r] + X
l i m e " 1 f (SF(Pt+e(r1

=

l i m e " 1 [ {S(E[F\Ft+t]){r1

+ \cf>))-SF(Pt(rl

+

+ \4>)-S{E[F\Ft}){r1

\(f>)))d\ + \(t>))d\=

0

because of the analyticity of SG for G G ££3 on Ec, where T is a simple closed curve on the complex number plane. By the Morera theorem (see 2 1 ) , C 3 A t-t Tp(t, 77 + A(p) is entire on C Letting e —>• 0 on each side of (4.5), we see t h a t \Tp(t, v)\ < c2

IFII.^,!.^*)! |/J,||_2m,-1-

< c*„„+l • WW

/ I JO •M2

Vi3)(s)ds (4.6)

142 by Proposition 2.3. Then TF(t, •) is locally bounded on Ec. Therefore, TF(t, •) is analytic on Ec. • Applying Theorem 3.7 to Proposition 4.3, there exists a unique generalized function Kp{t) in £ ^ such t h a t SKp(t)(r)) = Tp(t, rj) for each r) G Ec. By using (4.6), we follow the argument in the proof of Theorem 3.7 to see t h a t for any m > 0, nm^q(KF{t))<

C{m,q,F)<

+00,

(4.7)

provided t h a t q is large enough so t h a t e 2 ^ ^ . j A • < + 0 0 , where C(m, q, F) is a constant, independent of t h e choice of t, given by g(e2£A72<*-2>y,

C(m,q,F)=Mlc>

n=0 V

where Mm = mC:irn^

j=l

/

| | F | | _ 2 m _ i and c = s u p { n n + < 1 / 2 ) / ( e " n!) : n 6 N } .

D e f i n i t i o n 4 . 4 . Let G be a E^-valued tegral JQ G(t) dt as

function

lim n E(ti-*i-i)G(t;)

on [0,1]. We define the in-

in£i,

provided that the limit exists, where A = {0 = t0 < ti < • • • < i n = 1} is any partition of [0,1], ||A|| is
KF(t)dt, ^

= j ((KP(t), *>))„&.

It now follows from (4.4) t h a t we obtain SF{r,) = ({F, 1 ) ) ^ + S ( j

KF{t)dt\

(r,)

for F G £0* and r\ G Ec.

By Proposition 3.6, we obtain t h e Clark formula for a generalized function F G £0* below.

143

Theorem 4.5. Let F be in £$. Then there exists a unique KF(t) that S(KF(t))(v)=

£ £^ so

~S(E[F\Tt))(V)

for any t 6 [0,1] and r\ 6 Ec. Moreover, the integral JQ Kp{t) dt exists in £^ and F = ((F, 1)U + f Jo

KF{t) dt.

Corollary 4.6. For ip E £\, ip(x) = E[y>] + / E[Dip(x) l[t,i] \Ft] dB(t; x) [i-almost all x on C, Jo where the integral is the Wiener-Ito stochastic integral. Proof. First, we note that Dip(-) l[t]1] £ £i for t e [0,1]. Then the conditional expectation E[Dip(-) l[t,i] \?t\ lies in L2(C,n). By the Kubo-Takenaka formula (see [17]),

s(J

E[£W-)i[til]|:F,]dB(t)V)

(veEc)

=

f ((E[JD¥p(-)l[M]|^],^(r?)(-)l[t,1]))oo^. Jo By a direct computation, the equality (4.8) becomes f r,(t)S(E[D
f Jo

(4.8)

v(t)S{Dv(-)l[tA])(Pt(r,))dt

Applying Theorem 4.5 we see that / E[D
=

f KF(t)dt= Jo


• Remark 4.7. (a) For any F € £0*, the £^-valued function [0,1] 3 t ->• KF(t). Clark kernel function on [0,1] associated with F .

is called the

144 ( b ) T h e formula in Corollary 4.6 can be extended to all t h e generalized functions in £Q. Since t h e arguments are more involved, we shall prove this result in our forthcoming paper [18]. References 1. J. M. C. Clark: T h e representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970), 1281-1295; 42 (1971), 1778 2. M. de Faria, M. J. Oliveira, and L. Streit: A generalized Clark-Ocone formula, Preprint, 1999. 3. H.-C. Huang and Y.-J. Lee: Conditional expectation of generalized Wiener functionals, 2001, Preprint 4. G. Kothe: Topological Vector Space I, Springer-Verlag, New York/ Heidelberg/ Berlin, 1966 5. I. Karatzas, D. Ocone, and J. Li: An extension of Clark's formula, Stochastics and Stochastics Reports 37 (1991), 127-131 6. H. Korezlioglu and A. S. Ustiinel: New class of distributions on Wiener spaces, Lecture Notes in Math. Vol. 1444, Springer-Verlag, 1990 7. H.-H. Kuo: Gaussian Measures in Banach Spaces, Lectures Notes in M a t h . Vol. 463, 1975 8. H.-H. Kuo: White Noise Distribution Theory, CRC Press, 1996 9. Y.-J. Lee: Application of Fourier-Wiener transform to differential equations on infinite dimensional spaces I, Trans. Amer. Math. Soc. 262 (1980), 218-236 10. Y.-J. Lee: Sharp inequalities and regularity of heat semigroup on infinite dimensional space, J. Fund. Anal. 71 (1987), 69-87 11. Y.-J. Lee: Generalized functions on infinite dimensional spaces and its application to white noise calculus, J. Fund. Anal. 82 (1989), 429-464 12. Y.-J. Lee: On the convergence of Wiener-Ito decomposition, Bull. Inst. Math. Acad. Sinica 17 (1989), 305-312 13. Y.-J. Lee: Analytic version of test functionals, Fourier transform and a characterization of measures in white noise calculus, J. Fund. Anal. 100 (1991), 359-380 14. Y.-J. Lee: Positive generalized functions on infinite dimensional spaces, In "Stochastic Process, a Festschrift in Honour of Gopinath Kallianpur", Springer-Verlag, 1993, 225-234 15. Y.-J. Lee: Convergence of Fock expansion and transformations of Brownian functionals, in "Functional Analysis and Global Analysis", Edited by T. Sunada and P.-W. Sy, Springer-Verlag, 1997, 142-156

145 16. Y.-J. Lee: Integral representation of second quantization and its application to white noise analysis, J. Funct. Anal. 133 (1995), 253-276 17. Y.-J. Lee: Generalized white noise functionals on classical Wiener space, J. Korean Math. Soc. 35 No.3 (1998), 613-635 18. Y.-J. Lee and H.-H. Shih: Generalized Clark formula on the classical Wiener space, 2001, in preparation 19. P. A. Meyer and J.-A. Yan: Les "fonctions caracteeristiques" des distributionc sur l'espace de Wiener, Sem. P r o b a b . XXV, Lecture Notes in Math. 1485 (1991), 61-78 20. D. Ocone: Malliavin's calculus and stochastic integral representations of functionals of diffusion process, Stochastics. 12 (1984), 161-185 21. W. Rudin: "Real and Complex Analysis", 3d ed., McGraw-Hill Book Company, New York, 1987 22. A. S. Ustunel: Representations of distributions on Wiener space and stochastic calculus of variations, J. Funct. Anal. 70 (1987), 126-139 23. S. W a t a n a b e : Lectures on Stochastic Differential Equations and Malliavin Calculus, T a t a Inst, of Fundamental Research, Bombay, 1984

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Quantum Information IV (pp. 147-176) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

INVERSE S-TRANSFORM, WICK PRODUCT A N D OVERCOMPLETENESS OF E X P O N E N T I A L V E C T O R S NOBUAKI OBATA Graduate

School of Information Sciences Tohoku University Sendai 980-8579 Japan E-mail: [email protected]

By means of distribution theory over a complex Gaussian space we formulate a coherent state representation of a white noise function and a diagonal coherent state representation of a white noise operator. An inversion formula for the S-transform (the Segal—Bargmann transform) is written down in terms of coherent state representation. As a kernel distribution the S-transform is characterized by a diagonal coherent state representation of a Wick multiplication operator. The null kernel of the coherent state representation is described in terms of holomorphic distributions and the overcompleteness of the exponential vectors is explained clearly.

Introduction We start with the real Gelfand triple: E = S(R) CH = L 2 (R) c E* = <S'(R). The canonical bilinear form on E* x E, which is compatible to the inner product of H, is denoted by (•, •) and the norm of if by | • | 0 . The Gaussian measure on E* with variance a2, denoted by /J.^, is uniquely determined by the characteristic function:

e x p { - y \t\20} = J

e**^ ».,(dx),

^ £ .

The real Gaussian space is by definition the space E* equipped with // = Hi and the complex Gaussian space is Ec = E* + iE* equipped with v{dz) = Hi/2{dx)n1/2(dy),

z = x + iy.

(1)

Both are probability spaces. In Gaussian functional analysis an important role is played by an exponential vector (or also called a coherent vector) defined by 0{(a!) = e<"'«-«'«>/ J ,

£E£C,

xeE*,

(2)

where (-, •) is the C-bilinear form on Ec x Ec- It is known that {£ ; £ G Ec} is linearly independent and spans a dense subspace of L2[E*, [i). If £ G Ec in

148

(2) is replaced with z S E^, t h e function
p{z)<j>z u{dz)

(3)

JE-C

called a coherent state representation. In fact, it is more n a t u r a l and import a n t to consider operators rather t h a n functions for functions constitute a particular class of operators. W i t h each z 6 E]~. we associate an operator Qz defined by the formula:

Apart from normalization Qz is an analogue of one-dimensional projection on L 2 ( £ ' * , / i ) , but a precise meaning as a white noise operator is given only through distribution theory. It is then n a t u r a l to consider an operator version of (3), t h a t is, 3=

f

p{z)Qzu{dz)

(4)

JE-C

called a diagonal coherent state representation. This idea traces back to Klauder 1 2 in t h e case of finite dimension and has been extensively studied 1 ' 1 3 . Rigorous definitions of (3) and (4) based on the so-called white noise distribution t h e o r y 1 6 , 1 8 was first given by O b a t a 2 0 ' 2 1 . To b e more precise, we need a white noise triple: Wcr(t2(R))SI2(E*,/j) C W ,

(5)

where W and W* are, respectively, t h e spaces of test white noise functions and of white noise distributions on the real Gaussian space (E*,(J,). In this paper we adopt the CKS-space 6 . Then, in view of (5) and L2{E^.,u) = L2(E*,ni/2) ® L2[E*,/ii 12), which follows from (1), a CKS-space over the complex Gaussian space is constructed: VcL2(E*c,v)cV*. It will be shown t h a t exressions (3) and (4) are justified for any p 6 T>* and represent $ G W* and S £ £(W,VV*), respectively (Lemmas 3.4 and 4.2). Notice t h a t representation (3) is not unique due to the famous overcompleteness of exponential vectors; for example, for £ G Ec we have t= t

e(*<*)xv{dz)=

f

e<* l { > + < Z l £ >- < { l € > 0«i/(dz).

(6)

149 One of t h e main purposes of this paper is to clarify this phenomenon. A key role is played by t h e S-transform 1 4 which is a white noise extension of t h e famous Segal-Bargmann transform 8 . Let L2(EQ, I^)HOL denote t h e space of holomorphic L 2 -functions on t h e complex Gaussian space (EQ,V). T h e Segal-Bargmann transform is a unit a r y isomorphism from L2(E*,fi) onto L2(EQ,I/)HOL uniquely defined by t h e correspondence: ^ ( z ) = e<*'«>"«•«> /2

*»

e £ (z) = e,

£6

Ec.

On t h e other hand, t h e S-transform is defined for $ 6 W* by

S$(0 = « $ , ^ » ,

£e£c.

