Non-Commutativity, Infinite-Dimensionality and Probability at the Crosssroads
QP-PQ: Quantum Probability and White Noise Analysis
Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy
QP-PQ: Quantum Probability and White Noise Analysis
Vol. 16:
Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hora
Vol. 15:
Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg
Vol. 14:
Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola
Vol. 13: Foundations of Probability and Physics ed. A. Khrennikov
QP-PQ VOl.
10:
Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay
Vol. 9:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 8:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 7 :
Quantum Probability and Related Topics ed. L. Accardi
Vol. 6:
Quantum Probability and Related Topics ed. L. Accardi
QP-PQ Quantum Probability and White Noise Analysis Volume XVI Proceedings of the RIMS Workshop on Infinite-DimensionalAnalysis and Quantum Probability
n=Commutativity, Non-Commutativity, Non-Commutativity, Non-Commutativity, Non-Commutativity, Kyoto, Japan
20 - 22 November 2001 Editors
Nobuaki Obata Graduate School of Information Sciences Tohoku University, Japan
Taku Matsui Graduate School of Mathematics Kyushu University, Japan
Akihito Hora Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology Okayama University, Japan
r LeWorld Scientific
New Jersey London Singapore Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224
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U K ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library
NON-COMMUTATIVITY, INFINITE-DIMENSIONALITY, AND PROBABILITY AT THE CROSSROADS The Proceedings of the RIMS Workshop on Infinite Dimensional Analysis and Quantum Probability Copyright 0 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoJ 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 981-238-297-6
This book is printed on acid-free paper
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Preface Since the academic year of 1992 we have organized annually workshops on infinite dimensional analysis and quantum probability at Research Institute for Mathematical Sciences, Kyoto University. The papers in this volume are contributed by lecturers of the series of workshops and most of the papers were presented at the 10th workshop held during November 20-22, 2001. We would like to thank all the lecturers and participants who have contributed to creat a very stimulating atomosphere. The essential purpose of these workshops was to provide a forum to exchange new ideas emerging from various research areas and to promote collaboration among scientists with different backgrounds. We believe that such a basic idea is reflected in this volume. This volume consists of two parts: expository articles and research papers. We collected four expository articles: 0
Asao Arai: Mathematical theory of quantum particles interacting with a quantum field
0
fianco Fagnola: H-P quantum stochastic differential equations
0
Fumio Hiai: Free relative entropy and q-deformation theory
0
Un Cig Ji and Nobuaki Obata: Quantum white noise calculus
The topis discussed in the above papers, by no means covering all of our research interests, indicate some of the concrete outcome of our forum. The fourteen research papers deal with most current topics and their interconnections reflect a vivid development in this research area. From a technical point of view we are most grateful to the Research Institute for Mathematical Sciences for their constant supports and also acknowledge the support by Grant-in-Aid for Scientific Research from Japan Society for Promotion of Sciences. Finally we thank Professor W. Freudenberg for kind invitation to include this volume in the series of “Quantum Probability and White Noise Analysis.” We hope that this volume would contribute to the development of infinite dimensional analysis and quantum probability. Nobuaki Obata Taku Matsui Akihito Hora Editors October 2002
V
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Contents
Preface
V
Expository Articles Mathematical Theory of Quantum Particles Interacting with a Quantum Field A . Arai
1
H-P Quantum Stochastic Differential Equations F. Fagnola
51
Free Relative Entropy and q-Deformation Theory F. Hiai
97 143
Quantum White Noise Calculus U. C. Ji €4 N. Obata
Research Papers Interacting Fock Spaces and Orthogonal Polynomials in Several Variables L. Accardi €4 M. Nahni Can “Quantumness” Be an Origin of Dissipation? T. Arimitsu Eventum Mechanics as the Crossroad of Probability, Infinite-Dimensionality and Non-Commutativity V. P. Belavkin What is Stochastic Independence? U. Franz
vii
206
225 254 275
Creation-Annihilation Processes on Cellar Complecies Y. Hashimoto White Noise Analysis - A More General Approach T. Hida Characters for the Infinite Weyl Groups of Type B,/C, and for Analogous Groups T. Hirai €4 E. Hirai
192
288 and D,, 296
viii
Noncommutative Aspect of Central Limit Theorem for the Irreducible Characters of the Symmetric Groups A . Hora
318
Brownian Motion and Classifying Spaces R. Le'andre
329
Fock Space and Representation of Some Infinite Dimensional Groups T. Matsui €4 Y. Shimada
346
Cauchy Processes and the LBvy Laplacian N. Obata €4 K. Saito
360
Separation of Non-Commutative Procedures - Exponential Product Formulas and Quantum Analysis M. Suzuki
374
Free Product Actions and Their Applications Y. Ueda
388
Remarks on the s-Free Convolution H. Yoshida
412
Memorandum
435
Author Index
437
MATHEMATICAL THEORY OF QUANTUM PARTICLES INTERACTING WITH A QUANTUM FIELD ASAO ARAI Department of Mathematics Hokkaido University Sapporo, 060-0810 Japan E-mail: araiOmath.sci.hokudai. ac.jp
Dedicated to Professor Tukashi Ichiszose on the occasion of his sixtaeth birthday A survey on recent developments in mathematical theory of quantum particles interacting with a quantum field is presented. Contents 1 Introduction 2 Fock Spaces and Second Quantization
3
4
5
6
7
8
2.1 Two Kinds of Fock Spaces 2.2 Second Quantization Operators Boson Fock Space 3.1 The Creation and the Annihilation Operators 3.2 Representation of the CCR 3.3 Tensor Products of Boson Fock Spaces 3.4 Basic Estimates Fermion Fock Space Description of Models-An Abstract Form 5.1 Quantum Particle Systems 5.2 A Composed System of Quantum Particles and a Bose Field A List of Concrete Models 6.1 Non-Relativistic QED 6.2 The Nelson Type Model 6.3 The Generalized Spin-Boson Model 6.4 The Drezinski-G6rard Model 6.5 A Particle-Field Model in Relativistic QED Self-Adjointness of Hamiltonians 7.1 The Abstract Particle-Field Hamiltonian 7.2 Hamiltonians in Non-Relativistic QED 7.3 The GSB and the DereziriskkG6rard Harniltonians 7.4 The Dirac-Maxwell Hamiltonian Existence of Ground States
1
2 8.1 Definition of Ground States and Preliminary Remarks 8.2 An Example-The
9 10 11 12
1
Abstract van Hove Model 8.3 Infrared Singularity 8.4 Basic Strategies Absence of Ground States Embedded Eigenvalues, Resonances and Spectral Properties Scattering Theory Other Problems
Introduction
This paper is intended to be an introductory survey, mainly for non-experts and graduate students, on recent developments in mathematical theory of quantum particles (“quantum mechanical matters”) interacting with a quantum field. In this introduction we first describe some historical and physical backgrounds of important problems or issues to be investigated. As is well known, the spectrum of lights emitted or absorbed by an atom has a discrete distribution, which corresponds to the discreteness of the energy levels of the atom. In the case of a hydrogen-like atom with atomic number 2, which consists of a nucleus with electric charge e Z and one electron ( e is the fundamental charge), the main feature of the energy levels is given by the formula m~2e4
En ----
2n2
(n = 1,2,3,. . . )
where n E N (the set of natural numbers) and m denotes the electron mass.l) The sequence is called the principal energy levels. Theoretically, formula (1) can be derived as eigenvalues of the Hamiltonian2)
Hhyd
:=
--21m A,
Ze2 1x1
- -,
of the hydrogen-like atom acting on L2(R3), where L2(Rd)denotes the Hilbert space of square integrable functions on the d-dimensional Euclidean space Rd with respect to the d-dimensional Lebesgue measure and A, with x E Rd is l ) We use a unit system such that c(the light speed in the vacuum) = 1 , h := hl(27r) = 1 (h is the Planck constant). 2, The operator representing the total energy of a quantum system is called the Hamiltmian of the system.
3
the generalized d-dimensional L a p l ~ i a n . ~We ) remark that the eigenvalue En of H h y d is degenerate with multiplicity n2. The operator H h y d is an example of the so-called Schrodinger o ~ e r a t o r s . ~ ) Experimentally, however, the principal energy levels with n 2 2 have finer structures. These are explained as spectral properties of the operator
a relativistic version of H h y d , where Dj is the generalized partial differential operator in the variable xj, aj and p are 4 x 4 Hermitian matrices obeying the anticommutation relations
= 26jk, {aj,p} = 0, p2 = 1, j , k = 1,2,3 (4) ( { A , B } := AB + BA). The operator D h y d , acting in the Hilbert space e4L2(R3), is an example of the Dzrac per at or.^) To be concrete, D h y d has {ffj,Ctk}
eigenvalues
En,J
(5) where J = L f 1/2 are eigenvalues of the total angular momentum with 0 _< J 1/2 _< n and 0 5 L 5 n - 1 (1 is an eigenvalue of the orbital angular momentum).6) Formula (5) shows that, for each n 2 2, the energy level (eigenvalue) En,J are degenerate, since different values of orbital angular momentum L may give the same J. For example, the energy level E2,112 with n = 2 and J = 1/2 is degenerate, because there are two states with the same energy E2,1/2; the one is with orbital angular momentum L = 0 and the other with orbital angular momentum L = 1. We denote the former by 2S112 and the latter by 2Plp.
+
~~
~~
~
~
3)For a derivation of ( l ) , see, e.g., 511.3 in [87]. This example is one of typical examples which show that non-relativistic quantum mechanics is valid, to considerable degree, for atomic systems. 4, A d-dimensional Schr6dinger operator is given by an operator of the form -A, V (up to constant multiples) on L 2 ( R d )with V :Rd + R. 5, A general form of a three-dimensional Dirac operator is given by -i C,”,, aj Dj +Pm+V with V a function on R3 with values in the set of 4 x 4 Hermitian matrices. 6 , For a derivation of (5), see, e.g., p.35 in [30].
+
4
Remark 1.1 In the non-relativistic region (Ze2 << 1),7) we have
E,,J - m
-
En (Ze2 + 0 ) ,
obtaining, as the non-relativistic limit of en,^ - m, formula (1) in the nonrelativistic theory using the Hamiltonian Hhyd. Prior to 1947, formula ( 5 ) was in a completely satisfactory agreement with experimental spectral data of the hydrogen-like atom, except for the so-called hyperfine splitting of each energy level due to coupling between the electron and proton spins.*) However, in 1947, Lamb and Retherford [80] discovered an energy-shift between the 2S1p-state and the 2P1l2-state which, as remarked above, have the same energy in the Dirac theory. The energy level of the former is more than that of the latter, the difference being about 1OOOMHz in frequency. This energy level shift is called the Lamb shift. A theoretical explanation of the Lamb shift was first given by Bethe [28]. The basic idea is that the Lamb shift is due to an effect of the interaction between the electron and the quantized (quantum) radiation field. Using a non-relativistic model whose Hamiltonian is of the form Hhyd + Hrad H I , where Hrad is the free Hamiltonian of the quantum radiation field and HI is an operator denoting the interaction between the electron and the quantum radiation field (see Section 6.1 in the present paper), and taking into account a mass renormalization for the electron, he derived a formula for the Lamb shift which is in relatively good agreement with the experiment. The physical theory of quantum charged particles (e.g., electrons) interacting with the quantum radiation field is called quantum electrodynamics (QED). The original QED is relativistic. But, in considering atomic phenomena, it is not so unnatural to treat non-relativistically the quantum charged particles under consideration (note that the speed of an electron in an atom may be “small” in average compared with the speed of light). This type of QED is called non-relativistic QED. For example, the model used by Bethe [28] is a model of non-relativistic QED. Using the same model as that of Bethe, Welton [lo81 showed that the Lamb shift can be explained also as a fluctuation effect of the quantum radiation field in the vacuum. In this case too, the mass renormalization is needed.
+
7, In a unit system in which c (the speed of light) is restored as a parameter, Ze2 is equal to Ze2f c and hence the non-relativistic limit c -+ 00 gives Ze2f c -+ 0. A spin is an angular momentum of “internal rotation” of an elementary particle. The electron and the proton both have spin 1/2. In the case of the Dirac operator Dhyd, the spin degree of freedom is reflected through the matrices aj and p.
5
After Bethe’s work [28], calculations for the Lamb shift using relativistic QED also were made with renormalization theory. The results showed surprisingly good agreement with experiment (see, e.g., 57-3-2 in [74]). Calculations usually done in QED or in other quantum field models use formal perturbation theory, where “formal” means that no attention is paid on convergence of perturbation series. This applies to the case of Behte [28] and of Welton [lo81 too. In calculations of the Lamb shift, the following (a) and (b) seem to be assumed tacitly: (a) the Lamb shift is a shift of an eigenvalue EO of the unperturbed Hamiltonian due to the perturbation caused by the interaction of the electron with the quantum radiation field; (b) the shifted eigenvalue E has an asymptotic expansion E = EO eEl e2E2 . . . in the coupling constant e, a perturbation series, and E - EO may give the Lamb shift. However, these assumptions are not trivial at all. From a consistent view-point, they should be proved based on a model rigorously defined. In addition, assumption (a)may be physically dubious in fact. This is suggested from the existence of the spontaneous emission of photons from an atom in an excited.~tate.~)Namely, an atom in an excited state is not stable and, even without stimulation of external photons, after some time, a probabilistic transition of the excited state to a state with a lower energy can occur. This means that the energy of each excited state cannot be an eigenvalue in the strict sense of the term (recall that an eigenstate belonging to an eigenvalue of a quantum mechanical Hamiltonian describes a stable state). Therefore it is expected that the total Hamiltonian of a model in QED has no eigenstates perhaps except for a ground state, an eigenstate with the lowest energy. In most of the physics literature, QED is not treated in a mathematically rigorous way. In fact one faces some mathematical difficulties even in defining QED properly, which are connected with the trouble of infinite divergences in perturbative calculations of quantities of physical interests. Thus, in view of mathematical physics which should give a mathematically rigorous foundation to QED, one has to proceed as follows: (i) to define a model of QED properly; (ii) to analyze nonperturbatively poperties of the model with mathematical rigor and to show rigorous mathematical forms underlying physical phenomena such as the Lamb shift. This applies also to other models in quantum field theory (QFT). There are two categories in QFT: the one is relativistic and the other is non-relativistic. It is still an open problem to make it clear if any non-trivial completely relativistic quantum field model exists in the four-dimensional space-time, where a model in QFT is said to be non-trivial if it is not unitarily
+
The photons are the quanta of the quantum radiation field.
+
+
6
equivalent to a free field model, a model which describes no interactions of elementary particles. A typical class of non-relativistic QFT consists of models of non-relativistic quantum particles interacting with quantum fields. This class is the main object in this paper. From mathematical point of view, as suggested by the problems in QED mentioned above, models in QFT present challenging and important problems in infinite dimensional and non-commutative analysis. The present paper is organized as follows. In Section 2 we introduce Fock spaces which present mathematical frameworks for models in QFT. There are two classes of quantum fields: Bose fields and F e n i fields, corresponding to which there are two kinds of Fock spaces, i.e., Boson Fock spaces and Fernion Fock spaces respectively (e.g., the quantum radiation field is a Bose field, while the quantum electron field is a Fermi field). We describe them in the abstract forms. In the Fock spaces, second quantization operators are defined, which are used to define observables of quantum fields such as Hamiltonians and total momenta. In Section 3 we define basic objects on the abstract Boson Fock space and present elementary facts in the theory of Boson Fock space. For a comparison with the abstract Boson Fock space and for the (nonexpert) reader’s convenience in reference to Fermi fields, we give in Section 4 a brief description of basic objects on the abstract Fermion Fock space. Section 5 is devoted to an abstract formulation of some models of quantum particles interacting with a Bose field. In Section 6 we present a list of concrete models. From Section 7 to Section 11, we explain some fundamental subjects and problems with an overview of main results established so far. We do not go into technical details, but, try to concentrate our attention on ideas and methods for solving problems. In the last section we have a quick look at some of other important problems.
2
Fock Spaces and Second Quantization
An elementary particle with an integer spin (resp. a half-integer spin E { 1/2,3/2, . }) is called a boson (resp. fernion). For example, the electron is a fermion with spin 1/2 and the photon is a boson with spin 1. A quantum field whose quanta are bosons (resp. fermions) is called a Bose (resp. Ferni) field. In this section we introduce fundamental spaces, called Fock spaces, for mathematically rigorous descriptions of quantum fields. As mentioned at the end of Introduction, corresponding to the two classes of quantum fields, i.e., the Bose field and the Fermi field, there are two kinds of Fock spaces, which we first describe in this section.
--
7
2.1 Two Kinds of Fock Spaces Let 3-1 be a complex separable Hilbert space with inner product ( . , .)x and norm II-11x (we sometimes omit the subscript 3-1 if there would be no danger of confusion). Let @3,-1 (n E N) be the n-fold tensor product of 3-1 (go% := C ) . The infinite direct sum of these Hilbert spaces
F(3-1):= @go €9, 3-1
(6)
M
n=O
J
is called the full Fock space over 3-1. Let S, be the symmetric group of order n and S, be the symmetrization operator on @"X: 1 := u,, n!
c
s,
UES,
where U,, is a unitary operator on @"'ti such that U,,(~!Q@*--@Gn) = +,,(I) @ . @ &(,,), +j E 3-1, j = 1, . . ,n. It is easy to see that S, is an orthogonal Hence the subspace projection on @%.!
--
-
BF3-1 := Sn(@W) (8) is a Hilbert space. I t is called the n-fold symmetric tensor product of 3-1. We set @fX:= C . The Hilbert space
Fb(3-1):=o=@: @F 3-1 (9) is called the Boson Fock space over 3t or the symmetric Fock space over 3-1. is defined by The anti-symmetrization operator A , on
where sgn(n) is the signature of the permutation CJ E S,. The operator A, is an orthogonal projection on @n3-1. Hence one has a Hilbert space A"(3-1) := A,(@'%),
called the anti-symmetric tensor product of Hilbert space A('?!)
:= @",=o
X. We set
A" (31)
AO(7-f) :=
(10) C. The (11)
8
is called the Fermion Fock space over 3-1 or the anti-symmetric Fock space over 'U. In applications to QFT, the Boson (resp. Fermion) Fock space (resp. A(%)) is used to describe state vectors of a Bose (resp. Fermi) field. In this context, 3c is the Hilbert space of state vectors of one boson (resp. fermion) and is called the one-particle Hilbert space of the Bose (resp. Fermi) field under consideration. The Hilbert space 8a3-1 (resp. A"(%)) as a subspace of Fb(3-1) (resp. A('U)) is called the n-particle subspace of (resp. A(3-1)) . Example 2.1 A typical example of the Boson Fock space is given by the case 3-1 = L2(Rd).In this case we have the natural identification
&(x)
&(x)
@.,"L2(Rd) = LZy,(Rd") := {Ic, E
L2(Rd")lV0 E Sn1Ic,(k1,---,kn) = $ ( k , ( l ) , - - . ,k+))
=},
where "a.e." means almost everywhere with respect to the measure under consideration (in the present case, the Lebesgue measure on Rdn).Hence Fb
( L 2 ( R d )= ) @r=oL&,(Rd")
with LZYm(Rdn)Jn=o := C. Example 2.2 The Hilbert space of a non-relativistic particle with spin 1/2 moving in Rd (e.g., the electron) is given by L2(Rd;C2).Let S := {-l,l} (the configuration space of the spin degree of freedom) and po be the meausre on S with po((1)) = po({-l)) = 1. Then
L 2 ( R dC ; 2 )2 L2(Rdx S,dx 8 dpo) =
[
f : Rd x S + CI llf1I2
Ld
If(x,s)12dx <
:= s=fl
Hence, putting
Lg, (Rd"x S", dx" @ dpg ) := {Ic, E L2(Rdnx S",dX" @ d$)I
we can identify A" (L2(Rd; C2)) as
A"(L2(Rd; C2)) = LL(Rd"x S",dx" @ dpg).
m)
,
9
Hence
2.2 Second Quantization Operators
For a linear operator T on a Hilbert space, we denote its domain by D ( T ) . If T is densely defined, then we denote the adjoint of T by T*. For a subspace D c D ( T ) ,TID denotes the restriction of T to D . We denote by the closure of T if T is closable. The spectrum (resp. the point spectrum) of T is denoted 4 T ) b P . ap(T)). For a densely defined closed linear operator A on 7-t and j = 1,. .. ,n, we define a linear operator Aj on g1~7-tby j-th v
Aj := I @ * * - @ I @ A @I*--@I, where I denotes identity. For each n E (0) U N, we define a linear operator A(") on @n3c by n
A(') := 0, A(") := C A j 1 @& D ( A ) , n 2 1 ,
(12)
j=1
where @&D(A) means the n-fold algebraic tensor product of D ( A ) . The infinite direct sum of these closed operators
m ( A ) := cBZ=~A(~) on the full Fock space F(7-t)is called the second quantization of A. It is easy to see that m ( A ) is closed. The operator &(A) has an eigenvector
071 := { 1 , 0 , 0 , . ~ ~ }
(13)
with eigenvalue 0:
d ? ( A ) 0= ~ 0. (14) The vector 071 is called the Fock vacuum. The following theorem tells us basic properties of &(A) with A being self-adjoint. Theorem 2.1 Let A be self-adjoint. Then: (i) d ' ( A ) is self-adjoint.
10
(ii) If A is nonnegative, then so is d'(A). (iii)
where, for a set S
c R, 3
(iv) If a(A) n (-00, 0)
denotes the closure of S.
# 0 , then C ( A ) is not bounded below.
This theorem can be proven by the spectral theory for tensor products of self-adjoint operators.lO) The second quantization of A = I
N := C ( I )
(15)
is called the number operator. It follows that NI (W"')n D ( N ) = n, n 2 0. It is easy to see that &'(A) is reduced by the Boson Fock space &(%) and the Fermion Fock space A(X) respectively. We denote the reduced part of &(A) to Fb(31)and A(%) by &b(%) and &f(A) respectively. We set
Nb := &b(I), Nf := &f(I). (16) The operator Nb (resp. Nf) is called the boson (resp. fermion) number operator. Example 2.3 Consider the case where 3c = L2(Rd)(Example 2.1). Let w be a nonnegative Bore1 measurable function on Rd such that 0 < w ( k ) < 00 a.e. with respect to the Lebesgue measure dx on Rd. Then w defines uniquely a self-adjoint, nonnegative and injective multiplication operator on L2(Rd), denoted also w. Then
$ E D(nb(U)),
2 1.
(17)
In applications to QFT, w ( k ) usually denotes physically the energy of a free boson (a boson having no interactions with other objects) with momentum lo)
§VIII.lO in [94]. A detailed proof is given in Proposition 3.10 in 1141.
71 w(lcj) is a total energy of n free (wave vector) k . In this interpretation, bosons. The form of d I ' f ( w ) is the same as in (17), but with $ E D(dl?f(w)) C A (L2(Rd)). Typical examples of w are given as follows:
4 with m 2 0. This is the case where the boson under (i) w ( k ) = consideration is relativistic with mass m. If m > 0 (resp. m = 0), then the boson is said to be massive (resp. massless). k2
(ii) w ( k ) = - (rn > 0). This is the case where the boson under consid2m eration is non-relativistic with positive mass m.
3
Boson Fock Space
A natural dense subspace in the Boson Fock space
&(x) is given by
{ {$(n)}r=o&(x)there exists
Fo('?i):= $ =
E
a natural number no
such that, for all n 2 no, $(") = 0 } ,
(18)
called the subspace of finite particle vectors. We enumerate below basic linear operators on Fb(7-l).
3.1
The Creation and the Annihilation Operators
For each f E X,there exists a unique densely defined closed linear operator a(f) on &(%) such that the following (i)-(iii) hold: (i) For all f E 'U, Fo(7-l)c D ( a ( f ) )and
(iii) For all f E 'U and $ =
Fo(3C)is a core of a(f).
{$(n)}z=o E D(a(f)*),
12
The operators a(f) and a(f)* are called respectively the boson annihilation operator and the boson creation operator with "test vector" f . For all f E X,a(f)and a(f)*leave Fo(X)invariant obeying the canonical commutation relations (CCR)
b(f),a(g>*l= ( f , S ) N , [a(f),a(g)l= 0,
[a(f)*,a(g)*l= 0,
f,g E
(21) (22)
X
on Fo(X), where [ A,B ]:= AB - BA. Example 3.1 In the case of Fi,(L2(Rd)) (Example 2.1) we have for all f E L 2 ( R d )and Ij) E D(a(f))
(a(f)~j))(n)(kl,.-. ,kn)=
-1
f(k)*d("+l)(k,kl,-.-,k,)dk, n 2 0, R d
and, for all 4 E D ( a ( f ) * ) ,
where i j indicates omission of kj. Let S ( R N )( N E N) be the Schwartz space of rapidly decreasing C"functions on RN and SsYm(Rdn) be the set of symmetric elements in S(Rdn): Ssym(Rd")
:= {Ij) E S ( R d " ) I V a E
Sn,Ij)(kl,...,kn) = I j ) ( k u ( l ) , . . . ,ku(n))}-
Let
Ds(Rd) := {Ij) = {I~)'"'}:=o
E 30( L 2 ( R d ) )
E Ssym(Rdn),n 2 1).
Then, for each k E Rd,one can define a linear operator a(k) on F,,(L2(Rd)) with domain Ds(Rd) by
(a(k)~j))(")(kl,... ,kn):= ~ T ~ I ~ ) (,k,,), ~ +~j) ~E ) DS(Rd), ( In C 2 0. , It is easy to see that, for dl ?b, E Ds(Rd),the mapping : k continuous and
+ u(k)$
is strongly
for all f 6 L 2 ( R d )where , the integral is taken in the sense of strong Bochner integral. The operator a(k) is a rigorous version of the annihilation operator formally used in the physics literature. We remark, however, that a(k) is not closable. Indeed, the domain of the adjoint a(k)* as a linear operator on
13
&(L2(Rd))is (0) (Proposition 8.2 in [14]). But, noting the formal expression of a(k)*
where 6 ( - ) is the Dirac delta distribution on Rd,one can redefine a(k)* as a continuous linear operator from Ds(Rd) to the dual of D&,d) of Ds(Rd) with a suitable topoplogy. This idea leads one to white noise analysis.[91] It follows that
[a(k),ak-9*1= a(k - P), k,P E Rd. Moreover we have
In particular
as a sequilinear form on Ds(Rd). For each subspace V of X,we define
Fb,fin(v) := {$ = { $ ( " ) ) ~ = o E FO(x)I$(") E sn(@&v), n 2 1).
(23)
One can show that, for all f E D(A),
[flb(A)ia(f)] = -a(Af),
[mb(A),a(f)*]= a(Af)*
(24)
on Fb,fin(D(A)). This also is a basic relation. In the theory of the Boson Fock space, an important role is played by
14
called the Segal field operator. It is proven that, for each f E 31, +(f) is essentially self-adjoint on F'o(7-l) (e.g., Theorem X.41 in [95]).We denote its closure by the same symbol +(f). Usign (21) and (22), one easily shows that
[+(f),+(9)1 = iIm (f,g)x
on
FO(W.
(26)
Let
.(f)
:= +(if),
f E 31.
Then, for all f E 31,
a(f)=
1 J1z W ' + W ) ) ,4f)* = Jz(+(f' - i..(f))
(28)
on D ( a ( f ) )n D ( a ( f ) * ) . Hence the annihilation and creation operators are expressed in terms of the Segal field operators.
3.2 Representation of the CCR Let J : 31 -+ 31 be a conjugation, i.e., J is an antilinear mapping on 31 such that IlJfll = llfll for all f E 3t and J 2 = I . Then 31.1:= {f E X I J f = f} is a real Hilbert space and each f E 31 is written uniquely as f = f l i f 2 with f l , f 2 E 31.1.It follows from (26) that, all f,g E 31.1,
+
[9(f>,n(g)I = i ( f , g ) x J , [ d ( f ) , + ( g ) I = 0 = [n(f),n(g)] on Fo(31). (29)
In applications to construction of models of Bose fields, the operatorvalued functionals +(-) and n(.) on the real Hilbert space 31.1 are used, e.g., to give time-zero fields and to define a Hamiltonian. In connection with this aspect, we introduce a notion of representation of the CCR. Let X be a Hilbert space and 2) be a dense subspace of X. Let W be a real inner product space. Let IIw := {+(w),7r(w)lw E W } be a set of self-adjoint operators on X. Then we say that the triple {X, D ,IIw} is a representation of the CCR indexed by W if the following hold: (i) for all a, b E R and v, w E W , F(av h)= aF(v) bF(w) on D,where F = 4, R ; (ii) for all v,w E W , D c D($(v)+(w)) nD(+(.>.(w)) n D ( 4 v ) + ( w ) ) n D(4v)n(w)) and
+
+
[+(v>,n(.w)l = i(v,w)w, [+(v), +(w>I = 0, Two representations { X ,23, IIw} and
[dv), .(.1>1 {X',D',I$,,}
= 0 on D.
(III1,
:=
{+'(w),n'(w)Iw E W } ) of the CCR indexed by W are said to be unitarily equivalent if there exists a unitary operator U : X + X' such that V+(w)U-l = +'(w), Un(w)U-l = n'(w) for all w E W .
15
a dense subspace of @:3c with DO := C and 2 0). Let V be a dense subspace of RJ.Then {-Tb(x),V,{~(f),.Ir(f)lf E V } } is a representation of the CCR Let
V
D, ( n E (0)
:= {II,=
U N) be
{II,(")}r=o E Fo(X)III,(")E D,,n
indexed by V . This representation is called the Fock representation of the
CCR. There are infinitely many representations of the CCR inequivalent to the Fock representation. 11)
3.3 Tensor Products of Boson Fock Spaces
In the case where a boson has an internal degree of freedom like spin or "isospin", one-particle Hilbert space is given by a direct sum of some Hilbert spaces.12) In treating a Bose field with such a boson, the following theorem may be useful (for proof of the theorem, see, e.g., $4.7 in [14]). Theorem 3.1 Let 3c1 and "2 be complex separable Hilbert spaces. Then ) F b ( % l ) @ F b ( % 2 ) such there is a unique unitary operator U : F b ( x 1 ~ ~ 3 - 1 2+ that the following hold:
6) Ufl?iH1e3Hz= flH1 @ flH2. (22)
U F b , f i n ( R l @ xZ)= F b , f i n ( x l ) @alg F b , f i n ( R 2 ) -
Moreover, i f Aj is a self-adjoint operator on Rj (j = 1,2), then UO!I'b(Al
@ A2)U-l = d?b(Ai) @ I
+ 1@ n b ( A 2 ) .
In the sense of the isomorphism described in Theorem 3.1, one sometimes writes
3.4 Basic Estimates
In mathematical analysis of Bose fields, the estimates stated in the following proposition are fundamental (for proof, see, e.g., Proposition 3.14 in [ll]or Proposition 4.24 in [14]). See, e.g., Theorem X.46 in [95], 54.8.3 in [14]. For example, L 2 ( R d ;C 2 )= L 2 ( R d ) @ L 2 ( R d(Example ) 2.2). Another example is given by the case of photons, see 56.1. 11)
12)
16
Proposition 3.2 Let A be a nonnegative self-adjoint operator on 3-1. Suppose that A is injective. Then, for all f E D(A-lj2) and 11, E D(m'b(A)'/'),
Ib(f>$II 5 IIA-1/2fl IIlmb(A) 'I2$'11 , ll4f>*11,ll2I llA-1'2f11211~~b(A>1~211,112 + lf1121111,112. Corollary 3.3 Let A be as in Proposition 3.2. Then, for all D(A-1/2) and 11, € D(m'b(A)),
4
E
(31) (32)
> 0, f
E
Fermion Fock Space
Objects analogous to those on the Boson Fock space &(%) may be defined on the Fermion Fock space A(%) too. For each f E Z,there exists a unique bounded linear operator b ( f ) on A(%) such that the following hold (e.g., 85.2 in [32], Chapter 5 in [14]).
+ P g ) = a*b(f)+ , P b ( g ) . (ii) For all f E % and 11, = {ll,(n)}r=o E A(%) (i) For all f,g E %, a,P E C, b(af
(b(f)*11,)(')= 0, (b(f)*$)'") = fiAn(f 8 $("-')),
n 2 1.
(34)
The operators b ( f ) and b ( f ) * are called respectively the fernion annihilation operator and the fernion creation operator with "test vector" f . E %} of operators obeys the canonical antiThe set {b(f),b(f)*lf commutation relations (CAR): for all f , g € %, {b(f),b(g)*} = (f,g), { b ( f ) , b ( g ) I= 0, {b(f)*,b(d*I = 0(35) Example 4.1 In the case of A(L2(Rd;C2))(Example 2.2) we have for all f E L2(Rd;C2j and, for all 11, E A(L2(Rd; C2)),
(b(f)11,)'"'(zl,sl;*- ;zn,sn)
The operators b ( f ) and b ( f ) * as well as the second quantization dl?f(-)are used to define models of Fermi fields such as a quantized Dirac field describing electrons and positrons [14,107].
17
5
Description of Models - An Abstract Form
In this section we present in an abstract form a class of models of quantum particles interacting with a Bose field. 5.1
Quantum Particle Systems
The Hilbert space of state vectors of a system of quantum particles is taken a measure space. in an abstract form to be L 2 ( X , d v )with (X,v) Example 5.1 (X,dv) = (RdN,dx)(N E N). This case describes, in the coordinate representation, the Hilbert space of state vectors of a system of N particles without spin moving in Rd. Example 5.2 (X,dv) = (In,dvitint), I,, := { l , . . .,n},where vcOunt (n) is the counting measure on I,, with v~,"?,({j}) = 1, j = 1 , . . . ,n. Then L 2 ( X ,dv) = L2(In,dvi:int) E C". This case describes a system of a "particle" with n internal degrees of freedom and without external degrees of freedom. For example, the case n = 2 corresponds to that of spin 1/2.
Example 5.3 (X, dv) = (RdNx I f , dx 8 ( ~ 3 ~ d v i : i ~In~this ) ) . case we have the natural unitary equivalence
L 2 ( X , d v ) L2 (RdN,dx)8 ( B N C n )"= L2 (RdN,dx;g N C " ) , where, for a Hilbert space M , L 2 ( X , d v ; M )denotes the Hilbert space of M-valued square integrable functions on (X,v). Therefore this case describes a system of N particles each of which has n internal degrees of freedom. A concrete example is given by a system of N non-relativistic electrons with spin 1/2, for which the Hilbert space of state vectors can be taken to be L2 (RdN,dz; B N C 2 )or AN (L2 (Rd; C2)) (Example 2.2). A Hamiltonian of the particle system is given by a self-adjoint operator K. Example 5.4 In the case of a spin 1/2 particle without external degrees of freedom, the Hilbert space for the system is taken to be C2. The spin degree of freedom is described by the Paula matrices u := (61, u2, u3) with
It is easy to see that
18
A Hamiltonian of the spin system is given by Sh := vo(T1
+ h63,
(36)
where vo > 0 and h E R are constants. The operator SO(Sh with h = 0) has eigenvalues f v o whose eigenstates describe respectively the state with spin f 1 / 2 along an axis. This model may describe a “virtual” atom with only two energy levels. For objects T = (TI,. . . ,T d ) and S = (s1,. . . ,s d ) consisting of d components, we define T S := T . S := TiSi if the products TiSi ( i = 1, ,d) and their sum are defined. We write T 2 := T . T . Example 5.5 A Hamiltonian of a system of N non-relativistic particles with internal degrees of freedom may be given in the form
xf=l
N
H S :=
1 C -(pi 2mj
-
- a j ) 2+ v
j=1
on L 2 ( R d N ; B N C nwhere ), m j > 0 is a constant denoting the mass of the j-th particle, p j = -iVZj with VZj being the gradient operator (in the generalized sense) in the coordinate variable Xj E Rd of the j-th particle (z = ( X I ,... , I N ) E ,Rd x .; x RY, aj : RdN + R (which may denote a N factors
vector potential of an external magnetic field), and V : RdN+ B N C ” (which may depend on a = Under suitable conditions for ( ~ j ) and ~ ! V ~ , Hs is essentially self-adjoint [34].
(aj)gl).
5.2 A Composed System of Quantum Particles and a Bose Field
We consider a composed system of quantum particles with L2( X ,dv) being its Hilbert space of states and a Bose field with one-particle Hilbert space 31. Then the Hilbert space F for the composed system may be taken to be the tensor product of L 2 ( X ,dv) and the Boson Fock space &,(3t)
where the last expression means the constant fibre direct integral with base space ( X ,v) and fibre Fb(31) (e.g., see Chapter XIII.16 in [97]). Let K be a self-adjoint operator denoting the Hamiltonian of the particle system and S be a self-adjoint operator on 31 denoting the one-particle Hamiltonian of the Bose field. Then the unperturbed Hamiltonian of the composed
19
system is defined by
HO := K 8 I
+I@
&rb(S).
(38)
We assume that S is injective. To introduce interactions of quantum particles with the Bose field, let g : X 3 3c be an %-valued measurable function on X . Then we can define a decomposable operator
i.e.,
(4g1cI)W = 4(9(z)Mz),a.e.z, 1c, E
w%J).
Let gj and hj be %-valued measurable functions on X ( j = 1,.. . ,J, J E N), Bj be a symmetric operator on L 2 ( X ,dv), and V j k ( j ,k = 1,- .* ,J) be a bounded linear operator on 7 such that vTk = V k j , j , k = 1 , . . . ,J. Then we define the following operators on 3: J
j=l
Y
j,k=l
With these operators, a total Hamiltonian of the composed system is defined bY
H(X1,Xz) := Ho
+ X1H1 + X2H2,
(41)
where Xj E R ( j = 1,2) are coupling parameters. As is shown below, this model gives an abstract unification of Hamiltonians of quantum particles interacting with a Bose field. We call a Hamiltonian of the form H(X1, X 2 ) a particle-field Hamiltonian. 6
A List of Concrete Models
In this section we present a list of important concrete models which are realized as special cases of the abstract model given in the preceding section.
20
6.1 Non-Relativistic QED
As is explained in Introduction in the present paper, QED is one of the primarily important models in QFT. Completely relativistic QED is still very difficult to handle in a mathematically rigorous manner; the existence of it in the four-dimensional space-time is not yet proven. But non-relativistic QED may be more tractable than relativistic one. Here we present a mathematical description of non-relativistic QED. A basic object in QED is the quantum radiation field which is defined depending on the choice of the gauge.13) In non-relativistic QED, one usually takes the so-called Coulomb gauge. In this gauge, the Hilbert space of onephoton states is taken to be xph:= L2(R3)C3 L2(R3).
Then the Hilbert space of state vectors of the quantum radiation field is given by the Boson Fock space over x p h Frad
:= F b ( x p h ) = Fb(L2(R3)) '8 Fb(L2(R3)),
(42)
where the second equality is taken in the sense of the natural isomorphism given in Theorem 3.1. For T = 1,2, we fix a Borel measurable R3-valued function e(') on R3 satisfying
e(')(k) . e(")(k)= S,,,
-
e(')(k) k = 0 ,
a.e.k E R3, T , s = 1,2.
(43)
The vectors e ( , ) ( k ) , r = 1,2, are called the polarization vectors of a photon with momentum k. The energy Wph(k) of a free photon with momentum k E R3 is given by Wph(k)
:= Ikl.
Let x be a Borel measurable function on R3 satisfying x ( k ) * = x ( - k ) , a.e.k E R3,
and define G; : R3 + x p h by
13)
For physical discussions, see, e.g., [55].
21
Let
Aj(z;X) := ~(G:(z)), j = 1,2,3, (45) where +(.) is the Segal field operator on Frad. Then the quantum radiation field with momentum cutoff x is deifned by
4; x ) := ( A l h XI, Az(z; XI, A 3 b ; XI).
(46)
This is a vector Bose field. By (43), we have for all $ E FO(xph)
which means that A(-;x) satisfies the Coulomb gauge condition. Let
a'"(f) := a ( f , o ) ,
a ' 2 ' ( f ):=
f), (f,o>,(0,f) E x p h -
As in Example 3.1, we can define for each k E R3 and operator a(')(k) on 3rad such that
T
(48)
= 1 , 2 a linear
in the sense of sesquilinear form on DS(R3)galg DS(R3), where the sesquilinear := (a(.)(k)$, 4), a(')(k)($,4) := form a(.)(k)# is defined as ~(~)(k)*($,r$) ('$~a'~)(k)$)4 , $E, DS(R3)@alg DS(R3)The quantized magnetic field is defined by
B(z;x) := rot A(z; x).
(49)
The free Hamiltonian of the quantum radiation field is defined by Hrad := d?b(Uph@ Uph)
wph(k)a(')(lc)*a(')(k)dk,
= 1R3
where the second equality is taken in the sense of sesquilinear form on DS(R3)@alg DS(R3).
As for the particle system interacting with the quantum radiation field, we consider a system of N quantum charged particles with mass m > 0, charge
22
q E R and spin 1/2. Suppose that the particles move under the influence of a scalar potential U : R3N+ R which is in Lt0,(R3N):= {j : R3N+ ClVR > 0, lf(z)I2dz< 00). If no magnetic fields exist, then the Hamiltonian of the particle system is given by the Schrodinger operator
hXlsR
where p j := 4 V Z j (zj E R3)and Axj := Vxj-Vxj,the generalized Laplacian in the variable xj. Example 6.1 Consider the case where there exist N electrons and a nucleus with charge Z e at the origin. Then the potential energy of the Coulomb force is given by
U ( z ) = Ul
+
c -, I N
Ul(2):= -
u2,
j=1
c 1% N
Ze2
U2(z):=
Izj
j
e2
- 211 '
where Ul (resp. U2) is the Coulomb potential between the electrons and the nucleus (resp. the electrons). We set j-th
r$):=I@...@I@
@ I @ - - . @ I ,j = l , - . - , N , a = l , 2 , 3 ,
acting on mNC2,which describes the spin of the j-th particle. The Hilbert space of the composed system is FPF
(@ N C2 ) @ F r a d d z .
:= L2(R3N; B N C 2 )@ Frad =
(51)
L 3 N
The Hamiltonian of the system is defined by
(52) on FPF, where A , ( z j ; x) and B,(zj; x) (a = 1,2,3) are abbreviated notations for the direct integrals A , ( z j ; x)dz and B,(zj; x)dz respectively and 9 E R is a coupling parameter of the spin with the quantized magnetic field (physially it denotes the so-called g-factor). The model defined by the Hamiltonian Hpp(A) is called the full Pauli-Fzerz modeZ with spin [93].