2

If $ 6 L (E*,n), t h e S-transform S $ is extended to a holomorphic L2function on (EQ, V) a n d coincides with t h e Segal-Bargmann transform. Moreover, for $ 6 W* t h e S-transform 5 $ is viewed as a holomorphic distribution on EQ and one comes to t h e following T h e o r e m 3 . 3 The S-transform extends the Segal-Bargmann is a topological isomorphism from VV* onto 2?JJOL •

transform

and

T h e inverse S-transform is given by the coherent s t a t e representation. T h e o r e m 3.5 For any $ E W*, the S-transform 5 $ being regarded as a holomorphic distribution on the complex Gaussian space, it holds that

$ = f In particular, sentation.

S${z)4>zu{dz).

every white noise distribution

(7)

$ admits a coherent state

repre-

However, since a coherent state representation is not unique due to t h e overcompleteness of exponential vectors, there arises a question of characterization of t h e S-transform as a kernel in (7). To answer this question, we focus on uniqueness of t h e diagonal coherent state representation for white noise operators, t h a t is, T h e o r e m 4 . 3 Every white noise operator 3 S C(W,W*) admits diagonal coherent state representation as in (4) with p £ V*.

a unique

Applying this to a Wick multiplication operator, we come to t h e following T h e o r e m 5.2 Let W
W*= I JE'C

operator by $ 6 W*.

S(z)Qzv(dz).

Then

(8)

150 T h u s t h e S-transform 5 $ is characterized as a unique kernel distribution of W$. T h e inverse S-transform is again obtained from (8) by using simple relations W^4>o = 3> and Qzo = <j)z. Now, the first integral in (6) corresponds to the Wick multiplication: W+t

c<*- c >Q I i/(dz) I

= f JE'C

while the second identity: e<*'« + <*-«>-<«'«Q z i/(dz)

0C = f

gives the diagonal coherent state representation of the pointwise multiplication operator by £. Finally, we investigate the null kernel of coherent state representations. Let D A H b e the space of all anti-holomorphic test functions and T>AH denote the space of all p £ V* which annihilate X>AH> t h a t is, all p £ V* such t h a t ((p, <j>f)u = 0 for all HOL- (By our convention ((-, •)),, stands for the canonical C-bilinear form on V* x V.) T h e o r e m 5.4 Let $ £ W*. representation:

Then p £ T>* is a kernel of its coherent

$ = /

p(z)(j)z

state

u(dz)

JE-C

if and only if P = Pi + P2,

with pi(z)

Pi £ T>*AR,

P2 G £>AH>

— S$(z).

We have thus seen t h a t complex white noise theory established in this paper sheds some light on questions concerning the over completeness of exponential vectors. In particular, it seems worthwhile to note t h a t the S-transform is characterized in terms of coherent state representation. Complex white noise dates back to the early stages of white noise theory and has been studied from different aspects. Among others, a white noise approach to the Segal-Bargmann transform, discussed by Kubo-Yokoi 1 5 and Yokoi 2 8 , has a close relation to the present work. While, a new direction of characterization theorems is proposed by J i - O b a t a 1 0 where we enjoy an interesting encounter with theory of infinite holomorphy ' .

151

1 1.1

White Noise Functions Weighted Fock space

Let % be a (real or complex) Hilbert space. For n > 1 the n-fold symmetric tensor power of H is denoted by Ji0n and by definition H®0 is one dimensional. Given a sequence a = {a(n)}^_ 0 of positive numbers we put

ra(H) = U = (/„)£=„; / » e W S B . IHI2 = E n ! a W I / - l 2 < °°} • In an obvious manner r a ("H) becomes a Hilbert space and is called the weighted Fock space over ti. The Boson Fock space is by definition the special case of a{n) = 1 and is denoted by T(H). We need some general notion for positive sequences. A positive sequence a = {a(n)}£?_0 is called log-concave if a(n)a(n + 2) < a(n + l)2,

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

Two positive sequences a = {a(n)}^L0 and (3 — {/3(n)}£L0 are called equivalent if there exist positive constants K\,K2,M\, Mi > 0 such that KiMfain)

< f3(n) < K2M?a(n),

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

From now on we fix a weight sequence a = {a(n)}^=0 conditions:

satisfying the following

(Al) a(0) = 1 and inf crna(n) > 0 for some cr > 1; n>0 1/n

(A2) l i m { ^ }

" = 0;

n-l-oo [ n ! J

(A3) a is equivalent to a positive sequence 7 = {l(n)} such that J7(n)/n!} is log-concave; (A4) a is equivalent to another positive sequence 7 = {7(n)} such that {(n!7(n))"" 1 } is log-concave. For example, (n!)^ with 0 < (3 < 1 and the Bell numbers of order A; satisfy the above conditions. The exponential generating functions of a and 1/a are defined by Ga{t) = £ ^ t " , n*—' =0

n!

{t)

G

'

= £ -Jp^t-, *—' n i Q l n l

152 respectively. Both are entire holomorphic functions by ( A l ) and (A2). We next define

S l ,„ ( ( ) = £^{ a 5^}. n=0

*•

-*

It is known 2 t h a t (A3) and (A4) are necessary and sufficient conditions respectively for Ga and for G i / a to have positive radiai of convergence. These functions will play a crucial role in norm estimates. Moreover, the next fact is known 2 . L e m m a 1.1 Let a = {a(n)} (1) There exists a constant a(n)a(m)

be as above. C\ > 0 such that

< C"+ma(n

(2) There exists a constant

+ m),

=

0,1,2,....

n,m

=

0,1,2,....

C2 > 0 such that

a(n + m) < C™+ma{n)a(m), (3) There exists a constant

n,m

C3 > 0 such that a(n) < C ™ a ( m ) ,

n < m.

Then, by a simple calculation we have P r o p o s i t i o n 1.2 Let a — { a ( n ) } be as above and Ga(t) generating function. Then, for s, t > 0 we have: (1) Ga(0)

= 1 and Ga(s)

< Ga(t)

for

the

exponential

s
(2) G Q ( S ) G Q ( i ) < G Q ( C 1 ( S + *)). (3) Ga{s + (4) e°Ga(t)
t)
and M _ 1 = inf n >o

crna(n).

153 1.2

CKS-Space

As usual, we start with the real Gelfand triple: E = S{R)

CH = L2{K)C

E*=S'(R),

(9)

where the canonical bilinear form on E* X E is denoted by (•, •) and t h e norm of L2(R) by | • | 0 . For p > 0 let E±p b e the Hilbert space obtained by completing E = «S(R) with respect to the norm |£|± p = |"A ±p £|o) where A = 1 + t2 - d2/dt2. We then have E S proj lim Ep,

E* S ind lim £ _ p .

p->oo

p->oo

In general, for a real vector space X the complexification is denoted by XcFor notational convenience, t h e C-bilinear extension on f?£. x Ec is denoted by t h e same symbol. It is then noted t h a t |£|Q = (£, £} for £ G HeLet r a ( £ ' p ) b e t h e weighted Fock space over Ep. Then W = ra(£)=projlimrQ(£;p) p—>oo

becomes a nuclear Frechet space and we obtain a Gelfand triple: W = Ta{E)cT{Hc) cTa{E)* = W*,

(10)

which is called the Cochran-Kuo-Sengupta space6 or t h e CKS-space T h e topology of W is defined by the family of norms:

for short.

oo

71=0

As a consequence of s t a n d a r d argument, ra(i5),^indlimrQ-i(£_p), p—K30

where r a (i*7)* carries the strong dual topology and = stands for a topological linear isomorphism. T h e canonical C-bilinear form on W* x W is denoted by «-, •))• T h e n OO

p , )) = Y, m (Fn, /„>,

* = {Fa)eW,

n=0

and it holds t h a t

|(($,0))|<||$||_ p ,_IHI P , + ,

4>=(fn)eW,

154 where OO

j

V

n=0

>

Conditions (A1)-(A4) are sorted out by A s a i - K u b o - K u o 2 from many similar ones t h a t have been introduced to keep "nice" properties of a CKSspace. On the other hand, it is also possible to start with a generating function Ga or another function controlling growth rate. This reversed approach is concise and useful for some questions 2 , 7 , 1 1 , however, we prefer to the explicit description for our later calculation. 1.3

Wiener-Ito-Segal

Isomorphism

and Exponential

Vectors

Let fj.^2 be the Gaussian measure with variance a2 defined by t h e characteristic function: 2

e-
\Z\Z/2=

f c <(«.C) / i ( r l ( d a .) > J E*

£

£ E

.

(11)

We p u t fi = [j,! for simplicity and let L2(E*,fi) denote the Hilbert space of C-valued L 2 -functions on E*. T h e celebrated Wiener-Ito-Segal isomorphism is a unitary isomorphism between L2(E*,fj.) and T(Hc) uniquely determined by the correspondence

u>

V'-'^r--)'

UEc

-

T h e same symbol ^ is used for the right h a n d side. We call
P r o p o s i t i o n 1.3 { ^ ; £ G Ec}

spans a dense subspace ofW

T h e proof is well known. 2 2.1

Complex White Noise Complex

Gaussian

Space

We define a probability measure v on EQ = E* + iE* by v(dz) = H\/2{dx)n1/2{dy),

z = x + iy e E*c,

=

(12) exponential

Ta(Ec)-

155 where ^i/2 is the Gaussian measure on E* with variance a2 = 1/2, see (11). After Hida 9 the probability space (EQ , v) is called the complex Gaussian space. We write z = x — iy for z = x + iy £ E* + iE*. Here are basic formulae: e{~z'V+{z<")v(dz)

= e{t'r>),

Z,V£EC,

J E:

(13)

{z®m, 6 ® • • • ® U)(z®n, m ® • • • ® »j«> K ^ )

/ J B:

= < J m „ m ! ( £ i ® . . . ® f m , »7i®...®»j m ), -2.-2

CKS-Space

over Complex

Gaussian

£i,r)jeEc-

(14)

Space

By modifying the Wiener-Ito-Segal isomorphism (12) we have a unitary isomorphism: L2(E',vi1/2)°
(15)

uniquely specified by the correspondence: Ve(z)=e^<*'«>-<^>/2

(

£®2

£®n

\

Identifying functions on £ £ and on .E* x E* in a canonical manner: 0 ® ^ ( a 5 + itf) = 0 ( x ) ^ ( y ) ,

x,y£E*,

4>,V> € L 2 (£*,M1/2),

(16)

we have an isomorphism: L2(£^)-L2(iT,Ml/2)®L2(£;%/i1/2).

(17)

Then, combining (15) and (17) we obtain a unitary isomorphism l:T{Hc)®T{Hc)->L2{E*c,v).

(18)

Moreover, applying this I to the Gelfand triple (10), we come to a Gelfand

triple: VcL2{E*c,v)cV*,

(19)

which is referred to as t h e CKS-space over the complex Gaussian space. By construction, I becomes topological isomorphisms (denoted by the same symbol) from W ®W onto V and (W ®W)* onto V*. T h e canonical C-bilinear

156 form on T>* x T> is denoted by ((-, •)),, and the norms induced by Ta(Ep) denoted by || • H^ , ± for p > 0, i.e.,

|| 1 ( 0 ® V) | | „ i ± P i ± = II ^ ll ± p ,± IIV l l ± P l ± ,

are

0, V e w .

For £ £ S c we define

By a simple calculation we come to the following L e m m a 2.1 77ie isomorphism X is uniquely determined

by 2

2(0? ® 0„)(z) = ^ . ^ / ^ J ^ + ^ / ^ ^ e - ^ ' ^ / - ^ " ) / 2 , where f, 77 r u n over i?c 5.5

Holomorphic

I?

-Functions

By means of t h e orthogonal relation (14) one easily obtains t h e following L e m m a 2.2 For any 0 = (fm) € T(HC) OO

^( Z ) = Y, <*8ra. /•»>.

^ 4

m=0

is defined in the sense of L2(Ec,v) and <j> i-> w^ is an isometric T(HC) into L2{E*c,v). We define a closed subspace of L2(EQ,U) by

map

from

L2(££,z/)HOL = -K;0er(tfc)}. Then, in view of the W i e n e r - I t 6-Segal isomorphism L2(E*,^j.) obtain a unitary isomorphism L2{E\n)

= T(Hc)

we

L2{E*c,v)n0L-

=

This is the famous Segal-Bargmann transform 8 . L e m m a 2.3 Under the identification L2(EQ,V) = T(Hc) I we have the following correspondence:

\

W

/ 0

®T(Hc)

given by

( iC \<8>(m-k)

(j)

)®(o,...,m

-,

••°'-)'

,0,...),

157 where ( G Ec • PROOF.

It follows from L e m m a 2.1 t h a t l{4>i®r})(z)=EC{z)ec{z)e-«<<'\

where =

P 4- i n

Z + iy V2 '

S

, = £P -— ir)in V2 '

T h e n the assertion follows by Taylor expansion. L e m m a 2.4 For £ G Ec Then, for p >0 we have

I

and m = 0 , 1 , 2 , . . . we pw£ w m ,f(z) = (z, C) •

•x\\l,P,+

=rn<\C®-\l±±(f)a(k)a(rn-k)

(20)

fc=0 PROOF.

For

simplicity we

put

fc

MC) = (o,...,C® ,o,...),

CeEc,

fc>o.

Obviously, hfc(C) G W and

ll^(C)||p I+ =fe!a(fc)Kl". It follows from L e m m a 2.3 t h a t

rl w=

' S&)' i '(^) 8 ' , ^( ;<M'

Since the right h a n d side is an orthogonal sum with respect to the n o r m i i P , + > we have

hk [ j ^ j ® /.—*, ( ^

°m,C ll„.)Pi +

p,+

fc=o m

/

2

\ '

^"•m. — k. \

fe=o v

7

( - y/2")

= E r; ««(*)0 fc=o

fc=0 ^

as desired.

P,+

i—

V2

p,+ 2(m-fc)

2fc

(m — k)\a(rn — k) A/2

'

p

v/2

158 P r o p o s i t i o n 2 . 5 For <j> = ( / m ) G W let w^ £ L2(E(,,v) Lemma 2.2. Then for any p > 0,

be defined as in

i «• C , + = E ™! i /»e ^ E I r ( f e ) a ( m -fc)m=0 PROOF.

For fm

fc=0

G Eg"

1

^

we p u t wm(z)

(21)

' = (z®m,

fm).

T h e n , by t h e

polarization formula a n d Fourier expansion we see from L e m m a 2.4 t h a t

II wra \\lipi+ =ml\fm\l—Y^[?)a(k)a{m-k),

p>0.

Since u>^ = ^ m = o w"» *s a n orthogonal sum, (21) follows immediately.

2-4

Holomorphic

Test

|

Functions

We define

VnoL=VnL2(Ec,u)EOLLet <j) i-> o;,^ b e t h e unitary m a p from r(.ffc) onto L 2 ( . E C , I > ) H O L defined in L e m m a 2.2. P r o p o s i t i o n 2 . 6 TVie map w^, induces a topological isomorphism W

from

onto X>HOL-

P R O O F . Let 0 = (fm) G W a n d s t a r t with (21). Since a(k)a(m C™a(m) by L e m m a 1.1 (1), we have

IIu* t,p,+ = E

m!

- k) <

I /»lp ^ E ( 7 ) Q ( fc ) a ( m - fc)

m—0

fc—0

^

'

00


(22)

m=0

Choose q > 0 in such a way t h a t C\p2q Then, (22) becomes

< 1, where p = ||^4 _ 1 ||op = 1/2.

00

II <** \\ltPi+ < E m=0

ml

C^(m)p2n

fm \2p+q < II cf> 15+,,+ , ^ G W. (23)

159

A similar argument using Lemma 1.1 (2) yields

||0|| p , + <||^IL, p+7 ., + ,

0£W,

(24)

where r > 0 is chosen in such a way that C^p2* < 1. The assertion is a consequence of two norm estimates (23) and (24). I

3 3.1

Representation of White Noise Functions S-transform and Characterization

Theorem

The S-transform of $ £ W* is denned by

5#(0 = «*, «)),

ZeEc.

It follows from Proposition 1.3 that the S-transform determines a white noise function uniquely. In fact, for $ = (Fn) we have oo

S

*(t) = £ >

^G£c

(25)

n=0

Moreover, we have the following fundamental result known as the characterization theorem for S-transform 6,25 . Theorem 3.1 A C-valued function F defined on Ec is the S-transform of a white noise distribution $ £ W* if and only if (Fl) for any £,£i £ Ec, z H-> F(Z£ + £i) is entire holomorphic on C; (F2) £/iere exist some C > 0 anii p > 0 such that \F(t)\2
t€Ec.

In that case,

\\*\\2-{p+q),- 1/2 such that G a (||A~ 9 ||jj S ) < oo. (This choice is always possible since ||J4 _ ? ||HS —>• 0 as q —> oo.)

160 3.2

Holomorphic

Distributions

For <j) E W t h e S-transform S is naturally extended to a function defined on

50(z) = ((0„0)),

z e 4

where ®2

(

0n

\

and is also referred t o as t h e exponential vector. Then, for
n=0

by L e m m a 2.2. In other words, t h e S-transform, which is defined for all $ £ W* and S$ is an entire function on Ec, and t h e Segal-Bargmann transform, which is defined for all £ F(Hc) and o ^ is an entire function on EQ, coincide on a common domain. Moreover, we shall prove t h a t a formal notation oo

W*(z)

= Y, <*®n. Fn),

* = (^») € W*,

(26)

n=0

is given a meaning as an element of V*. (Recall t h a t (26) converges if $ e W or if $ S T(Hc)-) By a similar argument as in t h e proof of Proposition 2.5 we come to P r o p o s i t i o n 3.2 For ({> = (fm) G W /e£ w^ be defined as in (26). Then for any p > 0, °°

II " • Hi-p,- = £

1

!

™ I / » I-P ^

m

£

/

\

1

( T j Q (fc)a(m-fc)-

V

v

(27)

v

m=0 fc=0 ' ' ' Then, obviously t h e m a p I / I I - > ^ extends an isomorphism from W* into 2?* a n d we adopt a formal notation for uoHOL = {"*;

* e w*}.

T h e n we have already seen t h a t <$ H-> ai # is a topological isomorphism from >V* onto X'HOL- In t h e sense t h a t t h e m a p $ i-> w$ is an extension of t h e S-transform S<& we may state T h e o r e m 3 . 3 The S-transform extends the Segal-Bargmann is a topological isomorphism from W* onto PjjoL •

transform

and

161 A holomorphic distribution 12 e T>^OL feels only anti-holomorphic test functions. Define t h e space of anti-holomorphic L 2 -functions by L 2 ( B £ , ^ ) A H = {(/> e L2(E*c,v);

L2(E*C,V)UOL}

0 €

and t h e space of anti-holomorphic test functions by £>AH=2>nL2(££,i/)AHT h e n for fi £ 2?HOL

anc

^

w

£ £>AH

w e

have oo

«fi, "))„ = $ > ! < F „ , / „ ) ,

(28)

where

5.5

Coherent

oo

oo

71=0

71=0

State Representation

and Inverse

S-transform

By definition, for £ G .Ec we have e € (z) = e<2'«> = ipi/y/2{x)ipii/Vz(y),

z = x + iy G £ £ .

In other words, J

( ^ / V 5 ® ^ / v ^ ) = e«

(29)

and, in particular, e^ £ 23. L e m m a 3 . 4 For any p 6 T>* there exists a unique $ g W* suc/i £/ia£ 5$(e) = ((P)Ce»l/,

f £ E c

(30)

P R O O F . Denote by F ( £ ) t h e right h a n d side of (30). It is obvious t h a t F satisfies condition ( F l ) in Theorem 3.1. We shall prove (F2). Choose p > 0 such t h a t || p H^ _ < oo. Then in view of (29) we see t h a t

WOI2 < IMI',-

\P\\u-V,-

KP, l^/v^llp,+ ll^i«/^llp,+ it £ Ga V2 V2

In view of Proposition 1.2 (2) we have G„

€ V^

GQ

tf v/2

< G « ( C l | £ l p ) < G « ( C V « | £ lp+g)

(31)

162 where p = | | A _ 1 | | 0 p = 1/2. come to

Choose q > 0 such t h a t Cxp2* < 1. Then we

\F(t)\2 <\\p\\l,-P,-Ga(tt\l+g), and condition (F2) is fulfilled. Thus, by T h o e r e m 3.1, F is the S-transform of a white noise distribution in VV*. I T h e white noise distribution $ defined as in (30) is denoted by $ = /

p(z)<j>zv{dz)

JE-C

and is called a coherent state

representation.

T h e o r e m 3.5 For any $ £ VV*, the S-transform S$ being regarded as a holomorphic distribution on the complex Gaussian space, it holds that $ = / JE'C In particular, sentation. PROOF.

S$(z)<j>zv{dz).

every white noise distribution

$ admits a coherent state

repre-

It follows from Theorem 3.3 t h a t OO

p(z) = S*(z) = 5 > ® » , F„) 71 = 0

belongs t o V* a n d by L e m m a 3.4 t h e r e exists a unique \P G VV* satisfying S * ( f ) = {(p, e^)),, for f G Ec. Note t h a t

n=0

X

' '

Hence by (28),

n=0

which coincides with S$(£).

*

' '

n=0

Therefore $ = $ .

| 3

A relavant formula was obtained by Berezansky-Kondratiev by means of duality argument over complex Gaussian space in a different realization. T h e inverse S-transform was also discussed within the real Gaussian integral 1 5 , 1 7 .

163

4 4-1

Representation of White Noise Operators White Noise Operators,

Symbols and

Kernels

A continuous operator from W into W* is in general referred t o as a white noise operator. Let £ ( W , W*) denote t h e space of white noise operators. T h e

symbol of 5 £ C(W,W*) is by definition a C-valued function on Ec x Ec defined by

T h e symbol uniquely specifies a white noise operator by Proposition 1.3. Moreover, by t h e kernel theorem there exists a unique s.K G (W <8> W ) * such that

H(£,r,) = «H6 0,)) = « 5 * , 04 ® „)),

(,ri££c.

This H ^ is called t h e kernel of S. We now state t h e characterization for operator symbols 4 ' 1 8 . T h e o r e m 4 . 1 A function © : Ec x £?c -> C « the symbol of an operator E€C(W,W*) if and only if ( 0 1 ) for any £, £i,?7,771 G - E c ( z > u ; ) '""*' ®(z£ + £i,wr] + rji) is entire morphic on C X C ; ( 0 2 ) there exist constant

holo-

numbers C > 0 and p > 0 swcft £/ia£

|©(£,r?)| 2 < C G Q ( | ^ ) G Q ( | r , | 2 ) ,

£,r, G £ C -

/ n £/ia£ case

\\m\2-(P+q)l- < CGHWA-'W^ml^,

4> G W,

(32)

where q > 1/2 is tofcen as Ga(||-<4 _ , ||]j S ) < 00. ^.2

Diagonal

Coherent

State

Representation

W i t h each z E Ec v?e associate Qz G £ ( W , VV*) by t h e formula:

Qz4> = ((4>z,4>))<j>z,

0GW.