JzdN
JzdN
23
Remark 6.1 If one treats the charged particles as fermions (since they have spin 1/2), the particle Hilbert space should be replaced by AN(L2(R3; C2)) (Example 2.2). The statistics of the charged particles play an important role, e.g., in stability of matter [41,81,83,84]. One may also consider the operator HPF(A) without A2-term
A simplified version of HPF(A) is given by
(54) This is called a Pauli-Fierz Hamiltonian in the dipoZe approximation. A physical picture of this model is that the particles interact with the quantum radiation field only near the origin in the coordinate space R3. We may expect that, if U ( z ) 7 00 (lzl + 00) and IU(0)l << U ( Z ) ~ ( ~ ~ I = ~ ~with , ~ = TO ~ , be...,N ing the atomic radius (TO M 1OW8cm),then Hdipole(A)approximates H ~ F ( A ) in a suitable sense. Remark 6.2 Gauge covariance. Let A be an operator-valued function on R3 with values in the set of (not necessarily bounded) self-adjoint operators on .&ad such that (i) for all z, z' E R3,A(z) and A(.') strongly comrnute,l4) and A,(z; x) ( a = 1,2,3) strongly commutes with A(z'); (ii) for all a, b = 1 , 2 , 3 and z E R3,D,A(z) and Aa(z; x) commute on a suitable subspace. Let A j := J:3N A(zj)dz. Then, for all j , Z = 1 , . . . ,N , Aj and AI strongly commute. We call the unitary operator
uA:= e--iqAi . ..e--iqAN on 3 p the ~ quantum gauge transformation with gauge operator A. It is easy to see that uAHPF(A)Uil = HPF(A')
+ dAHrad
(55)
on a subspace, where A'(Z;X) := A(z;x) - VZA,
bAHrad := UAHrad(A)Uil
- Hrad.
14)Two self-adjoint operators on a Hilbert space are said to strongly commute if their spectral measures commute.
24
The transformation property (55) is called the gauge covariance of the Hamiltonian HPF( A ) . For the gauge transformed quantum radiation field A'(.;x ) to satisfy the Coulomb gauge condition, it is necessary and sufficient that A,A = 0 on a suitable subspace in FPF. Note that, if each A(z) is a constant multiple on the identity operator on F r a d , then UAHPF(A)U;' = HPF(A')on a subspace. A useful example is given in [51] with A(z) = z . A ( 0 ; x ) . In this case we have A'(z; x) := A ( z ;x>_- A(0;x). The Hamiltonians H P F ( A )and &ipole(A) do not have the gauge covariance.
6.2 The Nelson Type Model This model describes N non-relativistic particles interacting with a scalar Bose field on the d-dimensional Euclidean space Rd [go] (originally d = 3). The Hilbert space of states for the model is taken to be
The Hamiltonian is of the form
where M > 0 is the mass of the particle, V : RdN+ R is a scalar potential, w : Rd + [O,m) (Example 2.3), and g : RdN+ L2(Rd). An example of g is given by N
g ( z ) ( l c ) = x x j ( k ) e - i k z j , k E Rd,z= ( z ~ , . . ., z ~ E) RdN j=1
with x j , x j / f i E L 2 ( R d ) .In addition, if V = 0 and w ( k ) = d w (rn > 0 is a constant denoting the mass of a boson), then this is the case of the original Nelson model [go]. 6.3 The Generalized Spin-Boson Model
A model whose Hamiltonian is of the form HGSB:= HO+
x 1
Ba 8 +(fa)
a=l
(57)
25
on 3 is called a generalized spin-boson (GSB) model [18,19,35], where HOis defined by (38), B, is a symmetric operator on L 2 ( X ,dv), fa E 31 and 1 E N. Example 6.2 The Hamiltonian of the standard spin-boson model ([31,52,72,103,106] and references therein) is given by
HSB:= sh 8 I
+I
8 m'b(w)
+ Xa3 8 @ ( g )
on C2 8 31(L2(Rd)), , where s h is given by (36), w : Rd + [0,co), X E R is a constant, g E L2(Rd)with g / f i E L 2 ( R d ) This . is a special case of H G ~ B and describes a two level atom interacting with a scalar Bose field. For recent studies of the spin-boson model, see [31,46,56,57,73,106]. Example 6.3 The Pauli-Fierz model in the dipole approximation without A2 term and spin is described by the Hamiltonian N
where Hatomis given by (50). This also is a special case of the GSB model.
6.4
The Dereziriski-Gdrard Model
An extended version of the Nelson type model and the GSB model (57) with B, bounded can be defined in an abstract form [37]. Let K and 31 be complex separable Hilbert spaces, decribing a "matter" system and a one-boson Hilbert space respectively, so that the Hilbert space for the composed system is given by
K 8 &(%) = @:=OK 8 (8r'H)
(59)
We denote B ( K , K 8 31) the set of bounded linear operators from K to K 8 31. For each w E B ( K , K 8 . 3 1 ) , one can define a linear operator iT*(v) on K@Tb('?d) by (;i*(V)$)'O'
:= 0,
(60)
( ~ ( v ) $ ) ( n ):= f i ( ~ K 8 s,,) (w 8 1 , - - 1 ~ )
$(n-l),
n 2 1,
(61)
11, E DG*(v))
{
:= $ = {$(n)}:=o
m
C
E K 8 F,,(-X)~ Il(iT*(~)$)(")11~ n=O
1
< 00 , (62)
where, for a Hilbert space M , Im denotes the identity operator on M [note that 2) 8 18:-lR is in B ( K 8 (@.,"-l31),K @ (8.,"31))].
26
Remark 6.3 The operator Z*(v) is an extended version of the creation operator u(f)*,where “test vectors” f are replaced by bounded linear operators from K to K 8 X. Indeed, for each f E U and each bounded linear operator B on K ,one defines V ~ , BE B(K, K 8 X)by v f , ~ ( u := ) Bu 8 f, u E K. Then we have
Z * ( V ~ , B= ) B 8 a(f)*. The case B = IK gives the trivial extension of a(f)* to The adjoint
(63)
K 8 Fb(X).
-u(v) := (u*(v))*
(64)
of Z*(v) gives an extended version of the annihilation operator u(f ) (f E N). An analogue of the Segal field operator is defined by 1 &v) := -(Z*(v) Z(v)). (65)
fi
+
Let A be a non-negative self-adjoint operator on N. Then a total Hamiltonian of the composed system is defined by
HDG:= K 8 I + I 8 Clrb(A) + &v). We call it the Derezin’ski-Girard Hamiltonian [37].
(66)
6.5 A Particle-Field Model in Relativistic &ED A relativistic charged particle with spin 112 is called a Dirac particle. In view of the Dirac theory mentioned in Introduction, it is natural to consider the composed system of a Dirac particle interacting with the quantum radiation field and to investigate effects on the Dirac particle due to the interaction with the quantum radiation field. A Hilbert space of the composed system is taken to be
We denote the mass and the charge of the Dirac particle by m > 0 and q E R respectively. Then a natural total Hamiltonian is given by 3
HDM:=
C ctj(-iDj - q A j ( Z , ;x)) + pm + V + Hrad,
(68)
j=1
where a j ,p and Dj are as in Introduction [see (3) and (4)] and V is a function on R3 with values in the set of 4 x 4 Hermitian matrices, denoting an external field. The operator HDM is called a Dirac-Maxwell operator or a DiracMaxwell Hamiltonian [15,17].
27
7
Self-Adjointness of Hamiltonians
In the rest of the present paper, we explain some fundamental problems in mathematical analysis of particle-field interaction models and present a brief overview of results established so far. As is seen from the unified model H(X1,Xz) given by (41) or some of its concrete realizations as discussed in the preceding section, a particle-field Hamiltonian may contain coupling constants (parameters) (e.g., q, g in the Pauli-Fierz Hamiltonian H ~( AF) ) . The modulus of a coupling parameter measures the magnitude of the coupling between particles and a quantum field. In what follows we say that a result (or a property, a fact) is “perturbative” (resp. “non-perturbative”) if it holds in a “small” (resp. a not necessarily small ) neighborhood of the orign in the space of coupling parameters. Of course, non-perturbative results are desirable. In this section we discuss self-adjointness of particle-field Hamiltonians. According to the axiom of quantum mechanics, every observable of a quantum system should be represented as a self-adjoint operator on the Hilbert space of state vectors of the quantum system under consideration. But, usually, for an operator which is a candidate of an observable, one only knows the symmetricity of it at first. It is a non-trivial problem in general to prove the (essential) self-adjointness of such an operator or to define a suitable self-adjoint extension of it. This class of problems is called the self-adjointness problem in mathematical quantum theory. Naturally this problem applies to Hamiltonians of quantum mechanical models. For Hamiltonians of quantum particles (non-relativistic or relativisitc) without interaction with quantum fields, techniques for proving their selfadjointness have extensively been developed [95,107].For particle-field Hamiltonians, however, the methods as just mentioned seem not to be so useful. It would be an interesting problem to investigate if there exist yet unknown general abstract theorems which can apply to various particle-field Hamiltonians.
‘7.1 The Abstract Particle-Field Hamiltonian As for the Hamiltonian H(X1,XZ) given by (41), the following can be proven [13] Theorem 4.10:
Suppose that K i s bounded from below. Then, under suitable conditions on B j , g j , hj and V j k , H ( X 1 , X z ) is self-adjoint on D(Ho), essentially selfadjoint on every core of D(Ho), and bounded from below for all SUBciently small I X j 1, j = 1,2.
28
The idea of proof is to formulate a sufficient condition for each Hj ( j = 1,2) to be relatively bounded with respect to Ho, where estimates (31), (32), (33) and their variants are used. Thus, in the case where the unperturbed particle Hamiltonian K is bounded from below, the self-adjointness problem of particle-field Hamiltonians of the form H(X1,Xa) is affirmatively solved at least for a small coupling region. This result is perturbative, but, if one drops the term X 2 H 2 in H(X1,Xa) and imposes some stronger conditions, one can prove the following nonperturbative fact [13]:
+
Theorem 7.1 Let H A := H(X, 0) = Ho AHl. Assume the following: (i) K is bounded from below; (ii) each Bj ( j = 1, ... ,J ) is bounded; (iii) there exists a core D of &(S) such that D c njJ,lD(&(S)Bj) and [clr(S),Bj]lD is dl?(S)-bounded; (iv) for a.e. z E RdN,g j ( z ) E D(S-1/2) and that ess.supllgj(z)ll < 00, ess.supIIS-'/2gj(z)II < 00, where ess.sup means essential supremum. Then, for all X E R, H A is self-adjoint on D ( H o ) , essentially self-adjoint on every core of Ho, and bounded from below. The method of proof of this theorem is to use (33) to show that H I is infinitesimally small with respect to HO (then one can apply the Kato-Rellich theorem (Theorem X.12 in [95]) to conclude the desired result). As for some concrete models, non-perturbative results on self-adjointness problem of Hamiltonians have been obtained, which we briefly review below.
7.2 Hamiltonians in Non-Relativistic QED A . Existence of Self-Adjoint Extensions Let D := Cr(R3N)@alg ( R N C 2 )@,,1,[3~(~.ph)nD(Hrad)l. Suppose that Hatomis self-adjoint and bounded from below. Then the Pauli-Fierz Hamiltonian HPF(A)is bounded from below. This follows from the diamagnetic inequality [99,100] (cf. Theorem 11.1 in [ll],Theorem 13.26 in [14])
and the infinitesimal smallness of B , ( z j ; x ) with respect to Hrad which follows from (33). Hence H P F ( A ) I Dhas a self-adjoint extension H F ( A ) as the Friedrichs extension of it. Another self-adjoint extension f i p ~ ( Aof) H ~ F ( A )
29
may be defined via the sesquilinear form
) the form domain of SPF. Namely fip~(4) is a selfwhere Q ( s ~ F denotes adjoint operator such that D(lI?pF(A)11/2)= Q(SPF) and ( $ , H P F ( A ) $ )= SPF($, 4) for all $ E Q ( s ~ F and ) 4 E D ( f i p F ( A ) )But . it seems to be an open problem to clarify if H F ( A )= f i p ~ ( Aor ) not.15) In [17] a new self-adjoint extension of H ~ F ( Awith ) N = 1 is defined. This is connected with a non-relativistic (scaling) limit of the Dirac-Maxwell Hamiltonian HDM.
B. Essential Self-Adjointness Under some conditions for x and U , the Hamiltonian Hdipole(A)given by (54) is self-adjoint for all q E R [3,6]. The method is to use a unitary transformation (called physically a “dressing transformation”) and Nelson’s commutator theorem (Theorem X.36 in [95]). By using a unitary transformation, we can also prove, for all q E R, self-adjointness of &ipole(A) defined by (58) with a mass renormalization term added [9]. Non-perturbative results on essential self-adjointness of HP F( A )for a class of ( U , x ) have been established by Hiroshima [64,65], who uses a functional integral representation [60] for the heat semi-group generated by a self-adjoint extension of H P F ( A with ) g = 0.
7.3 The GSB and the Dereziriski-Gkrard Harniltonians (Essential) self-adjointness of these Hamiltonians can be discussed in a way similar to that in Theorem 7.1. But we omit the details. See [18,19] for the GSB model and [37] for the Dereziriski-GQard model. 15) The F’riedrichs extension of a symmetric operator of the form ‘2’1 coincide with the form sum T1/T2, see p.329 in [75].
+T2
does not necessarily
30
7.4 The Dirac-Maxwell Hamiltonian In the case of the Dirac-Maxwell Hamiltonian HDMgiven by (68), a different aj ( - i ) D j pm kind of difficulty appears, since the free Dirac operator is neiter bounded below nor bounded above. As for self-adjointness of HDM, only partial (but nonperturbative) results have been established [15].
+
8
Existence of Ground States
8.1 Definition of Ground States and Preliminary Remarks Let H be a self-adjoint operator on a Hilbert space and bounded from below. Let Eo(H) := inf u ( H ) ,
(69)
the infimum of the spectrum of H. The quantity Eo(H) is called the lowest energy or the ground state energy of H . We say that H has a ground state if ker(H -Eo(H)) # (0). In that case, each non-zero vector of ker(H -Eo(H)), which is an eigenvector of H with eigenvalue Eo(H), is called a ground state of H . If H is the Hamiltonian of a quantum system, then a ground state of H describes a state with the lowest energy and ensures a stable existence of the quantum system. Hence, from the theoretical point of view, it is very important to investigate if a given quantum system with Hamiltonian H has a ground state and, in that case, how many ground states exist (the multiplicity of the ground state = dimker(H - Eo(H))). To discuss the existence of ground states of a quantum field system, it is necessary to distinguish two cases; the one is the case where the quantum field is massive, i.e, the associated quantum (boson or fermion) has a positive mass, and the other is the case where the quantum field is massless. For the massive case, which may be more tractable than the massless one, some general methods to prove the existence of ground states have been established in the course of developments of constmctzve quantum field le) Originally
constructive quantum field theory [40,49,50,53,98,101] aimed at proving, by mathematically rigorous construction, the existence of a non-trivial, completely relativistic quantum field model in the four-dimensional space-time as well as deriving rigorously various properties of the model. This program, however, has not yet been completed. But mathematical results brought by studies of constructive quantum field theory, which include fucntional analysis, probability theory, infinite dimensional analysis and related fields of mathematics, are magnificent.
31
In the massless case, however, one has to take into account some additional subtlety which is connected with the so-called infrared divergence, a class of phenomena due to infinitely many massless bosons (photons in the context of &ED) of low frequencies, called “soft bosons,” with the total energy being finite.17) To illustrate this, we present here a simple example. 8.2 An Example-The
Abstract van Hove Model
Let 7-l be a complex separable Hilbert space and S be a nonnegative selfadjoint operator on 3t. Suppose that S is injective. Let g € D(S-lI2) and define
H S ( g ) := mb(s)-k 4 ( g ) on &(?f). As in the case of H A ,one can prove that H s ( g ) is self-adjoint with D ( H s ( g ) ) = D(QI‘(S))and bounded from below. We call the model whose Hamiltonian and time-zero field are given by H s ( g ) and +(.) respectively the abstract van Hove model or the abstract jixed source model.18) We say that the abstract van Hove model is massless (resp. massive) if &(S) = 0 (resp. EO(S)> 0). We can prove the following theorem (Theorem 12-12 in [14];cf. also Theorem 6.1 in [19]). Theorem 8.1 Let g E D ( S - l I 2 ) . (i) c ( H S ( g ) ) =
(A -k E S ( g ) ) A E c(flb(S))), -llS-1/2g112/2 is the ground state energy of H s ( g ) .
where
ES(g)
:=
(ii) Suppose that g E D(S-l). Then H s ( g ) has a unique ground state (up to constant multiples) given by
(iii) Suppose that 5’ is absolutely continuous, i.e., the spectrum of S is purely absolutely continuous. Then H s ( g ) has no ground states if and only zf g @ D(S-l>. The result (iii) is related to the infrared divergence of the model as explained below. For a more detailed physical description, see, e.g., Chapter 4, $4-1-2 in [74]. The original concrete model with a massive Bose field is discussed in [71,76,78,79,88,89]. In [71,88,89], discussed are problems of ultraviolet divergences of the model, which appear in removing the ultraviolet cutoff (in the present context, it is to take g to be “outside” of 17)
w.
32
Let g E D(S-1/2) and S be absolutely continuous. We denote by Es(-) spectral measure of S. Then, condition g $ D ( S - l ) is equivalent to that, the for all R > 0,
where X[g,R] is the characteristic function of the interval [ u , R ] . Hence, if g 6D(S-'), then Eo(S) = 0, i.e., the model is massless. Let u > 0 and
H 3 9 ) := + 4(9u), where g,, = x[,,,-)(S)g. The parameter u is called an infrared cut08 and H,(g) is called a Hamiltonian with infrared cutoff u. Note that g,, E D(S-'l2) n D(S-'). Hence, by Theorem 8.1-(ii), H;(g) has a ground state fls(g,,). One can easily show that H,"(g) converges to H s ( g ) in the norm resolvent sense as u + 0 and lim,,o Es(g,,) = E s ( g ) , but, w-limRs(g,) = 0, rJ.0
where w- lim means weak limit. This is an expression of the infrared divergence of the abstract massless van Hove model. The infrared divergence is reflected also in the fact that the expectation value of the boson number in the ground state diverges in the limit of removing the infrared cutoff, i.e.,
8.3 Infrared Singularity For a massless quantum field model with one-particle Hamiltonian S on the one-particle Hilbert space 3t and a "cutoff vector" g E 3t, condition (70),i.e., g 6 D(S-') is called an infrared singdarity condition. In this case, we say that the model is infrared singular or, by abuse of terminology, it is without infrared cut08 On the other hand, we say that, if g E D(S-'), then the model is infrared regular. As we have seen, in the case of the abstract massless van Hove model, the infrared singularity condition exactly corresponds to the absence of ground states. Example 8.1 (i) In the Pauli-Fierz Hamiltonian HPF(A),the infrared regularity reads:
33
(ii) In the Nelson type model, the infrared regularity reads:
(iii) In the GSB model, the infrared regularity reads: ,1.
fa
E D(S-'),u =
1 , s . .
8.4
Basic Strategies
We consider an abstract form of particle-field Hamiltonians:
+
H := HO XHI where HO is given by (38), H I is a symmetric operator on coupling parameter. We assume the following:
(71)
F and X E R is a
Hypothesis (I) The Hamiltonian K of the particle system is bounded from below and C := inf aess(K)> Eo(K), where aess(K) means the essential spectrum of K , i.e., the complement of the discrete spectrum ad(K) := { E E a p(K)IEis an isolated eigenvalue of K with a finite multiplicity}.
Hypothesis (11) H is self-adjoint with D(H)= D ( H 0 ) and bounded from below. Hypothesis (I) implies that Eo(K) is a discrete eigenvalue of K , i.e., K has a ground state and any eigenvector of K with eigenvalue in the set
< E z ( K ) < . ' . < c) {Ej(K)}y=l:=ad(K)n (Eo(K),C) (E1(K)
5
describes an excited bound state of the particle system unless , empty. The problem of proving the existence of a ground state of H is part of the spectral analysis of H . Some spectral properties of HO may be inherited by H and the others may not. Hence it is important to grasp first the spectral property of the unpertuibed Hamiltonian Ho. But this follows from the theory of tensor products of self-adjoint operators (e.g. '$VIII.lO in [94]): (n
00)
a d ( K )n (Eo( K ) C) is
c(H0)= {k f slk E a ( K ) ,s E a(mb(S))}, ap(H0) = {k 4-slk E a p ( K ) ,S E a p ( f l b ( S ) ) } -
(72) (73)
34
If U k E ker(K - k) \ (0) with k E op(K) and t,bs E ker(dI'b(S) - s) \ (0) with s E a,(dI'b(S)), then Houk €3 t,bs = ( k S ) U k €3 t,bs. In particular, each k E ap(K) is an eigenvdue of HO with eigenvector ?Jk @ ax.Hence
+
gP(K)
c .P(HO).
(74)
Example 8.2 A typical situation is given by the following case: a(S) = [m,co),op(S)= 0 with m 2 0. Then
oeSs(Ho)= [ E o W+ m, m),
This shows, in particular, that, if m = 0 (massless), then all the eigenvalues of HO are embedded in its continuous spectrum, i.e., they are embedded eigenvalues (see Figure 1 ) . Thus, in that case, regular perturbation theory can not be applied directly and perturbative methods are not valid in its original forms. embedded eigenvalues
Figure 1. The spectrum of Ho (the case where 0
< m < E i ( K )- Eo(K))
Remark 8.1 Let a(S) = [ O , c o ) . Then, under a fairly general condition for H I , one can prove that a ( H ) = [Eo(H),m) [13] Theorem 1.3. In what follows, we describe nonperturbative methods for proving the existence of a ground state of H . Of course, one may use perturbative methods based on the regular perturbation theory (e.g., [75], Chapter XI1 in [97]). But, as the above example suggests, they may be applied only to the following cases: (i) the Bose field is massive; (ii) an infrared cutoff is introduced in the interaction HI or S if the Bose field is massless. In perturbative methods in
35
“naive” forms, it turns out that results hold only for small 1)ll’s depending on m > 0 (the mass of the boson) or the infrared cutoff parameter D > 0 in such a way that X + 0 as m -+ 0 or as o + 0. Clearly this is unsatisfactory.
A . Functional Integral Methods As is shown in constructive quantum field theory [40,50,98], functional integral methods can be very powerful for models which have functional integral representations. These methods may be applied to the Hamiltonian H too, since the Boson Fock space Fb(31)is unitarily equivalent to the L2-space, L2(Q, dpo), of a probability measure space (Q, po) such that {q5(f)If E Xflreal} is represented as a family of jointly Gaussian random variables on (Q,po), where ‘Elreal is a real Hilbert space whose complexification is equal to 31 (e.g., 53.3 in [40], $1.3 in [98]). In this approach, one uses some abstract general theorems. Definition 8.2 Let (M,p) be a o-finite measure space. A bounded linear operator T on L2(M,dp) is said to be positivity preserving (resp. positivity improving) if ( T f ) ( x )2 0, a.e.z E M (resp. ( T f ) ( x )> 0, a.e.z E M ) for all f E L2(M,dp) with f(z)2 0 a.e.2 E M and f # 0. Basic important theorems on a positivity preserving operator are the following: Theorem 8.3 [53] (Ezistence of the mazimum eigenvalue) Let (M,p) be a probability measure space and T be a bounded self-adjoint operator o n L2(M, dp) which is positivity preserving. Suppose that there exist constants p > 2 and c > 0 such that
Then IlTll is a n eigenvalue of T with finite multiplicity. A detailed proof of this theorem is found in [40]. Theorem 8.4 [53] (Uniqueness of the maximum eigenvalue) Let (M, p ) be a o-finite measure space (not necessarily a probability measure space) and T be a bounded self-adjoint operator o n L2(M, dp) which is positivity improving. Suppose that is a n eigenvalue of T . Then the multiplicity of the eigenvalue is one and the eigenfunction (up to constant multiples) can be chosen to be strictly positive a.e. For proof of this theorem, cf. also [40,50,98]. If H is represented, by a unitary transformation, as a self-adjoint operator fi on L2(M,dp) with (M, p ) a o-finite measure space such that the heat semigroup e-tii (t > 0 ) generated by fi is positivity preserving, then it is worth examining if one can apply Theorems 8.3 or Theorem 8.4 with T = e d t f i
36
(note that [le-tfill = e-tEo(fi)= e-tEo(H)).Using this method, Hiroshima [63] proved the uniqueness of the ground state of a self-adjoint extension of the Pauli-Fierz Hamiltonian HPF( A ) without spin. Another abstract fact which may be useful for functional integral methods is the following: Proposition 8.5 Let X be a Hilbert space and A be a self-adjoint operator on X bounded from below. Suppose that there exists a vector $0 E X , $0 # 0 such that the following (a) and (ii) hold: -TA
(i) the weak Cmt Q := w-limT+os 1 %
exists and i s not zero.
Then !J? i s a ground state of A . Remark 8.2 In the case where ($,e-tAq!J) ($,+ E X ) is represented as a functional integral, the assumption of Proposition 8.5 may be easier to check than in purely operator theoretical approach. In addition, one may incorporate the theory of positivity improving heat semi-groups into this approach. For concrete applications, see [104,105].
B. Operator Theoretical Approach We consider the case where the Bose field is massless, i.e., Eo(S) = 0. In this case one proceeds as follows.
Step 1: One first replaces S by a self-adjoint operator S, with m > 0 a constant (a mass parameter) having the following propoerties: (i) E,-,(S,) = m > 0; (ii) there exists a common core D of {S,}, and S such that, for all $JE D ,Sm$J+ S$J(rn + 0) (e.g., S, = S m or S, = d w ) ; (iii) the operator
+
H , := Ho,m
+
+ AH1
with Ho,, := K @ 1 I @ Car(S,) is self-adjoint with D ( H m ) = D(Ho,,). The operator H, is a “massive” version of H. By applying standard methods from constructive quantum field theory [49,101],which use lattice (finite volume) approximations (or their abstract versions) of the quantum field under consideration, one proves nonperturbatively the existence of a ground state 0, of H , [18,19,22,23,25]. Recently new methods to prove the existecne of ground states of massive particle-field Hamiltonians have been presented by Dereziriski and GBrard [37] (in the case of the massive Dereziriski-GBrard Hamiltonian) and by Griesemer, Lieb and Loss [51] (in the case of the massive Pauli-Fierz Hamiltonian H&(A)
37
which is H ~ F ( Awith ) Wph replaced by the function ~ , ( k ):= d-, lc E R3 ). The former [37] establishes a theorem which is a Fock space version of the HVZ theorem in quantum mechanical many body systems [97]Theorem XIII.17. On the other hand, the method of the latter [51] is based on the following fact. Proposition 8.6 Let A be a self-adjoint operator o n a Hilbert space X and bounded from below. Put A := A-Eo(A) 2 0. Suppose that, for all normalized sequences C q(A1I2) fll$nll = 1) with property w-limn-boo$n = 0, one has liminfn,,(A1/2$n,A1/2?,bn) > 0. Let be a minimizing sequence of A, i.e., a sequence satisfying that, f o r all n E N, Il&ll = 1 with +n E D(A112) and
{?,bn}r=l
{&}z=l
Then there exists a subsequence {+nj}gl of {q5n}F=l such that 40 := w-limj.+m +nj is a ground state of A. In the context where A is a massive particle-field Hamiltonian, the assumption of Proposition 8.6 may be proven by localization methods in Fock space [37,51]. In this way, a ground state of the massive Pauli-Fierz Hamiltonian HFF(A)is shown to exist [51]. This method may be extended to the Hamiltonian H,. With the normalization llRmll = 1, there exists a subsequence { R m j } g 1 such that m j -1 0 ( j + 00) and R := w- limj+, Rmj exists. Then one expects that R be a ground state of H . Indeed, the following holds: Proposition 8.7 Let D ( K ) ~ 3 1 F,-,,fi,(D) ~ 1 ~ be a core of H , and H for all
Step 2
suficiently small m > 0 . Suppose that E = limj+, E0(Hmj) exists and that
R # 0. (78) Then R is a ground state of H and E is the ground state energy of H . This follows from an easy application of Lemma 4.9 in [18]. Thus, for proof of the existence of a ground state of H employing Proposition 8.7, it is a key ingredient to show (78). In fact, this may be a most difficult part in the method under consideration. Here we briefly describe some methods to prove (78). Let Q6 := EK([EO(K), C
- 61)
with 6 E (0,C) and PnX be the orthogonal projection onto the onedimensional subspace { z R x I z € C}, the Fock vacuum sector. Suppose
38
that, for some constants 6 and small,
Qa
EO
> 0,
independent of m sufficiently
@ Pi~.wQm) 2 EO.
(79) Then R # 0. Indeed, if {I)~}Y!~ is an orthonormal basis of Ran(Q6) (the range of Q6) (which is of finite dimension), then the left hand side of (79) is equal to l(fl,,$l @ Hence, taking the limit j + 00 in (79) with m replaced by m j , we have I(R, ljfl @ Rx)I2 2 E O , which implies that R # 0. Thus the problem is reduced to showing (79). But, generally speaking, this also is a difficult problem. One way is to note the operator ineqaulity (Qn,
El”=,
El“=,
which is easy to prove. Hence Q6
C3 pixw 2 I - I C3 Nb
- Qb @ Pn,
(81)
where Qf := I - Q6. Then we need upper bound estimates for (Om, I @ NO,), the ground state expectation of the boson number operator, and (R,, Q ~ @ P ~ X ~uniformly R,) i n m . This method is used in [18,19]for the GSB model, and in [22,23] for a version of the Pauli-Fierz Hamiltonian. See also [61-63] for the Pauli-Fierz Hamiltonian. But this method has a defect in that it may be valid only when the modulus of the coupling constant is “small” and the cutoff vector in the quantum fields in H I is infrared regular. In [25], a new method is presented to prove (78) in the case of the Pauli-Fierz Hamiltonian HPF( A ) without infrared regularity condition. (ii) In the case of the massless Dereziriski-GBrard Hamiltonian, GQard [47] gave a new method to prove (78) for all values of the coupling constant. But, in this case also, an infrared regular condition may be necessary. (iii) Griesemer, Lieb and Loss [51] discovered new mathematical structures of the Pauli-Fierz Hamiltonian HPF( A ) which yield, without infrared regularity condition, the existence of a ground state of H P F ( A )for all values of the coupling constants q and g. The key ingredients of their method are : (i) infrared bounds (uniform in the mass m > 0) for the expectation value of the photon number operator and of the photon derivative with respect to a ground state amof the massive Pauli-Fierz Hamiltonian H?F(A); (ii) an exponetial decay property of a ground state am in the configuration space of the particles; (iii) application of RellichKondrashov theorem (Theorem 8.9 in [82]). As for the existence of a
39
ground state of the Pauli-Fierz Hamiltonian H P F ( A ) their , result [51] Theorem 2.1 is the best among the existing ones.
For a more detailed review on analysis of ground states, see [67]. As other interesting problems concerning ground states, we mention only the following two problems. (a) Enhanced binding. Suppose that K has no ground states. Then, under what conditions does H with X # 0 have a ground state? Physically this asks if a coupling of a particle-system to a quantum field makes a ground state existent. This problem was considered in [68] for the dipole approximation Pauli-Fierz Hamiltonian Hdipole(A)with N = 1 and g = 0 and affirmatively solved, and in [54] for the Pauli-Fierz Hamiltonian H ~ F ( Awith ) N = 1 and g = 0 (the spinless case). (b) Degeneracy of ground states. If a particle-system has an internal degree of freedom like spin, then there may be a chance for the ground state of the particle-field sytem to be degenerate. This aspect was discussed in [20] as a stability problem and the existence of degeneracy of ground state of the Wigner-Weisskopf model was shown. Hiroshima and Spohn [69]proved degeneracy of ground state of a Pauli-Fierz model with spin. 9
Absence of Ground States
As we have already seen, the abstract massless van Hove model (88.2) has no ground states if it is infraredly singular. It is an interesting problem to investigate under what conditions a Hamiltonian of the form H defined by (71) with a massless Bose field has no ground states. This problem was discussed in [21] for the massless GSB model. For earlier work, see [42,43],which discuss the existence or the absence of a dressed one-electron state-a ground state of a Hamiltonian, called a non-relatiwzsitac polaron Hamiltonian-associated with the translation invariant massless Nelson model (HNelson with V = 0), and [103], where the absence of ground states of the standard spin-boson model with a massless boson is considered (cf. also [106]). In [85] it is shown that the massless Nelson model H N with~ N =~1 and ~ d =~3 has~ no ground states if it is infraredly singular (no infrared cutoff is made). As already remarked, a physical reason why the absence of a ground state of a massless particle-field Hamiltonian may occur is related to the existence of infinitely many soft bosons which strongly condense in such a way that the condensed state cannot exist in the Hilbert space under consideration. A method to overcome this kind of difficulty is to change the representation
40
of the CCR which gives the time-zero fields of the model under consideration.lg)This idea was applied to the massless Nelson model without infrared cutoff in [16] and shown that, in a non-Fock representation of the time-zero fields, it has a ground state, where the non-Fock representation is inequivalent to the Fock one if no infrared cutoff is made. See also [86] for a functional integral approach.
10 Embedded Eigenvalues, Resonances and Spectral Properties As remarked in Example 8.2, the unperturbed Hamiltonian Hc, may have embedded eigenvalues. It is a fundamental important question to ask how they behave under the perturbation XHI. In view of the spontaneous emission of photons from an atom (instability of excited states of an atom under the influence of the quantum radiation field), one expects that the embedded eigenvalues describing excited states of the unperturbed system disappear under the perturbation XHI and that, for all normalized states q5j equal or “approximately equal” to a normalized excited state +j = U E ~@ RH of Ho with embedded eigenvalue Ej := E j ( K ) of HO ( j 2 l ) ,
(4j,e - i t f f 4 j )
-
const.e-i(Ej+6Ej)te--tl(arj)
as t + w (in fact, t an “intermediate time” scale), where 6Ej is a real constant and rj is a positive constant. Physically 6Ej should give the “Lamb shift ” of Ej,the energy level shift of Ej under the perturbation XHI, and rj > 0 the life-time of the state q5j. Therefore the complex number zj := 1 Ej 6Ej - imay characterize the Lamb shift and the decay of the excited 27,. state $ j or &. By some heuristic arguments (see, e.g., sXII.6 in [97]), the complex number zj is expected to be given by a pole of an analytic continuation of the
+
Roughly speaking, a quantum field model with a Hamiltonian may be defined by a set {H,$(f),n(f)lf E W } (W is a re01 inner product space) of algebraic objects with the following properties: (i) IIw := { 4 ( f ) , ~ ( f ) l E f W } obeys the CCR: for all f , g E W , [ 4 ( f ) , ~ ( g )= l i ( f l g ) w I [$(f),$(g)I = 0 = [ ~ ( f ) t ~ ( g ) (ii) I i the objects H,4(f) and T ( g ) satisfy some commtation relations (these commutation relations characterize the model). ) called the The object H is called the Hamiltonian of the model, while 4(f) and ~ ( f are tirne-zem fields of the model. Representing IIw as a set of self-adjoint operators on a Hilbert space X is called a representation of the CCR on X (Section 3.2). The representation of IIw on X gives a representation of the model on X. In this way a quantum field model may have (infinitely) many representations. An important point here is that some of them may be unitarily equivalent each other, but others may not. 19)
41
resolvent form ( $ , ( H - z)-l$) with some $ M $ j onto the second sheet (across the real axis from the upper half-plane of the first sheet). Such a pole (if it exists) is called a resonance pole unless it is a pole of analytic continua- z)-l$). Thus one is lead to the need of careful analyses of tions of ($, (Ho analytic continuations of the function ($, ( H - z)-'$) with $3 in a suitable subspace of 7 . Such analyses were done in [4,5] for a model of a quantum harmonic oscillator coupled to a massless scalar Bose field (a concrete example of the GSB model) and in [6] for the dipole approximation Hamiltonian Hdipole(A) with N = 1,g = 0 and U = mwzx2/2 (wo > 0 is a constant). In these models one has explicit representations of some of the resolvent forms, which make it easier to locate resonance poles. A new development was given by Okamoto and Yajima [92] who extended the dilation analytic methods in the case of Schrodinger operators (e.g., SXII.6 in [97]) to analysis on the Boson Fock space over L2(R3)and proved the existence of resonance poles of the massive Pauli-Fierz Hamiltonian with N = 1 and without spin. Recently thorough investigations have been made for the Pauli-Fierz model in [23-251 with further developments of dilation analytic methods together with invention of new methods and techniques of renormalization groups, presenting mathematically rigorous descriptions of the conventional or heuristic pictures as outlined above. For the details we refer to [23-261. Cf. also [36].
Remark 10.1 Generally speaking, the behavior of embedded eigenvalues of HOunder the perturbation AH1 is subtle. Namely they do not necessarily disappear under the perturbation. It may depend on the range of the partameters contained in H . A simple example demonstrating such a subtle nature is given in [7] (see also [19] Theorem 6.3). Resonance poles of this model were investigated by Billionnet [29]. We also remark that there is a class of exactly soluble models in which embedded eigenvalues disappear under perturbations [8,10,12].
Another important problem in the spectral theory of particle-field Hamiltonians is to prove or disprove the absence of singular continuous spectrum. As for this problem too, there has been much progress through extensions of Mourre theory ([77], Chapter 4 in [34]) in such a way that it can be applied to particle-field Hamiltonians [23,26,31,36,37,72,102].
42
11
Scattering Theory
Scattering of light at an atom is one of the important phenomena in quantum mechanics. In particular, the scattering of photons with "long" wave length by bound electrons, called the Rayleigh scattering, is a basic one. A physical picture of the Rayleigh scattering is as follows: first, an atom is lifted into an excited state through the absorption of incoming photons by some of the bound electrons, where the electrons still remain bound to the nucleus, since the total energy is assumed to be below the ionization threshold. Then, as time goes on, the excited state relaxes to a ground state of the atom by spontaneous emission of photons, which, in the far future, move to spatial infinity. Thus, asymptotically in time, the state of the total system (the atom plus the quantum radiation field) becomes a state which describes an atom in its ground state and a cloud of photons moving to spatial infinity with the speed of light. To give mathematically rigorous basis to the above picture or to formal perturbative scattering theory usually used in the physics literature, one has to develop mathematical scattering theory for self-adjoint operators of the form H given by (71), in particular, for the Pauli-Fierz Hamiltonian
HPF (A). Roughly speaking, a mathematical theory for the Rayleigh scattering consists of two steps. Let H be given by (71) and, for a vector f € 3c, define a measure p ; on R by $(-) := llE~(.>f11~. (i) Suppose that there exists a constant CO > E o ( H ) , called an ionizution threshold, with the property that, for all closed interval A c (--00, CO), Ile"l"lEH(A)ll < -00 with some a > 0. Let X < Co and, for f E 3-1, M f := sup suppp;. Then prove the existence of asymptotic field operators:
a+(f)#1~) := s- lim e'Hta(je"'t)#e-'Ht$ t+m
for all II,E E H ( ( - - ~ o X])F , and f E D (D is a dense subspace of 3c) such that X M f < CO,where a ( - ) # denotes either a ( - )or a ( - ) * ,and, for all n>L
+
a + ( f i ) *.--a+(f,,)*l~)= s- lim eiHta(fie-iSt)* t+m
.a(fne-ist)*e-iHt+ (82)
for
43
(ii) Let D+ be the subspace algebraically spanned by vectors of the form (82) with E X,,(H) (the set of eigenvectors of H ) and (83). Then prove that D+ is dense in EH((-oo,Co))3. This property is called asymptotic completeness of the Rayleigh scattering for the Hamiltonian H [45]. This is one of the most difficult problems in analysis of the mathematical scattering theory.
+
Problem (i) can be solved by applying the idea of Cook’s method [96] Theorem XI.4 (for earlier work, see [78,79,70,3];cf. also [61] as one of recent studies). For some models, asymptotic completeness for the Rayleigh scattering has been proved : (a) the Pauli-Fierz model in the dipole approximation Hdipole(A) with N = 1,g = 0 and U = mw;x2/2 (WO > 0 is a constant) [6]; (b) the same model as in (a) but with U = mw~xc”/2 V(z), where V is a “small” perturbation [104]; (c) a massive Derezinski-GCrard model [37], where new methods and techniques were invented (cf. also [46]); (d) a Nelson type model with infrared cutoff in the interaction [44,45], where methods and techniques used in [37] are further developed; (e) massless Nelson models [48]; (f) the renormalized massive Nelson model without ultraviolet cutoff [l];(g) a spin-fermion model [2]. The method in [37] have been extended to a relativistic model with spacial cutoff [38].
+
12
Other Problems
There are interesting problems and subjects besides those described above. We list only some of them (see also [67]). (i) Scaling limits of a particle-field Hamiltonian. These are to derive effective particle Hamiltonians which incorperate effects of the quantum field under consideration [9,59,66]. This method gives a rigorous mathematical description of Welton’s picture [lo81 for the Lamb shift and may be useful to “extract” non-perturbatively various quantum effects on particlesystems due to interactions with quantum fields. (ii) Stability of matter interacting with the quantum radiation field. This is a fundamentally important problem to understand stable existence of the world or the universe from quantum mechanical point of view. See [33,41,81,83,84]and references therein. (iii) Removal of ultraviolet cutoffs in non-relativistic QED or other models. Consider the Pauli-Fierz Hamiltonian H P F ( A )and take the momentum
44
cutoff function x to be the function X A ( ~ ):= x[o,~](JkJ) with A > 0, where X [ o , A ] is the characteristic funciton of the interval [0,A]. We denote by H A the operator H P F ( A with ) x replaced by XA and with m replaced by m A a constant depending on A (mass renormalization). Let HT" := H A - EA with a constant En depending on A (energy renormalization). Then it is an interesting (but very difficult) problem to prove or disprove the existence of the limit s-limA+m e-iHynt (t E R). Preliminary discussions are found in [33,81,83,84]. We remark that, in the case of the Nelson model in the three-dimensional space, such a limit exists with only energy renormalization [go]. Recently Hirokawa, Hiroshima and Spohn [58] proved the existence of a ground state of the Nelson model without both infrared and ultraviolet cutoffs.