Note here t h a t b o t h maps 2 4 ^ £ W * a n d z 4 continuous. T h e symbol of Qz is given by Qz{Z,r1) = qLri{z) = e^+^\

Q 2 £ C(W,W*)

zeEc,

(,r, € Ec.

are

(33)

164

Then, for z = x + iy we obtain q^{x

+ iy) = e ^ ^ e ^ - ^ " ' )

and hence qitV = e«- n)X(4>li+Jl)/^

® 0l(_€+„)/v/^

(34)

In particular, q^iV £ V. L e m m a 4 . 2 For p 6 V* there is a unique operator 3 G £ ( W , W*) swc/i that « S ^ , 0,)) = ((p, g £ i „»„,

Uefic-

(35)

T h e proof is similar to t h a t of L e m m a 3.4 with t h e help of Theorem 4.1. T h e operator S defined by (35) is denoted by

a= f

p{z)Qzy{dz)

(36)

JE-C

and is called t h e diagonal coherent state representation. T h e next result was proved first by O b a t a 2 0 and we give a different proof here. T h e o r e m 4 . 3 Every white noise operator S G £ ( W , W*) admits a unique diagonal coherent state representation. PROOF. specified by

Qfa

0

We shall first define an operator Q G £(W>V, W®W)

4>n) = e -«,«/»-(^)/»0 ( { + . ) j ) M ® ^-ivy^

uniquely

f- ^ e ^ c

Since t h e exponential vectors are linearly independent, so are {£ ®4>n', f,») G i ? c } and ^ is well defined on a subspace spaned by such tensor products of exponential vectors. It is then sufficient t o check t h a t t h e function

0 ( £ ^ ; £',*?') = {{G{t ® 4>r,), <^< ® <M> = ^ ^ • s > / 2 " < 7 7 " ' > / 2 ( { ^ + I , ) / ^ H-))Uu-iv)/^

<M>

(37)

satisfies t h e conditions in t h e characterization theorem in t h e case of multivariables (see Theorem A.6). In fact, condition ( 0 1 ) is rather obvious. To see ( 0 2 ) we take an arbitrary p > 0. We see easily t h a t

Ktyc-H.,)/^' *€'»|2 ^ ll*(€-H,)/^l£,+ H^' ll-p,2 •

= 0.(|i^|JGl/.(iri".,,
165

Then by Proposition 1.2 (3) we obtain l « * K + « , ) / ^ . ^ ' » | 2 < Ga{C2 11 \2p)Ga(C2 | „ g ) G 1 / a ( | £' | % ) .

(38)

Similarly, \((^-in)/V2'

^))\2

< Ga(C2 \i\2p)Ga{C2

\l\2p)G1/a(\v'

tp).

(39)

On the other hand, it follows from Proposition 1.2 (4) that |e-<€.€>/2,2 < e |«,€>| < c t C a p ^ l e t J ) ,

(40)

where p — \\ A-1 ||op = 1/2, and similarly, \e~<™)/*\*
(41)

Combining (38)-(41), we obtain an estimate for (37) as follows: l©(^;£', V)l2 < Ga(c21 e| 2 )G Q (G 2 h|J)G 1 / Q (|e' |%)

xGa(C 2 |e|J)G a (C 2 h|^)G 1/Q (h'| 2 _ p ) x G Q (C 3 P 2 p | ££)G a (C 3 p 2 * | 7712)

(42)

By virtue of Proposition 1.2 (2) there exists some C > 0 such that Ga{C21 i | 2 )G Q (C 2 | £ | 2 )G a (C 3 p 2 " | £ | 2 ) < G a ( C | £ | 2 ) . Now choose g > 0 satisfying Cp2q < 1 to have Ga{C2 | i \2p)Ga(C2 K \2p)Ga(C3p*r \ £ | 2 ) < G a ( | £ | 2 + q ) . Together with a similar inequality for 77 we finally conclude that (42) becomes | 6 ( £ r?; £',r/)| 2 < G a ( | £ | 2 + , ) G t t ( | 77 | 2 + , ) G 1 / a ( | £' |l p )G 1 / c ,(| r/ | 2 _ p ), which shows the desired condition (02). Thus Q G C(W ® W, W ® W). Similarly, V. G £(W ® W, W ® W) is uniquely specified by H(0 ? ® 4>r,) = e^'iUu+r,)/^ Moreover,

® <J>n-i+n)/y/2>

9U{.4>t ® 0,) = H g ( 0 ? ® v) = 0£ ® 0 „

trj€

EC-

£, ry G £ C -

_1

Therefore, 5 is an automorphism of W <8> W and 6 = %. Now, suppose we are given an arbitrary operator H G £(W, W*). Then « H ^ , ^ ) ) = ( ( S * , ^ ® „))

= ((s Jf ,aw(^0^)» = ((e*sjr,«(0€®0,)» = ((0,S*,^+I,)/V5®^_€+1l)/V5»e«.''>.

166 Using (34) we see immediately t h a t

This means t h a t S admits a diagonal coherent s t a t e representation:

E= f

P{z)Qzv(dz),

P=

ig*zKev*.

JE'C

T h e uniqueness of p is obvious since IQ* is an isomorphism from ( W ® W ) * onto V. I

^.5

Examples

P r o p o s i t i o n 4 . 4 (Resolution of the identity) It holds that

1= [

Qzu{dz).

JE* For t h e proof we need only to compute the symbols of b o t h sides. T h e prototype of the above formula was given by Klauder 1 2 , see also K l a u d e r Skagerstam 1 3 for various developments. From Proposition 4.4 a unitarity criterion is derived 2 2 and is applied t o q u a n t u m stochastic differential equations involving square of white noise 2 3 . m,

W i t h each KitTn <E (Ec formal integral expression: "(,m(K(,m)

=

/

y

K

l,m\sli a

we associate a white noise operator by a

• • • > sh

*Sl---

a

^1) • • • i ^m)

a , a « i • • • atmds1

... dsidti

...dtm.

(43)

4 18

This is called an integral kernel operator. It is known ' t h a t for any m Kl,m £ (Ec )*, t h e integral kernel operator Sj i m («;i i m ) always belongs to C(yV,W*). T h e symbol is easily obtained:

Moreover, every S E £ ( W , W*) admits an infinite series expansion: oo

H

= X ] El,™(Kl,m), l,m=0

where the right hand side converges in £ ( W , W*).

(44)

167 P r o p o s i t i o n 4.5 For F e {E®m)*

we have

H0,m(F)= f

{z®m,F)Qzv(dz),

(45)

{z®m,F)Qzv{dz).

(46)

JE'C

Era,o(F)=/"

P R O O F . P u t p(z) = (z® m , F ) . As is easily verified by definition, p GT>* and a white noise operator S=

/

(z®m,F)Qzi/(dz)

is defined. Then t h e operator symbol is given by S & r , ) = «P, g Cl „» = /

( ^ m , F)

e<*'«>

+

<*'"> ^ ) -

Then using t h e orthogonal relation (14), we obtain with no difficulty m,v)

= (F, £®m>e<«'"> =

E0,m(Ff(tv),

which proves (45). By duality (46) is obtained immediately.

I

For F - 5t G E* we write naturally (z, F) = (z, 5t) = z(t). Then {z(t)} is nothing b u t the complex white noise 9 . As a special case of Proposition 4.5, we have the following P r o p o s i t i o n 4.6 ( Q u a n t u m white noise) at=

z(t)Qzv{dz),

a* = /

JE'C

5 5.1

zJi)Qzu(dz).

JE'C

W i c k P r o d u c t and Overcompleteness of Exponential Vectors Wick Product of White Noise

Functions

For $ ! , $2 £ VV* there exists a unique *& € VV* such t h a t 5*(0 = 5*1(0-5*2(e),

f6£c.

T h e verification is simple with the help of Theorem 3.1. In t h a t case we write $ = $1 0 $ 2 -

L e m m a 5.1 For $ S VV* fixed, the map W$ : H-> $ o is a linear operator from W into W*.

continuous

168

PROOF.

We compute the symbol. ({Wth,

<^» = « * o 4>(, 0,» = 5 ( $ o ^)(r?) = 5 * ( r , ) - 5 ^ ( r , ) = ((*,0,))e«'">.

Choosing p > 0 such that || $ ||_

(47)

< oo, we come to

<\\n2-P,-Ga{\v\l)e^+M',

(48)

where we used a simple inequality: 2| (f, 17) | < | £ | 0 + 177 | 0 < | £ L + | >7 LNow, with the help of Proposition 1.2 we see that (48) becomes l((^0|,^»|2<||$l|2-p,-Ga(2|r?|2)GQ(|£|2) 0 is taken in such a way that 2 | r\ \ < 2p2q \ r\ | . < | r\ \p+q- Then the assertion follows from the characterization theorem for operator symbols (Theorem 4.1). I The operator W$ G C{W,W*) defined in Lemma 5.1 is called the Wick multiplication operator associated with $ G W*. Theorem 5.2 Let W& be the Wick multiplication operator by $ G W*. Then W*=

f

S${z)Qzv(dz).

(49)

JE>C PROOF.

Let $ = (F m ). Then OO

S*{Z) = J2 (*0m. M m=0

as an element of T>*, see also the proof of Theorem 3.5. Consider the diagonal coherent state representation: J E:

S${z)Qzv(dz).

It then follows from Proposition 4.5 that 00

2 = 2_^ Z'mfiiFm),

169 and therefore, oo

m=0

This coincides with (47) and hence IE = W$ as desired.

I

T h u s t h e diagonal coherent s t a t e representation of a Wick multiplication operator is directly related to the S-transform. As an immediate application, the inversion formula for the S-transform is obtained. Since WQ<J>Q — $ and Qz(f>o = 4>z, it is sufficient to apply (49) to 4>Q. 5.2

Overcompleteness

of Exponential

Vectors

Let X ^ H denote the space of all p G V* which annihilate X'AH) t h a t is, all p eV* such t h a t ((p, 0)) = 0 for all G £>HOL = v n L2{Eh' ^ ) H O L - Recall t h a t ((•, •)) is a C-bilinear form by our convention. T h e o r e m 5.3 For pEV* put $ = /

p(z)<j>zu(dz),

- = /

JE-C

p(z)Qzv{dz). JE'C

Then the following four conditions

are

equivalent:

(ii) <* = 0; (iii) S annihilates

the vacuum;

3o = 0;

(iv) 3 is a quantum stochastic integral against the annihilation there exists L G E^® £ ( W , W*) such that

process,

i.e.,

3 = / L(t)at dt. Jn PROOF,

(i) = >

(ii). By L e m m a 3.4,

s*(0 = «P,e«»,

£e£c.

Since e^ G X>HOLi the assertion follows immediately. (ii) = > (i). It is easily verified t h a t t (->• ((p, e^)) is entire holomorphic on C . Then from 0 = S $ ( t £ ) = {(p, eti)) for all z G C it follows t h a t ((p,

Wm»

= 0,

Um(z)

= (z®m,

em)

•

This being valid for all £ G EQ, we conclude t h a t p G 2?AH #

170 (ii) •*=>• (iii) is obvious from Qz<po = z(iii) •£=£• (iv). Let S = ^T,i m=o —/,m(Kf,m) b e the expansion of the form (44). Then condition (iii) is equivalent to ((Ho, ,,)) = H(0,rj) = 0 for all r] 6 Ec, t h a t is,

1=0

This is equivalent to K/(O = 0 for all / > 0. Therefore S is a sum of integral kernel operators involving one or more annihilation operators. We then see t h a t such an operator is a q u a n t u m stochastic integral (in a broad sense) against the annihilation process 1 9 . I As an immediate consequence, we come to the following T h e o r e m 5.4 Let <£ 6 W*. Then p € T>* is a kernel of its coherent representation: $ = /

state

p{z)(j>z u(dz)

JE-C

if and only if P = P\+

with p\{z) =

P2,

Pi G V*AH, p2 E Z>AH,

S$(z).

Acknowledgments This work is supported in p a r t by Grant-in-Aid for Scientific Research No. 12440036, Ministry of Education, J a p a n . Appendix: Multi-Variable W h i t e Noise Functions In the proof of Theorem 4.3 we used a result on white noise functions of two variables. More generally, taking the n-fold tensor power of the one-variable CKS-space yVcL2(E*,fi)cW*,

E*=S'(R),

(50)

we obtain a Gelfand triple of n-variable functions: n times

W ® n C L2{E*

x . . . x Ei,'n

n times

x ••• x fl) c (W®")*.

(51)

171

In this Appendix we mention characterization theorems for (51). In fact, owing to nuclearity of W, most results concerning the one-variable CKS-space (50) admit straightforward generalization to the (finitely many) multi-variable case (51). However, for its importance in applications and lack of written literatures it seems worthwhile to give a brief account. Throughout this section we fix an integer n > 1 and consider a finite set T = { 1 , 2 , . . . , n} equipped with the counting measure. By the standard construction 18 with the help of the operator A ® 1 on L 2 (R x T) = L2(TV) ® R® n , where A = 1 + t2 - d2/dt2 as in Section 1.2, we obtain a real Gelfand triple: S{R x T) C L 2 (R x T) c S'{R x T).

(52)

Using nuclearity of iS(R), one sees easily that n times

5(Rxr)^E®R"^£ffi-®£,

E = S{R),

(53)

and hence n times n

<S'(R x T) S E* ® R

3* £ * © • • • © £ * ,

E* = S'(R).

(54)

We denote elements of 5 ( R x T) and «S'(R X T) by £ = ( & , . . . , £ n ) and x = (x\,..., x n ), respectively. Then n

(*,0 = £>*,&>• fc=l

Here it is important to note that the direct sum E © • • • © .E is obtained from a single space R x T and a single operator A® 1 = A® • • • @ A acting on L 2 (R x T) £ L 2 (R) ® R n £ L 2 (R) © • • • © L 2 (R) (n times). Let pi be the standard Gaussian measure associated with the Gelfand triple (52), i.e., determined by ei<"'c>pi(das),

e-\C\l/2=f

£eS(RxT).

(55)

JS'CRxT)

Recall that the one-variable CKS-space (50) is constructed from L2(E*,fj,) with the help of p-norms and a given weight sequence a. In the multivariable case, the p-norm is defined by the second quantized operator T(A (g> 1) and in the exactly same manner we come to the Gelfand triple: W C L2{S'(R

x T), Ji) ~ T(L 2 (R x T)) c W*.

(56)

172 On t h e other hand, by the uniqueness of the characteristic function, we see from (55) t h a t p, = fi x • • • x fj, ( n times) according t o the identification (54). Hence L 2 (<S'(R x T),JZ) =

L2(E*,»)®n.

T h e n by nuclearity of W one sees from (51) t h a t

In other words, the Gelfand triple (51) coincides exactly with the CKS-space constructed from L 2 ( R X T ) . Since structure of the underlying space R has no effect on the proofs of characterization theorems for the S-transform and for the operator symbol, the same argument can b e applied in the case where the underlying space is R x T. We only note t h a t || {A ® 1)-" HOP = || A~" | | O P ,

|| (A ® 1)~« | | | s = n\\ A~" \\2HS.

Thus, as literal translation from Theorems 3.1 and 4.1 (the case of n = 1) we obtain the following T h e o r e m A . l A function F : Sc(R x T ) - t C is the S-transform $ S (W®n)* if and only if ( F l ) z H-» F(z£+£')

is entire holomorphic

in z G C for any£,£'

of some

£ «Sc(RxT);

(F2) there exist C > 0 and p > 0 such that \F(S)\2
£eSc(RxT).

in that case,

HIIV^.-^^HI^'IIHS),

(57)

where q > 1/2 is taken in such a way that G Q ( n | | A~q H^g) < oo. T h e o r e m A . 2 A function 6 : 5 C ( R x T ) x 5 C ( R x T ) -+ C is the symbol of an operator E € £ ( W ® " , (W®n)*) if and only if ( 0 1 ) forany$,Z',Ti,T}' € 5 c ( R x T ) the function is entire holomorphic on C x C ; ( 0 2 ) there exist constant

(z,w) i-> 0(z£+£',

numbers C > 0 and p > 0 such that

!©(*,r,)\ 2 < CGa{\i

\2p)Ga(\r,

\2p),

€ , ^ 5

c

( R x T).

WTJ+TJ')

173 In that case || H0 || 2 _ ( p + ? ) i _ < CGl(n\\

A-" \\2ns) || \\2p+qt+ ,

G W®»,

(58)

where q > 1/2 is ta&en as G Q ( n | | A - ' | | | s ) < oo. Characterizations for W ® n and for £ ( W ® " , W ® n ) are also literal translation from known results for t h e case of n = 1. Instead of the translations we shall give more practical statements. We change notation. An exponential vector for (56) is given by ^ ( * ) = e<"'*>-«'*>/ 2 ,

£G<Sc(RxT),

W i t h the identification £ = ( £ i , . . . , £„) G S(R «S'(R x T ) , (59) becomes

M*)=n

n

xeS'c(RxT).

(59)

x T ) and x — ( a ? i , . . . , xn) G

e<-*-^>-«-«»>/a=n ^t(xfc). n

fc=i

*;=i

Hence, t h e S-transform of $ G (W®™)* is identified with a function on £J C in such a way t h a t S*{£u-

. . , £ » ) = « * , 4>tx ® • • • ® ^ € - » . n

6 , •••,&. G £ C -

n

Similarly, the symbol of S G £ ( W ® , (W® )*) is defined by E(£n---.C, l ; 7 ?i.---,»7n) = ((H((/»5l --- ® £„), m ® • • • & 0i,„)>After this change of notation Theorems A . l and A.2 are paraphrased. We only need to recall Hartogs' theorem of holomorphy and multiplicative properties of Ga described in Proposition 1.2. It is convenient to say t h a t a C-valued function F defined on Xn, where X is a complex topological vector space, is Gentire (Gateaux-entire) if z i-> F(£\,. .. , £k + z£',. .. £„) is entire holomorphic in z G C for any £ i , . . . , £ n , £' G X and 1 < fe < n. T h e o r e m A . 3 A function F : EQ —• C is £/ie S-transform of some <J> G (W® n )* i/ and only if ( F l ) F is

G-entire;

(F2) t/iere exisi C > 0 anrf p > 0 such that 71

2

1 ^ ( 6 , • • • , £«)| < C n

G«(| & Ip),

£l, • • • , in G -Be-

k=l

T h e o r e m A . 4 A function 0 : U c X 15 c —> C is £/ie symbol of an S G £ ( W 0 T \ (W® n )*) i/ and onfy if

operator

174 ( 0 1 ) 0 is

G-entire;

( 0 2 ) there exist constant

numbers C > 0 and p > 0 such that n

Wti,---,U,m,---,rh)\2
for£i,...,£n,r)i,...,T)n

e

T h e o r e m A . 5 A function i/ and on/y i/ ( F l ) F is

EC-

F : £Jg -> C is the S-transform

of some $ e W ® "

G-entire;

(F2) for any p > 0 i/iere exists C > 0 sztc/i £/ia£ n 2

1^(6, • • •,6.)l < c- J ] G 1/Q (| a 1%),

6 , • • •,£n e Ec.

T h e o r e m A . 6 A function 0 : E^. x E^. -> C is £Ae symbol of an E G £ ( W ® n , W ® n ) i/ and onfy if ( 0 1 ) 0 is

operator

G-entire;

( 0 2 ) / o r arai/ p > 0 there exist C > 0 and g > 0 such that n 2

l©(6, • • •,£»;m, • • • , ^ ) l < c J ] G a ( | a \2P+q)G1/a(\vk

tp),

fc=l /or^1,...,^n,rji,...,J7„ G

Ec.

References 1. S. T. Ali, J.-P. Antoine and J.-P. Gazeay: "Coherent States, Wavelets and Their Generalizations," Springer-Verlag, 2000. 2. N. Asai, I. Kubo and H.-H. Kuo: General characterization theorems and intrinsic topologies in white noise analysis, to appear in Hiroshima M a t h . J. 3 1 (2001). 3. Y. M. Berezansky and Y. G. Kondratiev: "Spectral Methods in InfiniteDimensional Analysis," Kluwer Academic Publisher, 1995. 4. D. M. Chung, U. C. Ji and N. Obata: Higher powers of quantum white noises in terms of integral kernel operators, Infinite Dimen. Anal. Quant u m P r o b a b . Rel. Top. 1 (1998), 533-559.

175 5. D. M. Chung, U. C. Ji and N. O b a t a : Quantum stochastic analysis via white noise operators in weighted Fock space, to appear in Rev. M a t h . Phys. 6. W. G. Cochran, H.-H. Kuo and A. Sengupta: A new class of white noise generalized functions, Infinite Dimen. Anal. Q u a n t u m P r o b a b . Rel. Top. 1 (1998), 43-67. 7. R. Gannoun, R. Hachaichi, H. Ouerdiane a n d A. Rezgui: Un theoreme de dualite entre espaces de fonctions holomorphes a croissance exponentielle, J. Funct. Anal. 1 7 1 (2000), 1-14. 8. L. Gross and P. Malliavin: Hall's transform and the Segal-Bargmann map, in "Ito's Stochastic Calculus and Probability Theory (N. Ikeda, S. W a t a n a b e , M. Fukushima and H. K u n i t a (Eds.)," p p . 73-116, SpringerVerlag, 1996. 9. T. Hida: "Brownian Motion," Springer-Verlag, 1980. 10. U. C. J i and N. O b a t a : A role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space, preprint, 2000. 11. U. C. Ji, N. O b a t a and H. Ouerdiane: Analytic characterization of generalized Fock space operators as two-variable entire functions with growth condition, to appear in Infinite Dimen. Anal. Q u a n t u m P r o b a b . Rel. Top., 2001. 12. J. R. Klauder: The action option and a Feynman quantization of spinor fields in terms of ordinary C-numbers, Ann. Phys. 1 1 (1960), 123-168. 13. J. R. Klauder and B.-S. Skagerstam: "Coherent States," World Scientific, 1985. 14. I. K u b o and S. Takenaka: Calculus on Gaussian white noise I-IV, P r o c . J a p a n Acad. 5 6 A (1980), 376-380; 411-416; 5 7 A (1981), 433-437; 5 8 A (1982), 186-189. 15. I. K u b o and Y. Yokoi: Generalized functions and functionals in fluctuation analysis, in "Mathematical Approach to Fluctuations II (T. Hida, Ed.)," p p . 203-230, World Scientific, 1995. 16. H.-H. Kuo: "White Noise Distribution Theory," CRC Press, 1996. 17. Y.-J. Lee: Analytic version of test functionals, Fourier transform and a characterization of measures in white noise calculus, J. Funct. Anal. 1 0 0 (1991), 359-380. 18. N. O b a t a : "White Noise Calculus a n d Fock Space," Lect. Notes in M a t h . Vol. 1577, Springer-Verlag, 1994. 19. N. Obata: Integral kernel operators on Fock space - Generalizations and applications to quantum dynamics, Acta Appl. M a t h . 4 7 (1997), 49-77. 20. N. Obata: A note on Hida's whiskers and complex white noise, in "Analysis on Infinite Dimensional Lie Groups a n d Algebras (H. Heyer and J.

176 Marion, Eds.)," pp. 321-336, World Scientific, 1998. 21. N. O b a t a : Coherent state representations in white noise calculus, Can. M a t h . Soc. Conference Proceedings 29 (2000), 517-531. 22. N. Obata: Coherent state representation and unitarity condition in white noise calculus, J. Korean M a t h . Soc. 3 8 (2001), 297-309. 23. N. Obata: Unitarity criterion in white noise calculus and nonexistence of unitary evolutions driven by higher powers of quantum white noises, to appear in P r o c . Mexico G u a n a j u a t o Conference Proceedings, 2001. 24. H. Ouerdiane: Fonctionnelles analytiques avec condition de croissance, et application a I'analyse gaussienne, J a p a n . J. M a t h . 2 0 (1994), 187-198. 25. J. Potthoff and L. Streit: A characterization of Hida distributions, J. Funct. Anal. 1 0 1 (1991), 212-229. 26. Y. Yokoi: Simple setting for white noise calculus using Bargmann space and Gauss transform, Hiroshima M a t h . J. 2 5 (1995), 97-121.

Quantum Information IV (pp. 177-185) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co. TOPICS ON COMPLEX GAUSSIAN R A N D O M FIELDS

SI SI Faculty of Information Science and Technology Aichi Prefectural University Aichi-ken 480-1198, Japan WIN WIN HTAY Department of Computational Mathematics University of Computer Studies Yangon, Myanmar

Mathematics

0

Subject Classification

(2000):

60H40

Introduction

We are interested in the complex random field which can be expressed in terms of complex white noise as

Z(C)=

[ 7(c)-

F{C;u)z(u1)--.-z(un)dun,

where C runs through t h e class C ~ = {C; C diffeomorphic to S

, convex }.