(iv) Studies of the full relativistic QED, i.e., a model of a quantized Dirac field interacting with the quantum radiation field. This direction of research is taken in [39,27]. Acknowledgments
This work is supported by Grant-In-Aid 13440039 for scientific research from the JSPS. References
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39. M. Dimassi and J.-C. Guillot: The quantum electrodynamics of relativistic bound states with cutofls. I, mp-arc 01-325, preprint, 2001. 40. H. Ezawa and A. Arai: “Quantum Field Theory and Statistical Mechanics,” Nihon-hyouronsha, Tokyo, 1988. (in Japanese) 41. C. Fefferman, J. Frohlich and G. M. Graf: Stability of ultraviolet-cut08 quantum electrodynamics with non-relativistic matter, Commun. Math. Phys. 190 (1997), 309-330. 42. J. Fkohlich: O n the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. Henri PoincarQ 19 (1973), 1-103. 43. J. Frohlich: Existence of dressed one electron states in a class of persistent models, Fortschr. der Phys. 22 (1974), 159-198. 44. J. Fkohlich, M. Griesemer and B. Schlein: Asymptotic electromagnetic fields in models of quantum mechanical matter interacting with the quantized radiation field, Adv. Math. 164 (2001), 349-398. 45. J. Frohlich, M. Griesemer and B. Schlein: Asymptotic completeness for Rayleigh scattering, Ann. Henri PoincarQ 3 (2002), 107-170. 46. C. GQrard: Asymptotic completeness for the spin-boson model with a particle number cutoff,Rev. Math. Phys. 8 (1996), 549-589. 47. C. GCrard: O n the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri PoincarC 1 (2000), 443-459. 48. C. GBrard: O n the scattering theory of massless Nelson models, mp-arc 01-103, preprint, 2001. 49. J. Glimm and A. Jaffe: The X((p4)2 quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. of Math. 91 (1970), 362401. 50. J . Glimm and A. Jaffe: “Quantum Physics (Second Edition),” Springer, New York, 1987. 51. M. Griesemer, E. H. Lieb and M. Loss: Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (2001), 557-595. 52. E. P. Gross: Ground state of a spin-phonon system I. Variational estimates, J. Stat. Phys. 54 (1989), 405-427. 53. L. Gross: Existence and uniqueness of physical ground states, J. F’unct. Anal. 10 (1972), 52-109. 54. C. Hainzl, V. Vougalter and S. A. Vugalter: Enhanced binding in nonrelativistic QED, mp-arc 01-455, 2001. 55. W. P. Healy: “Non-relativistic Quantum Electrodynamics,” Academic Press, New York, 1982. 56. M. Hirokawa: An expression of the ground state energy of the spin-boson model, J. F’unct. Anal. 162 (1999), 178-218. 57. M. Hirokawa: Remarks on the ground state energy of the spin-boson model
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H-P QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS FRANC0 FAGNOLA Universitd d i Genova Dipartirnento di Matematica Via Dodecaneso 35, I-I6146 Genova, Italia E-mail: fagnolaOdima.unige.it We discuss the main results on Hudson-Parthasarathy quantum stochastic differential equations dUt = Fp"UtdAE(t) and d v t = VtGzdA!(t) with Fp" and GZ operators on the initial space. We prove the existence and uniqueness theorems and study the conditions on the Fp" and G; for the solutions to be a process of isometries or coisometries. As an application we show the dilation of two classes of quantum Markov semigroup one arising in quantum optics and the other in the construction of a quantum diffusion process.
1
Introduction
Quantum Probability has grown in the last two decades as a discipline at the crossroad between functional analysis, probability theory and mathematical physics with a number of physical applications (Accardi, Lu and Volovich [5], Alicki and Lendi [6], Barchielli [7,8], Barchielli and Lupieri [9], Belavkin [12], Ohya and Petz [39], von Waldenfels [45], ...). The understanding of classical probabilistic notions and theories like independence, conditioning, Brownian motion and Markov processes has often been a major research line in the field. This led to the developement of several tools of classical stochastic analysis in the non-commutative framework. Quantum stochastic differential equation (QSDE) are, in short, stochastic differential equations driven by non-commutative noises. The overwhelming variety of non-commutative noises (Boson [29], Fermion [lo], free [32], boolean processes [13] like Brownian motions, Poisson processes, Levy processes) and their stochastic calculi leads to several classes of QSDE. Most of them, however, are not a merely mathematical object because they arise through suitable scaling limits of Hamiltonian evolutions in quantum mechanics. The various types of noise usually do not lead to different methods therefore QSDE can be divided into two classes: QSDE for operators on Hilbert spaces and QSDE for maps on operator algebras (C' algebras or von Neumann algebras). Fkom an analytic point of view the two classes correspond, roughly speaking, to stochastic equations on Hilbert spaces and stochastic equations on Banach spaces.
51
52
We shall be concerned only with linear equations since most important equations in QM are linear. Moreover we shall discuss only QSDE of the type of Hudson and Parthasarathy. Let ( A $ ; t 2 0 ) , ( A t ; t 3 0) and ( & ; t 2 0) be the creation, annihilation and number process on the Boson Fock space on L2(R+). Hudson and Parthasarathy [29] introduced stochastic integrals with respect to the three processes and developed a stochastic calculus which is a non-commutative analogue of the classical It6 calculus. They also solved the QSDE of the form
+
dUt = (FldA: Fzdht + F3dAt + F4dt) Ut dV, = vt (GidA$ Gad& + G3dAt G4dt)
+
+
where F1,. . . ,F4, G I , .. . ,G4 are bounded operators on a Hilbert space h called the initial space and the families of operators (Vt;t >_ 0 ) , (vt; t >_ 0) are the solutions. We shall call the first (resp. second) equation the right (resp. left) equation following a usual terminology (see Meyer [36]). The U,and are related to the evolution of a quantum mechanical system therefore it is a natural requirement in all the applications for them to be unitary or, at least, isometries. This leads to some algebraic conditions on the operators F1,. . .,F4, G I , .. . ,G4 playing the role of a priori estimates on the solution. Under these conditions a QSDE appears as a stochastic generalisation of the Schroedinger equation. The family of operators (Ut;t 2 0), (V,;t 2 0) provide homomorphic dilations j t ( a ) = U:aUt, kt(a) = V , a v of a completely positive evolution on the von Neumann algebra B(h) of all bounded operators on the Hilbert space h and can be regarded as quantum stochastic processes in the sense of Accardi, Frigerio and Lewis [2]. Generdisations to unbounded F1,.. .,F4,G I , . . .,G4 are necessary to include several interesting QSDE arising in naturally in the applications both mathematical (construction of dilations and quantum stochastic processes) and physical (irreversible evolutions of open systems (see Accardi, Lu and Volovich [5], Alicki and Lendi [6], Davies [16])). Existence and uniqueness results are not too difficult. However the algebraic conditions on the operators F1,. .. ,F4,GI,.. .,G4 alone are no more sufficient to obtain isometries or unitaries and a new condition which is a quantum stochastic analogue of non explosion (i.e. reach infinity in finite time) of trajectories of the classical Markov process associated with a minimal semigroup arises. The paper is organised as follows. In Section 2 we recall the basic definitions and results of quantum stochastic calculus in Boson Fock spaces. Then
53
we introduce, in Section 3, the left and right H-P equations, define what we mean with “solution” and give the first existence and uniqueness results for bounded Fl, . . . ,F4,G I , . . . ,G4. Moreover we find the necessary and sufficient conditions for the U t , Vt to be isometries, coisometries or unitaries. In Section 4 we discuss the most important algebraic property of solutions: the cocycle property. This follows easily because the F1,.. . ,F4, G I , . . . ,G4 act in the initial space h and do not depend on time. However, this property, together with the notion of dual cocycle, establishes a useful connection between solutions of QSDE and strongly continuous semigroups on a Hilbert space. In Section 5 we essentially characterise the families of operators G I , . . . ,G4 for which there exists unique contractions Vt solving the left equation. Then we characterise in Sections 6 and 7 unitary solutions via the socalled minimal quantum dynamical semigroups. Finally we state in Section 8 the main results on the right QSDE. As an application we study in Section 9 two types of right QSDE. The first arises from a physical model in Quantum Optics and the second is motivated by the realisation of diffusion processes as quantum flows in Fock space. We do not discuss QSDE of the Evans-Hudson type for lack of space. 2
Fock Space Notation and Preliminaries
Let h be a complex separable Hilbert space, called the initial space, and d 2 1, the number of dimensions of quantum noise. Let ‘?-t= h @ F , the Hilbert space tensor product of the initial space and F = r ( L 2 ( R + C ; d ) ) ,the symmetric Fock space over L 2 ( R + ; C d ) .The symbol @ denotes the tensor product for Hilbert spaces and their vectors. We shall denote the algebraic tensor product by @. We shall omit these two symbols whenever this does not lead to confusion. Moreover we identify bounded operators defined on a factor of a tensor product Hilbert space with their ampliation. Put
M
= L 2 ( R + ;C d )n LEc(R+;C d ) , and & = lin{e(f) : f E M }
where e(f) = ( ( n ! ) - 4 f B nis) the exponential vector associated to the test function f . The notion of adaptedness plays a crucial role in the theory of quantum stochastic calculus as developed by Hudson and Parthasarathy [29]. This is expressed through the continuous tensor product factorisation property of Fock space: for each t > 0 let
F~= r(L2([o,t);cd)),F~= r ( L 2 ( [ t0,0 ) ; cd)).
54
Then F I Ft €3 P via the continuous linear extension of the map e(f) I+ e(fl[O,t))€3 e(fl[t,m)),and Ft and P embed naturally into F as subspaces by tensoring with the vacuum vector. Let D be a dense subspace of h. Vectors ue(f) and ve(g) with u, w E D and f,g E M, f # g are linearly independent and the set h 0& is total in 31 i.e. the linear span of h O & is dense in Z. Therefore we can determine linear (possibly unbounded) operators on 31 by defining their action on h o &. For us an operator process on D is a family X = (X,; t 2 0) of operators on 31 satisfying:
n,
om (x,) 3 D o E , (i) (ii) t H Xtue(f) is strongly measurable, (iii) xtue(fl[o,t)) E h €3 Ft, and Xt.e(f) = [xt.e(fI[o,t))l @ e(fl[t,m)),for all uED,f EMandt>O. Any process satisfying the further condition 11X,ue(f)112 ds < 00 for all t > 0, (iv) is called stochastically integrable. It is for these processes that Hudson and Parthasarathy [29] defined the stochastic integral s," X, dA;(s) where A; is one of the fundamental noise process defined with respect to the standard basis of C d . The integral has domain D O E and the map t I+ X, dA;(s) is strongly continuous on this domain. The quantum noises in 31, {A; 1 0 5 a,P 5 d } are defined by $(t)
= Af(l(o,t)8 lep))
= Ai(t)
= N l ( 0 , t ) €3 lep)(eml) = A"(t) At(t) = A(l(o,,) €3 (e,l) n;(t) = t n
P > 0, if (Y,P > 0, if > 0, if
(Y
where A+, A, A denote respectively the creation, annihilation and gauge operators in F defined, for each u E h and each exponential vector e(f) by
A(l(0,t) €3 (eel>.e(f) = (etl(o,t),f) ue(f> ( e l , . . . ,ed) being the canonical orthonormal basis in C d .
We refer to the books of Meyer [36]and Parthasarathy [40] for the theory of quantum stochastic calculus.
55
A stochastic integral satisfies
IF = :1
+ S,’X, dAz(s), with :1
operator on h
(ve(9>,I t x 4 f ) ) t
= (ve(9), Iox4f)) +
SP(S)fQ(S)(ve(g),Xs.ue(f))ds
(1)
for all u E h,v E D , f , g E M and t > 0. Here f1,... ,f d are the components of the Cd-valued function f , by convention we set f o = 1 and fa(s) = fa(,). The identity (1) is called the first fundamental formula of quantum stochastic calculus. The second fundamental formula, the It6 forwith another stochastic mula, gives the product (ITve(g), IFue(f)) of integral IF =:1 s,” Y, dAE(s) as
IF
+
+ (Ysve(9)7 I,Xue(f)>9,(s)f”(s)
+ $;(ysve(9), A
A
where 6 is the matrix defined by d; = 1 if The It8 formula is written shortly as
(Y
X s 4 f ) ) 9 & ) f ” ( s ) } ds =p
(2)
A
> 1 and 6; = 0 otherwise.
dALdAp” = ggdA;. The following inequality can be proved by a simple application of the It6 formula together with the Gronwall lemma. It plays an important role in the construction of the solution of the simplest QSDE. Proposition 2.1 Let X: (0 5 a,@ 5 d ) be stochastically integrable processes. For each f E M , and t > s 0 we have
>
where c“,f, d) is the constant
PROOF.Let ZI; denote the left-hand side stochastic integral. By the It6
56
formula we have
Let
c
=
fp(.,X,P(r).e(f).
olP9
The Schwarz inequality for the scalar product (., in 7-l and for the double integral given by sum on 0 I a I d together with the integral on [ s , t ] shows that the 2%(-. .) is not bigger than a)
+
lf"(r)l2 = (1 l f ( r ) I 2 )the , elementary inequality for Thus, since Co
+
Ilz:uecf)ll'
I
/
t
llZs'W)Il2(1+ If(dl2)dr+ 2/'
c
' O
Il.a(r)ue(f)l12dr
and then, by Gronwall Lemma, lIziue(f)
[l2 5 2eS:(l+If(~)I~)d~ [O ~ d l l Z o ( ~ ) ~ e ( f ) 1 1 2 ~ ~ .
Now the norm inequality
11~a(~)Il2 = (d+
1)
c Ilx,Pc~>.e(f>l121fp(r)12
OlPld
implies the desired inequality.
I
It is worth noticing that a similar inequality can be obtained also for infinite dimensional quantum noises by taking functions f E M with only a
57
finite number of nonzero components. This number will replace d and the inequality holds for processes satisfying a summability condition of the form
for all t > 0 and all p.
3
The Left and Right H-P Equations: Preliminaries
We study the left and right QSDE:
dV, = V,G; d A t ( t ) , Vo = 1,
(3)
dUt = FpQUtdAP,(t), Uo = 1,
(4)
where G = [Gz]t,p=oand F = [FpQ]d,,p,O are matrices of operators on h, and Einstein's summation convention for repeated indices applies, with greek indices running from 0 to d and roman indices running from 1 to d. In what follows we will look for contractive solutions of these equations, that is V or U such that Vt or Ut is a contractive operator for each t. Let D C h be a dense subspace. An operator process V is a solution of (3) o n D O & for the operator matrix G if ( ~ i )D c om (Lii) the linear manifold Uff,pG;(D))0 & is contained in the domain of V, for all t 3 0 and the processes (V,Gp*; t 3 0 ) are stochastically integrable, (Liii)
n,,p
(q),
(
for all t 2 0. For the right equation (4)the situation is in general more complex since there is no reason to expect that, for any solution U ,the range of each lJt should lie in an algebraic tensor product of the form D' 03. For this reason we only define solutions of (4)when each component FpQof F is closable. In this case it can be shown (e.g. Fagnola and Wills [26],Section 1) that the standard ampliation FPQ0 1 to ?t is closable. Let D be as above, then a process U is a solution of (R) o n D O & for the operator matrix F if ut(oo E ) c om (FPQo 11, (Rii) each process FpQ 0 1U is stochastically integrable, and
(wut
na,P
58
(Riii)
ut=n+Jdt -Fp" 0lU, dAc(s). Quantum stochastic calculus and QSDE can be written also in the language of white noise analysis through Wick products (see Obata [38]). We prefer to use the original version of quantum stochastic calculus which is more familiar to us. It is easy to prove the following Theorem 3.1 Suppose that the operators GpQ,Fp" are bounded operators on h. There exzst operator processes ( & ; t 2 0 ) and ( U t ; t 2 0 ) solving (3) and (4) on h 0E .
PROOF.Both the operator processes ( K ; t 2 0) and ( U t ; t 2 0) will be constructed by the Picard iteration method. We first consider the left equation (3). Define by recurrence the sequence of stochastically integrable processes on h 0E
It is easy to prove by induction that, for all u E h, t 2 0 and f E M , the following inequality holds
I I v,'"'ue(f 1I l2 where ch( f , d) is the constant as in Proposition 2.1. Therefore the series n>_O
is convergent in the norm topology on H ' for all u E h and f E M . By defining &ue(f) its limit we find an operator process V. It is easy to check that it is stochastically integrable on h 0E . Moreover, for all n 2 0, we have
m=O
m=l
Letting n tend to infinity it follows that the process (Vt;t2 0) is a solution of (3) on h 0E .
59
I
The proof for (4) is similar. We omit it.
If we knew that the solution of (3) for Gp”= (Fp”)*is bounded we could find a solution of (3) simply by taking the adjoint Ut = (&)*. Unfortunately, in general, there is no reason for h @ & to be contained in the domain of (I$)*. The natural uniqueness result is, perhaps surprisingly, slightly different for the right and left equation. Theorem 3.2 Suppose that the Gp”, Fp” are bounded operators o n h. Then: (1) the operator processes (Ut;t 2 0 ) o n h @ & solving (4) o n h @ & is unique, ( 2 ) the operator processes (&; t 2 0 ) on h 0& solving (3) on h 0& is unique among the operator processes satisfying
w, t ,f ) = SUP OlsSt,
IlVsue(f)l12 < +m
1141<1
for all f E M and t 2 0. PROOF. Let (Ut(l);t2 0) and (Ut(2);t2 0) be two operator processes solving (4) and let Zt = U i - U:. Then
lo rt
Zt =
By Proposition 2.1, for all u E h, t
Fp”ZsdA$(s).
> 0 and f E M
we have
It follows then, from Gronwall’s inequality, that IlZtue(f)ll = 0. We prove now the second statement. The difference 2, of two solutions (5“); t 2 0) and (&(2); t 2 0)of (3) satisfies now rt
zt = j0 Z,G;;~A$(S) Thus, for all u E h, t
> 0 and f E M , we have
60
An n-times iteration of this formula shows that IlZtue(f)l12is not bigger than ( c k ( f ,d))" times
By the initial space boundedness condition this is not bigger than
The conclusion follows then letting n tend to infinity.
I
Remarks (a) Note that the solution is unique when the initial conditions Vo, VOare arbitrary operators on h. (b) An operator process satisfying condition 2. is called initial space < ~ <<~ +oo bounded. If a process (Xt;t 2. 0) is locally bounded, i.e. S U _~ _~ llXsll for all t > 0, then it is obviously initial space bounded. We now start studying the properties of solutions. An operator process X = (Xt;t 2 0) is bounded (resp. a contraction, a n isometry, a coisometry, a unitary) process if each operator Xt is bounded (resp. a contraction, an isometry, a coisometry, a unitary). Let G = [G;] be a matrix of operators on h. Then given any self-adjoint operator X on h and subspace D c Dom (X) r l Dom [GI such that G$.(D) c Dom (X) for all L 2 1,p 2.0,
((uy), ( v v ) )I+ (Xua,GzvP) + (GP,u",XvP) + ( G ~ U " , X G $ . W ~ )(6) defines a sesquilinear form on ( E B ( ~ + x~ )( D @ )( d + l ) DWe ) . denote this form by t9G(X), and say that &(X) is defined as a f o m o n D if we need to make precise the domain of definition. Note that if X E B(h) then Oc(X) is welldefined as a form on Dom [GI and, if also the operators G," are bounded, then
61
O c ( X )is a bounded operator on ( @ ( d + l ) Dx) ( @ ( d + l ) D )In. this case the linear map
+ Md+l(B(h)) is given by eG(x)= (xB I ~ + ~ )+GG * ( X B
OG : B(h)
+ G*A(X)G,
where A(X) = diag(0, X , . . . ,X} E Md+l(B(h)) and l d + l is the identity matrix in C d + l . Properties of solutions U and V like boundedness, contractivity, isometry and unitarity are closely related to properties of OF(1) and e G ( 1 ) . Indeed, we can prove the following Theorem 3.3 Suppose that F = [Fp"]is a matrix of bounded operators o n h. Let U be the unique solution of (3). Then the following are equivalent: (i) the process U is a contraction (resp. a n isometry), (ii) we have OF(1) 5 0 (resp. OF(ll) = 0). and
PROOF.(i) + (ii). Let {fk} of M . Then
1{
c = Ckuke(fk) for some finite sets { u k }
of h
t
0 3 IlUtS1I2- llE1I2 =
(UScp"(S),
0
FpQUScpP(4) + (F,P~scp*(s), ~scpP(s>>
+ (F~Uscp"(S),F;;Uscp~(S))} ds
(7)
by (2), where cp*(s) = Ckft(s)uke(fk) is the a-th component of a vector cp(s) in (@(d+l)h) @ .F = @(d+l)(h @ F).If we choose the f k to be continuous then we can differentiate the above at 0 to get
0L ~F(~)(cp(O),cp(O)). Varying the fk and 'ilk then gives the result, and note that if V is an isometry process then the inequality in (7) becomes an equality. (ii) + (i). For all 5 as above, by the It6 formula, we have rt
(here the scalar product is in e c d + l ) ( h@ 3) and the operators Us act as Uscp(s) = (Uscpo(s),. . . , Uscpd(s))).The conclusion is immediate. I By taking the adjoints we find a similar result also holds for the left equation. Corollary 3.4 Suppose that G = [GpQ]is a matrix of bounded operators o n h. Denote by Gt the transpose matrix Gt = [(Gg)*]. Let V be the unique initial space bounded solution of (3). Then the following are equivalent: (i) the process V is a contraction,
62
(ii) we have OG+(I) 5 0.
PROOF.If the operators V, are contractions then the adjoint operators Ut = are also contractions and satisfy the right QSDE dUt = (Gt)*UtdAt. Therefore, by Theorem 3.3, we have OGt (I)5 0. Conversely, if (ii) holds, then the unique contraction operator process satisfying the right QSDE dUt = (Gt)*UtdAE with Uo = 1 is contractive. The adjoint family (U,")t,,-, is also a contraction (thus initial space bounded) process on h 0E and satisfies (3). By uniqueness it follows that the operators Vt = U,*axe also contractions. I The reader noticed a lack of symmetry between Theorem 3.3 for the right equation and Corollary 3.4 for the left equation. Indeed, the symmetric condition of OF(1) 0 is OG(I) 0. It can be shown (see e.g. Fagnola [20] Proposition 3.1 p.150) that, for a matrix G of bounded operators on h, conditions OG(1) I 0 and OGt (I) I 0 are equivalent. The proof involves the notion of dual cocycle (see JournB [30]). In spite of the fact that this is a fact on operators on d d + l ) h , we do not know a proof independent of QSDE. We now characterse isometry processes solving (3). Proposition 3.5 Suppose that G = [G;] is a matrix of bounded operators on h. Let V be the unique initial space bounded solution of (3). Then the following are equivalent: (i) the process V is an isometry, (ii) we have OG(1) = 0.
<
<
PROOF. (i) + (ii). This follows from a differentiation at t = 0 argument as (i) + (ii) in Theorem 3.3 (see e.g. Fagnola and Wills [26]). (ii) + (i). Fix g, f E M . For all v ,u E h let $t(v,u) = (ve(g),ue(f)) (vtve(g), V,ue(f)). Condition (ii) implies that the maps $ t ( . , -) satisfy the integral equation
We now show that $t(v,u) = 0 for all v , u E h. Indeed, iterating the above equation, the same arguments of the proof of Theorem 3.2 yield the inequality
for all n 2 1 where IC is a constant depending on t , f,g, d , G. The conclusion follows letting n tend to infinity. I
63
It can be shown immediately by taking the adjoints that the process V is a coisometry if and only if e G t ( n ) = 0 and the process U is a coisometry if and only if e,t(l) = 0. The above results allow to prove immediately the characterisation of unitary solutions of (3) and (4) Theorem 3.6 Suppose that G = [GpQ] and F = [FF]are matrices of bounded operators on h. Let V be the unique initial space bounded solution of (3) and let U be the unique solution of (4). (1) The process V is unitary zf and only if @,(I)= 0 and eGt (1) = 0. (2) The process U is unitary if and only if @,(I)= 0 and e F t (1) = 0. Remarks (a) It is easy to show that the process U is bounded and IlUtll 5 exp(bt) (where b is a constant) if the matrix OF(1) is not bigger that the (d 1) x ( d 1) matrix of operators on h with the OO-entry equal to bll and all the other entries equal to 0. We do not know, however, if this condition is necessary. (b) If all the GZ but Gt vanish then the solution of (3) is unitary if and (Gt)* = 0 i.e. G = iH with H self-adjoint (and only symmetric only if G: if Gt). The same conclusion holds for the right equation. Thus (3) and (4) are quantum stochastic generalisations of the Schroedinger equation.
+
+
+
4
The Cocycle Property
We now discuss the most important and useful property of contraction processes U and V : the cocycle property. This is the key ingredient (see Accardi [l])for constructing homomorphic dilations of quantum Markov (dynamical, in the physical terminology) semigroups by unitary conjugation with U or V . Indeed, this was the main aim of Hudson and Parthasarathy [29] for studying unitary solutions of QSDE and is still the main application of QSDE driven by boson or other noises. We start by recalling the definition of operator cocycles. For each t 2 0 let at be the right shift on L2(R+;C d ) by (atf)(x) = f ( x - t ) if x > t and (atf)(z)= 0 if x 5 t. Let r ( a t ) be the operators in F defined by second quantization of ot, r(at)e(f) = 4atf 1
(8)
for all f E L2(Ft+;C d ) . The operators at and r ( a t ) are isometries for every t 2 0. Notice that, for all s,t 2 0 we have
r(as)*wt+,) = r ( a t ) , r(as)r(at) = r(a,+t). For each s 2 0 and each bounded operator X on ?f the operator r(a,)XI'(a,)*
64
maps h @ 3' into itself. Indeed we have the diagram
The canonical extension of r ( a , ) X r ( a , ) * to 3-1 via ampliation will be denoted by O,(X). Clearly (O,),>O is a semigroup of identity preserving *-homomorphisms on B(3-1). Moreover, for all x E B(h) and all s 2 0 , we have O , ( x ) = x and
O,(At(t))= A t ( t
+
S)
- At(s)
on the domain h 0 E for t , s 2 0. A bounded operator process ( X t ;t 2 0 ) on h is called a left cocycle (resp. right cocycle) if for every t , s 2 0 we have
xt+s = XSOS(Xt), (resp. xt+s = O S ( X t ) X , ) (9) QSDE are a natural tool to produce cocycles. Indeed, we can easily prove the following Proposition 4.1 Let D be a dense subspace of h and let G = [GpQ](resp. F = [F;]) be a matrix of operators on h. Suppose that there exists a unique bounded processes V solving (3) (resp. (4)) on D 0 E . Then V (resp. U) is a left (resp. right) cocycle. PROOF. (Sketch) Fix s defined by
> 0. Let X and Y the be bounded processes
xt = &+',
Y, = V,OS(v,).
The processes X and Y satisfy QSDE
dXt = X,GpQdAP,(t+ s ) ,
+
d Y , = YsGpQdAP,(t s ) ,
Xo = V,, Yo = V,,
+
where dAE ( t s ) is the stochastic differential of the translated quantum noise A t is the Fock space F".The two equations can be regarded as left QSDE of our type with initial space h replaced by h @ F,.By Theorem 3.2 and the following remark we have then X t = Y, for all t 2 0. I The cocycle property, on the other hand, turn out to be a useful tool in the analysis of QSDE. Indeed, by combining the cocycle property and the time reversal, we can establish a simple and nice relationship between the QSDE
d& = &GpQdAt and dXt = Xt(Gt)*dA!
dUt = UtFp"dAt and d Y , = (Ft)*Y,dA!.
65
Since conditions for the existence of a (isometric, unitary, ...) solution for (4) are naturally stronger, this turns out to be a useful tool allowing to shortcut several domain problems. We now introduce the precise definition of time reversal on 'l-l. Let pt be the unitary time reversal on the interval [O,t] defined on L2(R+;C d )by s5t
(ptf)(s) = f ( t - s) if
and
f(s) if
s
> t.
Let r ( p t ) be the operator on r(L2(R+;C d ) )defined by second quantization r(Pt)e(f) = 4 p t f 1. The operators p t are self-adjoint and satisfy
= 1,
V P t ) r ( P t ) = 1. Let Rt be the operator on B(31)defined by PtPt
Rt :~('l-l) -+ B(W,
R t ( z )= r(pt)sr(pt)*.
It can be shown (see e.g. Meyer [36] or Fagnola [22] Sect 5.2) that when (K; t 2 0 ) is a left (resp. right) cocycle then the operator process t 2 0) defined by
(c;
R = Rt (v;)
(10)
is a left (resp. right) cocycle. The cocycle ? is called the dual cocycle of V. When the cocycle V is the unique bounded solution of a QSDE the dual cocycle also satisfy a QSDE and the relationship between the two is given by the following Proposition 4.2 Suppose that the G ; , Fp" are bounded operators on h and that (3), (4) have unique bounded solutions V , U . Then the dual cocycles ?, fi are the unique bounded solutions of the QSDE
d c = E(GP,)*dAt,
dct = (F,P)*fitdAc.
PROOF. (Sketch) Differentiating (ve(g), cue(f)) it is not hard to find the QSDE satisfied by ? (see Fagnola [22] Prop. 5.12). I The dual cocycle allows, roughly speaking, to take the adjoint of a left (resp. right) QSDE and end up in a left (resp. right) QSDE without exchanging the operator coefficients GPQ,F and the solution. This turns out to be a useful feature because facts on the two equations often are not symmetric. Remark We are now in a position to show the equivalence of fJG(1) 5 0 and fJGt(1) I: 0 for bounded operators G; on h. Indeed, if @,(I) I: 0 then the
66
right QSDE dUt = GzUtdAE (UO= 0) has a unique contractive solution by Theorem 3.3. This is a right cocycle by Proposition 4.1. Therefore the dual The cocycle 3 satisfies the right QSDE dct = (G{)*UtdAc = (Gt);gtdA:. process is obviously a contraction. Therefore, again by Theorem 3.3, we have OG+(I) 5 0. The converse follows exchanging G and Gt . I
5
The Left H-P Equation with Unbounded GZ
This section is aimed at illustrating the theory for the left QSDE (3) with unbounded G;. We shall show that the algebraic conditions O,(l) = 0 and OG+(1) = 0 are no longer sufficient for V to be a unitary cocycle as symmetry is by far not sufficient for self-adjointness (see Remark (c) after Theorem 3.6). A natural necessary condition on the G; for V to be a contraction process solving (3) is easily deduced by the differentiation argument of Proposition 3.3. Proposition 5.1 Let G = [GZ]be a mat% of operators on h, and suppose that there exists a contraction process V that is a strong solutaon to (L) o n some dense subspace D C h for this G. Then OG(1) 5 0 as a form o n D . If V is an isometry process then &(I) = 0 on D . We refer to Fagnola [20], Mohari and Parthasarathy [37] for the original proofs or Fagnola and Wills [26] for a proof with the same notation we use here. An alternative proof via the characterisation of the generators of completely positive contraction flows is given in Lindsay and Parthasarathy [34] and Lindsay and Wills [35]. Remarks (a) If G is a matrix of operators on h such with Dom (GZ) 2 D such that OG(1) 5 0 then [ah1 GL]:,m=l defines a contraction on c&h, and so in particular each GL has a unique continuous extension to an element of B(h). If OG(1) = 0 then [&1+ G&]:,m=l is an isometry. (b) The inequality &(I) 5 0 yields (put ue = 0 for L = 1 , . . . , d and uo = u E D in (6)
+
d e=i
Therefore, by the Schwarz inequality, for all z E Cd and u E D,we have
67
for all E E]O, l[.Thus '& zeGfj is relatively bounded with respect to GO, with relative bound less than 1. Note that a contraction process V solving (3) is strongly continuous on 7-l (i.e. the maps t + &< are continuous for E %). Moreover conditions on the GE for a unique bounded solution V of (3) imply essentially that V is a (strongly continuous) left cocycle (see Proposition 4.1). Therefore we can find a family of strongly continuous semigroups on h associated with V in a natural way. For all f , g E & and all t 2 0 define bounded operators P&>f on h by
<
(v,p&Ju) = e-(gJ)(ve(g), &ue(f))
(11)
for all w ,u E h. Indeed, since le-(gJ) (ve(g>,&ue(f))
1 5 e-~(g,fl[0.t1)+IIgl[0,t1II~/2+IIflr0.t1II~/211~ll .1. - eII(g-f)l[o.t]IIZ/2llvll -
. lull,
( l f ~denotes , ~ ] the indicator function of the interval [0,t ] ) it follows that there exists bounded operators P87fon h with 11 5 exp(ll(g- f ) l ~ ~ , ~ 1 1 )such ~/2) that (11) holds for all v,u E h. Moreover, if the cocycle V is a solution of (3) on D 0 €, then
IIPt'f
( v , P & '=~(v,u) ~) +
I'
(v, P,"7f{G;g,(s) f P ( s ) } u ) d s .
Therefore, when the operators G; are bounded, and the functions axe constant in a right neighbourhood [0,R] of 0 this means that the operators Pa1*for 0 5 t 5 R) are the operators at time t of the uniformly continuous semigroup generated by G;ga(O) f P ( 0 ) . We shall now generalise this to strongly continuous contractive left cocycles. Lemma 5.2 Let V be a left cocycle, X a bounded operator on h and Y, a bounded operators on 3,. For all t > 0 and all f ,g E & constant on [0,t s] we have
+
(ve(g),VtOt(X 8 Y s ) u e ( f ) )= (ve(g), P & ' ~ (8 x Y,)ue(f)) for all v,u E h.
PROOF.Denote by l p t ] , l p M [ the indicator functions of the intervals [O,t],[t,co[ and by It the identity operator on Ft. By the definition of the
68 P f Y fand the continuous tensor product factorisation property of Fock space (ve(g), V,Ot(X 8 Ys)ue(f)) is equal to (veb), vt(X 8 @t(Ys))ue(f)) = (ve(gl[~,tl), vt(X 8 lt)ue(fl[o,tl)>(e(gl[t,oo[), @t(Ys)e(fl~t.oo[>> = (ve(gl[o,t]),vt(xu)e(fl[o,t]))(e(afg), Yse(aff 1) = e-(gl[t-[J)(ve(g), vt(Xu)e(j))(e(a;g), Yse(otf)>
- ((P&J )*v, (~u))e(gl[o.tlJ '[o,tl)(e(a;g), Y,e(c:f)) Since the functions f,g are constant on [0, t -t s] and Y, acts in a non-trivial way only on 38we have e(gl[o~tl~fl[o.tl)(e(o~g>, yse(a,*f)) = (e(g), Y,e(f)).
I
This proves the Lemma.
Proposition 5.3 Let V be a left cocycle, let f , g E & constant on an interval [O,r] and let P f Y f be the bounded operators on h defined by (11). For all t , s > 0 such that t + s 5 r we have sJ
- p g Jp g J .
pt+s -
t
8
Moreover, if the cocycle V is strongly continuous on 7-l then the map t is strongly continuous on h.
3
P!f'
PROOF.By Lemma 5.2 and the a-weak density of operators of the form X 8 Y, in B(h 8 F8)we have (ve(g>,Wt(v,)ue(f>> = (ve(g>,
vsue(f>)
pflf
for all w,u E h. Therefore, by the cocycle property,
(v,P,";~SU) = e-(gJ)(ve(g), VtOt (V,)ue(f)) = e-(gIf)(ve(g),
~
f
V8ue(j)) 3
~
= e-(gJ) (ve(g), pfJ P:J ue(f))
= (v,P&'fP,gJu). Thus Pf;; = P&lfPj>f for all t , s > 0 with t Moreover, for such s, t , we have
+s
(v,(~,";fs- ~ / * f ) u=) e-(g~fl[o,a])((p!lf)*ve(s), then, by elementary inequalities,
(v, - n ) u e ( f ) ) ,
69
Therefore, if V is strongly continuous on X,then the map t continuous on h.
+ P,S”
is strongly
I
Remarks (a) It is not hard to show that V is strongly continuous on 31 if and only if the maps t + Pa” for g, f constant in a neighbourhood of 0 are strongly continuous. Moreover, by a well-known property of semigroups (see e.g. Pazy [42]), strong continuity is equivalent to weak continuity. (b) Semigroups similar to the Pg,f were introduced in Fagnola and Sinha [25] to study QSDE for flows and extensively used by Lindsay and Parthasarathy [34], Lindsay and Wills [35] and Accardi and Kozyrev [4]. (c) The set M of test functions for exponential vectors e(f) in & could be replaced by another set M s such that lin{e(f) : f E M s } is dense in 3. A set with this property is called totalizing. A convenient choice for the totalizing set M s is the set of step (i.e. constant on the intervals of a partition of R+) functions f with values in (0, l}d.M. Skeide [44]has shown that this set is totalizing. Exploiting this fact one could show that the family of semigroups (P,SYf; t 2 0 ) with g, f E M s determine a unique cocycle V . From the above discussion and the Remark after Proposition 5.1 it is clear that the following is a natural hypothesis on the G;. Hypothesis HGC (i) The operator G t is the infinitesimal generator of a strongly continuous contraction semigroup on h and D is a core for Gg contained in Dom (G;) for all a,p. (ii) For all L, m E { 1 , . . . ,d } the operator GL is bounded. (iii) We have @(l) 5 0 on D i.e. (uQ,G;up)
+ (G:ua, up)+ (GLua,G$up)5 0
(12)
foralluO,...,u d E D.
Remarks (a) The hypothesis HGC and the Remark (b) after Proposition 5.1 imply, by and a well-known perturbation result in semigroup theory (see Dunford and Schwartz [18], Th. 19, p. 631) that also Gt + CeztG6 generates a strongly continuous semigroup on h with D as a core. The same conclusion holds for sums of the above perturbation and scalar multiples of bounded operators as the Gem (L, m = 1 , . . . ,d ) . (b) It is easy to show that the semigroup Poi0 corresponding to the dual cocycle ? is the adjoint of Pogo.Thus the adjoint operator (G;)* for the dual cocycle ? plays the role of Gt for V and there is a natural dual to the above perturbation result. There is no reason, however, for G: and (G:)* to have a common essential domain.
70
We now show that, under the hypothesis HGC, there exists a contraction process solving (3). The idea of the proof, as in the Hille-Yosida theorem, is to take bounded approximations of the unbounded operators GZ by means of the resolvents R(n;GO,) = (nll - GO,)-1. The well-known properties of resolvent operators for all u E h, w E Dom (Gg) yield lim nR(n;G;)u = u,
n+m
lim nGER(n;G8)v = Ggw
n+oo
in the norm topology on h. We first prove a preliminary Lemma (see Fagnola [20] Proposition 3.3). Recall that ( e l , . . . ,ed) denotes the canonical orthonormal basis of Cd. L e m m a 5.4 Let G = [G;] be a matrix of operators o n h satisfying the hypothesis HGC. For each n 2 1 let In, Gn be the operators o n d d + l ) h with domain d d + l ) D defined by
nR(n;GE)@ B($(d)h), G ( n )= I:GIn (13) where I($(d),,) is the identity operator o n d d ) h . Then the matrix G ( n ) of operators o n h has an extension which is a bounded operator o n @(d+l)h and (i) IlG(n>II5 2 (n+ 3 f i + 1 1 7
In
(ii)
eG(n)(l)
1
5 0,
(iii) f o r all E E
dd+l)~ we,
have n+m lim
G(n)<= G<.
(14)
PROOF.The matrix of operators G ( n )is given by n2R(n;(G!)*)G!R(n; G!) nR(n;(G!)*)G: .. . nR(n;(Gg)*)G;
1
nGkR(n;GO,)
G:
...
;I
G? ... G: nG,dR(n;Gg) The right lower corner [G',] is a bounded operator on d d ) D by HGC (ii). Indeed, taking uo = 0 and ue E h for 1 = 1,. .. ,d, we find that the matrix of operators [ d i G',] is a contraction on @(d)h. Taking uo = u E D and ue = 0 for 1 = 1,. . . ,d , we have llG6~11~ 5 -2&(u,Ggu). Since D is a core for GO,,it follows that the operators G6 can be extended to Dom (GO,)and the inequality still holds for all u E Dom (GOO). Moreover, replacing u E D by nR(n;Gg)u E Dom (GO,),we have also
+
EL
d
IlnG;R(n;G;)u1I2 5 -2&(u,n2R(n; (G!))*G;R(n; G:)u). e=i
(15)
71
By the well-known properties of resolvents nR(n;G;) is a contraction and G;R(n;G;) = nR(n;G:) - 1, therefore we have IIG:R(n;G:)ll 5 2 and d
As a consequence all the operators in the first column of the matrix G ( n ) have a bounded extension to h and the inequality (12) also holds for all uo E Dom (G:), u l , .. .,ud E D . Let T E R, v E h, u1,...,ud E D and denote u = (0,ul ,...,ud). The .. ,. wd) inequality (12) with vector (uo,.. .,u d ) equal to (nR(n;G:)w, ~ d , yields
+ 2 ~ %((ue,nG;R(n;G;)v) + ( ~ , n R ( G;)*G:d)) n; + C I I ~ G ; RG( ;~) V; I ~+~2 ~ 4 ~ , n ~( G~ ;() )n* G; ; R ( G~ ;; ) ~ )5 0.
T2(u,eG(l)u)
d
e=i
+
+
Therefore we have ar2 2br c 5 0 for all T E R where a, b, c are the above real constants with a 5 0 by (12) and c 5 0 by (15). It follows that lbI2 I lacl. Then, since,
we have
72
Thus each operator nR(n;Gg)*Gi has a bounded extension to h. This shows that the G(n) are matrices of bounded operators. Let E denote the orthogonal projection of d d + l ) h onto d d ) h given by E(uO,u l , . .. ,u d ) = (O,ul,... ,ud) and let El denote the orthogonal projection. The above estimates also show that IIELG(n)EII 5 4 6 , IIElG(n)E*ll 5 2n. IIEG(n)Elll 5 2 6 , IIEG(n)EII 5 2 ,
It follows that IIG(n>II5 IIEG(n)EII + IIJ+G(n)EII 5 2(n 3 4 i 1).
+
+
+ IIEG(n)E*II + IIJ+G(n)ElII
Now, replacing (uo,ul, ... ,ud)by (nR(n;Gt)uo,u l , . ..u d ) ,the inequality (12) and the density of D imply immediately that 8q,)(1) 5 0. Finally (14) follows from the properties of resolvent operators. I We now state the existence theorem Theorem 5.5 Let G be a matrix of operators on h satisfying the hypothesis HGC and let (G(n);n 2 1) be a sequence matrices of bounded operators on h such that O G ( ~ ) ( 5 ~ )0 and, for all E E dd+l)D,(14) holds. For all integer n let V(n) = (&(n);t2 0 ) be the unique contraction process solving (3) on h O & with G = G ( n ) . There exist a weakly convergent subsequence ( n k ) k ? l such that the contraction process (6;t 2 0 ) defined by
solves the
QSDE (3) on D 0 E .
PROOF. (Sketch) The sequence of contraction processes (&(?I); t 2 0),>1 is equicontinuous in t on D O E . Indeed, for all u E D , f E &, t > s > 0, by Proposition 2.1 we have
73
The sequences (G(n);u)nll in h are bounded by (14) therefore the functions t + (3,vt(n)ue(f )>, for u E D , f E E are equicontinuous and equibounded. Hence, by the Ascoli-Arzela theorem and the standard diagonalisation argument, there exists a subsequence converging on a countable subset of D 0 E uniformly for t in any bounded interval. The proof can be easily completed because 'H is separable (see Fagnola [20] Prop. 3.4 for the details).
I Under the hypothesis HGC the solution is also unique (see Mohari and Parthasarathy [37] under more restrictive hypotheses). Theorem 5.6 Let G be a matrix of operators on h satisfying hypothesis HGC. There exists a unique solution of the QSDE (3) on D 0 E . PROOF.A solution V of on D 0E the QSDE (3) can be constructed by taking the G(n) as in Lemma 5.4 and applying Theorem 5.5. We now prove uniqueness. Let (Xt;t 2 0) be the difference of two contractions solving (3) on D 0€. For all u, w € D ,g, f E € we have then
Since both the sides of this identities are analytic functions in g and f, for all integers n, m 2 0 we have (USBrn,X t u f @") =
1' (
(wg@rn, XSG~uf@")
+ (vg@rn,X,G~uf@("-l))f k ( s ) + (wg@("-'),X,G~uf@") ge(s) + (wg@("'-'), X,Giu f @("-l))gt(s)fk(s))ds
(18)
for all u , v E D and all pairs m , n of integer numbers with the convention = 0 if n < 0 and g@O = f @ O = e(0). g@" = f@" We now prove that the left-hand side vanishes by induction on p = n +m. Let p = 0. For every X > 0 the bilinear form on h (w,u) +
is bounded because llXsll
Lrn
exp(-Xt> (we(o>,xtue(o>>dt
5 2 and
74
Hence there exists a bounded operator Rx on h such that
(ZI, Rxu) =
Lm
exp(-At) (we(O),Xtue(O))dt.