T h e z(u),u G R , denotes the complex white noise: x(u) + iy(u), where x(u) and y(u) are mutually independent real white noises. It is observed t h a t the complex r a n d o m field Z(C) is harmonic in the sense of the Gross Laplacian, and further the innovation of Z(C) is established by taking its variation. 1

Background

We are going to discuss complex Gaussian random fields given by

Z(C)=

f F(C;u)z(u1)---z{un)durl, 7(C)"

(1.1)

178 which is parameterized by C in C

1

:

C d _ 1 = {C; diffeomorphic to 5 d _ 1 , convex}, where (C) denotes the domain enclosed by C. T h e z(u),u G Rd denotes the d-dimensional parameter complex white noise which is expressible as z{u) — x(u) + iy(u), x , y being mutually independent real white noises. As t h e background, we first recall the complex Gaussian systems and the complex white noise. 1.1 C o m p l e x G a u s s i a n S y s t e m Let {Q., B, P) be a probability space and let Z(w) be a complex-valued r a n d o m variable on (f2, B, P), such t h a t E(X(u>)) — m, m G C( complex numbers ), Re(Z{u)) = X(u) and Im(Z(u)) = Y(iv). D e f i n i t i o n If X(LJ) and Y(w) are independent and they have the same Gaussian distribution with mean zero, then

Z(u>) = m + X(v) is called complex

+ iY(uj)

Gaussian.

D e f i n i t i o n Let Z = {Z\(tv); X £ A} b e a system of complex-valued r a n d o m variables. If any linear combination £ ? • CjZ\ , Cj G C, is complex Gaussian, then Z is called a complex Gaussian system. T h e o r e m For any vector m = (m.\;A G A) and positive definite matrix V = ( V A , ^ ; A , ^ G A) on A, there exists a complex Gaussian system Z = {Z\ : A G A} with a mean vector and a covariance matrix which coincides with the given vector m and the matrix V , respectively. Let the random variable Z(LJ) be complex Gaussian, z £ C b e a complex variable, and t and s be arbitrary complex numbers. We have 1. E[expi(tZ

+ sZ)] = exp[i(tm+sm)

— tsa2], where m = E(Z)and

2

E[\Z-m\ }. 2. If E[Z] = 0, then we have

E[exP(tZ)}

= 1, E[Zn]

= 0, n > 0, E[ZnZm)

=

8n!mn\E[{\Z\2}n].

a2 —

179 3. Let Hp,q{z,z) rnite polynomial

= (-l^+'el^^f^eH2!2, p,q > 0, b e the complex of degree (p, q). Then, the system of functions of Z(ui)

Her-

Up\ql)-SHPtq(Z(U),2(u>));p,q>0}, is an orthonomal system in L 2 (fi, P). (See e.g. [2].) 1.2

C o m p l e x white noise

Let Ec = E + iE and E* = E*+iE* with a nuclear space E{c L2(Rd)). Denote £ 6 Ec by £ = £ + irj, £,rj £ E. Then, z e E* is of the form z = x + iy,x,y € E*. Set v = /ii x /i2, where /xi and /^2 \ on E*(= iE*)such t h a t

exp[i(x,£)]dfj,j(x) JE

are

white noise measures with variance

= exp

-\m\2

, J = 1,2.

Let B be the smallest sigma field generated by the cylinder subsets of E*. Then, (E*,B,i/) is called a complex white noise. T h e class

H° = { ( z , C ) ; C e . E c } is a complex Gaussian system on (E*, B, is), and so is 7i° = {(z, £); £ £ We have

*.C>]= /

Ec}.

(z,0d„(z) = 0, £[2] = ||C||2,

JE'C

where || • || denotes the L2(i?)-norm.

T h e covariance m a t r i x of "H0 is given by

E[(z,(i)KGJ] = {(!,&)• 1.3. C o m p l e x m u l t i p l e W i e n e r i n t e g r a l of d e g r e e

(m,n)

Denote the subspace of {L2C) spanned by t h e Fourier-Hermite polynomials of degree (m,n) based u p o n a complete orthonormal system {ivn} by T-L(m,n)Some known results are now in order. 1. T h e family of all functionals of the form

180 where the product is finite and the _/Vs are distinct, is a complete orthonomal system in (L2.). 2. An element of %(„,_„) is now called a complex multiple Wiener integral of degree (771,71). Set n n — k,k) 5

fc=0

which is the collection of members of degree n. T h e space (L2) has therefore the direct sum decomposition

(E © n

fc=0

\

W

<-M> /

00

= nE= 0© W »

3. A member ip(z) in T-L(n^) has an integral representation of the form
F(u)z(Ul)---z(un)dun,

where F is a symmetric L 2 (iZ n )-function. (For simplicity d is taken to be 1.) A generalization of such a representation can be obtained for any H(m,n) -functional. 2

C o m p l e x G a u s s i a n r a n d o m field

Define a complex Gaussian r a n d o m field Z(C) Z{C)=

f

by

F(C;u)z(Ul)---z{un)dun,

(2.1)

He) where C"*"1 = {C; C S C 2 , diffeomorphic to S^1},

(2.2)

and z = x + iy is a complex white noise. Let (i) Let Lc be t h e closed sbspace of L2(E*,v) volving the (z,0, supp(() C (C).

generated by t h e algebra in-

(ii) the sigma field B( C j is the smallest sigma field of subsets of E* with respect to which all members in Lc are measurable.

181 Obviously Z(C)

lives in the space Lc

and

LcCL2(B{c),v)c(L2c).

Proposition. Set L = \jcLc (the lattice sum). Then, we have a direct sum decomposition of the space L:

n

As in [3], we can introduce the differential operators where t h e white noises x(u) and y(u) are taken to be variables. Namely, d dx(u)'

d dy(u)

as well as the higher order partial derivatives. Indeed, they can be defined rigorously by using our standard technique of white noise analysis, although one might think the variables have a formalsignificance. Moreover, we can introduce the Gross Laplacian A Q with or without restriction of the domain of the parameter u. For instance, r g2 Q2 A

°(C) = /

7TT^\dud-

[FTU +

J(C) dx{uY

dy(uy

By expressing z as x + iy, we can easily see t h a t

f * y z ( C )+f * Vz(c)=0 \dx(u)) holds for Z(C)

K

K

^ydyiu))

>

given by (2.1). Thus, we have (AG)(C)(Z(C))

= 0. d

W i t h a particular choice of (C) — R

we can define AG(Rd)

which is

simply denoted by A c T h e o r e m 1. For any C the random field Z{C) = Z(C, z) is harmonic

in the

sense of the Gross Laplacian A g . R e m a r k Before closing this section we shall make a short, general remark on the representation of a stochastic process or a random field which is expressed

182 as a stochastic integral with respect to a r a n d o m measure. T h e notion of the canonical kernel was first introduced when t h e canonical representation of a Gaussian process was introduceed. This particular property of a representation means t h a t , intuitively speaking, the random measure and the represented process have exactly the same information until any fixed time. Hence, there is no need to restrict the process to be Gaussian to define the canonical property of the kernel. W i t h this understanding, we have the freedom to use the established criterion for t h e kernel to b e canonical, in the case where t h e given process or the r a n d o m field is expressed as a stochastic integral. W i t h this remark, we assume t h a t the kernel F in the integral of the form (2.1) is always taken to b e canonical, in what follows.

3

Innovation

It is known t h a t the infinitesimal equation (the stochastic variational equation) for a real Gaussian random field X(C) can be defined as a generalization of the Levy's infinitesimal equation for the variation 5X(t) of a stochastic process X(t). For the case of a random field X(C), the equation for the variation 5X(C) may be expressed in the form

5X(C) = $(X(C") C < C, Y(a), seC,C,

SC),

(3.1)

where C < C means t h a t C" is inside of C, t h a t is, the domain ( C ) enclosed by C is a subset of (C), and where
183

6Z(C) = $(Z{C);

C < C, z(s), z(s), s e C,C, 5C),

where C < C means, as before, that C" is inside of C, and where $ is a nonrandom function. The system {z(s), z(s), s G C;C} is the innovation in the sense that the family of the pairs {z(s),z(s),s 6 C,} is independent of the past {Z(C'),C < C} and that it describes the new information gained by Z(C) in the infinitesimal neighbourhood of C. Remark. For the innovation it suffices for us to obtain either one of z(s) and z(s) from 5Z(C), since the other is complex conjugate. Theorem 2.

The innovation for the random field Z[C) given by (3.2) is

l ,M z(s) = ^ {

rSZ(C)-E(5Z(C)\Z(C'),C'
(,)}, s G C,

(3.2)

where
5Z(C) = n[ f Jc J{c)"-1

+ f f F'n(C,u;s)zn®{u)8n{s)dunds, n Jc J(c)

(3.3)

where v\ = (ui,U2,--- ,u„_i) and F'n denotes the functional derivative of F{C\ u) evaluated at C. Now take the weak conditional expectation E which is defined by the orthogonal projection: E(SZ(C)\Z(C'),C


f

f

F'n{C,u;s)zn®(u)5n(s)dunds.

Then, we have 5Z(C) - E(6Z\Z(C),

C < C)

= n f [ F{C,v1;s)z(n-1)^{v1)z(s)dv1l-16n{s)ds. l Jc J(c)"~

(3.4)

184 Let 5n vary all over in the class of positive C°° —functions in such a way t h a t 5C is taken outward and t h a t the integrand over C is determined as a function of t h e variable s. Then, the right hand side of (3.5) will determine the following

z(s)

F{C,v1]a)z^-1^{v1)dvr1.

[

(3-5)

./(C) —

Let it be denoted as

z(*M«)

(3-6)

and use the same technique as in one dimensional parameter cace (cf. [4]). Thus, we know the value ip{s)2.

(3.7)

We may ignore its sign to determine <£>(s) . Divide (3.7) by f(s) to obtain the generalized innovation z(s) for a moment. Since the representation is cannonical, it can b e regarded as the same as the original complex white noise z(s). This means t h a t it is the real innovation (not in a generalized sense). T h e Theorem is therefore proved. It should be noted t h a t the equation (3.2) gives the original complex white noise z in (2.1), because the kernel F is canonical. However, if the kernel F is not cannonical, we can see an example such t h a t the following inequality holds. 5Z{C) - E{SZ\Z(C), n f

[

1

C


F(C,v1;s)z{n~1)®{v1)z{s)dv^-15n(s)ds.

(3.8)

Jc J(c)"-

Hence, in such a case we are given a generalized innovation which may be different from t h e original z. R e m a r k . T h e innovation z(s) can not be obtained in LQ, however it can exist in the space (L^). Now, as an appendix, we wish to note some elementary, but useful relationships among t h e real stochastic processes arising from the Fourier transform of white noise.

185 Take a real white noise x(t). can b e expressed in the form

Then, the Fourier transform x(A) of

x(X) = xr(X) + iii{\) where xr(X) the form

x(t)

(3.9)

and X{(X) are real valued. Then, we have formal expressions of 1

xr(X) =

f°° / cosXux(u)du V27T J-ca

(3.10)

i r00 £i(A) = _ / sin Xux(u)du. V27T 7-cx)

(3-H)

x(X) = x(-X).

(3.12)

Thus, we see t h a t

F a c t . {x,.(A), A > 0} and {x;(A), A > 0} are (real) white noises and {£(A), A > 0} is a complex white noise. A c k n o w l e d g e m e n t s T h e authors are grateful to t h e organizers of the Quant u m Information Conference who gave the opportunity to present the results reported in this paper.