The identity (18) for n = m = 0, U,ZI E D, yields
A(v,Rxu) =
XJd
t
00
exp(-At)dtl (ve(O),X,Ggue(O))ds 00
=
(ve(O),X,G~ue(O))ds
= (v,RxG:u)
r
exp(-At)&
We have then Rx(X1- Gg)u = 0 for every u E D. Since D is a core for Gg, the linear manifold ( A 1 - Gg)(D) is dense in h. Thus Rx vanishes. Therefore (ve(O), X(t)ue(O)) also vanishes for every t 2 0. This proves our statement for p = 0. Suppose that it has been established for a positive p . Then, for all rn, n with m + n = p + l , the induction hypothesis allows us to write (18) as Ft
The same argument of the case n = m = 0 then shows that (vg@'", XtufBn) = 0 for all t 2 0. This completes the proof. I
It is worth noticing here that, contrary to the case in which G is bounded and &(l)= 0 on D,the unique solution of the QSDE (3) needs not to be an isometry process even if 6~(1)= 0 (see Fagnola [19] for a simple example related to birth-and-death processes). The additional condition needed will be discussed in the Section 7. The uniqueness result allows us to prove as Proposition 4.1 the following Corollary 5.7 Let G be a matrix of operators on h satisfying hypothesis HGC. The unique solution V of the QSDE (3) on D 0 E is a left cocycle. 6
H-P Equations and Dilations of Quantum Markov Semigroups
In this section we (try to) discuss briefly the application of QSDE to a dilation problem. Indeed, this was one of the most important original motivations for studying QSDE.
75
A quantum dynamical semigroup on the von Neumann algebra B(h) of all bounded operators on a Hilbert space h is a w*-continuous semigroup 7 = (5;t2 0) of completely positive, normal maps Z on B(h) such that Z(1) 5 I. A quantum dynamical semigroup is called Markov or identity preserving if Z(1) = 1. We recall that, since B(h) is the dual space of the Banach space of all trace class operators on h, the w*-continuity of 7 simply means that the maps t + trace(pT(z)) are continuous for each trace class operator p on h and all x E B(h). Moreover “normal” means that, for every increasing net (z,), in B(h) with least upper bound x E B(h), the least upper bound of Z(zLI)is Complete positivity is a stronger property than positivity. It means that, for all integer n 1 1,given a matrix X = [zt,,] of operators on enhsuch that X is positive, then the matrix [ Z ( Z ~ ,of~operators )] on $“h is also positive. Quantum Markov semigroups are the natural mathematical model in the study of irreversible evolutions of quantum open systems (see Davies [IS]). Irreversibility, from the mathematical point of view, means that the maps 5 describing the evolution of the system with state space h are not automorphisms of B(h). A natural question arises: is it possible to realise a quantum Markov semigroup as the “projection” of another reversible evolution of a bigger system? More precisely: do there exist a bigger space K ,a projection E : K + h and a family (Ict;t 1 0) of automorphisms of B(K)such that
-m).
Z(x) = Ekt(x)E* for all z E B(h)? This is (a formulation of) the dilation problem (see Bhat and Skeide [14] for a detailed discussion and recent results). The Hilbert space K is usually taken as the tensor product of the system space h with a noise space (or heat bath in the physical terminology) given as a Fock space. Accardi, Lu and Volovich [5] gave physical reasons for this choice by the theory of the “stochastic limit.’’ Quantum Markov semigroups are also a natural generalisation of classical Markov semigroups on the L” of some measurable space ( E , E ) with a gfinite measure. Indeed, the above definition can be given in an arbitrary von Neumann algebra. However we choose the simplest non-commutative framework by taking B(h) having in mind the applications to quantum open systems. The following result, a quantum analogue of the classical Feynman-Kac formula, was first proved by Accardi [l].It can be proved essentially by the
76
same arguments of the proof of Proposition 5.3 (see e.g. Fagnola [22] Sect 2.3). Theorem 6.1 Let V be a strongly continuous (left or right) unitary cocycle on 3c. The maps T , % on B(h) defined by
( v , T ( ~ b=) (V,ve(O), (z8 1 ~ ) V , u e ( o ) > , (v,X(z)u>= (y*ve(O),( X B 1 F ) v u e ( O ) ) (1, denotes the identity on 3)for u , v E h are quantum Markov semigroups on B(h). The unitary cocycle V then solves the dilation problem for 7 (resp. ;I? with Hilbert space K: = 3c = h 8 3,projection Eue(f) = exp(llfl12)ue(0) and automorphisms Ict(z)= V ( z8 17)K (resp. & ( z )= V , ( X8 17)V). When the quantum Markov semigroup is given through its generator the construction of its dilation via a unitary cocycle solving a QSDE is straightforward. We sketch the idea in the simplest case of a norm-continuous, i.e. such that lim
sup
llZ(z) - zll
= 0,
t--torEB(h) 11z11<1
quantum Markov semigroup. In this case, by Lindblad's [33] theorem, the infinitesimal generator C can be represented in the form
where Le, K E B(h) and the series Ce>lL;Le is strongly convergent (i.e. CeL1 JILev112converges for each v E h) and L(1) = 0. This representation of the infinitesimal generator follows from complete positivity and normality. Suppose, for simplicity, that LL= 0 for L > d and let
GL = 0 for L,m E { 1,. . . , d } .
It is not hard to check that e G ( 1 ) = 0 = = 0 and there exists a unique unitary solution V of (3). This is a left cocycle dilating the quantum Markov semigroup 7 as we outlined in the above discussion. It is worth noticing here that the choice of the matrix of operators G is not unique. Indeed, we could take GO" its above for (Y E { 0 , 1 , . .. ,d}, take operators GL on h such that the matrix of operators [ d& GL] on d d ) h is unitary and define GO, = -(GF)* - (Gt)*GL. Remark The action of a quantum Markov semigroup on a commutative subalgebra (an LM space) of B(h) may coincide with the action of a classical and
eGt(1)
+
77
Markov semigroup. In this case one can use the tools of classical stochastic analysis to study the behaviour of the classical Markov semigroup which often gives valuable hints on the behaviour of the original quantum Markov semigroup (see Parthasarathy and Sinha [411 for jump processes, Fagnola [21], Fagnola and Monte [23] for diffusion processes). This idea plays an important role in Section 9. The relationship between the quantum Markov semigroup 7 and the cocycle V is established through the generator. If 7 is not norm-continuous its generator is unbounded, however, in many interesting cases it can be represented in a generalised Lindblad's form (see Davies [17]). This happens for the class of w*-continuous quantum Markov semigroups on B(h) whose generator is associated with quadratic forms €(z) (z E B(h)) 00
€(z)[v, u] = (Kv, zu)
+ C ( L e v ,zLeu) + (v, z K u ) e=i
where the operators K , Lc satisfy the following hypotheses:
Hypothesis HQDS (i) the operator K is the infinitesimal generator of a strongly continuous contraction semigroup (Pt),>0on h, (ii) LC are operators on h with 6om (Le) _> Dom ( K ) , (iii) €(1) 5 0, B being the identity operator on h. These semigroups arise in the study of irreversible evolutions of quantum open systems (see Accardi, Lu and Volovich [5], Alicki and Lendi [6],Gisin and Percival [28], Schack, Brun and Percival [43]). Often there are only finitely many non zero Le. It is well-known (see e.g. Davies [17] Sect.3, Fagnola [22] Sect. 3.3) that, given a domain D E Dom ( K ) ,which is a core for K , it is possible to built up a quantum dynamical semigroup, called the minimal quantum dynamical semigroup associated with K and the Le, and denoted T("'"),satisfying the equations: (v, X(.>U)
= (v, 221) +
I'+ c1
€(Z(z))[v,u]& t
(v, ~ ( z > u=>( ~ t v , z ~ t u ) eLi
(LePt-sv,K(z)LePt-,u)ds
(20)
0
for u,v E D. Indeed, the above equations are equivalent. More precisely a w*-continuous family ( X t ; t 2 0) of elements of B(h) such that llXtll 5 1 1 ~ 1 1 for a fixed z E B(h) satisfies the first equation if and only if it satisfies the
78
second. The idea of the proof is simple: differentiate s + (Pt-,v, X,Pt-,u) and integrate on [0,t] (see Fagnola [22]Prop. 3.18). The minimal quantum dynamical semigroup associated with K and the Le can be defined on positive operators x E B(h) as follows:
7 p i n )(x)= sup
p (x)
n>l
where the maps '&("I
are defined recursively by
(v, T'"+l'(x)u) = (Ptv,Z P t U )
+
L(LePt-,v, x(n)(Z)LePt--su)ds (21) e=i
for x E B(h), u,v E D . The equations (20),however, do not necessarily determine a unique semigroup. The minimal quantum dynamical semigroup is characterised by the following property. Proposition 6.2 Suppose that the hypothesis HQDS holds. Then, for each positive x E B(h) and each w*-continuous family ( X t ) -t > ~of positive operators 5 Xtfor all t 2 0. on B(h) satisfying (201, we have '&(min)(x) PROOF. Immediate from the inequality
T(")(z) 5 X t for n, t 2 0.
I
The above proposition allows us to establish immediately another simple characterisation of the minimal quantum dynamical semigroup that will be applied in the study of the left QSDE. Proposition 6.3 Suppose that the hypothesis HQDS holds and that E(1) = 0. For all q ~ ] 0 , 1the [ minimal quantum dynamical semigroup T(q)associated f o r all positive with the operators K , and qLe satisfies T(')(z) 5 5 E B(h) and all t 2 0.
T("'")(x)
PROOF. The minimal quantum dynamical semigroup 'T('7) associated with the operators K , and qLe is defined on each positive x E B(h) as the least upper bound of the sequence defined recursively by (21) with qLe replacing Le. It is easy to show by induction that,
T("'"'(x)
for all n 2 1, q ~ ] 0 , 1 [The . conclusion follows letting n tend to
00.
I
79
Proposition 6.4 Suppose that the hypothesis HQDS holds and that €(1) = 0 . For all q ~ ] 0 , 1the [ minimal quantum dynamical semigroup 7 ( 7 ) associated with the operators K , and VLe is the unique quantum dynamical semigroup satisfying
+
(v,$q'(x)u) = ( ~ t vx ,~ t u ) 9 2
c1 ezi t
(LePt-,Zt, x(')(x)LePt-,u)ds
O
for all positive x E B(h) and all t 2 0 and %("'")(z) = sup 7 p ( x ) , '€IOJ[
PROOF. The minimal quantum dynamical semigroup T(7)associated with the operators K, and qLe is defined recursively by (21) with Le replaced by QLe. Therefore, for all t 2 0 and all positive x E B(h), %(')(z) is the least upper bound of the increasing sequence n 2 1). It is easy to show by induction that
(eq'n)(x);
%('+)(x)
5 T(n)t(z)
[ moreover, 7('1v")t(x)5 T ( q 2 > " ) t ( z )Letting . n for all n 2 1, q ~ ] 0 , 1and, tend to infinity, it follows that, for all t 2 0 and all positive x E B(h), the map q + %('1 (z) is also increasing. Since s ~ p ~ ~ ] ~ , ~ [ 7 ; satisfies ( ' ) ( x ) (20), by Proposition 6.2 we have %("'")(x)= SUP^,=]^,^[ 'j$')(x). We now prove uniqueness. Suppose that ( S t ; t 2 0) is another quantum dynamical semigroup satisfying the same integral equation as 7. Then we can prove by induction (again!) on n that %(''n)(x) 5 & ( x ) for all n 2 1, q ~ ] 0 , 1 [t, 2 0 and all positive x E B(h). Indeed, it is clear that 7 ( " 1 ~ ) ~_<( z St(.). ) Suppose that the desired inequality has been established for an integer n. We have then (21,
+7 2
%"++1)(x>u>= ( ~ t uz ~, t . >
c/ L>l
t
(LePt-,u, K("n)(x)LePt-,u)ds
O
= (u,St(z)u).
It follows that e q ' n + l ) ( x )5 & ( x ) and, letting n tend to infinity, we obtain 7 p ( z ) 5 St(.).
80
Let D t ( x ) = & ( x ) - ~ ( " ( x )and f ka t u E h and all positive x E B(h) we have
> 0. Clearly, for all
s E [O,t],
It follows that SUP 11Dr(x)ll I 72
O
( SUP
O
and, since 0 < q < 1, D p ( 2 ) = 0 for all T 5 1 and all positive x E B(h). Therefore Dr vanishes on B(h) since every operator x can be decomposed as the sum of four positive operators. I Proposition 6.5 Suppose that the hypothesis HQDS holds and €(I) = 0 o n D . Then the following are equivalent: (i) the minimal QDS is Markow (i.e. 7,'"'"'(1) = 1,11 (ii) f o r all X > 0 there exists n o non-zero x E B(h) such that E ( x ) = Ax. We refer to Davies [17] Th. 3.2 or Fagnola [22] Prop. 3.31 and Th. 3.21 for the proof. Condition (ii) is a quantum analogue of Feller's non-explosion condition for the minimal semigroup of a continuous time classical Markov chain. Proposition 6.5 shows, in particular, that the minimal classical Markov semigroup is not Markov, then also the naturally associated QDS will not be Markov. Simple applicable (sufficient) conditions for uniqueness and Markovianity have been obtained by Chebotarev and Fagnola [15]; the following result (Th. 4.4 p.394 in their paper) will be sufficient for our purposes. Theorem 6.6 Suppose that the hypothesis HQS holds and suppose that there exists a self-adjoint operator C in h with the following properties: (a) Dom ( K ) is contained in Dom (C112)and is a core for C1/21 (b) the linear manifold Lc(Dom ( K 2 ) )is contained in Dom (C'12),
81
(c) there exists a self-adjoint operator 8 , with Dom ( K ) C Dom (@1/2) and Dom (C) C Dom (a), such that, f o r all u E Dom ( K ) , we have
-2%(u, K u ) =
C I I L I u ~=~ll@1/2~112, ~ I
(d) f o r all u E Dom (C1l2)we have ((@1/2u(( 5 llC1/2ul(, (e) f o r all u E Dom ( K 2 )the following inequality holds 00
2R(C1/2u,C 1 / 2 K u+ )
IJC1/2Leu112 5 bllC1/2u]12 (22) c= 1 where b is a positive constant depending only on K , Le, C. Then the minimal quantum dynamical semigroup is Marlcov. As shown in Fagnola [22] Sect. 3.6 the domain of K 2can be replaced by a linear manifold D which is dense in h, is a core for C1j2,is invariant under the operators Pt of the contraction semigroup generated by K , and enjoys the properties: Lt (R(A;G)) C Dom ((?I2)
R ( A ; G ) ( D )E Dom (C'/'),
where R(A;G) (A > 0) are the resolvent operators. Moreover the inequality (22) must be satisfied for all E R(A; G ) ( D ) . We can now return to the study of the left QSDE.
7 The Left H-P Equation with Unbounded GZ: Isometry In this section we suppose that the hypothesis HGC holds and we discuss conditions for V to be an isometry. Besides e G ( 1 ) = 0 on D as we found in Proposition 5.1 an additional condition is needed: the minimal quantum dynamical semigroup associated with Gg and the Gk must be Markov. We shall divide the proof in several steps. Proposition 7.1 Suppose that the hypothesis HGC holds and @,(I) = 0 o n D . For all q E]O,1[ let G(q) be the m a t r b of operators o n d d + l ) h defined by 19
=I
h @ (q1(@d1h))
7
G(q) = I;GIq-
(23)
The left cocycle V(9) solving the left QSDE dV,(9)- V,(9)G ( q ) z d A E ( t ) ,
V$) = 1
(24)
on D 0 & dilates the (minimal) quantum dynamical semigroup ' T ( 7 ) associated with Gg and the qG6.
82
PROOF. It is easy to see, by our definition, that the matrix G(q) of operators d d + l ) h defined by (23) satisfies the hypothesis HGC for all q E ]0,1[. In particular we have OG(,,)(I) 5 0 and, by Theorem 5.6, there exits a unique contraction process V(q)solving the left QSDE (24) on D 0 E . The contraction process V(q)is a left cocycle by Corollary 5.7. Therefore the identity (w,St(x)u)= (K‘”)ve(o), (zB 1T)K(q’ue(o)>
(w,u E h, x E B(h)) defines a quantum dynamical semigroup on h by Theorem 6.1. Moreover, by the quantum It6 formula (2), we can see immediately that S satisfies (v, s t)I(.
= (v, 221)
for all positive x E B(h) and all t 2 0. This equation can be written in the equivalent form (20) with Le = G t , K = G: and P the semigroup generated by GE. Therefore, by Proposition 6.4, S coincides with the unique minimal quantum dynamical semigroup T(q)associated with GO, and the qG6. This completes the proof. I
Theorem 7.2 Suppose that the hypothesis HGC holds and &(I) = 0 on D . Then the unique contraction V solving (3) is a left cocycle dilating the minimal quantum dynamical semigroup 7 associated with G: and the Gt.
PROOF.(Sketch) By Proposition 7.1 for all q ~ ] 0 , 1 we [ have (v, @q)(x)u) = (K(”we(O),(zB 1F)K(q)ue(O)) for all w,u E D. We can show, by the argument of the proof of Theorem 5.5, that there exists a sequence ( q k ; k 2 1) converging to 1 such that the contractions & ( “ k ) converge weakly to the unique solution V, of the left QSDE (3) for k going to infinity uniformly for t in bounded intervals. Therefore, for all u E h and all positive x E B(h) we have
83
Moreover, since V dilates a quantum dynamical semigroup associated with G: and the GC and 7(min) is the minimal one, it follows from Proposition 6.2 that the converse inequality also holds. This proves the theorem. I We can now prove the charactersation of isometries solving of the left QSDE (3). Theorem 7.3 Suppose that the hypothesis HGC holds and &(l)= 0 o n D and let V be the unique contraction solving (3). T h e following conditions are equivalent: (i) the process V is a n isometry, (ii) the minimal quantum dynamical semigroup associated with G: and the Gk is Markov. PROOF. Clearly, by Theorem 7.2, (i) implies (ii). We will prove the converse by showing that (Vtwg@m,&uf@") = (vg@m,up'") (25) for all m,n 2 0, t 2 0, w,u E h and f , g E E. The above identity holds for n = m = 0. Indeed, V dilates the minimal quantum dynamical semigroup associated with Gg and the G; by condition (i) and this semigroup is Markov. Suppose that the identity has been established for all integers n,m such that n+m 5 p . Then, for all n,m with n+m = p + l arguing as in the proof of Theorem 5.6 and using the induction hypothesis, we have
+
+
( V , G ~ v g @ m , V s G ~ u f @ n (V.G:wg@",V,uf@"))ds )
(26)
L
Let X > 0 and define an operator Rx E B(h) by
(w,Rxu) =
Irn
exp(-At) ( K v g a m , %uf@") d t .
( v , u E D ) . Multiplying by Xexp(-At) both sides of (26), integrating on [0, +oo[ and changing the order of integration in the double integral as in the proof of Theorem 5.6 we find X(W, Rxu) = (VgBm, ~f@") €(RA)[W, u].
+
Letting c = X-l(g@*,f@"), since E(1) = 0 we have €(Rx - c l ) = X(Rx -cl). It follows then from Proposition 6.5 (ii) that XRx = (gBm, f@")l so that X
lrn
exp(-At) (%vgBm, &uf@") d t = (wgwm, uf@")l
84
for all X > 0. Now the uniqueness of the Laplace transform leads to (25). This completes the proof. I The above theorem characterises isometries V solving the left QSDE (3). In order to study when V is (also) a coisometry (then a unitary), it is not possible to write the QSDE satisfied by V *and apply the above results because this is a right equation and we do not know whether a solution exists. It seems more reasonable to study the dual cocycle p which is a candidate solution of another left QSDE and, of course it is an isometry if and only if V is an isometry. Unfortuna_tely we do not know the most general conditions allowing to deduce that, V satisfies the left QSDE
d 6 = R(G;)*dA;(t)
(27)
on some domain 5 O E if and only if V satisfies (3) on D O E when the G; are unbounded. We bypass this difficulty by first regularising the G;, for example by multiplication with some resolvent operator, writing the left QSDE satisfied by the cocycle and, finally removing the regularisation. Proposition 7.4 Suppose that the hypothesis HGC holds. Let V be the unique contraction cocycle solving (3) o n D o E and let Gt = [(Gt);] be the matrix of operators o n d d + l ) h such that
(mp”= (GP,>*IB
where 5 is a dense subspace of h which is a core f o r (G;)*. Suppose that there exists a sequence (R,;n 2 1) of bounded operators o n h such that the operators G;R,, are bounded for all n 1 and lim Riv = v
>
n-+m
f o r all u E 5 and all v E h in the weak topology o n h. T h e n the dual cocycle ? is the unique contraction process satisfying (27). PROOF. The bounded processes (V&;t 2 0) satisfy the left QSDE d&R, = &G;&dAt(t) with initial condition R,. The time reversed processes (FtR:;t 0) satisfy the QSDE dVtRE = &(GpQ&)*dA;(t) with initial condition R:. This can be checked by differentiation as in the proof of Proposition 4.2. Therefore, for all 21 E h, u E 5, f , g E & e have
>
( e v e ( g ) , Ri‘ue(f)) = (ve(g), Riue(f))
85
The conclusion follows letting n tend to
00.
I
It is possible to prove that the dual cocycle satisfies the expected QSDE by other regularisations of the G; (see e.g. Fagnola [22] Prop. 5.24). The more convenient one usually depend on the special form of the G; appearing in the QSDE. 8
The Right H-P Equation with Unbounded FF
In this section we outline the main result for proving the existence of solutions to the right QSDE (4). It is clear form Theorem 3.3 that conditions for the existence of a solution must be stronger. Indeed, arguing as in the proof of (ii) + (i), if O F ( . l ) = 0 then the solution must be an isometry. When a cocycle V solves a left QSDE we need OG(1) = 0 and the additional condition on an associated quantum dynamicd semigroup that turns out to be satisfied when we can apply Theorem 6.6. This suggests that it should be possible to show the existence of isometries solving the right equation assuming an (a priori) inequality like (22) not only on the single operator Gg (the K in ( 2 2 ) ) but on the whole matrix of operators [GpQ]. This has been done by the author and S. Wills [26] who proved the following result. Theorem 8.1 Let U be a contraction process and F an operator matrix, and suppose that C is a positive self-adjoint operator o n h, and 6 > 0 and b l , b2 2 0 are constants such that the following hold: (i) There is a dense subspace D c h such that the adjoint process U* is a strong solution of dU: = U,*(Ff)*dAE(t) o n D 0E, and is the unique solution f o r this [ ( F t ) * ]and D . (ii) For each 0 < E < 6 there is a dense subspace D, c D such that ( C E ) 1 / 2 ( DcED ) and each (FF)*(C,)1/21~c is bounded. (iii) Dom (C1/2) c Dom [Fp"] for all a ,p. (iv) Dom [F]is dense in h, and for all 0 < E < 6 the f o r m OF(C,) o n Dom [F] satisfies the inequality OF(C,) Ib14CE) + b 2 l
+
where L ( C ~is)the ( d 1) x ( d + 1) matrix diag(CE,. .. ,C,) of operators o n h. Then U is a strong solution to the right QSDE (4) on Dom(C1/2) f o r the operator matrix F . We refer to Fagnola and Wills [26] for the proof.
86
As a Corollary we can give immediately conditions under which we can prove that U is an isometry or a coisometry process. Corollary 8.2 Suppose that the conditions of Theorem 8.1 hold and let U be the solution to (4) on Dom (C'/2)for the given matrix F. If either (i) Dom (C'l2)n Dom [F]is a core for C1/2 and 13~(1) = 0, or (ii) %(I) = 0, then U is an isometry process . In order to show that U is a coisometry process note that the adjoint process U* satisfies a left QSDE. Therefore it suffices to apply the results of Section 7 to obtain the following Corollary 8.3 [20,22] Suppose that the conditions of Theorem 8.1 hold and let U be the solution to (4) f o r the given matrix F. Suppose further that (F,O)* is the generator of a strongly continuous contraction semigroup, that the subspace D is a core for (F,")*,and let 7 be the minimal QDS with generator
+ ((F;)*u,X U )+ C;=,((F:)*U,
(u,L ( X ) V )= (u,x(F;)*v)
x(F:)*v).
The following are equivalent: (i) U is a coisometry process. (ii) OF. (1) = 0 on D and 7 is conservative. (iii) [ h f l + Fj]&=lis a coisometry on @:='=,h and 7 is conservative. A weaker notion of solution to a right QSDE, the mild solution, has been introduced by Fagnola and Wills [27] taking inspiration a from classical SDE. For U to be a mild solution we demand that Ut (D 0 E ) is contained in the Usds domain of all the Fp" with cr + p > 0 and that the smeared operator maps D 0 E in the domain of F:. Thus a mild solution is a process U such that
uU t ( D a€) n Dom(FpO C
€3 l),
a+P>O
t>O
u lo-Usds(D t>O rt
and
Ut = 1 + (F: 8 1)
1 t
0 E ) C Dom (F; €3 l),
Usds +
d a+P>O
/
t
(FPQ8 l)Vs dAt(s).
O
This is an important notion because, if we look for isometric solutions, as it is clear from the previous examples and the examples in Section 9, the operators F; are "less unbounded" than F: and, therefore, have a bigger domain.
87
An existence theorem for mild solutions inspired by Theorem 8.1 was proved in Fagnola and Wills [27] (Th. 2.3). 9
Dilation of Irreversible Evolutions Arising in Quantum Optics
In this section we show, as an application, that a class of right QSDE has a unitary solution. We first study a QSDE arising in the dilation of the quantum Markov semigroup in a model for absorption and stimulated emission introduced by Gisin and Percival [28]. Let h be the Hilbert space lz(N) of complex-valued square summable sequences (z,;n 2 0) with canonical orthonormal basis (e,;n 2 0). The annihilation, creation and number operators on h are defined by
{ I Dom(a*) = { u E h l x n l u n 1 2 < +m},
Dom(a) = u E h x n l u n 1 2 < +m},
aen = fie,,,
if n
> 0,
aeo = 0,
n/O
a*en = &Gien+i,
n20
Note that N = a*a and the above operators are closed. The evolution in the Gisin-Percival's model is given by the minimal quantum dynamical semigroup associated with
K = +<(a*
+ a) - -21 ( v 2 N 2+ p 2 N ) ,
L1 = vN, Lz = pa,
(28)
<
where p, Y > 0 and E R. Note that Dom ( K ) = Dom ( N 2 ) ,Dom (L1) = Dom ( N 2 ) ,Dom (L2) = Dom ( N ) and the operators K , L1, LZ are closed. In Fagnola and Rebolledo [24] Sect. 2.4 it has been shown that this quantum dynamical semigroup is Markov and, moreover, it has a unique faithful normal invariant state. Let F = [FPQ]be the matrix of operators on d 2 ) h given by
Ft = L1, Fi = La,
F: = -(L1)*, F i = -(L2)*,
F,O = K
and FA = 0 for all L,m E (1 ,... , d } . We choose as D the subspace of h of finite linear combinations of elements of the canonical orthonormal basis (en;n 2 0).
88
Let 3 be the symmetric Fock space over L2([0,+m[; C'). We now study the right QSDE
dUt = Fp" @ l U t d A t ( t ) , Uo = 1.
(29)
We apply Theorem 8.1. Therefore, as a first step, we prove the existence of a solution V to the left QSDE
dVt = &G;dAc(t),
V, = 1
(30)
where GZ = ( F t ) * . To this end we check the hypothesis HGC and apply Theorem 5.5 and Theorem 5.6. L e m m a 9.1 The operators F: and Gg generate strongly continuow contraction semigroups on h.
+
PROOF.The operator X = - ( v 2 N 2 p 2 N ) /2 with domain Dom ( N 2 ) is clearly negative self-adjoint. Therefore it is the infinitesimal generator of a strongly continuous contraction semigroup on h (indeed, an analytic semigroup). We now prove that -i<(a* a) is a relatively bounded perturbation of X with arbitrarily small relative bound. As a consequence, the closure of F: = X - i<(a* a) generates a strongly continuous semigroups by Th. 19 p. 631 in Dunford and Schwartz [18]. Indeed, for all u E Dom ( N ) we have
+
+
Il(a*
+ a)ull 5 lla*u11+ llaull = ( N + 1)U)'j2 + (u,Nu)'/2 I 2(u, + 11u1)5 211u111/2. llNu111/2+ 1(u11 (21,
I EllNull + k-' + 1)111 .1.
+
Therefore i<(a* a) is relatively bounded with respect to X with relative for all E > 0. bound The semigroup (Qt;t 1 0 ) generated by F: is contraction because
d dt
-(u, Qtu) = -2&(Qtu, F,OQtu)
and -2&(v, F t v ) 5 0 for all v E Dom (F:) = Dom (N'). The proof for F: = X i<(a* a) is identical.
+
+
I
The identity t9,(1) 5 0 follows immediately from our choice of the Fp". Thus the hypothesis HGC holds and Theorems 5.5, 5.6 yield the following Proposition 9.2 There exists a unique contraction process V solving (30) onD@E.
89
Having checked the hypothesis (i) of Theorem 8.1 we now choose as op6 = 1, D, = D for all E E]O,1[ and erator C the operator ( N
+
c, = ( N + q"(n + & ( N+
1)4)-2.
Clearly D is N-invariant, thus it is also C,'l2 invariant. Moreover, by functional calculus, the operator ( N + 1)4Ci'21~ is a contraction. Therefore, since (F,O)* = Gt is relatively bounded with respect to ( N it follows that (F,")*C:/2is bounded. It can be shown in the same way that the ( F J ) * , are relatively bounded with respect to N 1, thus the operators (Fz)*C,'/2 are also bounded. This shows that the hypothesis (ii) of Theorem 8.1 is fulfilled. The hypothesis (iii) is also because the domain of each operator Fp" is contained Dom (F:) = Dom ( N 2 ) . Finally we check the hypothesis (iv). We analyse each operator matrix element of O,(C,) and then we prove the inequality since are not interested in finding the best constants b l , b 2 . The operator matrix elements (eF)k(Ce)for C, rn E (1,. .. ,d } clearly vanish because FA = 0. We now compute ( e F ) t ( C , ) and show that it is not bigger that a constant times C,. Although there are at times a product of 12 operators in the working below it should be remembered that these expressions should strictly be interpreted as forms with domain D,and so everything is well-defined by the properties of the operators N , a*,a. Notice that (eF)g(z) (for suitable 2 E B(h)) is the sum of 3 terms (as forms on D)
+
+
(e)*
+zN2) , P" (a*az - 2a*xa + za*a), &(z) = -2 &(z) = it ((a*+ a ) z - %(a*+ a ) ) . Y2
L1 (z) = -- (N 2 z - 2 N z N 2
n
The action of L1 and L2 on functions of the number operator N is easily computed. Indeed, for suitable bounded functions f : N + R,we have
L2(f(N)) = P2N(f(N- 1) - f(N)), (here f(-1) can be defined in an arbitrary way since N ( f ( N - 1) - f ( N ) )= 0
Ll(f(N)) = 0,
on the spectral projection N = 0). Remark The restriction to the algebra the number operator (i.e. to operators of the form f(N)) of L1 L2 is the infinitesimal generator of a pure death
+
90
process on N with death rates p2n. It is well-known (see e.g. Karlin and Taylor [31])that the related classical minimal semigroup is identity preserving. It is natural then to expect, in spite of the commutator term L3, that the unique contraction process solving (29) is an isometry. A long but straightforward computation shows that & ( f ( N ) ) is equal to
+
+
+ +~ 28e2N6 N ~ + Ek2N5 + e2N4)(l+ E N ~ ) - ~+(EI( N+ l)4)-2.
p 2 N ( - I - 4N - 4N3 - 6 N 2 + ~ E ~ N '2' 2 ~ ~ N l '52e2N9 6 9 ~ ~ N '
+5
6 ~
Therefore a rough estimate shows that L 2 ( f ( N ) )is not greater than
+
+I ) ~ ) - ~ I 2 4 0 p 2 (+~ q 4 ( 1+ E ( N + I ) ~ )(-N~% ~ (+I EN^)-^) 5 240p2(N+ 1)4(1+ e(N + 1)4)-2= 240p2C,.
p 2 ~ 2 ~ 4 ( 2 4 0 ~+8&) (Ni ~ ) - ~ (E(N B
We now compute L 3 ( f ( N ) ) .Another long but simple computation using the commutation relations a f ( N ) = f ( N + l ) a , a * f ( N +I) = f ( N ) a *for suitable positive bounded functions f on N, shows that a * f ( N )- f ( N ) a *is equal to
( f (-~q1l2- f
+
( ~ ) l / ~ ) ~ * f ( ~f )( ~ l /) ~l
/ ~ a * ( f (~ f) l /(~+ ~
(where f(-1) = 0). This can be written in the form f(N)1/2{--.}f(N)1 where the operator within braces . } is {a
( f ( -~n)1/2f(iv)-1/2 - i)a* + a*(n - f
(+ ~ ~ ) l / ~ f ( ~ ) - l / ~ ) .
If we take f(n)= ( ~ 2 + l ) ~ ( l + ~ ( n + lthen ) ~ )the - ~ above operator is bounded. Indeed, writing the polar decompositions a = S * N 1 / 2 , a *= S ( N P)lI2 (where S is the right shift operator on h defined by Sen = e,+l), we have
+
Ila*(f(N
+ 1)1/2f(N)--1/2 - 1>41 = II(N + I ) ~ / ~ ( + ~ n)1/2f(iv)-1/2 (N
for all u E D. Notice that, for n E N,
5
+
+
( n 1 y 2 ( 2 n 3) (n 1)2
+
3,
-1)4
91
Moreover, since 0 < E
< 1,
It follows that
l l ~(* f (+~~ ) l / ~ f ( ~ )-- a)ull l / ~ 5 311~11. Similar computations yield
II(f(N - 1)1/2f(N)-1/2- l)a*uJI5 3112111. Therefore, for all u E h, we have
I@,
(a*f(N)- f(N)a*)u)II 6(u, C € 4 .
The same argument leads us to the same inequality for a f ( N )- f ( N ) a . Thus, for all u E D , we have then
IY (21, (a*+ 4 f ( N ) - f("a*
+ a>>.>l I 12151(u, C€.)
This we proves the following inequality ( @ F ) X E >5
(240P2 + 1215I)C€
(31)
We can now prove the following Lemma 9.3 For all E E]O,l[ we have
eF(c,) I ( 2 4 1 1 ~ 1+~12151+3 6 ) ~ ~ .
(32)
PROOF.Notice that (@F);(CJ = ( @ F ) ~ (=C0~and ) (@F)k(Ce) = 0 for all L,m E {1,2}. Therefore, for all u = (uo,u1,u2) E d 3 ) h , we have
+ (212, ( w ; ( c E ) u o ) + (210,(@F)%C&2).
(U,@F(C€)U> = (uo, ( @ F ) W € ) U O )
Remember that, as above, (eF)i(C,) and (eF)i(C,) can be written as pC:/2X,C:/2 and pC:/2Y,C,'/2 for some bounded operators X, and YEof norm less than 6, then
The desired inequality follows.
I
92
Proposition 9.4 There exists a unique unitary process satisfying (29).
PROOF.We have shown that all the hypotheses of Theorem 8.1 hold. Thus the adjoint process U to the unique solution V of the left QSDE (30) on Dom ( N 2 )0E is a solution to the right QSDE (29) on Dom ( N 2 )O E (and on D O E . The process U is obviously a coisometry. Hence it is contractive. Since OF(1) = 0 and Dom (C1/2)n Dom (F;) = Dom ( N 2 )is a core for C1l2,then it is also an isometry process by Corollary 8.2. The unitary solution to (29) is clearly unique since its adjoint is the unique solution to (30). I 10
Dilation of Classical Diffusion Processes
In Section 9 when constructing a dilation of a given quantum Markov semigroup we had some insight by looking at it as a perturbation of another quantum Markov semigroup which leaves the algebra of functions of the number operator invariant. Several quantum Markov semigroups admit an invariant abelian subalgebra i.e. an algebra of bounded functions on some measurable space. The restriction to this subalgebra is a semigroup of positive and identity preserving operators i.e. a classical Markov semigroup. Parthasarathy and Sinha [41] posed the following problem: given a classical Markov semigroup is it the restriction to an abelian subalgebra of a quantum Markov semigroup? The answer to this question would be step towards the understanding which classical processes can appear in quantum stochastics. We refer to Fagnola [22] and the references therein for some results on this problem. Here shall only construct a dilation (i.e. a quantum stochastic process in the sense of Accardi, F'rigerio and Lewis [2]) of a quantum Markov semigroup 7 on f?( h) that extends the Classical Markov semigroup T of a diffusion process i.e. such that
for all t 2 0 for all f E L"(Rd;C) where M f (resp. M T ~ is~ the ) multiplication operator by f (resp. Ttf) on L2(Rd;C). (The multiplication operator by f acts, of course, as M p = fufor all u E L2(Rd;C)). Let h = L2(Rd; C) and let 8~denote the partial derivative with respect to the l-th coordinate. Let OF : Rd + R (1 5 Lm 5 d ) and pe : Rd + R,
93
+ R (1 5 l ,m 5 d ) be bounded functions four times differentiable with bounded partial derivatives of the first four orders. Define the operators Le (with the sum convention)
qm : Rd
Leu = ( ~ 7 8 ,+ p i ,
i
HU = -5 (qmam+ amqm)
on the domain D of smooth functions on Rd with compact support. Then it can be shown that the closure of the operator
1 2
GE = --LiLe
+ iH
generates a strongly continuous contraction semigroup on h The adjoint F; also generates a strongly continuous contraction semigroup on h. Let
F,O = (GE)*,FJ = Le,
F: = -(Lm)*
and FA = 0 for all L, m E (1,... ,m } . It can be shown (also under some less restrictive hypotheses on the Oem, pe, qm) that there exists unique unitary solutions U , V to the right and left QSDE (29), (30) on D 0 E (see Fagnola [22] Sect.5.6 for the left equation and Fagnola and Monte [23] for the right equation). The right cocycle U dilates a quantum Markov semigroup on B(h) with infinitesimal generator ( O F ) : acting on multiplication operators by a smooth function f with compact support as ( O F ) t ( M f ) = M A J . The function Af is given by
where
Therefore the quantum Markov semigroup generated by ( O F) : extends the classical Markov semigroup of a diffusion process with covariance matrix a and drift b. It can be shown also that the family of homomorph'sms ( k t ; t 2 0) on B(h) with values in B(h 8 7 )defined by k,(z)= U;(z @ Br)Ut is the flow of quantum stochastic process extending a classical diffusion process.
94
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17. E. B. Davies: Quantum dynamical semigroups and the neutron diflusion equation, Rep. Math. Phys. 11 (1977), 169-188. 18. N. Dunford and J. T. Schwartz: “Linear Operators. I. General Theory,” Pure and Applied Mathematics 7, Interscience Publishers Inc., New York, London, 1958. 19. F. Fagnola: Pure birth and pure death processes as quantum flows in Fock space, Sankhys A 53 (1991), 288-297. 20. F. Fagnola: Characterisation of isometric and unitary weakly differentiable cocycles in Fock space, Quantum Probability and Related Topics VIII (1993), 143-164. 21. F. Fagnola: Diffusion processes in Fock space, Quantum Probability and Related Topics IX (1994), 189-214. 22. F. Fagnola: Quantum Markov Semigroups and Quantum Markov Flows, Proyecciones 18 (1999), 1-144. 23. F. Fagnola and R. Monte: Quantum stochastic differential equations of diffusion type, in preparation. 24. F. Fagnola and R. Rebolledo: Lectures on the Qualitative Analysis of Quantum Markov Semigroups, in “Quantum Interacting Particle Systems, QP-PQ: Quantum Probability and White Noise Analysis (L. Accardi and F. Fagnola Eds.),” World Scientific, 2002. 25. F. Fagnola and K. B. Sinha: Quantum frows with unbounded structure maps and finite degrees of freedom, J. London Math. SOC. 48 (1993), 537-551. 26. F. Fagnola and S. J. Wills: Solving quantum stochastic differential equations with unbounded coeficients, to appear in J. Funct. Anal. (2002). 27. F. Fagnola and S. J. Wills: Mild solutions of quantum stochastic differential equations, Electron. Comm. Probab. 5 (2000), 158-171. 28. N. Gisin and I. C. Percival: The quankm-state &Busion model applied to open systems, J. Phys. A: Math. Gen. 25 (1992), 5677-5691. 29. R. L. Hudson and K. R. Parthasarathy: Quantum Itb’s formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301-323. 30. J.-L. JournC: Structure des cocycles rnarkoviens sur I’espace de Fock, Probab. Th. Rel. Fields 75 (1987), 291-316. 31. S. Karlin and H. M. Taylor: “A Second Course in Stochastic Processes,” Academic Press, Inc., New York-London, 1981. 32. B. Kummerer and R. Speicher: Stochastzc integration o n the Cuntz algebra 0,, J. Funct. Anal. 103 (1992), 372-408. 33. G. Lindablad: O n the generators of Quantum Dynarnical Sernigroups, Commun. Math. Phys. 48 (1976), 119-130. 34. J. M. Lindsay and K. R. Parthasarathy: O n the generators of quantum
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stochastic flows, J. Funct. Anal. 158 (1998), 521-549. 35. J. M. Lindsay and S. J . Wills: Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory Related Fields 116 (2000), 505-543. 36. P.-A. Meyer: “Quantum Probability for Probabilists,” Lect. Notes in Math. Vol. 1538, Springer-Verlag, Berlin, 1993. 37. A. Mohari and K. R. Parthasarathy: On a class of generalizes EvansHudson flows related to classical markov processes, Quantum Probability and Related Topics VII (1992), 221-249. 38. N. Obata: Wick product of white noise operators and quantum stochastic differential equations, J. Math. SOC.Japan 51 (1999), 613-641. 39. M. Ohya and D. Petz: “Quantum Entropy and its use,” Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993. 40. K. R. Parthasarathy: “An introduction to quantum stochastic calculus,” Monographs in Mathematics 85, Birkhauser Verlag, Basel, 1992. 41. K. R. Parthasarathy and K. B. Sinha: Markov chains as Evans-Hudson diffusions in Fock space, in “Sdminaire de ProbabilitCs,” XXIV (1989), pp. 362-369, Lect. Notes in Math. Vol. 1426, Springer, Berlin, 1990. 42. A. Pazy: “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, Berlin 1975. 43. R. Schack, T. A. Brun and I . C. Percival: Quantum-state diflusion with a moving basis: Computing quantum-optical spectra, Phys. Rev. A 55 (1995), 2694. 44. M. Skeide: Indicatorfunctions of intervals are totalizing in the symmetric Fock space I’(L2(R+)),in “Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability (L. Accardi, H.-H. Kuo, N. Obata, K. Sait6, Si Si, and L. Streit Eds.),” pp. 421-424, Natural and Mathematical Science Series 3, Istituto Italian0 di Cultura (ISEAS), Kyoto, 2000. 45. W. von Waldenfels: Illustration of the quantum central limit theorem by independent addition of spins, in “SCminaire de ProbabilitCs,” XXIV (1988/89), pp. 349-356, Lect. Notes in Math. Vol. 1426, Springer, Berlin, 1990.