References 1. T. Hida, Cannonical representation of Gaussian processes and their applications, Mem. Coll. Sci. Univ. Kyoto, 33 (1960), 258-351. 2. T. Hida, Brownian motion, Springer-Verlag. 1980. 3. T. Hida et al. W h i t e noise. An infinite dimensional calculus, Kluwer Academic P u b . 1993. 4. T. Hida and Si Si, Innovations for random fields. Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics Vol 1, No. 4, World Scientific P u b . Co. (1998), 499-510. 5. Si Si, Gaussian processes and Gaussian random fields. Q u a n t u m Information II, ed. by T.Hida and K. Saito, World Scientific P u b . Co. 2000, 195-204. 6. Si Si, R a n d o m fields and multiple Markov properties. Supplementary Papers for the Second International Conference on Unconventional Models of Computation U M C 2 K , C D M T C S 147, Solvay Institute 2000, 64-70.

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Quantum Information IV (pp. 187-200) Eds. T. Hida and K. Saito © 2002 World Scientific Publishing Co.

Q U A N T U M INFORMATION IN SPACE A N D TIME A N D THEORY OF S T O C H A S T I C PROCESSES I G O R V. V O L O V I C H Steklov Mathematical Institute Gubkin St.8, 117966 Moscow, Russia E-mail: [email protected] Many important results in modern quantum information theory have been obtained for an idealized situation when the spacetime dependence of quantum phenomena is neglected. We emphasize the importance of the investigation of quantum information in space and time. Entangled states in space and time are considered. A modification of Bell's equation which includes the spacetime variables is suggested. A general relation between quantum theory and theory of classical stochastic processes is proposed which expresses the condition of local realism in the form of a noncommutative spectral theorem. Applications of this relation to the security of quantum key distribution in quantum cryptography are considered.

1

Introduction

Recent remarkable experimental and theoretical results have shown that quantum effects can provide qualitatively new forms of communication and computation, sometimes more powerful then the classical ones, for a review see e.g. [1]. Interesting and important results obtained in quantum computing, teleportation and cryptography are based on the investigation of basic properties of quantum mechanics. Especially important are properties of nonfactorized entangled states which were named by Schrodinger as the most characteristic feature of quantum mechanics. Modern quantum information theory is built on ideas of classical information theory of C. Shannon and on the notions of von Neumann quantum mechanical entropy and of entangled states as formulated by J. Bell, see [2] for a recent review. The spacetime dependence is not explicitly indicated in this approach. As a result, many important achievements in modern quantum information theory have been obtained for an idealized situation when the spacetime dependence of quantum phenomena is neglected. We emphasize the importance of the investigation of quantum information effects in space and time. a "The importance of the investigation of quantum information effects in space and time and especially the role of relativistic invariance in classical and quantum information theory was stressed in the talk by the author at the round table at the First International Conference on Quantum Information which was held at Meijo University, November 4-8, 1997.

188 In this paper entangled states in space and time are considered. A modification of Bell's equation which includes t h e spacetime variables is suggested. A general relation between q u a n t u m theory and classical theory of stochastic processes is proposed which expresses the condition of local realism in the form of a noncommutative spectral theorem. Applications of this relation to t h e security of q u a n t u m key distribution in q u a n t u m cryptography are considered. Entangled states, i.e. the states of two particles with the wave function which is not a product of t h e wave functions of single particles, have been studied in many theoretical and experimental works starting from works of Einstein, Podolsky and Rosen, Bohm and Bell, see e.g. [3]. Bell's theorem [4] states t h a t there are q u a n t u m spin correlation functions t h a t can not b e represented as classical correlation functions of separated random variables. It has been interpreted as incompatibility of the requirement of locality with t h e statistical predictions of q u a n t u m mechanics [4]. For a recent discussion of Bell's theorem see, for example [3] - [9] and references therein. It is now widely accepted, as a result of Bell's theorem and related experiments, t h a t "Einstein's local realism" must b e rejected. Let us note however t h a t , evidently, t h e very formulation of t h e problem of locality in q u a n t u m mechanics is based on ascribing a special role to the position in ordinary three-dimensional space. It is rather surprising therefore t h a t the space dependence of the wave function is neglected in discussions of the problem of locality in relation to Bell's inequalities. Actually it is the space p a r t of t h e wave function which is relevant t o t h e consideration of t h e problem of locality. We know t h a t the wave function of particle includes not only the spin p a r t but also the part depending on spacetime variables. Recently it was pointed out [9] t h a t in fact the spacetime part of the wave function was neglected in the proof of Bell's theorem. However just t h e spacetime part is crucial for considerations of property of locality of q u a n t u m system. Actually the spacetime part leads to an extra factor in q u a n t u m correlations and as a result t h e ordinary proof of Bell's theorem fails in this case. We present a modification of Bell's equation which includes space and time variables. We present a criterion of locality (or nonlocality) of q u a n t u m theory in a realist model of hidden variables. We argue t h a t predictions of q u a n t u m mechanics can be consistent with Bell's inequalities for some Gaussian wave functions a n d hence Einstein's local realism is restored in this case. Moreover we show t h a t due to the expansion of the wave packet t h e locality criterion is always satisfied for nonrelativistic particles if regions of detectors are far enough from each other. This result has applications to t h e security of certain

189 q u a n t u m cryptographic protocols. We will consider an important connection between q u a n t u m mechanics and theory of classical stochastic processes. Consider for example an equation cos(a - (3) = Eia-qp where £a and rjp are two r a n d o m processes [10] and E is the expectation. Bell's theorem states t h a t there exists no solution of t h e equation for bounded stochastic processes such t h a t |£ Q | < 1, \T)p\ < 1. T h e function cos(a —/3) describes the q u a n t u m mechanical correlation of spins of two entangled particles. It was shown in [9] t h a t if one takes into account the space p a r t of the wave function t h e n the q u a n t u m correlation in the simplest case will take the form 5rcos(a — /3) instead of just cos(a — /3) where t h e parameter g describes the location of the system in space and time. In this case one gets a modified equation g cos (a - 0) = E£ar)p One can prove (see below) t h a t if g is small enough then there exists a solution of the modified equation. It is i m p o r t a n t to study also t h e more general question: which class of functions / ( s , t) admits a representation of t h e form f(s,t)

=

Exsyt

where xs and yt are bounded stochastic processes and also analogous question for the functions of several variables f{t\,..., tn). Bell's theorem constitutes an important p a r t in q u a n t u m cryptography [11]. It is now generally accepted t h a t techniques of q u a n t u m cryptography can allow secure communications between distant parties [12] - [19]. T h e promise of secure cryptographic q u a n t u m key distribution schemes is based on t h e use of q u a n t u m entanglement in the spin space and on q u a n t u m nocloning theorem. An important contribution of q u a n t u m cryptography is a mechanism for detecting eavesdropping. However in certain current q u a n t u m cryptography protocols the space p a r t of t h e wave function is neglected. B u t just the space part of t h e wave function describes the behaviour of particles in ordinary real three-dimensional space. As a result such schemes can be secure against eavesdropping attacks in t h e abstract spin space but could be insecure in t h e real three-dimensional space. We will discuss how one can try to improve the security of q u a n t u m cryptography schemes in space by using a special preparation of the space p a r t of t h e wave function, see [18]. Note t h a t some problems of q u a n t u m teleportation in space have been discussed in [20].

190 2

Bell's T h e o r e m

In the presentation of Bell's theorem we will follow also more references. Bell's theorem reads:

[9] where one can find

cos(a - (3) ? EtaTip

(1)

where £a and r\p are two r a n d o m processes such t h a t |£ Q | < 1, |T?/31 < 1 ar >d E is the expectation. In more details: T h e o r e m 1. There exists no probability space (A, T, dp{\)) and a pair of stochastic processes £ a = £ a (A), r\p = ^ ( A ) , 0 < a,(3 < 2~K which obey lf«(A)| < 1, |»?/9(A)| < 1 such t h a t the following equation is valid cos(a -13) = E£arjp

(2)

for all a and f3. Here the expectation is EtaVfi = f

fa(A)^(A)dp(A)

JA

We say t h a t Eq.

(2) has no solutions (A, T, dp(\),

£a, r\p) with the b o u n d

16.1 < i, \nP\ < i. We will prove the theorem below. Let us discuss now t h e physical interpretation of this result. Consider a pair of spin one-half particles formed in t h e singlet spin s t a t e and moving freely towards two detectors. If one neglects the space p a r t of the wave function then one has t h e Hilbert space C 2 C 2 and the q u a n t u m mechanical correlation of two spins in t h e singlet s t a t e rpspin £ C 2 ® C 2 is Dspin(a, b) = {if>api„\a • a a • b\ipspin) = —a • b (3) Here a = (ai,a2,a3) and b = (b\,b2,b3) are two unit vectors in threedimensional space R 3 , o" = [<j\,o"2, cr3) are the Pauli matrices,

CTI =

(?U)'

cr2 =

(!7)'

CT3 =

(L°I)'

3

t=l

and

^" = 7l((?) 0 (o)-(o)K?))

191

If the vectors a and b belong to the same plane then one can write —a • b = cos(a—/3) and hence Bell's theorem states that the function Dspin(a, b) Eq. (3) can not be represented in the form

P(a,b) = J t(a,\)ri{b,\)dp{\)

(4)

i.e. Dspin(a,b)^P{a,b)

(5)

Here £(a, A) and -q(b, A) are random fields on the sphere, |£(a, A)| < 1, \rj{b, A)| < 1 and dp(X) is a positive probability measure, Jdp(X) = 1. The parameters A are interpreted as hidden variables in a realist theory. It is clear that Eq. (5) can be reduced to Eq. (1). To prove Theorem 1 we will use the following Theorem 2. Let / i , /2, g\ and g2 be random variables on the probability space (A,T,dp(X)) such that l/i(A)fli(A)| < 1, i,j = 1,2. Denote Pij=Efigj,

i,j = 1,2.

Then \Pn-Pi2\

+ \P2i + P22\<2.

Proof of Theorem 2. One has Pu - P12 = Efl9l

- Ehg2

= E(fm{l

± / 2 5 2 ) ) - E{hg2{l

±

f2gi))

Hence |Pn - P u | < E(l ± f2g2) + E ( l ± f29l)

= 2 ± (P22 + P21)

Now let us note that if x and y are two real numbers, then \x\<2±y

-•

H + |y|<2.

Therefore taking x = Pu — P\2 and y = P22 + P 2 i one gets the bound \Pll-Pl2\

+

\P2l+P22\<2.