FREE RELATIVE ENTROPY AND q-DEFORMATION THEORY FUMIO HIAI Graduate School of Information Sciences Tohoku University Aoba-ku, Sendai 980-8579,Japan E-mail: hiat8math.is. tohoku. ac.jp After giving short reviews on free independence, random matrices and free entropy, we introduce the free relative entropy for compactly supported probability measures on the real line based on a large deviation for the empirical eigenvalue distribution of a relevant random matrix, and show the perturbation theory for probability measures via free relative entropy. Next, we survey three interpolation/deformation theories of free group factors: interpolated free group factors introduced independently by Dykema and by RMulescu, free Araki-Woods factors in Shlyakhtenko’s functor and von Neumann algebras in Boiejko and S p e icher’s q-functor. We finally consider q-deformed Araki-Woods algebras combining Shlyakhtenko’s and Boiejko and Speicher’s functors
Introduction In early 198Os, Voiculescu [54] discovered a new concept of probabilistic independence (called “free independence” or simply “freeness”) which is formulated in noncommutative algebras and very closely related to free products of operator algebras, particularly to free group factors. Since then, a new probabilistic world based on this new concept of independence has extensively developed into a well-systematized discipline (called “the free probability theory”). Although the contents of free probability theory are very different from those of classical probability theory, there are many strong parallelisms between both theories. Indeed, there are so many free analogues of corresponding notions in classical probability; for instance, semicircle distribution vs. normal distribution, the free analogue of classical central limit, free entropy vs. Boltzmann-Gibbs entropy, free products vs. tensor products, etc. An important feature of free probability is its close connection with random matrix theory, which was first realized in [55]. Random matrix models of noncommutative random variables have played crucial roles in many stages of free probability. In this article we survey two subjects around free probability theory; the one is free (relative) entropy and the other is q-deformation of free group factors. Throughout the paper our main reference is [31] concerning random
97
98
matrix models, related large deviations and entropy in free probability theory; it also contains a detailed exposition on interpolated free group factors. (Amalgamated) free products of operator algebras and actions on them are central in applications of free probability to operator algebra theory. We do not enter into the details of this subject, but fortunately there is Ueda’s article “F’ree product actions and their applications” in this book; the reader is recommended to consult with it and references therein (and also in [24]). Sections 1-3 of this article are reviews on some basics of free probability theory. The free independence, the most fundamental notion in free probability theory, is introduced in Section 1 in contrast with the notion of classical independence. In Section 2 we review random matrices and related large deviations, including Voiculescu’s asymptotic freeness of random matrices [55] and the large deviation results due to Ben Arous and Guionnet [5] and Hiai and Petz [27]. Section 3 is a brief survey on Voiculescu’s free entropy [56]-[60]. Although the expositions in Sections 1-3 are not at all complete and they are just a few slices of free probability theory, we present them as a guidance to the theory for general reader. Section 4 is based on [26]. Voiculescu’s single variable free entropy C ( p ) is generalized in [35,26] in two different ways to the free relative entropy C ( p , v ) for compactly supported probability measures p,v on R. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. (Note that Biane and Speicher [S] introduced the notion of free relative entropy with respect to a function F while ours are defined with respect to a measure v.) Next, the perturbation theory for compactly supported probability measures via free relative entropy is presented on the analogy of the perturbation theory via relative entropy. We see that the free relative entropy C ( v h ,v) for the perturbed measure v” via free relative entropy is a normalized limit of the relative entropies of the distributions of random matrices perturbed according to h. This result provides one more evidence for close relation between free probability theory and random matrix theory that has been widely believed so far. Furthermore, as a consequence we determine the form of the Legendre transform of the single variable free entropy, which may be called the free pressure. In the last of Section 4 we introduce the multivariable free pressure and attempt to give a new definition of the multivariable free entropy via the Legendre transform of the free pressure. The details on the multivariable free pressure will be presneted elsewhere. Section 5 is mainly concerned with interpolation/deformation theories of free group factors. Firstly, we briefly review interpolated free group factors
99
t(F,) (1 < T 5 00) constructed independently by Dykema [20] and W u l e s c u [44], from which a major progress was made in structure theory of free group factors. However, the (non-) isomorphism problem of free group factors, one of the most famous problems in operator algebras, is left open. Secondly, we summarize free Araki-Woods factors constructed by Shlyakhtenko [50,51]. This theory is the type I11 deformation of free group factors and it is quite an interesting application of free probability theory to operator algebras. Thirdly, we recall the q-deformation (-1 < q < 1) of free group factors in the q-functor by Bozejko and Speicher [12,13] and also [lo]. The case q = 0 is Voiculescu's free Gaussian functors, and the limit cases q = -1 and q = 1 become the CAR and CCR functor, respectively, which are useful in quantum statistical physics. Our main purpose of Section 5 is to present q-deformed Araki-Woods algebras investigated in [25] in the functor combining Shlyakhtenko's and Bozejko and Speicher's. Note that the same construction in the limit case q = -1 provides usual Araki-Woods factors, as explicitly shown in Subsection 5.4. Finally, a little about q-deformed distribution is given from [12,48,49]. 1
Free Independence
In classical probability theory, we treat random variables (or measurable functions) on a probability space (R, F,P). For a real random variable X on R we define the expectation E ( X ) := J, X dP, the kth moment E ( X k ) ,the variance V(X):= E ( [ X - E(X)]2) and so on. The distribution of X is a probability measure px on R given by px(S) := P ( X - l ( S ) ) for Bore1 subsets S of R. In noncommutative probability theory, there is no real underlying probacp) of a (noncommutative) bility space so that we must begin with a pair (d, *-algebra A with identity 1 and a linear functional cp : A + C such that cp(1) = 1 and cp(a*a) 2 0 (a E A). A classical probability space ( O , F , P )can be considered in this setting by taking A = L"(R, P ) and cp = E ( - )= Jn .dP. d may be a purely algebraic *-algebra for a while though it must be a C*algebra or a von Neumann algebra in order to do functional calculus in A. Such a pair (d, cp) is called a noncommutative probability space or an algebraic probability space for it is given in an algebraic manner. An element a E A is called a (bounded) noncommutative random variable, and we have the expectation cp(a) and the kth moment cp(ak) for k € N. Moreover, for a multivariable ( a l ,. . . ,a,), ai E A, we have the joint moments cp(attafi - . . a t ; ) for aJl rn E N,1 5 il,. . . ,i, 5 n, k l , . . . ,k, E N. Let us regard these joint moments as the joint distribution of ( a l ,. . .,a,); in fact, for noncommutative a1 , .. . ,a, there is, in principle, no joint distribution as a probability measure.
100
The most fundamental concept in classical probability theory is that of independence. A family { X i } i E lof random variables is said to be independent if the sub-a-fields a(Xi) generated by X i , i E I , are independent, which means that, for each different il, .. .,in E I and each polynomials p l , . . . , p n , we have
( X i ,>PZ(Xi21 .. .pn( X i ,) = E@l(Xil 1)E(pz(Xi2) ) . * * E(pn( X i , ) ) * This is equivalent to saying that, for each different 21,. . . ,in E I and each polynomials p l , . . . ,p n , E(p1
E@k(Xik))= 0 (k = 1, * * . n)
* E(P1(Xil)p2(Xiz)." p n ( x i , > ) = 0.
(1)
The concept of free independence plays the same role in free probability theory as the usual independence does in classical theory. Let (d,cp)be a noncommutative probability space and {ai}iEr a family of noncommutative random variables in (d,cp).It is said that { a i } i E ~is free independent (or simply free) if, for each il # iz # . * # in (all neighboring pairs are different) in I and each polynomials p l ,. . . ,p,, we have
-
(k = 1 , . * ,n)=+- d ~ l ( a i , ) p 2 ( a i ,* )* *pn(ai,>)= 0. (2) Note for instance that X Y X Y = X 2 Y 2 for commutative X , Y while abab # a2b2 for noncommutative a,b. So it would be natural to allow, for instance, il = i3 for il # iz # -.. # in. For free independent a, b, since by (2) ~ @ k ( a i , )= ) 0
cp((a - cp(a)l)(b- cp(b)l)) = 0,
cp((a - cp(a)l)(b- cp(b)l)(a- cp(41))= 0 9
we get
cpW) = cp(a)cp(b)> cp(aW = cp(a2)v(b) ?
which are the same as in the case of classical independence; however the formula
cp(abab) = cp(a2)cp(b)2+ cp(42cp(bz) - cp(a>2cp(b>2 is different from the classical case E ( X Y X U ) = E ( X 2 Y 2 )= E ( X z ) E ( Y 2 ) . As is seen from the above formulas, we can regard the free independence of { a i } i E r as a rule of computing the joint moments of the ai's from their separate moments cp(af) (i E I , k = I,2,. ..). Although the above definition of free independence seems rather similar to the classical one (see (l), (2)), the contents of free probability are quite different from the classical theory. Nevertheless, it should be stressed that the structures of both theories are in quite parallel. A typical example of parallelism is the following free analogue of the central limit theorem. The central limit theorem is one of the most important theorems in classical probability
101
+
theory, and it tells that ( X I +. . X n ) / f i converges in law to a normal distribution as n goes to 00 when XI,X2,.. , are independent random variables with some condition; in particular, if the Xi’s are i.i.d. with E ( X i ) = 0 and E ( X ? ) = 1, then we have
”>‘ ) &
lim E ( ( ~ ‘.’. + ~ n+ca
--
7r
Jm
xke-x2/2
dx
-ca
( k - l)! ( k even). 2k/’-’(k/2 - l)! Let a l , a 2 , . . . be a free independent sequence of noncommutative random variables such that cp(ai) = 0, cp(aq) = 1 and supi Iq(af))l< 00 for each k E N. Then
Theorem 1.1 ([54])
=
\L( k / 2 + 1 k’ / 2 ) (kewen).
The semicircle ( Wigner) law of radius
T
is
2 := -d-xr-r,rl(x) dx (3) nr2 The limit distribution in the free central limit theorem is the standard semicircle law w2 while that in in the classical central limit theorem is the normal (Gaussian) law N(0,l) = -&e-xa/2 dz. The 2m-th moments of N(0,l) is Wr
(2m - l)!! :=*. which is the number of all pair partitions of a set of of 202 is the 2m elements. On the other hand, the 2m-th moment so-called Catalan number familiar in combinatorics theory as the number of all non-crossing pair partitions of 2m elements. The above comparison between two limit theorems strongly suggests that the semicircle law is the free analogue of the normal law; in fact, the semicircle law plays important roles in various places of free probability as the normal law does in classical probability. We now give two examples in which the free independence emerges naturally, and they show that it is a not curious but proper notion in mathematics. Example 1.2 One would first imagine free groups from the term “free.” Let Fn be the free group with n generators ( n = 2 , 3 , . . . ,00) and consider the
&(z)
102
Hilbert space
t2(F,) := {t : F, + Cc I
Cg@,1<(9>12< I. -
with the inner product ( 6 , ~ ) := CgEP, <(g)q(g). Let Lh (h E G ) be the left regular representation of F, on L2(U)defined by (Lh<)(g) := <(h-'g) for [ E 12(lF,) and h,g E F,. We then have a noncommutative probability space (L(F,), T ) consisting of the free group factor L(F,) := {Lh : h E 3,)" (see also Subsection 5.1) and the tracial state T := ( . & , 6 , ) where e is the identity of F,. Let 91,.. . ,g, be n generators of F,. It is easy to see that { L g l LgF1}, , .. ., {Lg,,,Lg,1} are free independent in (L(F,),T ) . In particular, ai := (Lgi Lg;1)/& (i = 1 , . . . ,n) are free independent with .(a;) = 0 and ~ ( a f=) 1 so that Theorem 1.1 implies that as n + 00
+
Pzm(n):= x-
((Lg, +Lg;l 1
+...+ Lgn +Lg;1)2mSe,6e)
(.'").m + l
(2n)2m m
When we consider the random walk on F, starting from e with transition probability 1/2n from g to hg for each h E (91,gT1,... ,g,, g;'}, the above Pz,(n) is equal to the probability of returning to e after 2m steps. In this way, we notice that the free central limit theorem has a relation with the asymptotic behavior of the returning probability in some random walk. The free central limit in this special case on free groups was found in [9]. Example 1.3 The Boson and Fermion Fock spaces are often used in quantum statistical mechanics. These are the direct sum Hilbert spaces of all n-fold ( n = 0 , 1 , 2 , . . .) symmetric or antisymmetric tensors of a given Hilbert space U . Here, the summand for n = 0 is the onedimensional space generated by the vacuum vector. Rather more simply, one can define the direct sum of all n-fold full tensors, i.e.,
with X@O := (GO, R being the vacuum vector, which is called the free (or f i l l ) Fock space. For each h E U the free creation operator a*(h)and the free annihilation operator a(h) on 3 ( X ) axe defined by
a* (h)R := h , a*(h)(f1 €3 . . .€3 f,) := h c 3 f1 €3 . . . @ f,
,
(4)
a(h)R := 0 , a(h)(f1€3 * * * €3 fn) := ( h , f I ) f 2 €3 - * * 8 f n .
(5)
103
(Note that the inner product above is conjugatelinear in the left variable as usual in treating Fock spaces.) Then Ila*(h)l(= llhll and a(h) = (a*(h))* are valid. When {ei}ieI is a complete orthonormal set in 31, the family ({a*(ei),a(ei)})iel is free independent with respect to the vacuum state cp := (a,.a),and furthermore (a*(ei) a(ei))i€lbecomes a semicircular system, i.e., a family of free independent noncommutative random variables all of whose distribution is the semicircle law w2. The semicircle system in the full Fock space model plays an important role in studying free probability theory (see also Subsection 5.2). The use of tensor product construction L"(Ri,Pi) = Lm(@ieI(i'2i,Pi))is familiar to make an independent family of random variables (or sub-a-fields) in classical theory. On the other hand, free product construction is useful to make a free independent family of noncommutative random variables. Given a family (di,cpi) (i E I) of noncommutative probability spaces, the free product noncommutative probability space *ieI(di, pi) = (*ieIdi, *ielcpi) is defined so that (di)ieI is free independent in (*ieIdi,*ieIcp;) (see [31] for details). Indeed, free independence is nothing but the notion extracted from properties satisfied by ai E di (i E I) in this free product probability space. In short, free probability theory is a sort of noncommutative probability theory based on free independence and free product.
+
2
Random Matrices
An n x n matrix X = [Xij]whose entries Xij are random variables on a probP) is called a random matrix. We assume that all X i j belong ability space (0, to fll
where tr, is the normalized trace on n x n matrices. The next theorem tells the limit distribution of random matrices with independent entries as n goes to 00; this is Wigner's classical theorem, supplemented by Arnold concerning the almost sure convergence. Theorem 2.1 ([64,3]) For each n E N let H ( n ) = [Hij(n)]be a selfadjoint random matrix such that Hij(n) (1 5 i < j 5 n ) are independent. Assume
104
that E(Hij(n))= 0, E(IHij(n)I2)= 1 for all 1 5 i sup{E(IHij(n)lk) : n E
< j 5 n and
N,1 5 i 5 j 5 n} < 00 for each k E N.
Then the distribution of H ( n ) with respect to rn tends to the semicircle law n + 00, that is,
202 as
Indeed, the empirical eigenvalue distribution of H ( n ) (i.e., the distribution of H ( n ) with respect to tr,) converges almost surely to w2, that is,
The semicircle law is the limit distribution in both Theorem 2.1 (Wiper’s theorem) and Theorem 1.1 (the free central limit theorem). This fact suggests that random matrices have some relation with free independence. Indeed, Voiculescu found the notion of asymptotic free independence connecting Wiper’s theorem and the free central limit theorem. Voiculescu’s asymptotic freeness tells us the limits of joint moments of multi-sequence of independent random matrices, and it is much stronger than Wigner’s theorem treating a single random matrix sequence. Before stating the theorem we give the precise definition of asymptotic freeness. Definition 2.2 For each n E N let ( X ( ~ , n ) ) .be ~ sa family of n x n random matrices. When {Sj : j E J} is a partition of S, we say that ( { X ( s , n ): s E S j } ) j c is ~ asymptotically free if the following conditions hold: (a) ( X ( ~ , n ) ) , ~has s the limit distribution p as n + 00, that is, p is a linear functional on C(Xsls E S) with p ( 1 ) = 1 such that, for each S1,...,SkESl
p(x81&2 *--xs.,,) = n+m lim Tn(X(s1’n)x(s2,n). - * x ( S k , n ) ) . (6) (b) (C(X,ls E
S j ) ) j Eis ~
free independent in (C(X,Is E S ) , p ) .
In the above, C(X,ls E S) denotes the noncommutative polynomial algebra of noncommutative indeterminates X, (s E S). A selfadjoint n x n random matrix H ( n ) is said to be standard Gaussian if {ReHij(n) : 1 5 i 5 j 5 n} U {ImHij(n) : 1 5 i < j 5 n} is am independent family of Gaussian random variables, the distribution of ReHii(n) for 1 i 5 n is N ( 0 , l/n) and that of ReHij(n) and ImHij(n) for 1 5 i < j 5 n is N(O11/2n). Here, the term “standard” means that r n ( H ( n ) ) = 0 and r n( H(n)2)= 1.
105
Theorem 2.3 ([55]) Let ( H ( s ,n)),€sbe an independent family of standard selfadjoint n x n Gaussian matrices. Let ( D ( t , n ) ) t € Tbe Q family of n x n constant matrices such that sup, IlD(t,n)ll < 00 (t E T ) (11 . 11 being the operator norm) and ( D ( t , n ) ,D ( t ,n)*)tEThas the limit distribution. Then
(({%
n)I)sES 9
{W, n ) ,D ( t ,n)* : t E TI)
is asymptotically free. Let ( u s ) s Ebe ~ a semicircular system in a noncommutative probability space (d,p) as in Example 1.3. Since by Theorem 2.1 the limit distribution of each H ( s , n ) above is w2 (the distribution of u s ) , the asymptotic freeness of ( { H ( s ,n ) ) ) s € s tells lim ~ , ( H ( s l , n ) H ( s 2 , n-). - H ( s k , n ) )= v(aslas2- - - a s , ) n-+m for all s1,.. . ,S k E S, that is, each joint moment of ( H ( s ,-n)),~stends to that of ( U ~ ) ~ E The S . inclusion of constant matrices in the above theorem is of some use in applications to operator algebras. Furthermore, the asymptotic freeness in the stronger sense of almost sure convergence (with tr, instead of T, in (6)) can be shown for a wider class of random matrices having unitary invariant distribution (see [30]). The asymptotic freeness of ( H ( s ,n)),€sin almost sure sense was proved independently by Thorbj0rnsen [53]. The joint density on Rn of the eigenvalue distribution of the above standard selfadjoint n x n Gaussian matrix H ( n ) is known to be
(See [34]for details on eigenvalue distributions of random matrices.) Now, let us consider a bit more general probability density on R". Let Q : R + R be a continuous function satisfying lim I zI exp(-eQ(z)) = 0 for every E
z-tfm
For /3
where
> 0 consider the probability density on R"
> 0.
106
We denote by M(R) the space of all Bore1 probability measures on R equipped with weak topology, which is a Polish space (with the so-called LBvy metric). The Dirac measure at z E R is denoted by 6(z). The next theorem is the large deviation principle of level 2 for the above (5,) due to Ben Arous and Guionnet [5], which is a matricial (or free) counterpart of the famous large deviation of Sanov. (See [16,17] for general theory of large deviations.) Theorem 2.4 ([S]) The finite limit B := limn++, n-2 log 2, exists. If is a random vector in R" distributed with the joint distribution in,then the sequence of random probability measures
satisfies the large deviation principle in the scale nd2 with the good rate finetion
I(P) := - P W )
+ /&(XI
~PL(z)
+B
P E MW,
where
There exists a unique minimizer U, O of I ( p ) with I ( p o ) = 0. The double logarithmic integral (10) is the free entropy introduced by Voiculescu [56] (see the next section for a review on free entropy). The above large deviation says that, for every closed F C M(R) and for every open G c M(R), we have
1 lim sup - log X, n+Do
n2
(~ ( z I+)
* *
+6(zn)
5 -inf{I(p)
: /I E
F},
The goodness of the rate function I ( p ) means that the level sets { p E M(R) : I ( p ) 5 c } are compact in weak topology for all c 2 0. Moreover, it is known that (P,) is exponentially tight. The assertion on the minimizer is a consequence of the fundamental result on weighted potentials (see [47:1.1.3 and 1.3.11). Moreover, Theorem 2.4 implies that P, converges almost surely to po as n -+ co in weak topology. The proof of this fact is a simple application of the Borel-Cantelli lemma as shown in [31: p. 2111.
107
In particular, when inis the joint eigenvalue distribution of H ( n ) , the random measure Pn in (9) is the empirical eigenvalue distribution of H ( n ) so that the large deviation holds for (Pn)with the rate function I ( p ) := -C(p)
+f
/
3 4
z2 dp(z) - - ,
and I ( p ) attains the minimum 0 at p = w2 only. This strengthens Theorem 2.1 saying that Pn converges almost surely to w2 as n + 03. Similar large deviation results are known for the eigenvalue distributions of random matrices of several types (see [6,27,29] and also Theorem 4.1 below). In the rest of the section let us present the large deviation for Wishart matrices. For each n E N let p ( n ) be a positive integer. Let T ( n )be a p ( n ) x n real random matrix whose entries are independent and have the same distribution N ( 0 , l ) . Then the n x n random matrix n-lT(n)tT(n)is the Wishart matriz with normalization; the Wishart matrix is familiar in multivariate statistical analysis. In the case p ( n ) 2 n, the joint probability density of the (random) eigenvalues of n-lT(n)tT(n)is known (see [l]p. 534) to be
where 2, is the normalizing constant. So the empirical eigenvalue distribution of n-lT(n)tT(n)is given as the random discrete measure
+
6 ( ~ 1 ) .* * + 6 ( z n ) , n where ( ~ 1 , ... , z n )is a random vector distributed under (11). In the other case when p ( n ) < n, the random matrix n-'T(n)V(n) is singular and its eigenvalues are those of n-lT(n)T(n)t plus n - p ( n ) zeros. The eigenvalue distribution of p(n)-lT(n)T(n)tis given by (11) with n , p ( n ) being interchanged, so the empirical eigenvalue distribution Pn of n-'T(n)T(n) when p(n) < n is Pn :=
where
(21,
. . . ,zp(,))is distributed under
108
It is known ([63]) that if limn+.oop(n)/n= X converges in weak topology to
> 0, then P,,
given above
p~ := ma{ 1 - A, 0}6(0)
almost surely. The distribution p~ is the so-called Marchenko-Pastur distribution after [33] and is also called the free Poisson distribution because it appeared as the limit in the free probabilistic analogue ([62]) of the Poisson limit theorem (see also Subsection 5.4). Let M(IR+) be the space of Bore1 probability measures on IR+ := [O,m) equipped with weak topology. The large deviation is satisfied for P, as follows. Theorem 2.5 ([27,31]) When p(n)/n + X > 0 as n + 00, the empirical eigenvalue distribution P, of the Wishart matrix n-lT(n)T(n)t satisfies the large deviation principle in the scale n-2 with the good rate function I ( p ) for p E M(IR+) defined as follows: when X 2 1,
1 I(p) := ---C(p) 2
+
1
--(3X
4
(X
- (A - 1) logs) d/.i(z)
- A2 logX
+ (A - 1)2log(X- 1)) ,
and when 0 < X < 1, X2 ---C(p) 2
1 4
-- (3X
I ( p ) := +m
XJ -
+5
(z- (1 - A) logx) dp(z)
+ (1 - X)2 log(1 - A)) if p = (1 - X)d(O) +
X2 log X
p E M(R+),
otherwise.
Moreover, the unique minimizer of I is p~ given in (12).
3 Free Entropy We begin with the classical case again. When a probability measure p on IR has the density f ( x ) , the Boltzmann-Gibbs entropy (or the differential entropy) is given by ~ ( p:= ) -
J f(x>logf(x) dx .
109
If the support of p is included in [-R, R], then we obtain the asymptotic limit expression
1 ;:yo "--too n
S ( p ) = lim lim -logL"({z
E [-R,R]"
:
(13) Imk(Kn(2)) - m k ( ~ ) II C, k I r } ), where L" is the Lebesgue measure on R", m k ( p ) is the kth moment of p and K,(X) :=
b(z1)
+ . . . + b(zn)
n By looking at the asymptotic behavior of the Boltzmann-Gibbs entropy of random matrices, Voiculescu [56] asserted that the free entropy of p should be the double logarithmic integral (10). To be more precise, C ( p ) is defined by (10) if log I z - yI is integrable with respect to dp(z)dp(y), and otherwise C ( p ) = -m. It is indeed the minus of the logarithmic energy of p familiar in potential theory (see [47] for example). The free entropy C ( p ) is upper semicontinuous in weak topology when the support of p is restricted in a fixed compact set, and it is strictly concave in the sense that W P 1
+ (1 - 4 p 2 ) > W P l ) + (1 - 4C(pa)
if 0 < X < 1 and p1, p2 are compactly supported probability measures such that p1 # pa, C(p1) > -m and C(p2) > -m (see [31]5.3.2). The space M z of selfadjoint n x n matrices is canonically isometric t o the Euclidean space Bn2, so the "Lebesgue" measure A, on M z is defined as the copy of the Lebesgue measure on P"'. Moreover, the measure A, induces on P" via the map of A E M i a to the permutation-invariant measure (Al(A) ,...,X,,(A)) E R" where A1(A) 5 . - . 5 X,(A) are the eigenvalues of A. When the support of p is included in [-R, R], the asymptotic limit expression 1 3 C ( p ) - log 2n 2 4
A,,
+
+
110
was given in [57]. A convenient way to prove this is the use of large deviation technique, although Voiculescu’s original proof had a lot of indications leading to large deviation. In this way, C ( p ) has a limit formula of the same type as that of S(p), and it suggests that C ( p ) is the free probabilistic analogue of the Boltzmann-Gibbs entropy S(p). When the variance is fixed, the semicircle law is a unique p maximizing C ( p ) while the normal law maximizes S(p). The free entropy for noncommutative multivariables was introduced in [57] by generalizing the above limit formula of C ( p ) . In what follows in this section, assume that ( M ,T ) is a W*-probability space consisting of a von Neumann algebra M and a normal tracial state T. Let (a1,. .. ,a ~be) an N-tuple of selfadjoint elements in M . For a muIti-index k = (kl,. . . ,ks),1 5 kj 5 N , write Ikl := s and denote the joint moment 7(ak1 . a k s ) by mk(a1,. .. , a ~ ) . Also, for Al,. ..,AN E M F write rnk(A1,. .. ,AN) for tr,(Akl . ’.Ak,). For each n,r € N,E > 0 and R > 0 define r ~ ( a 1.,.. ,U N ; n, r, E ) : IlAjll 5 R, := { (Ai,. . . ,AN) E Imk(A1,. . . ,AN) - m k ( a i , . - ., a ~ )Ll E ,
(15)
I4 L r } ,
which is a set of n x n selfadjoint matrices approximating (al, .. . ,U N ) in the sense of joint moments. Then, the free entropy X(a1,. .. ,U N ) of (uI,. . .,U N ) is defined as follows: XR(a1,
...,a
~ :=)
hl limsup r++o
? I + -
X(a1,. .. , a N ) := supXR(alr * R>O
,aN).
This ~ ( a l ,. . ,U N ) is sometimes called the microstate free entropy by regard) microstates. In particular, ing (Al,. . . ,A N ) approximating (al,. . . , a ~ as (14) says 1 3 x(a) = C ( p ) - log2a 4 2
+
+
for the one-variable case. The entropy X(a1,. . . ,a ~ has ) natural properties of subadditivity, upper semi-continuity, change of variable formulas, etc. (see [57]). In particular, the maximization and the additivity in the next theorem show that this entropy is very suitable in free probability theory.
111
Theorem 3.1 ([59])
+ + a x ) 5 C, then
(a) If ~ ( a : . . .
N 2neC X ( a l , - - - , a N )5 T l o g N , and equality occurs here if and only if al, . . . ,U N are free independent and the distribution of each ui is w 2 m . (b) If al, . . . ,U N are free independent, t h e n X(al,. . .,a N ) = x(a1)
+ .. . + x ( a N )
Conversely, if this additivity holds with ~ ( a , > ) -CQ for 1 5 i a1, . . . ,U N are free independent.
5 N , then
Furthermore, the free entropy dimension
was introduced in [57], where {Sl,.. . ,S N }is a semicircular system in M free independent of { a l , .. .,U N } . The free entropy dimension is closely related to the free independence of ul, .. . ,U N . Roughly speaking, 6(al,. . . ,U N ) means the “free dimension” of the von Neumann algebra {al, . . . ,U N } “ , and it relates with the (non-) isomorphism problem of free group factors. The free entropies g(u1, . . . ,U N ) for non-selfadjoint elements a1,. .. ,U N E M and xu(ul,... ,U N ) for unitaries 211,. . . ,U N E M can be defined by modifying the definition of ~ ( u l ., ..,U N ) above. For example, the definition of xu is as follows. Let yn denote the Haar probability measure on the n-dimensional unitary group 2-4,. Let ~ 1 , ... ,U N E M be unitaries. For n, T E N and E > 0 define
where
The free entropy xU(ul,. ..,U N ) is defined by
112
The relation among x, 2, xu under polar decomposition was established in [28] (see also [37]). The Fisher information of a classical random variable X is d 2 I ( X ) := da: log f ( a : ) ) f(z)da: = -da:,
/(
/
whenever X has the continuously differentiable density f(z). For any random variable X with finite variance, the differential formula in [4] can be reformulated as follows:
S ( X + fiZ)
-
;I'
I(X
+ J t Z ) dt = S ( X ),
s 2 0,
where 2 is a standard Gaussian random variable such that X and 2 are independent. In [56], for a E M"" whose distribution has the density f(z), the free Fisher information of a is defined as
4 2/ f ( a : ) 3
@(a):= 3
&,
and the free analogue of the above differential formula was given: C(a
+ 45')- 5
I'
@(a
+ A S ) dt = E(a),
where S is a semicircular element (having the distribution independent of a. The above formula implies
s 2 0, 202)
of M"" free
On the other hand, in [60],the multivariable free Fisher information @*(al,. . .,a ~was ) introduced by use of noncommutative Hilbert transform, and further another (microstates-free) free entropy x* ( a l ,... ,a ~was ) defined as
;1yiTi-
X*((al,..-,aN) := N
@* (al
+ 4S1, . . .,a N + ~
S N dt)
+ N log 2 x e ,
where 4, .. .,SN are as in the definition of free entropy dimension. Although @*(a)= @(a) and x*(a) = x(a) for the one-variable case were shown, it is open whether the equality x* = x is true or not for the multivariable case. But, it is worth noting that the inequality 5 x* is recently obtained in 171 as a consequence of a large deviation result for selfadjoint matrix valued Brownian motion towards free Brownian motion (also [14]).
x
113
It is remarkable to say that several long-standing open problems on free group factors were solved by use of the free entropy and/or the free entropy dimension (see [58,21,22,52] for example). 4 4.1
Free Relative Entropy for Measures
Definition of Free Relative Entropy
In classical probability theory, when p, v are probability measures on R, the relative entropy (or the Kullback-Leibler divergence) S(p,v) of p with respect to Y is defined as
if p is absolutely continuous with respect to v ; otherwise S ( p , v ) := +w. If p and v are supported in [-R,R], then the relative entropy S ( p , v ) has the asymptotic expression similarly to (13) as follows: 1
- s ( p , v ) = lim lim -logv"({z
12;
n--tw
n
E
[-R,R]" :
Im(&z(z>)- m h ) I
s
IC 5 r } ) (17) where v" is the n-fold product of v. This expression as well as (13) can be El
1
derived from Sanov's large deviation theorem for the empirical distribution of i.i.d. random variables (see [31]5.1.1 for details). Now, naturally arises the following question: What is the free analogue of the relative entropy S(p,v)? It turned out ([35]) that the free relative entropy Z ( p , v) of p with respect to v can be defined as
which is the logarithmic energy of a signed measure p - v. Here, the following two definitions may be available for precise meaning of Z ( p , v):
(A) Define X ( p , v) by the above double integral if log )z-yI is integrable with respect to the total variation measure dip - vI(z)dip - v l ( y ) ; otherwise E ( p , v) := +w.
(B) Based on the fact that E > 0 I+- JJlog((z-y(+e) d ( p - v ) ( z ) d ( p - v ) ( y ) is increasing as E 4 0 ([35]Lemma 3.6), define
114
Note that if loglz - yI is integrable with respect to dlp - vl(z)dlp - vl(y), then the definitions (A) and (B) are the same; this is the case in particular when C ( p ) > -m and C ( v ) > -00. Let R > 0 and Q be a real continuous function on [-R,R]. For each n E N define the probability distribution i n ( & ; @ on B" as in (7), (8) by
n n
x
X [ - R , R ] (2;)dzldz2
* * *
dxn ,
i=l
where Zn(Q : R ) is the normalizing constant:
Moreover, let X,(Q; R ) be the probability distribution on Mza which is invariant under unitary conjugation and whose joint eigenvalue distribution on B" is X,(Q; R ) , that is, the permutation-invariant probability measure X,(Q; R ) is induced by X,(Q; R). More explicitly, Xn(Q; R) := (dU @ Xn(Q; R ) ) o @il,
(21)
where dU is the Haar probability measure on the n-dimensional unitary group : U, x R" + M: is defined as
U,,and @,
a n ( U ,(q,. . . ,zn)):= Udiag(z1,. . . , z n ) U * . One can consider X,(Q; R ) as the distribution of an n x n random selfadjoint matrix, or more explicitly X,(Q; R ) itself as a random matrix. The support of X,(Q;R) is
(M;)R := { A E M F : IlAll 5 R } . We now have a modification of Theorem 2.4 as follows; its proof is essentially same as [31: 5.4.3,5.5.11. Theorem 4.1 Let Q and Qn (n E N) be real continuous functions on [-R, R] such that Qn(z) + Q(z) uniformly on [-R,R]. For each n E N define the probability distribution in(&,; R) supported on [-R, R]" by (19) and the normalizing constant Zn(Qn;R ) by (20) with Qn in place of Q. Then the finite limit 1 B(Q; R) := lim -log Zn(Qn;R ) n+m n2
115
exists, and i f (XI,. ..,x,) E [-R, R]" is distributed with the joint distribution in(&,; R), then the empirical distribution A(b(z1) . .+ S(zn))satisfies the large deviation principle in the scale n-2 with the good rate function
+.
I(P) := --C(pL)+ AQ)+ B(Q;R )
PE
R1)-
There exists u unique minimizer p~ of I with I ( ~ Q=)0 and B(Q;R ) is determined only by Q independently of (9,). Furthermore, the above empirical distribution converges almost surely to p~ QS n + 00 in weak topology. Let v be a compactly supported probability measure on R, and assume that the function r
is finite and continuous on R. For R > 0 define the probability distribution X,(v; R ) on Mza by putting Q = Q y in (19)-(21), i.e., Xn(v; R ) := X n ( Q Y ; R). Then the next asymptotic expression of C ( p , v ) is the free analogue of (17) and the relative version of (14). The proof is an application of Theorem 4.1 for Qn= Q = Qy. Theorem 4.2 ([35]) Let p, v be compactly supported probability measures, and assume that Q y ( x ) is continuous on P. For any R > 0 such that suppp, supp v C [-R, R] (suppp denotes the support of p ) ,
1
- C ( p , v ) = lim lim -logX,(v;R)({A
;-+T~n + w
n2
E
M r : IlAll I R,
= lim lim -1l o g ~ , ( v ; R ) ( { z E [ - R , R ] " :
l-++"o n+w
n2
Imk(Kn(x))
- mk(~)I I k 5 r})
in either definition (A) or ( B )f o r C ( p ,v). The free relative entropy C ( p ,v) in (18) is symmetric unlike the relative entropy S ( p , v ) ; however C ( p , v ) shares other properties such as strict positivity, joint convexity and lower semi-continuity with S(p,v) (see [35] or [26] for details). 4.2
Free Perturbation Theory for Measures
In the rest of this section, for simplicity, let R > 0 be fixed and v be a probability measure supported in [-R, R] such that the function Q := Qv defined above in (23) is continuous on [-R, R]. Let M ( [ - R ,R])be the space
116
of all probability measures supported in [-R,R] and CR([-R,R])the space of all real continuous functions on [-R, R]. We adopt ( B ) as the definition of C(p, v); note by Theorem 4.2 that both definitions (A) and (B) are the same for all p E M([-R,R])whenever Q is continuous on R. For v E M ( [ - R , R])fixed as above, the Legendre transform of the function p E M([-R,R])I+ C ( p , v) is defined as
~ ( hV), := SUP{ -p(h) - C ( p ,V ) : p E M ( [ - R , R ] ) } for each h E Cw([-R,R]). Theory of weighted potentials ([47])plays a key role to prove the following theorems as well as the assertion on the minimizer in Theorems 2.4 and 4.1. Theorem 4.3 ([26]) (a) c ( - ,v) is a convex function on CR([-R,R ] )satisfying
-mIc ( h , v ) I llhll (in particular, c(0,v) = 0 ) where llhll is the sup-norm, and it is decreasing, i.e., c(h1,v) 2 c(h2,v) zf hl 5 hz. Moreover, Ic(h1,v)
- C(hZ,V)I Illhl - h211
f o r aZZh1,h~E &([-R,R]). (b) For every p E M([-R,R], C ( p , v ) =sup{-p(h)
- c ( h , v ) : h E Cpz([-R,R])}.
(c) For every h E &([-R,R]) there exists a unique v h E M ( [ - R , R ] ) such that -vh(h) - C ( v h , v) = c(h,v) ,
that is, vh is a unique m u i m i z e r of - p ( h ) - C ( p ,v) f o r p E M ( [ - R , R ] ) . Moreover, C ( v h ) is finite and
+
+
~ ( hV ), = C ( v h ) C ( Y )- vh(Q h) .
(d) For every h E CR([-R,R] and p E M([-R,R]),p = uh i f and only i f
c ( h + k , v ) > c ( h , v ) - p ( k ) for all ~ E C R ( [ - R , R ] ) . We call uh in Theorem 4.3 the perturbed probability measure of v by h (via free relative entropy). Clearly, vh+cr= vh and c(h a, v) = c(h, v) - a for a E R.
+
117
It is instructive to consider the perturbed measure uh in comparison with the similar perturbation via relative entropy. For any v E M ( [ - R , R ] )and h E Cw([-R, R ] ) ,it is well known that logv(e-h) = sup{-p(h) - S ( p , v ) : p E M ( [ - R , R ] ) } and the probability measure po := ( e - h / v ( e - h ) ) v (i.e., dpo/dv = e - h / v ( e - h ) ) is a unique maximizer of -p(h) - S(p, v) for p E M ( [ - R ,R ] ) . This can be easily verified by using the strict positivity of S(p,P O ) ;in fact,
0 5 S(P, Po) = P(h) + 1%
W h+)S b ,v)
and equality occurs if and only if p = po. M([-R,RI),
Moreover, for every p E
S(p,v) = sup{ - p ( h ) - log v ( e - h ) : h E Cw([-R,R])} .
The probability measure po perturbed from v via the relative entropy S(p,v) is the so-called Gibbs ensemble. The above c(h,v) is considered as the “free” counterpart of the pressure logv(e-h), and the characterization of vh in the above (d) is the “free” analogue of the so-called variational princaple for Gibbs ensembles ([46]).It is worth noting that this type of perturbation theory via relative entropy was developed even in the quantum probabilistic setting on operator algebras ([40],[18],[39] 512). We shall write vh*’ for the above po to distinguish it from vh. Theorem 4.4 ([26]) (a) For every p E M ([R,R ] ),
45 C ( P ,v) + A h ) + c(h,4 . Moreover, if supp p
c supp vh, then
%, 4 = %,
+ P ( h ) + c(h,
(b) For every h E Cw([-R,R]),
c ( h , v ) 3 -v(h) Furthermore, if supp v
+ C ( v h , v ) 2 - v(h) +2 vh(h)
c supp uh, then C(v” u) =
v(h)- vh(h) 2
7
118
c(h,v) = -v(h)
+ C ( v h ,v) = - v(h) +2 vh(h)
(c) Let h,lc E CR([-R,R]).I f Q v . ( x ) := 2Jloglx-yldvh(y) as well as is continuow on [-R, R] and supp ( v h ) kc supp v h , then h k - h+k ( v ) --y ,
c(h
Qv
+ Ic, v) = c(h,v) + c(Ic,v h ).
As for the perturbation Y c) uhiS via relative entropy, it is obvious that supp vhiS= supp v, and the following formulas generally hold: ~ ( pvhpS) , = S(p,v)
+ p ( h ) + log v ( e - h ) ,
vh,S k , S - ,,h+k,S 0 9
+
log v ( e - ( h + k ) )= log v(eVh) log vh,S(e--k). But, as for the free perturbation v c) vh,it is known ([26]) that the support assumptions in the above (a)-(c) itre essential. It is shown that if p E M([-R,R])satisfies p 5 QV for some constant (Y 2 1, then Q,(z) := 2Jlog Iz - yldp(y) is continuous on [-R,R] and there exists an h E Cw([-R,R]) such that p = vh. Next, to consider the continuity properties in h of the perturbation vh, we set Mc([-R,R]) := { P E M([-R,RI) : C ( P ) > -00)
7
and define d(p1,pz) := % w 2 ) 1 ' 2
(E [0,0O)) for p1,p2 E Mc([-R,RI).
It is known ([23]) that d(p1,pz) is a metric on Mc([-R,R]) and the &topology is strictly stronger than the weak topology restricted on Mc([-R,R]). Indeed, we note that ( M c ( [ - R , R J ) , dis) a non-compact Polish space. Hence, the convergence C ( p n , p ) -+ 0 implies p, -+ p weakly for pn, P E Mc([-Rt R]).
Theorem 4.5 ([26]) (a) If h, h, E CR([-R,R]), n E convergences hold: (i) c(h,,v) + c ( h , y ) .
N,satisfy Jlh, - hi1 + 0 , then the following
119
(ii) E ( v h - , p ) + C ( v h , p ) for alI p E Mc([-R,R]); in particular, E(vhn, v h ) + 0. (iii) v h n -+ vh weakly. (iv) vhn(hn)+ v h ( h ) . (v) C(&) + C ( v h ) . (b) Let pn,p E M([-R,RJ) f o r n E N, and assume that there as an a 2 1 such that pn 5 au for all n E N. Then p, + p weakly if and only if E(pn,p) + 0. In this case, C(p,) + C ( p ) and E ( p n , p f )-+ E ( p , p f )for all p' E ME([-R,R]).
As for relative entropy, it is known that if p,, v,, are probability measures on R such that llpn - pII + 0, IIv,, - vll -+ 0 and there is an a > 0 such that p n 5 av, for all n E N, then S ( p n , v,) + S(p,v). (This is true in the operator algebra setting, see [2]Theorem 3.7.) However, this fails to hold for free relative entropy; one can easily provide an example of p,, v, E ME([-& R]) such that llpn - vII + 0, llvn - v1I -+ 0 and pn 5 av, for all n E N,but E b n , vn) P 04.3 From Relative Entropy to Free Relative Entropy In this subsection, for each n E N we simply write X,(v) for the probability measure X,(v;R) = Xn(Q;R)on ( M z a ) given ~ in (19)-(21). Here note that (M:)R is a compact subset of M i a identified with a Euclidean space W"'. For a given h E &([-It, R])and n E N, let &(h) denote the real continuous function on ( M i a ) defined ~ by cj,,(h)(~) := n2trn(h(A)) for A E
( M ~ ) R ,
where h ( A ) is defined via functional calculus and tr, is the normalized trace on M,. Then one can get the probability measure on ( M z a ) ~ which is the perturbed measure of X,,(v) by q5,(h) via relative entropy; namely, X,,(~)bn(")*~ is a unique maximizer of the functional
-dcjn(h)) - s(71, Xn(v>) for 7 E M((M,?)R) 1 where M((M;")R)is the space of all probability Bore1 measures on (M?)R. In fact, as mentioned after Theorem 4.3,it is given by
which satisfies
120
The measure X , ( V ) + ~ @ ) > ~on ( M ~ ) may R be considered as an n x n selfadjoint random matrix which is a perturbation of X,(v) via relative entropy. The next theorem says that this perturbation of X,(v) via relative entropy on the matrix space approaches asymptotically as n + 00 to v h ,the perturbation of v via free relative entropy. In particular, it justifies our formulation of free relative entropy. In the theorem we actually treat a sequence of perturbed measures X,(v)+n(hn)-S determined by h, E Cw([-R,R])separately for each n satisfying llhn - hll + 0. The proof is based on the large deviation result presented in Theorem 4.1. Theorem 4.6 ([26]) Let v E M ( [ - R , R ] ) be as above. If h,h, E Cw([-R, R]),n E N,satisfy llh, - hll + 0, then the following hold: (i) The empirical eigenvalue distribution of X , ( V ) + ~ ( ~ - ) >converges ~ almost surely to uh as n + 00 in weak topology. (ii)
(iv) With B ( Q ; R )defined by (22) and B(Q place of Q,
+ h;R) similarly with Q + h an
1 c(h,v) = Jilp logX,(v)(e-4n(hn)) = B(Q h; R) - B(Q;R).
+
1
v(h) - v h ( h )- z ( v h ,v) = lim --s(x,(v), n+m n2 Hence (see Theorem 4.4 ( b ) ), if supp v
X , ( V ) + " ( ~ ~ ) V .~ )
c supp v h , then
Besides its conceptual importance, Theorem 4.6 supplies the asymptotic formulas of vh(h)and c(h,v) (when h, = h for all n ) ; thus we obtain the asymptotic formula of C(vh,v) = -vh(h) - c(h,v). In particular, if p, v are
121
Non-Commutativity, Non-Commutativity,
9 on [-R, R],C(CTR)= log and R C ( ~ , C T= R )- C ( p ) + log 2 for p E M([-R,R]).
Then QbR(z) 210g
On the other hand, if dx 12R, then
TIIR
is the uniform distribution on [-R,R], i.e., mR =
S(p,mR) = -S(p)
+ log(2R)
for p E M([-R,R]).
Thus, the arcsine law can be considered as the free probabilistic analogue of the uniform distribution. Since the minus free entropy is a special case (up to an additive constant) of free relative entropy, we can directly transform the perturbation results (Theorems 4.3 and 4.6 for instance) for free relative entropy to the case of the free entropy C ( p ) or ~ ( p ) Define . the Legendre transform of - C ( p ) for p E M ( [ - R ,R])as
n(h):= SUP{ - p ( h )
+ C(p):
ji
E
M([-R,R ] ) }
so that n(h)= c(h7OR)
R + log 2
for each h E Cw([-R,R]).Then the Legendre transform n(h)of -x(p) is 1 n ( h )= n(h) -log% 2
+
+ -34
for h E Cw([-R,R]),
122
and x ( p ) is the (minus) Legendre transform of r(h): x(p) = inf{ p(h) + 4 h ) : h E CR([-R, R ] ) }
(25)
for p E M([-R,R]). For every h E CR([-R,R])let uk denote the unique maximizer of -p(h) x ( p ) for p E M ( [ - R , R ] )so that
+
-o;(h)
+ x ( a i )= a(h) ;
in fact, O; is the perturbed probability measure of OR by h introduced in the previous subsection. The Boltzmann-Gibbs entropy of a probability measure q on (M:)R can be defined as
S(77):= - - S ( q , L )
7
where S ( q , A , ) is the relative entropy of q with respect to the “Lebesgue” measure A, on M Z (see Section 3). The entropy S(q) is indeed equal to the usual Boltzmann-Gibbs entropy of the measure on Rn2 (ZMz”) induced by q. The next theorem is an adaptation of Theorem 4.6 to the present situation. Theorem 4.7 ([26]) If h , h , E & ([-R , R ]), n E N, satisfy llhn - hll + 0, then the following hold: (i)
(ii)
In terms of statistical thermodynamics ([46])
/” /” . ..
-R
exp (-n
-R
2
h ( q ) )A(z) dz
i=l
is the partition function of n logarithmically interacting particle in an outer field h. So
is considered as the pressure in a one-dimensional Coulomb gas model.
123
4.4 Toward the Multivariable Case It is tempting to extend the expressions (26) and (25) to the multivariable case. Let C(X1,.. . ,XN) be the noncommutative polynomial algebra of noncommuting indeterminates XI ,. . . ,XN, where the *-operation is defined according to X: = Xi, (Xi, Xi, . . .Xi,)* = Xi, .. Xi,Xi,. The space of all selfadjoint polynomials in C(X1,. .. ,XN) is denoted by C(X1,. . . , X N ) ' ~ .For every p E C(X1,. . ., X N ) " we ~ have a mapping (Al,. . . , A N ) E ( M i a ) NH p(Al, . .. ,AN) E M,"". For each R > 0 we extend formula (26) to the multivariable case, and we define the "pressure function'' on C(X1, . . . ,XN)"" by T R @ ) := lim sup e
n+oo
" 1
+-1Ogn 2
.
(27)
For each linear functional p : C(X1,. . . ,X N ) " + ~ R with p (1) = 1, the (minus) Legendre transform of T R @ ) is defined by % ~ ( p:= ) inf{p(p) -I- T R ( P ) : p E C(X1,. . . ,XN)'"}
.
(28) In order to put two functions (27) and (28) in a proper setting of duality, we conveniently topologize C(X1, . . . ,XN) with a certain C*-norm. For each R > 0 and p E C(X1,. . . ,XN) define IbllR
:= sup{llp(Al,---,AN)II: A l , * . * , A NE M F , IlAill 5 R (1 5 i 5 N ) , n E N} .
Then it is easy to see that 11. I I R is a C*-norm on C(X1,. . . ,XN). So we have the C*-completion of C(X1,, . . ,XN) with respect to 11 . 1 1 ~ ; the C*-algebra thus obtained is denoted by dR(X1,. ..,XN) (or simply by dR). Notice that if M is a finite von Neumann algebra imbedded into the ultrapower product of the hypefinite 111 factor, then for any ul,. .. ,U N E M"" with lluill 5 R we have l b ( ~ l , - - . , ~ N5) lIlpllR, l
p E C(xl,.-.,xN)
so that the mapping p E C(X1,. . . ,XN) H p(a1,. . . ,U N ) E M can extend by continuity to a contraction h E dR(Xi,.. . ,XN) ++ h(a1,. . . , U N ) E M ,
(29)
124
which is obviously a *-homomorphism and is considered as the "continuous functional calculus" for noncommuting multivariable (al,.. . , a ~ ) .In fact, for the single variable case ( N = l), we have dR(X) = C([-R,R])and h E C([-R,R])I+ h(a) E M is the usual continuous functional calculus of a E M"" with llall I R. Lemma 4.8 (a)
TR
is a convex function.
(b) I r R b 1 ) - r R b 2 ) 1 I lip1 - m l l R f o r a l l p l , n E C(X1,---,XN)Sa. (c) If0 c R1 < R2, then r~~(P)I r ~ ~ b ) . (d) Ifpl E C(X1,... , X L ) and ~ ~p2 E C ( X L + ~.., .,XN)""with 1 5 L then pl + p2 E C(X1,... ,z , ) ' " and K R b 1 +P2)
5 rR(P1) + r R b 2 )
*
Thanks to (b) above T R can be extended by continuity to a function on of the C*-algebra d R = dR(X1,. . . ,XN), which is denoted by the same notation T R . The function T R on dg is convex and satisfies the same Lipschitz continuity as (b). Let T ( d R ) denote the set of all tracial states on the C*-algebra d R . The next theorem says that T ( d R ) is the essential domain of x ~ ( pand ) that the two functions K R ( ~ on ) Ag and z ~ ( pon ) T ( d R ) are the Legendre transforms of each other under the duality between dg and (dk)"". Theorem 4.9 (a) Let p : @(XI,. . . ,X N ) ~ "+ B be a linear functional with p ( 1 ) = 1, The value z ~ ( pgiven ) an (28) is -00 unless p extends to an element of
dg, the selfadjoint part
T(dR).
(b) For every p E T ( d R ) , k ~ ( p=) inf{p(h)
+ rR(h) : h E A:}.
Hence, the function g ~ ( p on ) ~ ( A Ris)concave and upper semicontinuous in weak* topology. (c) For every h E A:, rR(h) = suP{-p(h)
+ %R(P) :p E T ( d R ) } .
Now, let ( M ,T) be a tracial W*-probability space. For each ul, . .. ,UN E M"" the distribution p(al,...,a N ) : C(X1,... , X N ) ~+ " B is given by
...,a N ) b ) := ~ b ( a l , - - * , a* ~ ) )
p(a1,
125
We introduce a variant of free entropy for (ax, . . . ,a ~as)follows: %R(al,.. . 7 alv) := %R(&ai ,...,a N ) ) > %(Ul,
> 0,
R
... , a N ) := SUpiR(a1, ...,alv). R>O
If M is imbedded into the ultrapower product of the hyperfinite 111 factor and llaill 5 R, then we can write zR(U1,.
. . , U N ) = inf{T(h(al,. . . ,
+
a ~ ) ) T R ( ~:)h
E dg}
in terms of the functional calculus (29). Also, notice that P ( " ~ . . . , "extends ~) to an element of 7 ( d R ) (for any R > 0) if (al, . . . ,a ~has ) finite-dimensional approximants (see [61]). From (c) and (d) of Lemma 4.8 we immediately have:
< R1 < R2, then (ii) For 1 I L < N , (i) I f 0
%R(al,*
* 7
kRl(u,. . . , a ~ 5) %Rz(al,.. . , a ~ ) .
alv) 5 Z R ( a 1 , .. * > aL> + ?R(aL+l,-. . >alv) 9
It is quite an interesting problem to compare Z(a1,. . . , a ~ with ) X(al, ...,a N ) . Thanks to (25) and (26) we have %(a)= 2 ~ ( a = ) x(a) for every a E M"" with llall 5 R. Theorem 4.10 (a) For every al, . . . ,a N E M"" and R
> 0,
% R ( a l , . . .,alv) 2
XR(a1,. . . >a N ) 7
(b) I f a l , . . . , a E~M"" are free independent and llaill then Z(a1,.
5 R for 1 I i 5 N ,
. . ,alv) = ZR(U1,. . .,alv) = x(a1,. . . ,alv) .
The maximizations of f ~ ( a. .~. ,, a ~ and ) of Z(a1,.. ., a ~ are ) the same as those of XR(a1,. .. , U N )and X ( a l , . . . , a ~ as ) follows:
126
(i) Given R is
> 0, the maximal value of zR(a1,.. . ,aN) for al,. . , ,aN
E M""
and it is attained if al,. . . ,aN are free independent and the distribution of each ai is the arcsine law U R in (24).
+
+
(ii) When T(a: . . . a$) al,. .. ,aN E M"" is
5 C, the maximal value of %(all... ,aN) for
N 2neC 2 N ' and it is attained if al,. . .,aN are free independent and the distribution of each ai is the semicircle law w r in (3), r = 2 a .
- log -
5
q-Deformation Theory
+
For a real Hilbert space ?fawith its complexification ?fc := Ifla i?ffla let F(?fc)be the free Fock space over Re, and a*(h)and a(h) (h E 31~)be the free creation and annihilation operators on F(?f@) defined by (4) and (5) in Example 1.3. Then Voiculescu's C* -free Gaussian functor ([62]) is given as the C*-algebra l?(?fa)generated by s ( h ) := a*(h) a(h) (h E ?fa),and its W*variant is the weak closure l?(?fa)". The C*-algebra l?(?fa)is isomorphic to the reduced C*-free product *iEl(C[-l, 11, p ) with the semicircle distribution p := ~ d ~ x ~ -dx1and , 1 q 11 = dim7f.a while r(31R)'' is isomorphic to the free group factor L(Fdimxa). Indeed, if {ei}iel is an orthonormal basis of ?fa, then (s(e;))iEl forms a semicircular system as noted in Example 1.3 and each s(ei) generates Lm([-l, 11,p ) G L ( Z ) ,so we have
+
r(?fw)lt = {s(ei): i E 1)"
% *iEiL(Z) G
L(Fl1l).
In this way, the three concepts of free independence, free products and free groups are quite closely related and they altogether form the core of free probability world. This explains why free probability theory is especially useful in analyzing the structure of free group factors. A concise exposition on free product von Neumann algebras (as well as free product actions) is included in Ueda's article in this book (see also references therein). The free group factors L(F,) on natural numbers n = 2,3,. .. ,00 are interpolated with a one-parameter family of type 111 factors L(F,) parameterized by real numbers 1 < r 00 ([44], [20]). A brief survey on interpolated free group factors is given in Subsection 5.1. Moreover, there are two types of
<
127
interesting deformations of the free Gaussian functor; the one is the q-functor due to Bozejko and Speicher [12] and the other is the free “CAR” functor due to Shlyakhtenko [50]. The latter is introduced in Subsection 5.2 and the former is in Subsection 5.3.
5.1 Interpolated Free Group Factors Let M be a type 111 factor with the tracial state T and B(3t) be a type I , factor with the usual trace Tr. For each t E (0,m) choose a projection p E M BB(31) with ( ~ @ T r ) ( p=) t and define a type I 4 factor M t := p(M 8 B(3t))pwhose isomorphism class is determined independently of the choice of p . Since ( M t , ) t ,S M t l t z (t1,tz > 0 ) , the set { t E (0,m) : M t z M } forms a subgroup of the multiplicative group (0,m). This subgroup of (0,m) is called the fundamental group of M , which is of course an isomorphism invariant for type 111 factors. Riidulescu [44] and Dykema [20] independently constructed the interpolated free group factors L(F,) (1 < r 5 oo), a one-parameter family of type 111 factors having the following properties: (a) L(F,)E L(F,) if T = n E {2,3,. . . ,m}. (b) Compression formula: L(FT)t L(F1+(,-1)/t2) for all 1 < r 5 00 and O
L(F,) (1 < T 5
m) are all isomorphic and the fundamental group of
L(F,) is (0,m) for any 1 < r 5 00. (ii)
L(F,) (1 < r 5 00) are mutually non-isomorphic, and the fundamental group of L(F,)is (1) for any 1 < r < 00.
However, the (non-) isomorphism problem of L(F,) is still open. Very recently, Popa [41] succeeded in constructing a type 111 factor whose fundamental group is {l}, so the case (ii) above might be possible. The paper [19] says that free products of finite-dimensional algebras, the hyperfinite 111 factor and
128
interpolated free group factors are interpolated free group factors in most cases; the paper showed how to compute the parameter r (called the free dimension) of those interpolated free group factors.
5.2 Free Araki- Woods Factors Let 7 - l ~be a separable real Hilbert space and Ut a strongly continuous oneparameter group of orthogonal transformations on 'Uw. By linearity Ut extends to a one-parameter unitary group on the complexified Hilbert space 7 - t ~:= 7 - t ~ iR,. Write Ut = Ait with the generator A (a positive non) define an inner product singular operator, unbounded in general, on 7 - l ~ and (*,.)tJ on xc by
+
(z,y ) := ~ (2A(1+ A)-'z, 9) ,
Z,y
E 'Uc.
Let 7-l be the complex Hilbert space obtained by completing 'Uc with respect to (., .)u;then Ut extends on 3c by continuity. For each h E 3c the creation operator a*(h)and the annihilation operator a(h) on the full Fock space 3 ( ' U ) over 'U are defined as in Example 1.3, and set s(h) := a*(h) a(h). In [50], Shlyakhtenko introduced the C*-algebra r('Ua,Ut) := C * ( s ( h ): h E 'Uw) and the von Neumann algebra I'('Uflnr,Ut)''. The Hilbert space 'U being twisted from 'Uc in terms of the generator A , the vacuum state cpu := (52,. R) is no longer tracial on I'('Ua, UJ'. One can canonically extend Ut on 'U to a one-parameter unitary group (the so-called second quantization) F ( U t ) on F('U) by
+
F ( u t ) a : = R , F ( u t ) ( f l w - ~ j f , ) (vtfi)c+-~(utfnj. := Notice F(Ut)a*(h)&(Ut)*= a*(Uth)for h E 'U so that
F(Ut)s(h)F(Ut)* = s(Uth),
h E 'UR.
Thus, at := AdF(Ut) defines a strongly continuous one-parameter automorphism group on I'('UR, Ut)". Theorem 5.1 ([50]) (a) R is cyclic and separating for I?(?&, Ut)". (b) The vacuum state cpu on I?('Uw,Ut)'' satisfies the KMS condition with respect to at at /3 = 1.
Theorem 5.2 ([50]) I'('UHp, Ut) i s a simple C*-algebra whenever dim3Cw 1 3.
129
The C*-algebra I'(31R, Ut) with the state cpu is considered as a free analogue of the CAR algebra with a quasi-free state, and so cpu is sometimes called the free quasi-free state. On the other hand, the von Neumann algebra r(Xw, Ut)" is called a free Araki- Woods factor because it is indeed a factor (as stated in Theorem 5.4) and is considered as a free analogue of Araki-Woods factors. In particular, when Ut E 1, r(XR)'/ := l?(XR, U, 3 1)" is isomorphic as mentioned at the beginning of this secto the free group factor L(FdimXWin) tion. However, it is not known whether I'(XR) := r ( X R ,U, 1) is isomorphic to the reduced free group C*-algebra c,*(Fdim3Coc)or not. A fundamental example comes from a one-parameter rotation group on X R := R2:
u, :=
cos(t log A) - sin (t log A) sin(t log A) cos(t log A)
1
(with period 2n/ log A).
(30)
(b) (Tx,cpx) is a type IIIx factor whose core is L(F,) @ B ( X ) t i e . ,
Tx X,,VA T g L(F,)
@ B(X).
Shlyakhtenko classified free Araki-Woods factors r ( X R , Ut)" as follows: Theorem 5.4 ([50,51]) (a) r ( X R , Ut)" is a type IIIx factor zf Ut is periodic with period 2n/ log A (0 < x < 1).
(b) I'(XR, UJ' is a type 1111factor if Ut is not periodic. (c)
If Ut is almost periodic (i.e., the closed linear span
of the eigenvectors whose isomorphism class is determined by the subgroup of the multaplicative group (0,oo) generated by the point spectra of A. This subgroup of (0,oo) coincides with Connes' Sd invariant ([15]) of r('Ha,Ut)". of A is
X),then I'('Iiflw,Ut)'' is a full factor
(d) I n particular, for each 0 < A < 1 the above Tx gives a unique isomorphism class of type IIIx factors of the form r(3cR,Ut)".
130
In particular, the above theorem says that no type 1110 factor arises as I'(XR,Ut)''. The case U, : f t-) f(. + t ) on XR := L2(R;R)is a typically continuous example. It was shown in [51] that l?(Xa,Ut)" is a type 1111 factor whose core is C(iF,) €3 B(X),i.e.,
r(xR,UJ'
xCv R G L(F,)
€3 B ( X )
and the d u d action on the core coincides with a trace-scaling action constructed by W u l e s c u [43].
5.3 q-Deformed Araki- Woods Algebras
+
Let '?i~ be a real Hilbert space and 3cc := 3 c ~ i x ~the , complexification. For -1 < q < 1 the q-Fock space Fq(3cc)was introduced in [12,10] as follows. Let Pnite(X@) be the linear span of f 1 @ . ..€3 fn E (n = 0,1,. ..) where Xtl@""= Cs2 with vacuum $2. The sesquilinear form (., - ) q on pnite(31c) is given by
(h €3
* *.
€3 fn , 91 €3 . * . 63 g m ) q := L n
c
q""'(fl,gr(l))
. . . (fn, 97r(n)>>
iGL
where i(7r) denotes the number of inversions of the permutation T E 8,. For -1 < q < 1, ( . , - ) . q is strictly positive and the q-Fock space Fq('i4c)is the completion of Pnlte(Xc)with respect to (., . ) q . Given h E 3 c the ~ q-creation operator aG(h) and the q-annihilation operator aq(h)on Fq(X@) are defined bY
(h)(fl€3 . . . €3 fn) := h @ f l €3
af(h)R := h ,
* *
. €3 fn
3
aq(h)R:= 0, n
aq(h)(f1
:=Cq'-l(h,fi)fl@...€3fi-1
@.--@fn)
€3fi+l
€3*--@fn.
i= 1
The operators ai(h) and aq(h)are bounded operators on
.Fq(XC)
with
and they are the adjoins of each other. Furthermore, they satisfy the so-called q-commutation relation (see [lo]) a,(g)a;(h) - qa;(h)aq(g) = (9,h ) l >
9,h E XC.
131
In the limiting cases q = -1 and q = 1 this becomes the CAR on the Fermion Fock space and the CCR on the Boson Fock space. The free case q = 0 was given in Example 1.3. In this way, the q-deformed theory interpolates the Fermion ( q = - l ) , free ( q = 0) and Boson ( q = 1 ) cases. ~ define the C*-algebrarg(31Hp):= Set sq(h):= aG(h)+a,(h) for h E 3 1 and C"(s,(h) : h E 3 1 ~ )and the von Neumann algebra rq(31R)", which are the q-deformations of I'(3cflw) and r(3cR)" E L(IFd;, xR),respectively. Then: Theorem 5.5 ([13,10]) (a) The vacuum R is a cyclic and separating trace-vector for I',(UR)" so that T := (0,.0) is a faithful normal tracial state on r q ( 3 1 R ) f t . (b) I f - 1
< q < 1 and dim?& > 1 6 / ( 1 -
1q1)2, then r q ( 3 c R ) "
is not injective.
(c) If dim31R = 00, then rq(31fla)is a factor (of type I l l ) for aZZ -1
< q < 1.
Let (XR, U t ) be a real Hilbert space having a one-parameter group Ut of be the Hilbert space modified orthogonal transformations, and let (31, (. , from 3 1 = ~ U R i3cw as defined in the previous subsection. The q-creation a;(h), the q-annihilation a,(h) and sq(h) := aG(h) aq(h) for h E 3c are defined on the q-Fock space Fq(31), -1 < q < 1, as above (with 31 in place of XC).Then, combining Shlyakhtenko's construction and Boiejko and Speicher's construction we define the C*-algebra rq(31R,Ut) := C*(s,(h) : h E 3 c ~ )and the von Neumann algebra rq(3cR,Ut)" for -1 < q < 1. We call r q ( 3 c R , Ut)" a q-deformed Araki- Woods algebra. The following theorems go on the lines of Theorems 5.1, 5.5 and 5.4. Note that when Ut 1 or A = 1, the assertions (c) and (d) below reduce to (b) and (c) of Theorem 5.5. On the other hand, when q = 0, the assertions (d)-(f) below correspond to (a) and (b) of Theorem 5.4, though more decisive results such as (c) and (d) of Theorem 5.4 are not known in the q-deformed case. Theorem 5.6 ([25])
+
0
)
~
)
+
=
(a) The vacuum R is cyclic and separating for rq(3cR,Ut)" so that cp(= cp4,u):= (R,-R), is a faithful normal state on l?q(3cR,Ut)", called the q-quasi-free state. (b) Let .Fq(Ut) be the one-parameter unitary group on Fq(3c) (the second quantization of Ut, see the previous subsection). Then at := Ad.F,(Ut) defines a one-parameter automorphism group on Ut)" and the state cp on l?,(31Ip1Ut)" satisfies the KMS condition with respect to at atp=l. (c) Let EA be the spectral measure of the generator A (Ut = A i t ) . If there is
132
T E [1,00) such that
then r q ( 3 C R , Ut)" is not injective. In particular, jective if A has a continuous spectrum.
r 4 ( 3 C R , Ut)"
is not in-
(d) Assume that the almost periodic part of (31flp\,Ut) is infinite dimensional, that is, A has infinitely many mutually orthogonal eigenvectors. Then
( r q ( 3 t R , udft);n rq(RR, ut>"= a, where (r4(3Cflp\, Ut)'t)v is the centralizer of re(7-Ia,U,)"with respect to the vacuum state 9. In particular, r q ( 3 1 R , Ut)" is a factor. (e) Assume that A has infinitely many mutually orthogonal eigenvectors. Let G be the closed multiplicative subgroup of (0, .o) generated by the spectrum of A . Then r4(3Ca,Ut)" is a non-injective factor of type II1 or type IIIx (0 < X 5 1)) and
type II1 i f G = {l}, type IIIx if G = {A" : n E 9) (0 < X type 1111 i f G = B,.
< l),
(f) I f Ut has no eigenvectors, then r q ( 3 C R , Ut)" i s a type III1 factor. We end the subsection with new progress on q-deformed von Neumann algebras. The thesis [32] treated the Yang-Baxter deformation (more general than the q-deformation) of free group factors, and it was proved that the generated von Neumann algebra is a full type 111 factor if the underlying Hilbert space is infinite dimensional. On the other hand, it was proved in [38] that the generated von Neumann algebra in the Yang-Baxter deformation is not injective as soon as the underlying Hilbert space is at least two dimensional. A sufficient condition different from that in Theorem 5.6 (c) was given in [38] for the non-injectivity of a q-deformed Araki-Woods algebra.
5.4 The Case q = -1 It may be instructive to explain here what we get when the construction of the q-deformed Araki-Woods algebra r q ( 3 1 R , Ut)" is applied to the limit case q = -1. Let ('&, Ut),3 1 ~A,, (31, (., -)(I) be as in the previous subsection. Let
133
F-('H) be the Fermion Fock space (with vacuum 0) over 31, i.e., P-F(%) where P- is the orthogonal projection given by
F-(X)=
For each h E 31, the Fermion (CAR) creation and annihilation operators aE(h), a-(h) on F-(3c) are defined by
a*_(h)R: = h , ~*_(h)(fi A . - - A f n ) =:hAfi A - * . A f n ,
u - ( h ) 0 := 0 , n
a- (h)(fi A
.. . A fa) :=
-
( - 1)i - l (h,f;)ufi A . * A fi-
1
A fi+i A
. . . A fa .
i=l
We have Ila?(h)ll = Ila-(h)JI = llhllu and the CAR relations
+
=0, a-(h)a-(g) +a-(g)a-(h) = a*_(h)a?(g) af_(g)a*_(h) a-(h)a?(g)
+ a"g)a-(h)
= (h,g)ul
for all h, g E 31. Consider the von Neumann algebra r- (R,,Ut)" generated on F-(31) by s-(h) := aE(h) a-(h), h E 31,, and the vacuum state cp := (0;n)- on I'-(3cRI Ut)". Extend Ut on 3c to a one-parameter unitary group F-(Ut) on F-(31) by
+
F-(Ut)n:= 0 , F- (Ut)(fi
. . . A (Utfn) . A strongly continuous one-parameter automorphism group at := Ad F-(Ut) is defined on l?-(Xw, Ut)". Then we have: A
. . A fn) *
:= (Utf1) A
134
(a) fl is cyclic and separating for I?-(Xw, Ut)". (b) The state 'p on I?at at /3 = 1.
(Xw, Ut)" satisfies the KMS condition with respect
to
We determine the form of (I?- ('?f~,Ut)", 'p) in the following cases. Conse(XR, Ut)"is an Araki-Woods factor when (ZR, Ut) quently, we observe that is infinite dimensional and almost periodic. This is the reason why we call rq(XR,Ut)" a q-deformed Araki-Woods algebra. Case (i). Let (RR, Ut) = @:=l(Xw( k ) ,U,( k )) where Xf' := R2 and U j k )is as in (30) with A k E (0,1] in place of A. Let
so that {el"), e p ) } 1 5 k s nis an orthonormal basis of (X, (-, is 22n-dimensional and r-(XR,Ut)" is generated by xk
:= 5 1 (s-(fik')
a
)
~
)
.
Then,
.F-(X)
+is-(fik)))
Direct computations give
Set
Then it follows that { e $ ) } 1 5 i , j 5 2 (1 commuting 2 x 2 matrix units. Since
n
5 k 5 n) are n families of mutually
k-1 Zk
=
j=1
(eg)
- eg))eit),
1 5 IC
5 n,
135
we see that
Moreover, notice
where P(x:xl,. . . ,x:xn) is a polynomial of x ; q , . Hence it is obvious that p(eiljl (1) eiz (2) j2 . eLrin = o
. . ,z:x,
and # E (1, *}.
--
whenever i k # j k for some k. On the other hand, when i k = j k for all k, it is easy to see that p(e('! e!2! . . . e2!"! ) =KElK2---nn, Z l l l 2222 , 2 ,
where
Therefore, cp is the product state
Case
(ii). Let
(Xa,Ut)
=
@,id) @ @2=l(X,(k),Ut(k) )
where
$E=l(XR(k) ,Ut(k) ) is as in Case (i). Then, F-(X)is 21+2n-dimensional and I'-(?lR,Ut)'' is generated by xo := s - ( f o ) with f o := 1 in the first component P and Xk (1 5 k 5 n) as in (31). Write xo = el"' - e?) where e?),er) are orthogonal projections, and set {eij (k) }lsi,jj2 (1 5 k 5 n) as in (32) and (33). Since XoXk
-k
XkXo
=0,
z 0 . E
+ X;Zo
=0
(1 5 k
5 n),
136
F'rom these relations it is not difficult to see that ( 0 ) (1) . . .e.("1 2,3n : 1 I io,il,jl,. . . , i , , j , <_ 2, io il ..-+ i n is odd}, {eio eiljl {eio ( 0 )eiljl (1) . . . e("), fn3n : ~ ~ i o , i l , ..., j l ,i,,jn~2,io+il+...+iniseven}
+ +
,
are two disjoint 2" x 2" matrix units; so
C := C iO+il+...+i,
( 0 ) (1)
El :=
iO+il+...+i,
E2
("!
eio eilil. . .ein z n ' is odd
ei;)e!'! . . .e.anan ("1. 1121 is even
+
are central projections of I'-(Xa, Ut)" (El E2 = 1) and
r-(3cw, Ut)"El
2 M2n (C) ,
r-(& Ut)"E2 s M2n (c).
Therefore, it is seen that
r-(XR, UJ'
M2- (C) CB M p (C).
+
Moreover, since e p ) = ( 1 x 0 ) / 2 and e p ) = (1 - x 0 ) / 2 ,if follows that
Hence, we notice where $ is the product state given in (34). Case (iii). Let (7&,Ut) = @~=l(3cR( k ),U, ( k )) where 'Hc' := R2 and
Uik) is as above with Xk E (0,1]. Then, by (a) and Case (i), we see that (I?- (Xa, Ut)",'p) is nothing but
137
that is an Araki-Woods factor. In particular, I?-(3cR,Ut)'' is Powers' IIIx factor when X k = X E (0,l) for all Ic; it is the hyperfinite 111 factor when the Xk's are all 1. Case (iv). Finally, let (3cflw,Ut) = @,id) @ $&(3cc),Ut(k') where $El(3c$),U,(k)) is as in Case (iii). For n E N let M, be the subalgebra of I?-('HR,Ut)" generated by s(h), h E @,id) @ $i=l(3cf),U:k)). Then M, Z M p ( C ) @ M p ( C ) thanks to (a) and Case (ii). Furthermore, the argument in Case (ii) shows that the inclusion M , C M,+1 is given as
From this inclusion together with the form of cpl~,, determined in Case (ii) one can see that (r-(3cR, Vt)",cp)is the Araki-Woods factor (35)again. 5.5 q-Deformed Distributions This final subsection is not directly related to operator algebras, but we give a short summary on the distributions of q-deformed random variables on the q-Fock space. For a Hilbert space 3c and -1 < q < 1 let Fq(3C) be the q-Fock space, and for h E 3c with llhll = 1 define the q-creation operator ut(h) and the q-annihilation operator a,(h) as above. Then al;(h)f a,(h) is called a qGaussian random variable and its distribution (with respect to the vacuum state) is called the q-Gawsian distribution. This is supported on the interval [-2/fi, 2/,/=] with the density
(see [11,10] and [36]). When q = 0 this becomes the semicircle law, and the limit as q + 1 is the normal law. Moreover, the sequence of orthogonal polynomials with respect to the q-Gaussian distribution is the so-called qHermite polynomial sequence given by the recurrence relation
P,+l(Z) = zP,(.) For 0 < q
<
-
1 - qn
-P,-1(2)
1-q 1 and X > 0 the operator
z q ( h ,A) := (a;@)
(P-l(.) = 0 , Po(2) = l),
+ 6 1 ) (a,(h)+ 6 1 )
= a;(h)a,(h)
+ 6 ( a ; ( h )+a,@)) + A 1
138
is the q-Poisson random variable, which is a linear combination of a q-number operator, a q-Gaussian random variable and a scalar. In fact, ul(h)uq(h) deserves the name q-number operator since
u;(h)aq(h)h@"= [n],h@" where [nIqis the q-number, i.e.,
The distribution of xq(h,A) is called the q-Poisson distribution of the parameter A. It is known ([49]) that this distribution is the orthogonalizing probability measure for the sequence of polynomials given by the recurrence relation
Pn+1(2) = (2 - (A + [ n I q ) ) P n ( ~-)A[n]qPn-l(z) ( P _ l ( Z )= 0 , PO(2)= 1). The explicit forms of the support and the density of the q-Poisson distribution are in [48]. Note that the distribution tends as q 3 0 to the free Poisson distribution (or the Marchenko-Pastur distribution) introduced in (12). Also, the limit q + 1 gives the classical Poisson distribution
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QUANTUM WHITE NOISE CALCULUS UN GIG JI Department of Mathematics Research Institute of Mathematical Finance Chungbuk National University Cheongju, 361-763 Korea E-mail: [email protected] NOBUAKI OBATA Graduate School of Information Sciences Tohoku University Sendai, 980-8579Japan E-mail: obataOmath.is.tohoku.ac.jp Recent developments in quantum white noise calculus are outlined on the basis of various characterization theorems for operator symbols. A quantum stochastic integral and a quantum stochastic differential equation are generalized t o a white noise integral and a white noise differential equation, respectively. Unique existence of a solution is shown for a certain class of white noise differential equations. Their regularity properties are examined by means of operator symbols. Higher powers of quantum white noises, which are pathological objects in traditional theories, emerge naturally and are studied within the quantum white noise calculus. Cauchy problems associated with the infinite dimensional Laplacians are discussed by means of transformation groups on white noise functions.
1
Introduction
Since Hida [28] launched out the secalled white noise (Hida) calculus in 1975 quite a number of works covering many topics with various backgrounds have been published1) and it has become difficult to follow the whole in detail. Nevertheless, observing how the idea of white noise has broadened the spectrum of quantum stochastic analysis during the last decade since white noise encountered quantum probability, we are fairly certain that the “quantum aspect” of white noise calculus is creating a promising approach to nonlinear stochastic analysis. The purpose of this article is to outline the idea of quantum white noise calculus again from the beginning and to review the latest results on white noise differential equations. The term quantum white noise calculus being new, one may probably guess easily what it means and what it deals with. l ) More
than 500 relevant papers are listed in Kuo [52].
143
144
The origin traces back to operator theory on white noise functions developed in the early 1990s as is seen in the lecture notes by Obata [61] where the then obtained results are collected. Using more recent terminologies we shall explain our idea. Let us recall the celebrated Wiener-ItCSegal isomorphism:
L 2 ( E * , p )2 r(L2(R)),
(1)
where E* is the dual of the nuclear space E = S(R) and /I is the Gaussian measure on it; and r(L2(R))is the Boson Fock space over L 2 ( R ) .Here R is regarded as a time axis. Then we introduce a Gelfand triple (nuclear rigging)
W
c L 2 ( E * , p )E r(L2(R))C W*.
(2)
We adapt a CKS-space [18], which is explicit and sufficiently general among many variants discussed in literatures. The Brownian motion {Bt} is contained in the middle space; In other words, Bt is an L2-random variable. However, its time derivative called the white noise {Wt} is outside. The CKS-space is formulated so as to include {Wt} in W*. Tuning our attention to the Boson Fock space r(L2(R)),we notice that creation and annihilation operators play a fundamental role. According to standard understanding in quantum field theory the annihilation operator at and the creation operator a; at a point t E R are defined as (unbounded) operator-valued distributions. In contrast, within the CKS-space (2), at becomes a continuous operator on W and by duality, so does a; on W * . Moreover, regarded as multiplication operator acting on W , the white noise Wt is decomposed into a sum of them:
Wt = at
+ a,*.
(3)
Then, on the analogy of quantum field theory the creation and annihilation operators in (3) respectively creates and annihilates a virtual “noise particle,” and these actions are the origin of the white noise. The Wiener-ItCSegal isomorphism is an apt place where quantum aspect of white noise calculus emerges. The pair {at,a t } is called the quantum white noise. Elements in W and W* are called a white noise test function and a white noise distribution (or Hida distribution), respectively. An operator acting on white noise functions is generally called a white noise operator. Quantum white noise calculus aims at a systematic study of white noise functions and operators in terms of the quantum white noise. The paper is organized as follows; In Section 2 we review construction of a CKS-space with some technical modification. This space was introduced by Cochran-KueSengupta [181 as generalization of a Hida-Kubo-Takenaka space [47] and a Kondratiev-Streit space [43]. Further expressive refinement
145
has been made by Asai-KubwKuo [4] on which we rely too. It turns out recently that the scheme of infinite dimensional holomorphic functions developed by Gannoun-Hachaichi-Ouerdiane-Rezgui [22] is slightly more general and has independent applications. This topic is, however, somehow beyond our present purpose and will be discussed separately. In Section 3 we revisit the famous characterization theorems which are significant features of white noise theory. Following Ji-Obata [39] we prove characterization theorems for W * ,W , L(W,W * )and L ( W ,W ) in a unified manner. It was Potthoff-Streit [72] that first proved the characterization theorem for white noise distributions in terms of S-transform in the case of a Hida-Kubo-Takenaka space. Since then many relevant works appeared, among others, motivated by a CKS-space [18], Asai-Kubo-Kuo [4] tried to sort out minimum conditions in order to keep characterization theorem of the same type. The operator symbol, introduced by Berezin [6] as a fundamental concept in a study of Fock space operators, see also Kr6eRqczka [45], plays a role of the S-transform in white noise operator theory. The characterization theorem for operator symbols was first proved by Obata [60] and then the proof was simplified by Chung-Chung-Ji [8]. In Section 4 we discuss how basic concepts of (quantum) stochastic analysis are formulated in a CKS-space. Starting with a realization of Brownian motion in terms of a Gaussian space, we derive the fundamental identity (3). Then, following Obata [62] we introduce quantum stochastic processes and quantum stochastic integrals. In Section 5 we review an integral kernel operator, Fock expansion, Wick product and Wick exponential as central concepts in white noise operator theory, An integral kernel operator was first introduced by Hida-Obata-Saito [32] to study infinite dimensional Laplacians. Many interesting white noise operators are integral kernel operators. The Fock expansion theorem says that every white noise operator admits an infinite series expansion in terms of integral kernel operators. This was first proved by Obata [60] though similar results had been known in different contexts, see e.g., Berezin [6], Haag 1271. In Section 6 we introduce a Fock chain interpolating a CKS-space in order to discuss regularity properties of white noise operators. We give a characterization of white noise operators which map W into another Boson Fock space. Since nuclearity is broken, this characterization is in a completely different form and is based on Bargmann-Segal space. Our results are viewed as an operator version of the theorem of Grothaus-Kondratiev-Streit [26]. In Section 7 we discuss higher powers of quantum white noises which are beyond the traditional It6 theory. We shall observe how the difficulty of divergence occurs in compositions of such singular operators. Overcoming this
146
difficulty by renormdization Accardi-Lu-Volovich [3] introduced a renormalized quantum It6 formula by means of a formal algebraic computation. This formula is derived by Chung-Ji-Obata [13] by means of integral kernel operators with delta sequence kernels. A “regular part” of this formula was already noticed by Huang [34]. In Section 8 we formulate a general (not necessarily linear) white noise differential equation. Unique existence of a solution was first proved by JiObata [37] and a result on regularity of the solution was obtained by Ji-Obata [40]. Further study with concrete examples is an important problem. In Section 9 normal-ordered white noise differential equations (NOWNDEs) are discussed. These equations are first introduced by Obata [65] as a direct generalization of quantum stochastic differential equations of HudsonParthasarathy type [36]. Development in this line of study can be seen in a series of papers [14,15,66]. We shal! illustrate some latest results on regularity properties of solutions after Chung-Ji-Obata [16]. By classical reduction our technique covers, in principle, classical white noise equations developed by Holden-Bksendal-UbaeZhang [33], Kondratiev-Leukert-Streit [42], Kuo [51]. It is desirable to make an explicit connection with classical results. In Section 10 we discuss heat type equations associated with infinite dimensional Laplacians (the Gross Laplacian and the number operator) by means of transformation groups on white noise functions. This approach is motivated by the idea of infinite dimensional harmonic analysis, which traces back to Hida [28]. Quantum white noise calculus prompts a systematic study. Among others, motivated by KUO’SFourier-Mehler transform [48,49], Chung-Ji [9] first studied one-parameter transformation groups induced by the Fourier-Gauss transforms and obtained their characterizations. Generalization of the Fourier-Gauss transforms was also discussed systematically by Chung-Chung-Ji [7] and by Chung-Ji [ll]. Finally in Section 11 we discuss the LBvy Laplacian [56,57]. Many interesting results concerning the LCvy Laplacian have been recently investigated by many authors. We only mention a relationship with square of quantum white noises after Obata [68]. The square of quantum white noises and the LCvy Laplacian are known as pathological objects by tradition, however, we are now convinced that both of them emerge naturally and are found next to the traditional stochastic analysis. There are some topics that are closely related to our discussion but are omitted in this paper: for example, white noise approach to quantum martingales, chaotic expansion of a white noise operator, complex white noise and coherent state representation, unitarity of a solution, rotation groups and rotation-invariant operators, etc. In addition, a study of classical stochastic
147
processes by means of quantum white noise calculus is also interesting. 2
Standard CKS-Space
2.1 Standard Countable Hilbert Space We adapt the well known construction of a countable Hilbert space [5,23] for our study. Let H be a complex Hilbert space with norm 1. lo and T a selfadjoint operator with dense domain Dom ( T )c H such that inf Spec ( T )> 0. We note that T-’ becomes a bounded operator on H and the operator norm is given by IIIT-lII1 = (inf Spec (T))-’. Then, for each p >_ 0, the dense subspace D, z Dom(Tp) C H becomes a Hilbert space equipped with the norm IElp=lTPE1O,
EEDom(TP).