The theorem is proved. The last inequality is called the Clauser-Horn-Shimony-Holt (CHSH) inequality. By using notations of Eq. (4) one has \P(a, b) - P(a, b')\ + \P(a', b) + P{a', b')\ < 2

(6)

192 for any four unit vectors a, b, a', b'. P r o o f o f T h e o r e m 1. Let us denote /i(A) = e«.(A), gj(X) = r1l3j(X), for some ai,f3j.

i,j = 1,2

If one would have cos ( a ; - / ? . , ) =

Efigj

then due to Theorem 2 one should have | c o s ( a i - f3i) - cos(ai — /3 2 )| + | cos(a 2 - Pi) + cos(a 2 — /?2)| < 2. However for a.\ = vr/2, a 2 = 0, /3i = n/4,f32

— —n/4 we obtain

I c o s ^ — Pi) - cos(a! —/9 2 )| + | c o s ( a 2 —fix) + cos(a 2 - P2)\ = 2\/2 which is greater t h a n 2. This contradiction proves Theorem 1. It will b e shown below t h a t if one takes into account the space part of the wave function then the q u a n t u m correlation in the simplest case will take the form g c o s ( a — /3) instead of just cos(a — /3) where t h e parameter g describes the location of t h e system in space and time. In this case one can get a representation g cos(a -(3)

= E£ar]p

(7)

if g is small enough. T h e factor g gives a contribution to visibility or efficiency of detectors t h a t are used in the phenomenological description of detectors. 3

Localized Detectors

In t h e previous section the space part of the wave function of the particles was neglected. However exactly the space part is relevant to t h e discussion of locality. T h e Hilbert space assigned to one particle with spin 1/2 is C 2 ® L 2 ( K 3 ) and the Hilbert space of two particles is C 2 L2(M.3) ® C 2 L 2 ( R 3 ) . T h e complete wave function is i/> = (ipap(ri, r2,t)) where a and f3 are spinor indices, t is time and r i and r 2 are vectors in three-dimensional space. We suppose t h a t there are two detectors ("Alice" and " B o b " ) which are located in space R3 within the two localized regions OA and Og respectively, well separated from one another. Q u a n t u m correlation describing the measurements of spins by Alice and Bob at their localized detectors is

G(a, 0A,b,

GB) = (xl>\a • aPOA

® )

(8)

193

Here PQ is the projection operator onto the region O. Let us consider the case when the wave function has the form of the product of the spin function and the space function ip — ipapin
b)

(9)

where the function g(0A,0B)=

|0( r i ,r 2 )| 2 dndr 2

f

(10)

JOAxOB

describes correlation of particles in space. It is the probability to find one particle in the region OA and another particle in the region OB. One has

o
(n)

Remark. In relativistic quantum field theory there is no nonzero strictly localized projection operator that annihilates the vacuum. It is a consequence of the Reeh-Schlieder theorem. Therefore, apparently, the function g(0A, OB) should be always strictly smaller than 1. Now one inquires whether one can write the representation g(0A,0B)Dspin(a,b)=

U{a,0A,\)v(b,0B,X)dp{\)

(12)

Note that if we are interested in the conditional probability of finding the projection of spin along vector a for the particle 1 in the region OA and the projection of spin along the vector b for the particle 2 in the region OB then we have to divide both sides of Eq. (12) by g(0A, OB). Instead of Eq (2) in Theorem 1 now we have the modified equation g cos{a - P) = E^rtf

(13)

The factor g is important. In particular one can write the following representation [8] for 0 < g < 1/2: /

gcos(a-(3)=

v

Jo

2scos(a-A)v/2^cos(/3-A)—

(14)

27T

Therefore if 0 < g < 1/2 then there exists a solution of Eq. (13) where &»(A) = y/2gcos(a-

A), rj^(A) = y ^ c o s ^ - A)

and |£ Q | < 1, | ^ | < 1. If g > l / \ / 2 then it follows from Theorem 2 that there is no solution to Eq. (13). We have obtained

194 T h e o r e m 3 . If g > l / \ / 2 t h e n there is no solution (A, J-, dp(\), £a,77,a) to Eq. (13) with the bounds |£„| < 1, \qp\ < 1. If 0 < g < 1/2 t h e n there exists a solution to Eq. (13) with the bounds |£ Q | < 1, \r}p\ < 1. R e m a r k . A local modified equation reads |^(r1,r2,t)|2cos(a-/3) = ^(a,r1,t)r?(/a,r2,i). Relativistic Particles We can not immediately apply the previous considerations to t h e case of relativistic particles such as photons and the Dirac particles because in these cases t h e wave function can not b e represented as a product of t h e spin p a r t and the spacetime part. Let us show t h a t t h e wave function of photon can not be represented in the product form. Let Ai(k) be the wave function of photon, where i = 1, 2, 3 and k S M.3. O n e has t h e gauge condition klAi{k) — 0. If one supposes t h a t the wave function has a product form Ai(k) = if(k) then from the gauge condition one gets A{(k) = 0. Therefore the case of relativistic particles requires a separate investigation. N o n c o m m u t a t i v e Spectral T h e o r y and Local R e a l i s m As a generalisation of the previous discussion we would like to suggest here a general relation between q u a n t u m theory and theory of classical stochastic processes [10] which expresses the condition of local realism. Let % be a Hilbert space, p is the density operator, {Aa} is a family of self-adjoint operators in "H. One says t h a t t h e family of observables {Aa} and the state p satisfy to the condition of local realism if there exists a probability space (A, T, dp(\)) and a family of r a n d o m variables {£„} such t h a t the range of £Q belongs to the spectrum of Aa and for any subset {A{\ of mutually commutative operators one has a representation

Tr{pAil...AiJ

= EZil..4in

If t h e family {Aa} would be a maximal commutative family of self-adjoint operators then t h e previous representation would be nothing but the usual spectral theorem [21]. In our case the family {Aa} consists from not necessary commuting operators. Hence we can call such a representation a noncommutative spectral representation. Of course one has a question for which families of operators and states the noncommutative spectral theorem is valid, i.e. when we can write t h e noncommutative spectral representation. It would b e helpful t o study t h e following problem: describe t h e class of functions f(t\, ...,£„) which admits t h e representation of the form

f(t1,...,tn) where xt,..., 1.

=

Exh...ztn

zt are r a n d o m processes which obey the bounds |ast| < 1,..., | z t | <

195

From the previous discussion we know that there are such families of operators and such states which do not admit the noncommutative spectral representation and therefore they do not satisfy the condition of local realism. Indeed let us take the Hilbert space % = C 2 C 2 and four operators Ai,A2,A3,A4 of the form (we denote A3 = B\,A± = B2) =

/ s i n a i cosa, \ ycosai — s i n a j y

/ s i n a 2 cosa 2 \ 0 ycosa^ — sin 0:2/

f

and ^-cospi

sin/?! y

y-cosp2

sinft

/

Here operators A^ correspond to operators a- a and operators B; corresponds to operators a-b where a = (cos a, 0, sin a), b = (— cos/3, 0, — sin/3). Operators v4, commute with operators Bj, [Ai, Bj] = 0, i, j = 1,2 and one has {i>spin\AiBj\tj)1,pin) = cos(ai - f3j), i,j = 1, 2 We know from Theorem 2 that this function can not be represented as the expected value E£ii~)j of random variables with the bounds |&| < 1, |?7j| < 1. However, as it was discussed in this paper, the space part of the wave function was neglected in the previous consideration. It is tempting to speculate that in physics one could have only such states and observables which satisfy the condition of local realism. Perhaps we should restrict ourself in this proposal to the consideration of only such families of observables which satisfy the condition of relativistic local causality. Let us now apply these considerations to quantum cryptography. 4

The Quantum Key Distribution

There are quantum cryptographic protocols with one and with two particles, for a review see for example [19]. Here we shall consider the quantum key distribution with two particles. Ekert [11] showed that one can use the EinsteinPodolsky-Rosen correlations to establish a secret random key between two parties ("Alice" and "Bob"). Bell's inequalities are used to check the presence of an intermediate eavesdropper ("Eve"). There are two stages to the quantum cryptographic protocol, the first stage over a quantum channel, the second over a public channel. The quantum channel consists of a source that emits pairs of spin onehalf particles, in a singlet state. The particles fly apart towards Alice and Bob, who, after the particles have separated, perform measurements on spin

196 components along one of three directions, given by unit vectors a and 6. In the second stage Alice and Bob communicate over a public channel.They announce in public the orientation of the detectors they have chosen for particular measurements. T h e n they divide t h e measurement results into two separate groups: a first group for which they used different orientation of the detectors, and a second group for which they used the same orientation of the detectors. Now Alice and Bob can reveal publicly the results they obtained b u t within t h e first group of measurements only. This allows them, by using Bell's inequality, to establish the presence of an eavesdropper (Eve). T h e results of t h e second group of measurements can be converted into a secret key. One supposes t h a t Eve has a detector which is located within the region OE and she is described by hidden variables A. We will interpret Eve as a hidden variable in a realist theory and will study whether the q u a n t u m correlation can b e represented in t h e form Eq. (12). From Theorem 3 one can see t h a t if t h e following inequality g(oA,

oB)

< 1/2

(15)

is valid for regions OA and OB which are well separated from one another then there is no violation of t h e CHSH inequalities (6) and therefore Alice and Bob can not detect the presence of an eavesdropper. On the other side, if for a pair of well separated regions OA and OB one has 9(OA,OB)

>l/v / 2

(16)

then it could be a violation of the realist locality in these regions for a given state. Then, in principle, one can hope to detect an eavesdropper in these circumstances. Note t h a t if we set g(0A, OB) = 1 in (12) as it was done in the original proof of Bell's theorem, t h e n it means we did a special preparation of the states of particles to be completely localized inside of detectors. There exist such well localized states (see however t h e previous Remark) but there exist also another states, with the wave functions which are not very well localized inside t h e detectors, and still particles in such states are also observed in detectors. T h e fact t h a t a particle is observed inside the detector does not mean, of course, t h a t its wave function is strictly localized inside the detector before the measurement. Actually one has to perform a thorough investigation of the preparation and the evolution of our entangled states in space and time if one needs to estimate the function g(0A, Og).

197 4-1

Gaussian

Wave

Functions

Now let us consider t h e criterion of locality for Gaussian wave functions. We will show t h a t with a reasonable accuracy there is no violation of locality in this case. Let us take the wave function of the form <j> = V'i( r i)V'2(r2) where the individual wave functions have the moduli

Itfi Wl2 = (^) 3/2 e- mV/2 ,|<Mr)| 2 = A ^

e

- ^ " ^

(17)

We suppose t h a t the length of the vector 1 is much larger t h a n 1/m. We can make measurements of PQA and PQB for any well separated regions OA and OB- Let us suppose a rather nonfavorite case for the criterion of locality when the wave functions of the particles are almost localized inside t h e regions OA and OB respectively. In such a case the function g(0A, OB) can take values near its maximum. We suppose t h a t the region OA is given by \ri\ < 1/m, r = (7*1, r2,r3) and t h e region OB is obtained from OA by translation on 1. Hence I/>I(TI) is a Gaussian function with modules appreciably different from zero only in OA and similarly ^2(^2) ls localized in the region OB- Then we have

9(0Al OB) = ( - L f e-»2/2dx) \ V27T J-l

(18)

J

One can estimate (18) as

g(0A,0B)<(^]

(19)

which is smaller t h a n 1/2. Therefore the locality criterion (15) is satisfied in this case. Let us remind t h a t there is a well known effect of expansion of wave packets due to the free time evolution. If e is t h e characteristic length of the Gaussian wave packet describing a particle of mass M at time t = 0 then at time t the characteristic length et will be

I

h?t2

,v

It tends to (h/Me)t as t -4 00. Therefore the locality criterion is always satisfied for nonrelativistic particles if regions OA and OB are far enough from each other.

198 5

Conclusions

We have studied some problems in q u a n t u m information theory which requires the inclusion of spacetime variables. In particular entangled states in space and time are considered. A modification of Bell's equation which includes t h e spacetime variables is suggested a n d investigated. A general relation between q u a n t u m theory and theory of classical stochastic processes was suggested which expresses t h e condition of local realism in the form of a noncommutative spectral theorem. Applications of this relation to t h e security of q u a n t u m key distribution in q u a n t u m cryptography were considered. There are many interesting open problems in t h e approach to q u a n t u m information in space a n d time discussed in this paper. Some of t h e m related with t h e noncommutative spectral theory and theory of classical stochastic processes have been discussed above. In q u a n t u m cryptography there are important open problems which require further investigations. In q u a n t u m cryptographic protocols with two entangled photons to detect t h e eavesdropper's presence by using Bell's inequality we have to estimate t h e function g(OA,Og). To increase t h e detectability of t h e eavesdropper one has to do a thorough investigation of t h e process of preparation of t h e entangled s t a t e and then its evolution in space and time towards Alice a n d Bob. One has to develop a proof of t h e security of such a protocol. In t h e previous section E v e was interpreted as a n abstract hidden variable. However one can assume t h a t more information about Eve is available. In particular one can assume t h a t she is located somewhere in space in a region OE- It seems one has to study a generalization of t h e function g(OA,Os), which depends not only on t h e Alice a n d Bob locations OA a n d OB b u t also depends on the Eve location OE, a n d try to find a strategy which leads to an optimal value of this function.

Acknowledgments I a m grateful to Prof. T. Hida a n d Prof. K. Saito for t h e invitation to the stimulating and fruitful conference on Q u a n t u m Information at Meijo University. This work was supported in part by R F F I 99-0100866 a n d by INTAS 99-00545 grants.

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