Furthermore, we define D-, to be the completion of H with respect to the norm 15 I-, = I T-PE lo, E H . Then we have
<
I E ,1 5 IIl~-lIllq-p I E lq ,
5 E D,,
--oo
< P L Q < 00,
and a chain of Hilbert spaces: * * -
c D, c
* - *
C
Do = H c . * * c D-, C ... ,
(4)
where each inclusion is continuous and has a dense image. Moreover, for any p , q E R the operator T P - q is naturally considered as an isometry from D p onto D,. From (4) we obtain a countable Hilbert space and its dual space:
D m ( T ) = projlimD,,
DZ,(T) = indlimD-,.
P-+W
P+m
Note that the inductive limit and the strong dual topologies coincide. The space D,(T) is nuclear if and only if there exists p > 0 such that T - P is of Hilbert-Schmidt type. 2.2
Boson Fock Space and Weighted Fock Space
Let H be a Hilbert space with norm 1 . I. For n 2 0 let HGn be the n-fold symmetric tensor power of H and their norms are denoted by the common ! ~put symbol I for simplicity. Given a positive sequence (Y = { ( ~ ( n ) } :we
148
Then F,(H) becomes a Hilbert space and is called a weighted Fock space with weight sequence a. The Boson Fock space is the special case of a(n)G 1 and denoted by I?( H). For two positive sequences a = {a(.)} and p = { p ( n ) } we write p 4 a if there exists a positive number C > 0 such that P(n) 5 Ca(n)for all n 2 0. With these notation, we easily see the following Lemma 2.1 Assume that a Hilbert space H2 is densely imbedded in another Hilbert space HI and the inclusion map H2 C ) HI is a contraction. Let a = { a ( n ) } and P = { P ( n ) } be two positive sequences such that 1 4 p 4 a. Then we have continuous inclusions with dense images: ra(H2)
- Fp(H2)
rp(H1).
Moreover, the second inclusion is a contraction. 2.3 Standard CKS-Space Let T be a selfadjoint operator densely defined in a Hilbert space H. When a standard CKS-space is concerned, we always assume (H) inf Spec ( T ) > 1 and T-' is of Hilbert-Schmidt type for some r
> 0.
Given such a selfadjoint operator T , let
Dm(T) = D, = proj lim D p P+m
be the standard countable Hilbert space constructed from T . Let a = { a ( n ) } be a weight sequence satisfying the following conditions:
( A l ) a(0) = 1 and there exists some n 2 1 such that inf a(n)nn> 0; n/O
{
= 0; n. (A3) a is equivalent2) to a positive sequence y = { y ( n ) } such that { T ( n ) / n ! } is log-concave3);
(A2) lim
n+m
(A4)
0:
is equivalent to another positive sequence y = { T ( n ) } such that
{ (n!y(n))-'} is log-concave. 2, We say that two sequences {a(n)},{-y(n)} of positive numbers are equivalent if there exist K ~ , K ~ , M I ,>M0 ~such that K i M ? a ( n ) 5 -y(n)5 KzM;a(n) for n = 0,1,2, ... 3, A positive sequence P(n) is called log-concave if B(n)P(n 2) 5 P(n 1)' for n = 0,1,2,....
+
+
149
Define weighted Fock spaces I'a (D p ) by
and consider their limit spaces:
I',(DW) = proj 1imra(Dp), P+m
r,(Dm)* = indlimr,(D,)* = indlimI'l,a(D-p). p+m
(5)
p+m
By Lemma 2.1 we have I'a(Dm)
c r ( H ) c ra(Dm)*,
which is referred to as a standard CKS-space. The canonical C-bilinear form on r,(D,)* x Fa(Dm) is denoted by ((., .)). It is easy to see that ra(Drn)is a nuclear space.
2.4 Examples Here are basic examples of a = {a(n)}satisfying conditions (Al)-(A4) in 82.3 and corresponding CKS-spaces: (1) A CKS-space corresponding to a(n) E 1 is called the Hida-KuboTakenaka space [47] and is denoted by ( E ) C r ( H ) C (E)*. (2) A CKS-space corresponding to a(n)= ( n ! ) p with 0 5 p < 1 is called the Kondratiev-Streit space [43] and is denoted by (E)p C r ( H ) C (E);. (3) The k-th order Bell numbers {Bk(n)}defined by k-times
satisfies condition (Al)-(A4). The corresponding CKS-spaces are so large that Poisson measures on E* are regarded as white noise distributions [18].
2.5 Generating Functions Let a = { a ( n ) }be a weight sequence satisfying (Al)-(A4). The generating functions defined by
150
are entire holomorphic on C by (Al) and (A2). Next we define
00
G l / a ( t )E
(n) {in!?}. Ct"n2na n!
n=O
ea(resp.
It is proved [4] that condition (A3) (resp. (A4)) holds if and only if ella) has a positive radius of convergence Ra > 0 (resp. R1/, > 0). 3
Characterization Theorems
3.1 S- Transform and Operator Symbol Let ra(D,) = ra(D,(T)) be a standard CKS-space, see 82.3. With each 5 E D, we associate an exponential vector or a coherent vector defined by
n! Since
we see that q5t E r a ( D , ) . Moreover, Lemma 3.1 The exponential vectors {& ; t E D,} are linearly independent and span a dense subspace of r a ( D W ) . Definition 3.2 [47] For 0 E r,(D,)* the function on D , defined by
S W ) = ((a,4E)) 7
5 E Dm,
is called the S-transform of 0. Definition 3.3 [6,45,60] For 5 E ,C(ra(D,),r,(Dw)*) the function on D, x D , defined by A
=(t,d= ((54E, 4,)) ,
t777 E D,,
is called the symbol of Z. It is important to characterize the S-transforms and the symbols as functions on D , and on D , x D,, respectively.
151
3.2 Unified Characterization Theorem Theorem 3.4 [39] Let r a ( D m ( S ) )and l?p(Dm(T))be two standard CKSspaces. For a continuous operator Z E L ( r a ( D m ( S ) )l?p(Dm(T))) , put
e(t,~ =) {(~+y), +hT))),
< E Dm(S>, v E om(T)*
(7)
c$Y)
where and q5hT) are exponential vectors in F a ( D , ( S ) )and r p ( D , ( T ) ) , respectively. Then, 0 satisfies the following two conditions: (i) 0 i s a Ghteaux-entire function4) on D m ( S ) x Dm(T);
(ii) for any p 2 0 there exist q 2 0 and C 2 0 such that 2
I@([, V > II ~ C G ~ EOI ; + ~ ) G I , ~ (v/ I - ~ ) ,
E E om(S), v E D ~ ( T ) -
Conversely, i f a C-valued function 0 defined on D m ( S ) x D m ( T ) fulfills conditions (a) and (ii), it is expressed as in (7) with a unique continuous operator E E L(r,(Dm(S)),rp(Dm(T))). In fact, since (ii) implies that 0 is locally bounded, a function satisfying (i) and (ii) is entire [19]. For the proof of Theorem 3.4 we need some lemmas. Lemma 3.5 Let F : D m ( T ) + C be a Ghteaux-entirefunction. Assume that there exist an entire function G on C and p E R such that
I F ( OII ~ G(I E I;), Then the n-th Ghteam derivative
E E Dm(T)-
becomes a continuous n-linear f o r m on D m ( T ) satisfying
for all s 2 0 such that T-" is of Hilbert-Schmidt type.
PROOF. Note first that z Taylor expansion is given by
I+
F(z<) is an entire function on C. The
m
F(~E =)C F n ( E , - - . , E ) z n i
n=O
E
E Dm(T),
*) A C-valued function F defined on X1 x . . . x En, where x k is a complex topological vector 26,. . ,&) is spaces for all k, is called Gdteauz-entire if the function z H F ( ( 1 , . .. ,( k entire holomorphic in z E C for any choice of (1 E XI,. . . ,tn E X, and E Xkr 1 5 k 5 n.
+
.
152
and the coefficient is estimated by Cauchy's integral formula:
Then, (9) follows by the polarization formula and by explicit calculation with Fourier expansion of F' in terms of eigenvectors of T . I Lemma 3.6 Let F : D,(T) + C be a Gdteaux-entire function. Assume that there exist constants C _> 0 and p E R such that
For each n 2 0 let Fn be the n-th Gdteaux derivative defined b y ( 8 ) . Put 9 = (Fn).Then we have
In particular, 9 E
(D,(2')) * .
PROOF.From the definition of norms and (9),
which proves the assertion.
I
We note that ga(IIT-" /I&) < 00 for all sufficiently large s > 0 as guaranteed by lims4m 11 T-"1 1 =~ 0. ~ By a similar argument as in the proof of Lemma 3.6 we have Lemma 3.7 Let F :Dm(T ) + C be a Gdteaux-entirefunction. Assume that there exist constants C 2 0 and p E R such that
For each n 2 0 let Fn be the n-th Giiteaux derivative defined by ( 8 ) . Put 9 = (Fn). Then we have
153
It is obvious from (11) that 9 E I',(Dw(T))*.Moreover, if (10) holds for any p 5 0 with some C 2 0, then (11) means that 9 E r,(D,(T)).
PROOF OF THEOREM 3.4. Fix q E D,(T) and we consider a Giiteauxentire function F,(<) = O(<,q), E D,(S). Then by Lemma 3.6 there exists 9, E r,(D,(S))* such that F,(<)= ((9,, 4 ~ ) and )
<
II 9, I12_(p+q+s),-
-
I CGa(II s-" IIis)Gl/p(I711 5 ~ ) -
(12)
Next, for a fixed 4 E r, (D, (S))we consider H+(q) = ((9, , 4)) , q E D, (T). We see from (12) that
+
Moreover, H+(q zq') is a compact uniform limit of a sequence of entire functions which are linear combinations of H+,(q zq') = 4 ~ ) )= O(E,q zq'), where runs over D,(S), and hence H+ is Giiteaux-entire. Applying Lemma 3.7 to H4 we find Q+ E rp(D,(T)) such that H+(q) = ((Q4, 47)) for 77 E k ( T ) and
+
+
<
((+,+zqt,
which proves that Z is continuous. This completes the proof.
I
3.3 Characterization of S-Transform In the previous subsection we have already proved the following Theorem 3.8 [4,18,43,72] A C-valued function F o n D,(T) transform of some 9 E l?,(D,(T))* i f and only if (Fl) F is a G6team-entire function o n D,(T); (F2) there exist constants C 2 0 and p E R such that
IF(<)I'
I CGa(I E I:),
EE~rn(~>.
I n that case, for all s 2 0 satisfying 11 T-"Ilks < R,, 2 II 9 ll-(p+"),- < - Cz.la(IIT-"I I & ~ ) *
is the S-
154
Theorem 3.9 [4,46,54]A C-valued function F o n D m ( T ) is the S-transform of some E r,(Dm(T)) if and only if ( F l ) and (F3) For all p 2 there exists a constant C 2 0 such that
ImI2I CG1/,(1 5 tP),5 E ~ m ( T ) . In that case, for all s 2 0 satisfying 11 T-"
II a Il;-",-
< RII,,
L CG/,(Il T-" 11s;).
The above two characterization theorems axe included in Theorem 3.4. In fact, Theorem 3.8 is immediately obtained from Theorem 3.4 with D w ( T ) = ( 0 ) . Similarly, Theorem 3.9 is obtained by setting Dm(S)= (0).
3.4
Characterization of Symbol
Theorem 3.10 [7,61] A C-valued function 0 defined on D m ( T ) x D m ( T ) is the symbol of an operator E E L(l?,(Dm(T)),I',(Dw(T))) i f and onZy i f
(01) 0 is a Giteaux-entire function o n D w ( T ) x Dw(T); (03) f o r any p 2 0, there exist constant numbers C 2 0 and q 2 0 such that 2
1@(5,77)12 L ~ ~ ~ ~ l ~ l ; + q ~ ~ l / 5~7 7 ~7 E1Dm(T>. 771~p~~ I n this case, f o r each r, s 2 0 satisfying 11 T-' we have
< R, and 11 T-" Ilks < R1/,
II ~4 II:-~,+ I c ~ , ( I I ~ - ' I I ~ ~ ) ~ l , , ( I I ~ - " I I ~ ~ ) I I ~ I I ~ +4qE+rr ,a+(, ~ m ( ~ ) ) PROOF.This follows from Theorem 3.4 with Dw(S) = Dw(T).
I
Theorem 3.11 [13,60,61] A C-valued function 0 defined on D m ( T ) x D m ( T ) is the symbol of an operator E C(r,(D,(T)),r,(D,(T))*) if and only i f
(01) 0 is a Giteauz-entire function on D m ( T ) x Dw(T ); ( 0 2 ) there exist constant numbers C 2 0 and p 2 0 such that
5,77E L ( T ) .
l@(E,77)I2 I cG~(l5l;)Ga(l77l;), In that case, f o r all q 2 0 satisfying 2 II -=$ II-(p+q),-
11 T-q
< R, we have
2 I c a l l rq 11s;) II $ Ilp+g,+
7
4 E ra(Dm(T)).
155
PROOF.Let H be the Hilbert space where T acts. There is a canonical isomorphism r ( H @ H ) r ( H ) @ r ( H ) uniquely determined by the correspondence of exponential vectors q+b,, H q+ @ 4,,. By using the fact that there exist constants C1, C2 > 0 such that for any n,m = 0,1,2,. . *
+
a(n)a(m)I CF+ma(n m ) , a(n + m) 5 C~+"a(n)a(m),
(14) which respectively follows from (A3) and (A4) [4], we see that the isomorphism r ( H @ H ) S r ( H ) @ r ( H ) induces a topological isomorphism r a ( D r n ( T@) Dm(T)) ra(Drn(T)) ra(Dm(T))We next note that an isomorphism H @ C 2S H @ H induces a topological isomorphism D, (T) @C2 S D, (T) @Dm ( T ) .This isomorphism is given by a basis { v I , v ~ }of C2.Then, between functions on D,(T)@C2 D m ( T @ I )C H @ C 2and those on D m ( T ) x Dm(T) there is a one-tc-one correspondence determined by @(G@'1+52@)2)= F ( G , 5 2 ) , <1,<2 ED,(T). It follows from Hartogs' theorem of holomorphy that 0 is Giiteaux-entire on D m ( T @I ) if and only if so is F on D m ( T ) x Dm(T). Moreover, if there exist C 2 0 and p E R such that
/@(d12
f c Dm(T @ I ) , I CGa(I {I:), then there exist C' 3 0 and q 2 0 such that 5 1 , G E Dm(T)lF(t1,J2)12I C'Ga(lG l;+,)Ga(152 Similarly, if then there exist C 2 0 and p E R such that
1~(<11<2)1~ ICG~(IG I ~ G ~ I:(), I E ~51,1 E Drn(T), then there exist C' 2 0 and q 2 0 such that
i ' Dm(T ~ @ 1).
l@(d12 I C ' G ~ f(l Ii + q ) ,
The above equivalence is a simple consequence of
+
Ga(s)Ga(t) I Ga(Cl(s + t ) ) , Ga(s t ) I Ga(C2s)Ga(Gt), s , t 2 0 , which follows from (14). The statement of Theorem 3.4 remains valid if D m ( S ) and Dm(T) are replaced with D m ( T )8 C2 and { 0 } , respectively. On the other hand,
Thus, in this case, Theorem 3.4 is reduced ot our assertion.
156
R e m a r k 3.12 In a concrete computation we wish to have a practical form of the generating functions G,(t) and Gl/,(t). For the Hida-Kubo-Takenaka space (E) the generating function is simple: G l ( t )= e t . For the KondratievStreit space ( E ) p constructed from a(n) = ( n ! ) p with 0 < P < 1, no concise form of the generating function is known; However, in the conditions of characterization theorems G,(t) and Gl/,(t) can be replaced with g p ( t ) = exp((1- P)tl/(l-p)}and with g-p(t) = exp((1 + P ) t ’ / ( l + P ) } , respect i ~ e l y . ~See ) [4] for more details. R e m a r k 3.13 Characterization theorem for the S-transform was also discussed within the framework of infinite dimensional holomorphic functions by Lee [55] in case of Hida-Kubo-Takenaka space. The latest achievement by Gannoun-Hachaichi-Ouerdiane-Rezgui [22] is slightly more general than the framework proposed in [4]and seems more suitable for a study of white noise operators. This line of research is now in progress [39,41]. 4
4.1
White Noise Approach to Quantum Stochastic Analysis
Gaussian Space and Brownian Motion
We take the complex Hilbert space H = L2(R), where R stands for the time axis, and a selfadjoint operator
d2
A = 1+ t2 - -
dt2 * There exists an orthonormal basis ( e k } & ,of H such that Aek = (2k 2)ek for k = 0 , 1 , 2 , . . . . In fact, ek(t) being a Hermite function, we may assume that ek E HR. Obviously, inf Spec ( A ) = 2 and
+
1
“
1
k=O
<m,
q>
-.21
The standard countable Hilbert space Dm(A)obtained from A coincides with a well known space:
+
D m ( A ) = S(R) iS(R), where S ( R ) is the space of rapidly decreasing R-valued functions. For simplicity we write
N = Dm(A),
E = S(R).
5 , Two functions f ( t ) and g ( t ) are called equivalent if there exist constants u1, u2, b l , b2 > 0 such that u l f ( b 1 t )5 g ( t ) azf ( b2t)for all t 2 0. Then G , and G I / , are equivalent to gp and 9 - 0 , respectively.
<
157
Note that E* = S'(R) is the space of tempered distributions. We thus have real and complex Gelfand triples:
E = S(R) C H R = L k ( R ) C E* = S'(R)
(15)
N = E + i E C H = H R + i H R = L2(R) c N * = E* +iE*.
(16)
The canonical C-bilinear form on N* x N is denoted by -) and the norm 4) for E H . of H by 1 . lo. It should be noted that I 4 ;1 = The Gelfand triple (15) is suitable for infinite dimensional measure theory. By the Bochner-Minlos-Yamasaki theorem (see e.g., [29,77]), for each u > 0 there exists a probability measure plrz on E* uniquely specified by
(t,
<
(a,
where E* is equipped with the u-field generated by the cylinder sets.6) For simplicity we write p = p1, which is called the (standard) Gaussian measure. The probability space ( E * , p )is called the Gaussian space. Lemma 4.1 For E E we set Xc(x) = (x, 4). Then the distribution of the random variable X,, i.e., the image measure of p by x I-+ X,(x) E R is the Gaussian measure with mean 0 and variance ISl;. Moreover, f o r any pair < , q E E we have
<
E(XeX,) =
s,.
(x, <) (2, 17) Adz) = / p ( t ) dt = 677) .
PROOF.The first assertion follows from (17). For the second assertion we note first that X, and X, are independent if (4, q) = 0. Then, E(XcX,) is calculated for general <,q E E by means of orthogonalization. I Let L 2 ( E * , p )(resp. L & ( E * , p ) )be the Hilbert space of C-valued (resp. R-valued) square integrable functions on E* with respect to p. It follows from Lemma 4.1 that I-+ Xc is extended to isometric embeddings from H R into L & ( E * , p ) and from H into L 2 ( E * , p ) .For t 2 0 let l p t l be the indicator function of the interval [O,t].Then, 1pt] E H R and we define with the above embedding
<
&(x) = Xlro,t,(4 = (2,l [ O , t ] ) ,
2
E E*, t
L 0.
(18)
Then {Bt ; t 2 0) c L 2 ( E * , p )and satisfies
&(x) = 0,
E(Bt) = 0,
E(B,Bt) = min{s,t},
s,t
0,
6 , Namely, the smallest a-field such that I H (5,c) is measurable for all [ E E. This a-field is also generated by the open subsets of E', or by the weakly open subsets.
158 that is, { B t }is a (realization of) Brownian motion. The white noise {Wt} is formally defined as the time derivative of Brownian motion: d Wt(.) = -& W z ) = (2,&)
7
(19)
where the last expression is apparently ill-defined. In fact, as we see in the next subsection, a CKS-space gives a rigorous meaning of white noise. 4.2
White Noise Distributions
We first recall the celebrated Wiener-It6-Segal isomorphism between L 2 ( E * , p )and r ( H ) . Theorem 4.2 There exists a unitary isomorphism between L2( E * p , ) and the Boson Fock space r ( H ) , which is uniquely determined by the correspondence:
<
where runs over N . For an arbitrary (fn) E r ( H ) the corresponding function in L 2 ( E * , p )is expressed in the form: m
n=O
-
where : x8": is the Wick tensor product, for details see e.g., [31,51,61]. In particular, for the Brownian motion Bt we have
and hence for the white noise, Wt(z)
(O,l[O,t],O,.
. .I,
(0,&,0,...).
(21)
(22)
Obviously, ( O , & , O , . ..) is outside of r ( H ) ,as is expected from (19). The space corresponding to Pa ( N ) through the Wiener-ItGSegal isomorphism is also denoted by the same symbol but sometimes is denoted by W for brevity. Thus we come to the complex Gelfand triple:
w = r,(N)
c r(H) z L ~ ( E * c , ~r)a p ) * = w*,
which is called a (concrete) CKS-space [18].
(23)
159
Definition 4.3 Elements in W and W* are called a white noise test function and a white noise distribution, respectively. An element in L(W,W * )is called a white noise ~ p e r a t o r . ~ ) Proposition 4.4 Let { B t } be the Brownian motion defined in (18). Then the map t c) Bt E W* is infinitely many times differentiable in W * .
PROOF. By (21) and (22) it is sufficient to prove that t I+ bt E E* is infinitely many times differentiable. But this follows from the fundamental theorem of calculus. (The norm estimates are also known, see [64].) I Thus the time derivative of Brownian motion:
is defined in W * and t c) Wt E W* is also infinitely many times differentiable. In that case, {Wt} is called the white noise process. A simple model of a particle moving in an environment with friction and random force is given by m-dv = -yv
+ wt,
dt where v = v ( t ) is the velocity of the particle, m the mass and y > 0 a friction coefficient. (24) is a Langevin equation in the simplest form. Since t I+ Wt is a smooth map, there is no difficulty of using an elementary calculus to obtain the solution:
However, these are justified only in W*. The integral in (25) is understood as a white noise extension of a standard stochastic integral.
4.3 Quantum White Noise For t E R and 4 E W we put
We can check that the limit always exists and that at E L ( W , W ) and L(W*,W * ) .The pair {at,a;} is referred to as the quantum white noise
a; E
7, For two locally convex spaces X, 9 the space of all continuous linear operators from X into 9 is denoted by L ( X , Q ) and is equipped with the topology of uniform convergence on every bounded subset.
160
process. In quantum field theory at and a,* are called the annihilation operator and creation operator at a point t E R. By a straightforward computation we have at$€ = E(t)4€,
E E N.
Moreover,
at : ( 0 , . .. , o , < @ * , o , ...)
I-+
(o,,. . , ~ , n < ( t ) < @ ( * -.'.I,) , ~ , .
a: : (0,. .. , o , E @ ~ , o.,.) . I-+ (0,. . .,o,StS<@",o,.. . ). We also notice that both maps t differentiable.
I+
at and t
I-+
a; are infinitely many times
4.4 Stochastic Processes In general, a time-parametrized family of random variables is called a (classical) stochastic process. In the traditional quantum stochastic analysis [36,58,69], a time-parametrized family of operators acting in a Fock space is called a quantum stochastic process. Both (classical) white noise process { W t } and quantum white noise process { a t ,a,*}are outside of these usual definitions. Nevertheless, if we employ the framework of white noise calculus, as we have seen before, both { W t } and {at,a,*}behave in quite a nice manner. So we propose to use Definition 4.5 [62] A continuous map t I-+ Ot E W* defined on an interval is called a classical stochastic process. Similarly, a continuous map t I-+ Zt E C(W,W * )defined on an interval is called a quantum stochastic process. A criterion of the continuity of t I+ Zt E C(W,W*)in terms of operator symbols is known [37,66]. Moreover, more precise continuity property in C(W,Gp), which will be introduced in $6, is obtained [40]. Here we mention a simple example of a quantum stochastic process found in application (quantum dissipative systems). Given constant numbers y > 0 and w E R,we set - e-(iw+y)t
e-ist
bt=-flL
i(w-s)+y
a, ds.
Then bt E C(W,W ) . Moreover, =t I
- e-(iw+Y)tI + bt
satisfies a quantum Langevin equation:
dZ dt
- = -(iw
+ y)Z -
e--ista, ds, =
R
I
= I.
161
Note that during the above computation we require only usual calculus but the values vary over the white noise operators.
4.5
Quantum-Classical Correspondence
For 4, .1c, E W we write 4.1c,for the pointwise multiplication. Lemma 4.6 The pointwise multiplication yields a continuous bilinear map from W x W into W . PROOF. For 4 = (f,,) and .1c, = (gn) we write 4$ = (h,,). Then an explicit expression of h, is known (see e.g., [61]) and a direct estimate of 11 41) [Ip, + is possible. Another proof is based on the characterization theorem for symbols. I For E W* and 4 E W we define a4 = E W* by .1c, E W . (($49 $)) = ((a, 4+)) 7 Obviously, the map (@,4)I+ a4 is a separately continuous bilinear map. In particular, each a E W* gives rise to a multiplication operator Ma E L(W,W * )defined by M a 4 = a4. With this we have a continuous injection W* + L(W,W*).Note also that (Ma)* = Ma. Theorem 4.7 Every classical stochastic process {at}c W* corresponds to a quantum stochastic process by multiplication. Conversely, for a quantum stochastic process {Zt} C L ( W ,W * )a classical stochastic process is associated by at = Zt4o. In particular, Theorem 4.8 Regarded as a multiplication operator, the white noise process Wt is decomposed into a sum of quantum white noises:
w, = at +a;.
(26)
PROOF.By simple calculation we see that (26) is valid on the exponential vectors which span a dense subspace of W . Then the proof is completed by noting that both sides of (26) are continuous operators. I
4.6 Quantum Stochastic Integrals For a quantum stochastic process { L t } c L ( W ,W * )the integral t
Zt =
L,ds,
t E R,
(27)
is defined in a usual manner, e.g., through the canonical bilinear form and becomes a quantum stochastic process.
162
Theorem 4.9 [37,40] Let { L t } and {at}be two quantum stochastic processes related as in (27). Then the map t I+ E L(W,W*)is differentiable and
holds in L(W,W * ) . The annihilation process, the creation process, and the number (gauge) process, which play a fundamental role in the Hudson-Parthasarathy calculus [36,58,69], are quantum stochastic processes also in our sense and are expressed respectively as
1 t
At =
a, ds,
A; =
t
l
a: ds,
At =
a:a,ds.
(29)
Note that the Brownian motion is decomposed into a sum of the annihilation and creation processes:
Bt = At + A ; ,
(30)
where the left hand side is regarded as multiplication operator. Apparently, (30) is an integral form of (26). As is easily verified, if { L , } is a quantum stochastic process, so are both {Ltat} and { a t L t } . Therefore, for a quantum stochastic process { L t } the integrals
I’
a:L, ds, ds, are meaningful in L ( W ,W * )and become quantum stochastic processes. These axe called the quantum stochastic integrals against the annihilationprocess and against the creation process, respectively. The latter is also called the quantum Hitsuda-Skorokhod integral. The classical Hitsuda-Skorokhod integral is defined for {at} C W* by t
!Pt =
a:@.,ds,
see e.g., [31]. By the quantum-classical correspondence we have a quantum Hitsuda-Skorokhod integral:
These two integrals axe related in an obvious manner:
!Pt
= Ztq5o.
163
5
White Noise Operator Theory
5.1 Integral Kernel Operators and Fock Expansion
For K E (N@('+"))*we shall define an integral kernel operator formally expressed in the form: Z',rn(K>
=/
K.(Sl,...
,Sl,tl,.*.
RI+m
x a:,
,tm)
at,-.*at,,,dsl
To be precise we need the right m-contraction of N@("+") defined by
**.dsidtl 0
R
.
.
dt,.
(31)
E (N@('+"))*and f E
where e,- = ei, 8-.-@eim,e,- = ej, @. . .@ej,, e z = e k l 8 . .- @ e k , , are respectively a complete orthonormal basis of H@", H@' and H@". It is noted that nBrn is regarded as a continuous operator from NBrninto ( N @ l ) *determined by
Obviously, the map K. I+ K.@, induces the canonical isomorphism (N@('+"))*E C(N@", ( N @ ' ) * )which , is guaranteed by the kernel theorem. With these notation we define E',,(K)$ = (gn) by
Theorem 5.1 [13] For any n E (N@('+"))* the integral kernel operator El,m(K.) is always a white noise operator, that is, E l , m ( ~E) C(W,W*).Moreover, Z~,,(K)€ C ( W ,W ) i f and only if K € N@' @ (N@'")*. Integral kernel operators play a fundamental role in white noise operator theory as is suggested by the following Theorem 5.2 [13] For any 6 E C(W,W*)there exists a unique family of distributions ~ 1 E (N@('+m))&,(l,rn) , ~ such that
l,m=O
where the right hand side converges in C(W,W * ) .If 3 E C(W,W ) , so does Z',,(K~,,) for all 1, m and the series (33) converges in C ( W ,W ) .
Expansion (33) is referred to as Fock expansion [61].
164
5.2 Wick Product and Wack Exponential For two white noise operators Z I , E ~L ( W ,W * )there exists a unique operator 5 E L(W,W * )such that A
A
z(<,rl)= z 1 ( < , ~ ) ~ 2 ( < , r l ) e - ( € ~<,rl ~ ) ,E N .
(34)
.
In fact, it is not difficult to check that O(<,q) = Z l ( < , 7 7 ) ~ 2 ( < , r l ) e - ( € ”satis1) fies (01)and (02) in Theorem 3.11. The operator Z defined in (34) is called the Wick product of E l and Z 2 , and is denoted by S = El o E 2 . We note some simple properties:
-
lOz=sol==,
-
(=I 0 = 2 ) * =
- - =- -
-
(z10=2)053 = z l O ( z 2 0 = 3 ) ,
z; o q ,
=1 o z 2
=2 o z 1 .
As for the annihilation and creation operators we have a, oat = asat,
a: o a t = a:at,
a, oa,* = at+as, a: .a,*
= .:a,*.
More generally, it holds that
-
a:, .--a:l=at, - - - a t , = (a:, -..a:,at,
...at,)
oS,
E E L(W,W*). (35)
Obviously, equipped with the Wick product, L ( W ,W * )becomes a commutative *-algebra. However, it is not closed under the Wick exponential defined by
where Zon = Z o . . . o 2 (n times) and Zoo= I. In fact, wexp Z is defined in mot her CKS-space. Consider two weight sequences a = {a(.)} and p = {P(n)} satisfying conditions (Al)-(A4), the generating functions of which are related in such a way that
G d t ) = exp Y{Go(t) - 11,
(37)
where y > 0 is a certain constant.*) In that case, we have continuous inclusions:
w0c W , c r(H) E L ~ ( E * , , uc) W: c w; 8)In fact, for our purpose it is sufficient to assume that Gp(t) and e x p T ( G , ( t ) equivalent in the sense of footnote 5 ) .
- 1) are
165
and hence
L(Wa1W:)
c L(W/3,W;).
The relation given as in (37) is abstracted from the case of Bell numbers which satisfies the recurrence formula:
GBell(l)(t)= eti GBell(k+l)(t)= exp 'Yk {GBell(h)(t)- I} 1 k 3 where y k + 1 = exp ^/k for k 2 1 and y1 = 1. Theorem 5.3 Let a and p be two weight sequences satisfying (Al)-(A4) and assume that their generating functions are related as in (37). Then for any B E L(WalWz) the Wick exponential wexp Z defined in (36) converges in .C(Wl Recall that every Z E L(W,W*)admits a Fock expansion as in (33). Then the degree of E is defined by
q,.
degZ = max{Z
+ r n ; z l , m ( ~ l#, n0)a5)
00.
For the Hida-Kubo-Takenaka and Kondratiev-Streit spaces we have Theorem 5.4 For any E E L ( ( E ) ,( E ) * )with degS 5 2/(1 - p), the Wick , In particular, if degz 5 2, exponential wexpZ converges in L ( ( E ) p (B);). the Wick ezponential wexp Z converges in L ( ( E ) ,( E ) * ) . 6
Fock Chain Interpolating CKS-Space
6.1 Fock Chain The space of white noise operators L(W,W * ) is very wide and include L(W,W ) L(W*,W*) and L ( r ( H ) r, ( H ) ) as subspaces. From the structure of a CKS-space we see that M
L ( W ,W * ) =
(JL ( W ,W - p ) p=o
and characterization theorems discussed in $3 reflect the above stratification. However, this structure sends no message about the possibility of a white noise operator becomes an operator acting on another Boson Fock space which might be singular to the original one r ( H ) . Given a CKS-space W c r ( H ) c W * , we shall introduce another chain of Boson Fock spaces interpolating the CKS-space in such a way that
w c 0,
= projlimGp c
... c G p c
- - a
c r(H)
L2(~*,,u),
P+W
F(H)
L 2 ( E * , p )c
- - - c G-p c ... c indlimG-, P+W
= GL
c W*.
(38)
166
Let T be a selfadjoint operator in H such that inf Spec (T) >_ 1. Note that we do not assume condition (H) in $2.3 which is essential for a CKS-space. As usual, a Hilbert space D, is defined from the norm 1 = I T p C lo and their limit spaces by
D,
= projlimD, C H C
D L = indlimD-,. P+,
P+,
To make a connection with N
c H c N * we assume
(i) N is densely and continuously imbedded in Dp for any p 2 0; (ii) E (the real part of N ) is invariant under T. For p E R let Q, = l?(D,) be the Boson Fock space over D,. By definition, the norm of Q, is given by
c 00
II 4 Il&
=
n! I fn
4 = (fn) E 4 ,
fn
E Df".
(39)
n=O
We define Qm
= projlimQ, c r ( H ) c G& = indlimQ-,, P+,
(40)
P+,
where Qm becomes a countable Hilbert space equipped with the Hilbertian norms defined in (39). In general, Q, is not a nuclear space. Thus we obtain an interpolation of the given CKS-space as in (38), where the injections are continuous and with dense images. The canonical C-bilinear form on Q& x Gm is denoted by ((-, .)) again. The Schwartz inequality yields
I
4)) 1 5 11 a I l T , - p 11 + I l T , p ,
E Q&> 4
QW*
6.2 Bargmann-Segal Space Define a probability measure Y on N * = E* 4 d z ) = C11/2(W x P1/2(dY),
+ iE* in such a way that
z =2
+ iY,
Z,Y
E E*,
where p1/2is the Gaussian measure with variance 1/2, see (17). The probability space ( N * ,Y ) is called the complez Gawsian space [29]. Following [26]we define the Bargmann-Segal space, denoted by €2 ( u ) , to C such that be the space of entire functions g : H
167
where P is the set of all finite rank projections on HR with range contained in E. Note that P E P is naturally extended to a continuous operator from N * into H (in fact into N ) , which is denoted by the same symbol. For 4= E r ( H ) define
(fn)r=o
Since the right hand side converges uniformly on each bounded subset of H , J 4 becomes an entire function on H . Moreover, it is known [26] that J becomes a unitary isomorphism from r ( H ) onto E 2 ( v ) . In fact,
In particular, the Bargmann-Segal space E 2 ( v ) is a Hilbert space with norm 11. Ilez(v). The map J defined in (41) is called the duality transform and is related with the S-transform ($3.1) in an obvious manner: JdN
= s4,
4 E r(H)-
6.3 Characterization of S-Transform Theorem 6.1 [26] Let p E R. Then a C-valued function g on D, is the S-transform of some !D E 9, i f and only if g can be extended to a continuous function on D-, and g o TP E E2 (v). In this case,
We keep in mind that the countable Hilbert space 9, is not assumed to be nuclear and the argument in $3 is not applicable in this case.
6.4
Characterization of Symbols
It is remarkable that the symbol of a continuous (equivalently, bounded) operators from 9, into 9, is characterized by means of the Bargmann-Segal space. Theorem 6.2 [38] Let p , q E R. A C-valued function 0 on D, x D, is the symbol of some Z E L(GP,Gq)if and only if (i) 0 can be extended to an entire function on D, x D-,;
168
(ii) there exists a constant C 2 0 such that
for any k 2 1 and any choice of & E D, and ai E C , i = 1 , . .
+
,k.
Similar characterizations for L(G,, Gq),L(Gml G&) and L(Gm,G,) follow also from Theorem 6.2. Here we only record the following Corollary 6.3 A C-valued function 0 on D, x D , is the symbol of a bounded operator on r ( H ) if and only if (i) 0 can be extended to an entire function on H x H ; (ii) there exists a constant C 2 0 such that
for any k 2 1 and any choice of ti E H andai E C , i = 1 , - . . ,k.
A similar characterization of the symbol of an operator of HilbertSchmidt class is also obtained, see [38]. Finally we mention characterization of L(W,G,), p E R. Note first that the symbol of E L ( W ,G,) is extended to an entire function on N x D-,. Theorem 6.4 [40] Let p E R and let 0 a C-valued function defined on N x N . Then there exists E L(W,G,) such that 0 = E i f and only i f A
(i) 0 can be extended to an entire function on N x D-,; (ii) there exist q 2 0 and C 2 0 such that Il~(E,KP.)112EZ(”)
I CGa(l5l:)l
t
E N.
(43)
Corollary 6.5 ( 1 ) For any p 2 0, the map t I+ at E L(W,G,) is continuous. (2) If t I+ St E D-, is continuous with some p 2 0, so is t I+ at E C(W,G-,). Corollary 6.6 Let El,Z2 E L ( W ,G,). Then if there exist q 2 0 and C 2 0 such that
II %(<,
K P . ) ~ ~ ( ~ , K P - ) ~ - ( E ~ ~ll&(vl ~.)
o E2 E L(W,G,) i f and only
5 CGJI I:),
< E N.
169
7
Singular Quantum Stochastic Processes
7.1 Higher Powers of Quantum White Noises It is noticeable that quantum white noise calculus provides many different classes of very singular "noises" which can not be formulated within the traditional stochastic calculus. As we have seen in (29), the traditional calculus deals with only three different combinations of quantum white noises: at, a: and a;at. However, from the viewpoint of white noise calculus there is no reason to restrict ourselves to such "lower powers" of quantum white noises. Let us start with the following formal expression:
We note that { ~ ~E L~( Wa,W ~ * )is}a quantum stochastic p r o c e ~ s . ~ ) We first need to write (44) in terms of an integral kernel operator. For f E N * and n 2 2 we define ~ " ( fE) (N@*)*by
( T " ( f ) , ~ l @ . . . @ ~ " ) = ( f , ~ l . . , ~ , ,f)E, N * , 51,-.-,t" E N , where (1 . . -5" is a pointwise product. In short, T,( f ) is an n-variable distribution obtained by letting f be concentrated on the diagonal set in R". By definition TI(^) = f . Note that T,, : N * + (N@")*is continuous and its adjoint T: : N@" + N is defined by T:(& @-.-@&,) = (1-..&. For f E N * and 1,m 2 0 with 1 + m 2 1 we put ,-
Then we come to the following Proposition 7.1 [13] (1) Wl,m(f) E L ( W ,W * )for f E N * . Moreover, the map f I+ Wi,,,,(f)E L ( W ,W * ) is continuous and Wi,m E N @ C ( W ,W * ) . (2) Wqm(f) E L(W,W) for f E N * . Moreover, the map f I+ WO,,(f) E C(W,W ) is continuous and W0,, E N @ C(W,W ) . With each C-valued measurable function f on R we associate the multiplication operator M f . Let M ( N ,N * ) be the space of all such functions f such that M f E L ( N , N * ) and we identify it with a subspace of L ( N , N * ) through the injection f I+ M f . By definition, (72(f),
v @ < )= (MfS, V ),
E,v E N , f E M ( N , N * ) .
(46)
A different aspect was given to the integral (44) by Huang [34], where af'aFdt is studied under the name of quantum stochastic measure.
9,
170
Then M ( N , N ) = M ( N , N * )n L ( N , N ) coincides with the space of (3"functions of which the higher order derivatives are of polynomial growth,lO) see e.g., Chap. V of [73]. Proposition 7.2 [13] (1) Let f E M ( N , N * ) . Then, Wl,,(f) E L(W,W) if and only iff E M ( N , N ) . (2) Let 1 2 2 and f E M ( N , N ) . Then Wr,m(f)E L(W,W) occurs only when f = Q . We see from Propositions 7.1 and 7.2 that for any f E N * and g E M ( N ,N)the usual compositions of W O , ~and ( ~W1,m(g) ) belong to L ( W ,W ) . Here are some relations among those operators: [WO,m(fl), Wo,mt(f2)1 =0
[Wo,m(f), W1,mt(911= mWO,m+m'-l(fg) [Wl,m (91) W1,mr(g2)]= ( m - m')J+'1,m+m'-1(9192).
In connection with the Wick products we remark W0,m(f1)W0,m~(f2) = W0,m(f1) 0W
O ,(~ f 2)) (47) WO,m(f)Wl,m) (9)= Wo,m(f)0 Wl,m((g) + mWo,m+m~-l(fg) (48) Wl,m(91)Wl,mJ(92) = Wl,m(gl)0 Wl,m/(g2) + ~Wl,m+m~-1(9192).(49) For gE
f E N we have Wo,l(f) E L(W,W)n L(W*,W*);and hence, for any N*,[Wo,l(f), Wl,,,,(is g) well-defined ] in L ( W ,W * ) .In that case, [WO,l(f), Wl,m(9)1= W - l , m ( f g )
+ ZWl-l,m(fS).
W O , l ( f ) ~ l , m= M WO,l(f 0) Wr,m(g)
Finally we note that for
f1, f2
(50) (51)
E N*,
Wl,m(f1)W0,m((f2) = Wl,m(fi) oWo,ml(f2).
(52)
Formulae (51) and (52) give a meaning to the generalized quantum It6 formula obtained by Huang [34].
7.2 Composition Formula By Fock expansion (Theorem 5.2) the composition of two integral kernel operators is decomposed into a sum of integral kernel operators. We shall give an explicit formula. lo) A
c
measurablefunction f on R is called being of polynomial growth if there exist constants
2 0 and k 2 0 such that If(t)l 5 c(l + t 2 ) k for all t E R.
171
We need notation. For inner k-contraction by K,
Ok
x =~
K,
E N @ ( m + k and ) X E N@("+") we define the e i @ eg)(X, eg 8 ey)e;@ e,;
( I C ,
i,f,iE'
where e;, e3, eg are similarly defined as in (32). By definition IC 00 X = IC 8 A.
Proposition 7.3 For IC E N @ ( t + m and ) X E N@("+"') we have
where 1 ti-k
Sh-k,m'(ICOk A) =
c c(6,
et @ eF@ eg)
;,5,r7,?
k'
x (A, ez @ el; 8 e?)e;@ el; 8 e f @e?.
PROOF.The above formula (53) is well known (even at a formal level) [24,44] and the proof is a routine computation by repeated application of the canonical commutation relations. I 7.3 Renormalized Product We first note from Proposition 7.2 that if 1' 2 2, Wp,,r(g)+ 4 W in general even for a smooth g. Then there is no meaning of the composition: Wt,,,,(f)Wtt,ml (9). Roughly speaking, the ill-definedness is caused by higher powers of delta functions arising from successive application of the canonical = a( s - t ) . Nevertheless, it is possible to give a commutation relations [us,a,*] meaning to such an ill-defined composition by eliminating a certain divergent term. This procedure is called renormalisation and the resultant product will be called the renormalazed product. We propose to approximate ~l+,,,(f) E (N*('+m))* by sufficiently regular functions { f c } E > C ~ N@('+"). For example, fc is obtained by molifying ~ l + ~ ( for) from a partial sum of its Fourier series expansion. Then the composition Zt,m(fc)Ztt,m'( g r ) is well-defined but not converges to Wt,,(f)Wp,,t (9). So the renormalized product is defined as
[Wl,m (f)Wt',mt(g)]REN = ! z { z t , m (fc
)=t# ,mr(ge)
- zc},
(54)
172
where 2, is a diverging term. To specify 2, we recall the composition formula mentioned in Proposition 7.3:
x
1J'-k ~l+l'-k,m+m'-k(Sm-k,m'(fE
Ok
ge)).
(55)
The inner contraction fE Ok gE causes divergence. In fact, for k = 0,1, choosing suitable approximating sequences, one may prove the convergence. L e m m a 7.4 [13] Let f , g E N and m,n 2 0 . Take a n approximating sequence {fE} C N B m such that lim,+o 1 f, - ~ ~ (I-p f = ) 0 for some p 2 0. Let { g E } C N B n be a similar approximuting sequence converging t o r,,(g). Then, lim fE
e+O
00
ge = ~ ~ (8fT,(g) )
in (N@("+"))*.
L e m m a 7.5 [13] Let f , g E N and m , n 2 0. Take a n approximating sequence {fE}C N@("+') an such a way that for any p 2 0 there exists q 2 0 such that lim, +o I fc - ~ ~ + l Jl,m;p,--(p+g) ( f ) = 0. Let {gd} c N@("+l)be a similar approximating sequence converging t o T,+~ ( 9 ) . Then, lim fc
E+O
01
gc = Tm+n(fg)
in ( N @m( + n ) ) * .
On the other hand, due to higher powers of delta functions, the limit lim,,o f, O k gE does not exist in (N@(1+m+1'+m'-2k) ) * for k 2 2. Hence the correct term to be eliminated is
and the renormalized product is defined as in (54). Using an explicit expression, we come to the following Theorem 7.6 [13] Let f , g E N . Then, for l,m,Z',m' 2 0 we have [wl,m(f)Wlt,mt (g)]REN
+
= wi,m(f) 0 wit,m/(g) mlfW1+i'-i,m+m'-i(fg)-
(56) It is noted that (47), (48), (49),(51), (52) are included in formula (56) whereas no actual renormalization is needed. The renormalized quantum It6 formula due to Accardi-Lu-Volovich [3] is understood to be a symbolic notation for (56) in a differential form as following:
[dWL,mdWl',m']REN= ml'dWl+l'-l,m+m'-l
*
173
8
White Noise Differential Equations
8.1 General Solutions Let us consider the following white noise differential equation:
where F : [0,TI x L(W,W * )+ L(W,W * )is a continuous function and Zo is a white noise operator. A solution of (57) must be a C1-map defined on [0,TI with values in L(W,W * ) . Theorem 8.1 [37] Let cr = {cr(n)} and fi = { f i ( n ) } be two weight sequences satisfying conditions (Al)-(A4), and assume that their generating functions are related as in (37). Let F : [O,T]x L(Wp,W;) + L(Wp,W;) be a continuous function satisfying that there exist p 0 and a nonnegative function M E L'[O,T ] such that
>
(i) f o r all J,v E N ,
E L(Wp,W;), and s E [O,T]
=1,22
IPb,W L77) - R%=2)(5, dI2 A
I~ (ii) f o r all
~
~
~
~
a
~
-w l 54 ~ 12 l ;
~
J,v E N , Z E L(Wa,W:), and s E [O,T]
I%=)(5,v)l2
I M(s)Ga(I 5 I;)Ga(l
771;)u + 13&77)1'>.
Then, for any Zo E L(Wa,W:) the initial value problem (57)has a unique solution zt E L(Wp,W;), t E [O, TI. The proof is based on the Picard iteration adapted for the operator symbols and requires a tedius computation. A sharper and more practical statement is desirable. 8.2 Regularity of Solutions The solution mentioned in Theorem 8.1 is something like a distribution and is not grasped as an operator acting on a Hilbert space. We here give a sufficient condition for the solution Zt to be in L ( W ,&,). Theorem 8.2 [40] Let cr = {cr(n)} and ,L? = { f i ( n ) } be two weight sequences satisfying conditions (Al)-(A4), and assume that their generating functions are related as in (37). Let F : [O,T]x L(Wp,W;) + L(Wp,W;) be a continuous function and assume that there exist q 2 0 and a nonnegative function
~
174
Then, for any 20 E L(Wa,Gp)satisfying that there exist R 2 0 and q' 2 0 such that
I n!(RG,(lJI:!))"
Ilg(<, KP*)"~O(E, K'.)l:z(,)
, n=
.. ,
(59)
the initial value problem (57) has a unique solution St E L(Wp,G,), t E [0,TI. 9
Normal-Ordered White Noise Differential Equations
9.1 Passage from QSDE to NOWNDE
A typical quantum stochastic differential equation of It6 type [36,58,69] has the form: d X ( t ) = (LldAt
+ L2dAf + LsdAt + L4dt) X ( t ) ,
Xlt,o = I .
(60)
In fact, (60) is a short-hand notation for a stochastic integral equation of It6 type and is equivalent to
( ( X ( t > 4 €+TI)) , = ((4€,47J
+
where <,q run over a certain dense subset of L 2 ( R ) . We here suppose that <,q E N = S ( R ) iS(R).Since at& = <(t)&€, the integral in (61) becomes
+
175
Then by reversing the above argument and by using the smoothness of s a, E L ( W ,W ) we come to the differential equation:
I+
+
-dX ( t ) - L1X(t)% + LZa,*X(t) Lsa,*X(t)at+ L 4 X ( t ) , Xlt,o = I . (62)
dt Note that (62) is meaningful whenever t H X ( t ) E L ( W ,W * )is differentiable. Moreover, the right hand side is expressed by means of the Wick product. In fact, in view of (35) we see that (62) is equivalent to the following ordinary differential equation for white noise operators:
-d X ( t ) - (LlUt + L 2 4
+
+
Laa,*at L4) 0 X ( t ) , Xlt=o = I . dt More generally, it is natural to consider an equation of the form:
where { L t } is merely a quantum stochastic process, i.e., t F+ Lt E L(W,W * ) is continuous. Equation (63) is generally called a normal-ordered white noise differential equation (NO WNDE). Consequently, a quantum stochastic differential equation of It6 type (60) is brought into a normal-ordered white noise differential equation with coefficientsinvolving only lower powers of quantum white noises.
9.2
Unique Existence of a Solution as Wkite Noise Operators
Since the Wick product is commutative, a formal solution to (63) is given by the Wick exponential: E t = w e x p { l L , d s } =00~ $ ( l L , d s ) o n n=O
Applying Theorem 5.3, we come to the following Theorem 9.1 [16] Let cr and P be two weight sequences satisfying (A1)(A4) and assume that their generating functions are related as in (37). Let { L t } c L(Wa,W,t) be a quantum stochastic process, where t mlzs over [0,TI. Then (63) has a unique solution in L(Wp,W;) and is given by (64). Slightly more general, we have Theorem 9.2 Let cr and P be the same as in Theorem 9.1. Let { L t } ,{ M t } c L(Wa,WLf) be two quantum stochastic processes, where t runs over [0,TI. Then the initial value problem
d , =LtoZt+Mt, dt
-=t
-
==.lt=o = Eo
E L(Wa,WZ),
(65)
176
has a unique solution in C ( W p ,W i ) . Theorem 8.2 can be applied to look for the space L(Wp,Gp)where the solution to the equation (65) lies, see [37]. 9.3 Weighted Fock Spaces Interpolating CKS-Space Given a CKS-space W , c r ( H ) C W:, we consider an interpolating weighted Fock space. Let K* be Hilbert spaces with norms 1. and assume that the inclusions
If
N
c K+ c H c K - c N *
are all continuous, the inclusion K+ L) H is a contraction, and K* are dual each other with respect to H . For example, K+ = D, or N, has this property for all p 1 0. Let p = { @ ( n ) }be a positive sequence such that 1 4 /3 4 a. Then we have continuous inclusions with dense images:
W,
c rp(K+)c r ( H ) C rp-i(K-) c W:.
(66)
Moreover, rp(K+)and r p - l ( K - ) are dual each other with respect to r ( H ) . For simplicity the norms of rpi(K*) are denoted by 11 Ilk, namely,
-
n=O
We then ask how the intermediate space r p - l ( K - ) in (66) is characterized. For this question we have only a sufficient condition, which is, nevertheless, useful for our discussion later. Theorem 9.3 [16] Let p = { p ( n ) } be a positive sequence satisfying conditions (Al)-(A3). Let F be a G6teaux-entire function on N . If there exist C 2 0 and a bounded, non-negative sesquilinear form Q on K+ with TrQ < Rp such that
IF(0l2I CGp(Q(&C)),
< E N7
then there exists a unique @ E r p - 1 ( K - ) such that F = S@. Moreover, in that case,
II @ II? I CGp(nQ). Theorem 9.4 [16] Let (Y = { a ( n ) } and p = {p(n)} be two sequences satisfying conditions (Al)-(A3). Let 0 be a Giiteaux-entire function on N x N .
177
Assume that there exist C 2 0 , p 2 0 and a bounded, non-negative sesquilinear f o r m Q o n K+ with TrQ < Rp such that l@(Et~)l2 5 CGa(1EI~)Gp(Q(~t~))7E , 71E N -
Then there exists a unique E E L(Wa,rp-l( K - ) ) such that 0 = E. Moreover, in that case, for any q > 1/2 with 11 A-q Ilks < R, we have h
II Ed II? I CGa(IIA-q Il&s)GpWQ>II d l;+,+
.
Theorem 9.3 is proved with a simple modification of the usual argument for characterization theorems for S-transform (see $3) and then is used to prove Theorem 9.4. 9.4 Regularity Properties in Terms of Weighted Fock Spaces
Let us go back to the initial value problem (63) and discuss regularity properties of its solution. Consider the Fock expansion (Theorem 5.2) of the coefficient: W
Lt =
C Z,m(Xl,m(t>>-
(67)
l,m=O
Since t
I+ Lt
E L(W,W*)is continuous, SO are t I+ Al,m(t) E (N@('+"))*and
t
I+
t
q m ( t )=
Xr,(s)
ds E (N@('+"))*.
Recall that the (formal) solution of (63) is given by
Then, regularity properties of St is described in terms of q m ( t )instead of h,m(t)Theorem 9.5 [16] Assume that Lt as given by 1
n
l=O m=O
such that
=
/Q,m(t)
I'
X',,(S)
ds E
(K--)@l c3 (N@")*.
Then, the unique solution to (63) lies in L ( ( E ) I'(K-)) , i f 0 5 n 5 1; and in L((E)p,r ( K - ) ) with P = 1- l / n if n 2 1.
178
Next we state a result in the case when Lt involves a higher order creation part. For 1,m 2 0 let K+ be the space of K. E ( K - ) @8 [ (N@")* for which of K + , a non-negative sequence there exist a complete orthonormal basis {a,} E L1, and constant numbers C 2 0, p 2 0 such that
{cn}
I ( K 8 m ( ~ * , ~ ~ l 8 . . . 8 ICai;**ai,I
Lt =
n
C C si,m(A,,(t))
"t
with Kt,,,,(t)
k 0 m=O
L-
Xt,m(S)
ds E
G,m,
then the unique solution to (63) lies in L ( ( E ) pr, p - ' ( K - ) ) , where 0 I /3 < 1is chosen in such a way that max{k+l, k+n, 2n} = 2/(1-/3). Here p(n) = ( n ! ) P . A more concrete example where Ic = 2 will be discussed in 511.3. On the other hand, some results are known for the case where the coefficient { L t } is an infinite series of integral kernel operators, see [16,41]. 10
10.1
Transformation Groups and Cauchy Problems Gross Laplacian
Recall the canonical isomorphism ( N 8 N ) * E L ( N ,N * ) based on the kernel theorem. For L E L ( N ,N * ) we denote by TL E ( N 8 N ) * the corresponding kernel. We call TL the L-trace. In particular, for the identity operator I E L ( N ,N ) we write r = TI E N 8 N * , which is called the trace. Obviously, m
k=l
With each L E L ( N ,N ) E N 8 N' we associate an integral kernel operator
which is called the L-Gross Laplacian [ll]. It is known (Theorem 5.1) that A G ( L )E L ( W ,W ) . In particular,
,.
is called the Gross Laplacian [25]. These are second order differentialoperators acting on white noise functions.
179
10.2 Number Operator The number operator is defined by
r ( s ,t)a:atdsdt = It is known that N belongs to L(W,W ) as well as .L(W*,W * ) . A similar generalization as the L-Gross Laplacian is also possible and is more common to write
,.
This is called the differential second quantization of L. 10.3 Transformation Group Following [7,11] we shall introduce transformations @,,B and their duals 5 , , ~ acting on white noise functions, where K E ( N B m ) *rn , 2 0, and B E L ( N ,N ) . The @,,B-t?-anSfOrm is defined by We can easily show that the function @ ( < , q )= ((@,,B$E, &)) satisfies the condition (03) in Theorem 3.10. Therefore (69) is enough to define a continuous operator @,,B E L ( W ,W ) . Define
56,B =
a, ,@ :
Then 5 , , E ~ L(W*,W * )and is called the S,,B-transform. Remark 10.1 6 , , is~a generalization of Fourier-Gauss transform [51]. For 8 E R put ~ ( 0 = ) (i/2)eiesin87- and B(8) = eieI. Then Se = $,(e),B(e) is is the the Kuo's Fourier-Mehler transform [30,49] and, in particular, 5--rr/2 Kuo 's Fourier transform [48]. The actions of @,,B and S,,B are given explicitly as follows. Theorem 10.2 (1) For @ = (F,) E W* the n-th component of 5 , , ~ @ is given by
(2) For 4 = ( f n ) E W the n-th component of 6,,& is given by
180
en,l= e"o.m(n). Hence,
@O,B =
From (71) we see that
I'(B), the second quantization of B , and
@Zn,B = r(B) o
Moreover, for any
K , K'
e'o-m(n).
(72)
E (N@"),*,, and B, B' E L ( N ,N ) we have @ ,, , B # ~ ~=, @ B n + ( ~ * ) a m n ~ , ~ ~ ~ .
In particular, if B E G L ( N ) ,then
;a ,;
(73)
e j n 6, ~GL(W)l l)and
= @-((B-l).)amn,B-l.
Therefore, for m 2 0 fixed, 6 = ( B n , ~K; E ( N @ " ) * , B E G L ( N ) } is a subgroup of G L ( W ) . We assemble a few properties.
Theorem 10.3 [7,11] Let K E (N@m):ym, K' E (N@m')zym, B E G L ( N ) and L E L(N,N). (1) @jn,Bzo,m'(6')= =o,m.(((B-l)*)@m'K')6n,B. (2) ~f ( ~ @ ( ~ 8 - lL )* ) K = (YK, (Y E c and [B,L]= 0, then we have @K,B=l,lh)
(3) If
=
(=l,l(TL)
+ m a s O , m ( ( ( B - l ) * ) @ " K ) ) @n,B.
@ L * ) K= (YK, (Y E C with (Y # 0 and [B,L] = 0 , then we have
+
@ - l / ( m a ) n , B ( ~ O , m ( K ) E 1 , l (TL))= E1,l ( T L ) @ - l / ( m a ) n , B -
In particular, if ( I @ ( ~ - '8) L * ) K = 0, then
[ z ~ , ~ ( T L~) o ,
, ~ ( n=) 0. ]
Corollary 10.4 Let K E (E@m):ymand B E G L ( E ) . (1) @ n , B N = ( N m Z O , m ( ( ( B - l > * ) @ " K ) ) @jn,B. E c. (2) @ - l / ( m a ) n , B ( ~ O , r n ( K + . ) (YN) = aN@--l/(ma)n,B, for any 0 # The above relations are useful in some Cauchy problems involving the number operator. By taking the adjoint, we obtain immediately relations for X,,B-transforms.
+
Theorem 10.5 [7,11] Let K E (N@m):ym,B E G L ( N ) and L E L ( N ,N ) . (1) 5 n , B = m t , 0 ( ~ f= ) z~~,~((B*)@~'K~ for ) sany ~ , Btcf E ( N @ ~ ' ) : ~ , ; (2) If ( I @ ( m - l )8 L*)K= an, (Y E C and [B,L] = 0, then we have & C , B = l , l ( T L * )= ( E l , l ( T L * ) - m&n,o(4)
(3) If
8 L*)K= ( Y K , (Y E C with cr 1 * @m
5l/(ma)n,B(=o,m(((B11) The
1
5n,B;
# 0 and [B,L] = 0 , then we have
+ =l,l(TL*)) = =1,1(TL*)5l/(ma)n,B.
group of linear homeomorphisms on a locally convex space X is denoted by GL(X).
181
10.4 Infinitesimal Generators
A one-parameter subgroup {Re}ecR of G L ( N ) is said to be differentiable in 8 E R if there exists 2 E C ( N ,N ) such that
In this case Z is unique and is called the infinitesimal generator of a differentiable one-parameter subgroup { RB}@cR. Similar definitions are used for G L ( W )too. We have the following heredity of differentiable subgroup under the second quantization. Theorem 10.6 [7,63] Let L E L ( N , N ) . Then L is the infinitesimal generator of a differentiableone-parameter subgroup {Re}ecR c G L ( N ) if and only if &(L) is the infinitesimal generator of a differentiableone-parameter {r(Qe)lecR C G L ( W ) . ~~bgroup Theorem 10.7 [7,11] Let {Re}ecR c G L ( N ) be a differentiable oneparameter subgroup with the infinitesimal generator L € C ( N ,N ) . Assume that IE E (N@"):y, satisfy @ L*)K= 0. Then {C5jBn,Re}eER is a differentiable one-parameter subgroup of GL(W ) with the infinitesimal generator Z0,m I(. +W L ) . Theorem 10.8 [7,11] Let {Re}scR C G L ( N ) be a differentiable oneparameter subgroup with the infinitesimal generator L € L ( N ,N ) and let IE E (N@m):y, satisfy (I@(m-l) @ L*)K= ( Y K , a! E C with a! # 0. Then
e E R,
61/(,,)((R;)~m-l)n,ne>
is a differentiable one-parameter subgroup of G L ( W ) with the infinitesimal generator ZO, (6) &( L ) . Some interesting one-parameter subgroups are obtained by specializing parameters, see also [9]:
+
generator
one-parameter group Wiener group
eeAG
eeN
Ornstein-Uhlenbeck group
group of dialations
N iN
Fourier-Wiener group Fourier-Mehler group
AG
eiON
,ON
e$ie'e sin B A G
e~(e28-l)Aa
+AG . + iN AG+N
182
10.5 Cauchy Problems
Let Z E L(W,W) be the infinitesimal generator of a differentiable oneparameter subgroup {Re}eER of G L ( W ) and let # € W be given. Then the unique solution of the Cauchy problem
is given by u(f3,x)= Re$(z). The Cauchy problem associated with the Gross Laplacian traces back to Gross [25] and that with the number operator to Piech [70]. 11
LBvy Laplacian
11.1 Formulation The LCvy Laplacian [56,57] is defined to be the Cesko mean of the second derivatives:
A, = lim
N+w
62
1 n=l
e'
where (z1,22, .. .) is an infinite dimensional orthogonal coordinate. This Laplacian has been studied by many authors, see e.g., [20,51,71] and references cited therein, and is famous for some pathological properties. At the early stage Hida [28] brought the LBvy Laplacian into white noise calculus, and Kuo [50] formulated the LBvy Laplacian as an operator acting on white noise distributions as well as the Gross Laplacian and the Laplace-Beltrami operator (the number operator up to a constant factor). Since then the LBvy Laplacian has been studied within the white noise theory and many new results have come out, see e.g., [1,2,17,75]and references cited therein. In general, the LBvy Laplacian is defined depending on a finite time interval. For simplicity we take the unit interval [0,1]. We fix a countable subset C E S(R)satisfying the following properties:
{
{
(ii)
{
(iii)
{
183
Let F : E + C be a C2-function. Then the second derivative F"(<) E ( E C3 E)* is defined by
Then the LCvy Laplacian AL with respect to -
is defined by
N
whenever the limit exists. For a white noise function we define the LCvy Laplacian through the Stransform. For E W*the S-transform S@(<)= ((a, 45)) is an entire function on N = E +iE,the U v y Laplacian may act on Sa. If A,Sa(<) exists for all E E and is the S-transform of a certain white noise distribution P ! E W*, we define P! = A L ~that , is, ALSG = S A L ~Here . we use the same symbol AL to avoid notational complication.
<
11.&
Normal Functions
Consider a function F : E
F(<)=
Lk
-+ C of the form:
f ( t l , t Z , . . . , t k ) ~ ( t l ) " l. . - < ( t n ) n k d t l .**dtk,
(76)
where f E Lm(Rk) and nl,. . . ,nk are positive integers. A finite linear combination of such functions is called a normal function. It is easy to see that every normal function is the S-transform of some element in W*. Moreover, for a normal function the Lkvy Laplacian exists. In fact, for F(<) given by (76) we have
c k
ALF(<) =
nj(nj
- 1)
j=1
On the normal functions (and their completions with resoect to particular topologies) several properties of the LBvy Laplacian are derived. For example, a formula describing the LCvy Laplacian as a limit of the Gross Laplacian is obtained [17], see also [51].
184
11.3
Quadratic Quantum White Noise
We now focus on a normal-ordered white noise differential equation with quadratic quantum white noises as coefficients:
The unique solution is given by the Wick exponential: m
.
where
A proper Fock space where the solution lies is determined. Theorem 11.1 The unique solution { E t } to (77) Iies in L(W,r’(N-2)) and t I+Et is a C1-map therein. PROOF.By a simple computation we have
with an obvious estimate: @t(E,V)l
I ~ X {P I E 10
+ IS :1 + 177 I:} I exp ( 2 15 :I + 2 I r~ :1 }.
(81) We now need Theorem 9.4 with a particular choice of a = p = 1. Using the notation Q(q,V ) = 2 ( f j , q ) = 2 I 77 ;1 and the obvious inequality 1910
I 5 lo = I A-1/2A1’2Elo I lllA-1/2111I 5 l1/2 =
1
I E l1/2 ,
we see that (81) becomes
Igt(E,711 I ~ X {P I E 1:/2 + Q(v,7 7 ) ) Then the assertion follows from Theorem 9.4 provided Q is a bounded, nonnegative sesquilinear form defined on N2 such that TrQ < ( 2 e 2 ) - l . We shall prove this. Let { e k } g o be the usual orthonormal basis of H such that Aek = (2k+2)ek. Then, { f k E ( 2 I ~ + 2 ) - ~ e kbecomes } a complete orthonormal basis of N g , q 2 0. With this basis we have
185
<
where is Riemann’s zeta function. As is easily seen, there exists 3/2 < < 2 such that 2c(2q,)/4q* = 1/2e2. Then, for any q > q*, in particular for q = 2, Q becomes a bounded, non-negative sesquilinear form on Nq with TrQ < (2e2)-’. This is what we needed to show. I q*
11.4
Heat Equation Associated With the Ldvy Laplacian
Theorem 11.2 [68]Let {at} be the classical stochastic process associated with the quantum stochastic process determined by (77). Then 9t is an eigenfunction of the Livy Laplacian in such a way that AL9t =
{
0, tso; 2t9t, Ost 5 1; 29t, t 2 1.
(82)
PROOF.Since Gt = Zt40 by the quantum-classical correspondence, we have
where (80) is taken into account. Then by direct computation that
and hence by definition (75) we obtain
It is not difficult to show ALFt = 0, 2tFt and 2Ft for t 5 0, 0 5 t 5 1 and t 2 1, respectively. I Corollary 11.3 Let (Y E C and let {9’t}be the same as in Theorem 11.2. For any finite measure v on [0,1], the integral qt
=
6’
aseast v(ds)
belongs to r ( N - 2 ) and satisfies a heat type equation associated with the Ldvy Laplacian:
a CY at !l?t = ZALQt,
t E R.
(85)
186
PROOF.Since s e zsE L(W,r(N-2)) is continuous by Theorem 11.1, so is the map s I-+ as = ZS&, E r(N--2), and hence {as; 0 5 s 5 1) is a compact subset of r ( N - 2 ) . Therefore the right hand side of (84)is integrable and becomes a member of r ( N - 2 ) . The S-transform is given by F l
F1
where (83)is used. Then by the Lebesgue dominated convergence theorem we come to
In other words,
Applying the characterization theorem for S-transform to (86), we see that relation (87)is equivalent to the desired one (85). I For the Cauchy problem associated with LCvy Laplacian (85) with an initial value being in a particular space, the solution is obtained by means of a one-parameter group generated by the LCvy Laplacian, see [17]for details. Furthermore, associated stochastic processes are studied [75,76]. References 1. L. Accardi and V. I. Bogachev: The Ornstein-Uhlenbeck process associ-
ated with the Livy Laplacian and its Direchlet form, Prob. Math. Stat. 17 (1997),95-114. 2. L. Accardi, P. Gibilisco and I. V. Volovich: The Livy Laplacian and the Yang-Mills equations, Rend. Accad. Sci. Fis. Mat. Lincei 4 (1993), 201-206. 3. L. Accardi, Y.-G. Lu and I. Volovich: Non-linear extensions of classical and quantum stochastic calculus and essentially infinite dimensional analysis, in “Probability Towards 2000 (L. Accardi and C. C. Heyde, Eds.),” pp. 1-33, Lect. Notes in Stat. 128,Springer-Verlag, 1998. 4. N. Asai, I. Kubo and H.-H. Kuo: General characterization theorems and intrinsic topologies in white noise analysis, Hiroshima Math. J. 31
(2001),299-330. 5. Yu. M. Berezansky and Yu. G. Kondratiev: “Spectral Methods in Infinite-Dimensional Analysis,” Kluwer Academic, 1995.
187
6. F. A. Berezin: “The Method of Second Quantization,” Academic Press, 1966. 7. C. H. Chung, D. M. Chung, and U. C. Ji: One-parameter groups and cosine families of operators on white noise functions, J. Korean Math. SOC.37 (2000), 687-705. 8. D. M. Chung, T. S. Chung and U. C. Ji: A simple proof of analytic characterization theorem f o r operator symbols, Bull. Korean Math. SOC. 34 (1997), 421-436. 9. D. M. Chung and U. C. Ji: Transformation groups on white noise functionals and their applications, Appl. Math. Optim. 37 (1998), 205-223. 10. D. M. Chung and U. C. Ji: Some Cauchy problems in white noise analysis and associated semigroups of operators, Stochastic Anal. Appl. 17 (1999), 1-22. 11. D. M. Chung and U. C. Ji: Transforms o n white noise functionals with their applications to Cauchy problems, Nagoya Math. J. 147 (1997), 1-23. 12. D. M. Chung, U. C. Ji and N. Obata: Transformations on white noise functions associated with second order differential operators of diagonal type, Nagoya Math. J. 149 (1998), 173-192. 13. D. M. Chung, U. C. Ji and N. Obata: Higher powers of quantum white noises in terms of integral kernel operators, Infin. Dimen. And. Quantum Probab. Rel. Top. 1 (1998), 533-559. 14. D. M. Chung, U. C. Ji and N. Obata: Normal-ordered white noise differential equations I: Existence of solutions as Fock space operators, in “Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability (L. Accardi et al. Eds.),” pp. 115-135, Istituto Italian0 di Cuitura, Kyoto, 2000. 15. D. M. Chung, U. C. Ji and N. Obata: Normal-ordered white noise diferential equations 11: Regularity properties of solutions, in “Prob. Theory and Math. Stat. (B. Grigelionis et al. Eds.),” pp. 157-174, VSP/TEV Ltd., 1999. 16. D. M. Chung, U. C. Ji and N. Obata: Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), 241-272. 17. D. M. Chung, U. C. Ji and K. Saitb: Cauchy problems associated with the Livy Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 18. W. G. Cochran, H.-H. Kuo and A. Sengupta: A new class of white noise generalized functions, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998), 43-67.
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19. S. Dineen: “Complex Analysis on Infinite Dimensional Spaces,” SpringerVerlag, 1999. 20. M. N. Feller: Infinite-dimensional elliptic equations and operators of Le‘vvy type, Russian Math. Surveys 41 (1986), 119-170. 21. C. W. Gardiner: “Quantum Noise,” Springer-Verlag, 1991. 22. R. Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui: Un the‘oreme de dualite‘ entre espaces de fonctions holomorphes Ci croissance exponentielle, J. Funct. Anal. 171 (2000), 1-14. 23. I. M. Gelfand and N. Ya. Vilenkin: “Generalized Functions, Vo1.4,” Academic Press, 1964. 24. J. Glimm and A. Jaffe: Boson quantum field models, in “Collected Papers, Vol.1,” pp. 125-199, Birkhauser, 1985. 25. L. Gross: Potential theory on Halbert space, J . Funct. Anal. 1 (1967), 123-181. 26. M. Grothaus, Yu. G. Kondratiev and L. Streit: Complex Gaussian analysis and the Bargmann-Segal space, Methods of Funct. Anal. and Topology 3 (1997), 46-64. 27. R. Haag: On quantum field theories, Dan. Mat. Fys. Medd. 29 (1955), no. 12, 1-37. 28. T. Hida: “Analysis of Brownian F’unctionals,” Carleton Math. Lect. Notes no. 13, Carleton University, Ottawa, 1975. 29. T. Hida: “Brownian Motion,” Springer-Verlag, 1980. 30. T. Hida, H.-H. Kuo and N. Obata: Transformations for white noise functionals, J. Funct. Anal. 111 (1993), 259-277. 31. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise: An Infinite Dimensional Calculus,” Kluwer Academic Publishers, 1993. 32. T. Hida, N. Obata and K. Sait6: Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93. 33. H. Holden, B. Bksendal, J. Ub0e and T. Zhang: “Stochastic Partial Differential Equations,” Birkhauser, 1996. 34. Z.-Y. Huang: Quantum white noises - White noise approach to quantum stochastic calculus, Nagoya Math. J. 129 (1993), 23-42. 35. Z.-Y. Huang and S.-L. Luo: Wick calculus of generalized operators and its applications to quantum stochastic calculus, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998), 455466. 36. R. L. Hudson and K. R. Parthasarathy: Quantum I t 6 3 formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301-323. 37. U. C. Ji and N . Obata: Initial value problem for white noise operators and quantum stochastic processes, in “Infinite Dimensional Harmonic Analysis
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38. 39. 40. 41. 42. 43. 44.
45. 46. 47. 48. 49. 50. 51. 52.
53.
(H. Heyer, T. Hirai and N. Obata, Eds.),” pp.203-216, D.+M. Grabner, Tubingen, 2000. U. C. Ji and N. Obata: A Role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space, preprint, 2000. U. C. Ji and N. Obata: A unified characterization theorem in white noise theory, to appear in Infin. Dimen. Anal. Quantum Probab. Rel. Top. U. C. Ji and N. Obata: Segal-Bargmann transform of white noise operators and white noise diferential equations, RIMS Kokyuroku 1266 (2002), 59-81. U. C. Ji, N. Obata and H. Ouerdiane: Analytic characterization of generalized Fock space operators as two-variable entire functions with growth condition, to appear in Infin. Dimen. Anal. Quantum Probab. Rel. Top. Yu. G . Kondratiev, P. Leukert and L. Streit: Wick calculus in Gaussian analysis, Acta Appl. Math. 44 (1996), 264-294. Yu. G. Kondratiev and L. Streit: Spaces of white noise distributions: Constructions, descriptions, applications I, Rep. Math. Phys. 33 (1993), 341-366. P. Krde: La the‘orie des distributions en dimension quelconque et l’inte’gration stochastique, in “Stochastic Analysis and Related Topics (H. Korezlioglu and A. S. Ustunel eds.),” pp. 170-233, Lect. Notes in Math. Vol. 1316, Springer-Verlag, 1988. P. Krde and R. Rqczka: Kernels and symbols of operators in quantum field theory, Ann. Inst. Henri Poincarh Sect. A 28 (1978), 41-73. I. Kubo, H.-H. Kuo and A. Sengupta: White noise analysis on a new space of Hida distributions, Infin. Dimen. Anal. Quantum Probab. Rel. TOP. 2 (1999), 315-335. I. Kubo and S. Taken& Calculus on Gaussian white noise I, Proc. Japan Acad. 56A (1980), 376-380. H.-H. Kuo: O n Fourier transforms of generalized Brownian functionals, J. Multivariate Anal. 12 (1982), 415431. H.-H. Kuo: Fourier-Mehler transforms of generalized Brownian functionals, Proc. Japan Acad. 59A (1983), 312-314. H.-H. Kuo: On Laplacian operators of generalized Brownian functionals, in “Stochastic Processes and Applications (K.Ito and T.Hida, eds),” pp. 119-128, Lect. Notes in Math. Vol. 1203, Springer-Verlag, 1986. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. H.-H. Kuo: A quarter century of white noise theory, in “Quantum Information IV (T. Hida and K. Sait6, Eds.),” pp. 1-37, World Scientific, 2002. H.-H. Kuo, N. Obata and K. Sait6: Livvy Laplacian of generalized f i n c -
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tions on a nuclear space, J. Funct. Anal. 94 (1990), 74-92. 54. H.-H. Kuo, J. Potthoff and L. Streit: A characterization of white noise test functionals, Nagoya Math. J. 121 (1991), 185-194. 55. Y.-J. Lee: Analytic version of test functionals, Fourier transform and a characterization of measures in white noise calculus, J. Funct. Anal. 100 (1991), 359-380. 56. P. L6vy: “LeGons d’Analyse Fonctionnelle,” Gauthier-Villars, Paris, 1922. 57. P. LBvy: “Probkmes Concrete d’Analyse Functionalle,” GauthierVillars, Paris, 1951 58. P. A. Meyer: “Quantum Probability for Probabilists,” Lect. Notes in Math. 1538, Springer-Verlag, 1993. 59. N. Obata: Rotation-invariant operators o n white noise functionals, Math. Z.210 (1992), 69-89. 60. N. Obata: An analytic characterization of symbols of operators o n white noise functionals, J. Math. SOC.Japan 45 (1993), 421445. 61. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. 62. N. Obata: Generalized quantum stochastic processes o n Fock space, Publ. RIMS 31 (1995), 667-702. 63. N. Obata: Constructing one-parameter transformations o n white noise functions in terms of equicontinuous generators, Mh. Math. 124 (1997), 317-335. 64. N. Obata: Integral kernel operators o n Fock space - Generalizations and applications to quantum dynamics, Acta Appl. Math. 47 (1997), 49-77. 65. N. Obata: Quantum stochastic differential equations in terms of quantum white noise, Nonlinear Analysis, Theory, Methods and Applications 30 (1997), 279-290. 66. N. Obata: Wick product of white noise operators and quantum stochastic differential equations, J. Math. SOC.Japan. 51 (1999), 613-641. 67. N. Obata: Unitarity criterion in white noise calculus and nonexistence of unitary evolutions driven by higher powers of quantum white noises, in “Modelos Estochticos I1 (D. Hernhdez, J. A. L6pez-Mimbela and R. Quezada) ,” pp. 251-269, Aportaciones Matematicas Investigacih 16, Mexican Math. SOC.,2001. 68. N. Obata: Quadratic quantum white noises and Ldvy Laplacian, Nonlinear Analysis, Theory, Methods and Applications 47 (2001), 2437-2448. 69. K. R. Parthasarathy: “An Introduction to Quantum Stochastic Calculus,” Birkhauser, 1992. 70. M. A. Piech: Parabolic equations associated with the number operator,
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INTERACTING FOCK SPACES AND ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES LUIGI ACCARDI AND MARCOLINO NAHNI Centro Vato Volterra Universitd da Roma “Tor Vergata” E-mail: accardiOvolterra.mat.unirornal.it Web Page: http://volterra.rnat.unarorna&.it We extend to polynomials in several variables the Accardi-Bozejko canonical isomorphism between one-mode interacting Fock spaces and orthogonal polynomials in one variable. This gives a constructive rule to write down easily the quantum decomposition, as a sum of creation, annihilation and number operators, of an arbitrary vector valued random variable with moments of any order. In the multimode case not all interacting Fock spaces are canonically isomorphic t o spaces of orthogonal polynomials. We characterize those which enjoy this property in terms of a sequence of quadratic commutation relations among finite dimensional matrices.
1
Introduction
In the past years the theory of interacting Fock spaces has been used in a multiplicity of different contexts (cf. [5-7, 9-11]). In many of these papers the natural correspondence between interacting Fock spaces and orthogonal polynomials played a relevant role. This correspondence was proved, for one-mode interacting Fock spaces by Accardi and Bozejko who showed that the theory of one-mode interacting Fock spaces is canonically isomorphic to the theory of orthogonal polynomials in one variable, i.e. with respect to a probability measure on the real line with finite moments of any order. The canonical feature of this isomorphism is exhibited by the fact that it maps the multiplication operator by the independent variable into a linear combination of the creation, annihilation and number operators of the corresponding interacting Fock space. The problem to extend this isomorphism to polynomials in several variables has been open for a few years. In the present paper we discuss a solution of this problem. The main new feature with respect to the one-mode case is that, in the multi-mode case, not all interacting Fock spaces are canonically isomorphic to spaces of orthogonal polynomials. We characterize those which enjoy this property in terms of a sequence of quadratic commutation relations among finite dimensional matrices. This gives in particular a constructive rule to write down easily the quantum decomposition, as a sum of creation, anni-
192
193
hilation and number operators, of an arbitrary vector valued random variable with moments of any order. Let d E N. We denote p a probability measure on Rd with finite moments of any order; z = (21, . .. ,Z d ) any element of Rd;xj denote the coordinates . . .,X,“) : (Rd,p ) + Rd the Rd-valued in the canonical basis; X o = (X,”, coordinate random variable, characterized by: j = 1 , . ..,d. X ~ ” ( Z=xi, )
Definition 1.1 The complex *-algebra P = P d , with identity, generated (algebraically) by the X ; ( j = l,.. . ,d) and the constant functions with pointwise addition and multiplication and f*(z)= f(z)-complex conjugate) is called the polynomial algebra in d indeterminates. In the following, the dimension d will be fixed, so we will frequently omit the super-script d and write, for example, P instead of P d . The p-integral defines a state on P and the G N S construction, applied to the pair {P,p } [17] gives a Hilbert space X,,a representation
n, : P
+ B(31,)
(the bounded linear operators on 31,) and a unit vector l,, cyclic for x(P). 31, is a closed sub-space of L2(Rd,p ) and coincides with it if and only if the measure p is uniquely determined by its moments. Thus the space 31, gives in some sense a measure of the non uniqueness of the moment problem for p.
Definition 1.2 We will say that p is generic if (i) the union of the coordinate hyperplanes measure zero;
xj
= 0 (j = 1,.. . ,d) has p-
(ii) x, is injective (notice that this implies that also the map p E P x,(p) .1, is injective).
+
This is surely the case if n, has a density with respect to the Lebesgue measure. In the generic case the *-algebra
P, := n(P)
(1)
is isomorphic to P and also the elements of P, will be called polynomials. They are bounded operators on 7c, if and only if p has compact support. Define the coordinate multiplication operators
Xj”f ( ~ := ) ~
j ,= j f ( ~ )
1,. . .,d
194
and their action on
Up: 7r
j = 1, . . . ,d.
( Xj" ) 13-1, =: X j ,
(2)
Condition (i) above implies that the operators X j ( j = 1,.. . ,d) are invertible in Up.In the following we will identify the elements of U p to elements of L2(Rd,p ) so that, for each j = 1,.. . ,d, X j acts by multiplication on U :
X j f ( 4 =z j m , f E 3c (3) and defines a symmetric pre-closed operator on 31 on the dense invariant domain Pp .1,. The algebra P, is generated by the monomials X;l . . .X F d ,
V n l , . . . ,n d E N
(4)
and the vectors
X,"l . . . X? . 1,
(5)
are total in U pby construction. In the following, when no confusion can arise, we shall denote with the same symbol both the vectors ( 5 ) and the corresponding multiplication operators (4)in P,. Definition 1.3 For each n E N define d
pi:=linearspanof
{
x ; ~ . . . x F ; ~ ~ E N ,c n j
Sn}.
j=1
It is clear that P," is the vector space of all polynomials in the variables X I ,.. . ,x d of degree at most n, where the degree of the monomial X,", . . .X? is C,",, nj and the degree of a polynomial P is defined to be the highest, among the degrees of the monomials which appear in P with a non zero coefficient. Notice that, for each j = 1 , . . .,d,
xj P:
c_ P:+l.
Define inductively
Lemma 1.4 For each n E N , n
P,". 1, = @v$ k=O
195
I n particular:
v:+,
=
(6vt)
I
n linear span of
k=O d
nj
E N (j= 1 , . . . ,d),
j=1
J
I
PROOF.Clear.
+ . . .+
.
Cnj = n + 1}.
n d = n, denote :X,"l . . . X :" : the orthogonal projection of in V k = P,-l .1,. By linearity the symbol : p , ( X ) : is defined for any polynomial p , of degree n. The vectors :XT' ... X T : span V,, but in general they are not orthogonal. Lemma 1.5 (Linear independence) For any j k e d n, the vectors : X y l .. . X F : with C djnj = n are linearly independent. In particular, f o r n > 0, V," 5 Crt with
For nl
X;'
.. . X i d . 1 ,
ek<,
r::= and the vectors :X?'
(n + d - 1 )
. ..Xdn, : are a basis
(9)
of V,".
PROOF.Fix n E N. The genericity of p implies that the map p , e : p , : is injective (p, polynomial of degree n). Therefore, for all nj E N with C; nj = n, the :XT1.. . X F : are linearly independent because, by the genericity condition, such are the Xrl . . . X i d .1,. Therefore the cardinality of the set
is
("+:-').
Since dimPf =
(":d)
[15], using the identity
we can conclude that d = dim V," = dim P,d - dim Pn-l
This proves the lemma, since the set {: Xyl . ..X:. :; nj E N, Cj"nj = n } generates V," . I
196
Remark Since the elements of any orthonormal basis of V, are in one-to-one correspondence with the solutions (in N) of the equation n1 + - - - 4-nd = n we will often use the notation IA) = Inl,. .
. ,nd)
(10)
to denote an arbitrary orthonormal basis of V,. When confusion can arise we will use the more explicit notation
46 = 47%l,...,7%d The recurrence relations among orthogonal polynomials are consequences of the following result: Theorem 1.6 For any n E N , if k 4 { n - 1, n, n + 1) then for each j = 1,.. . ,d,
xjvn I Vk. PROOF.Let IA) E V, and f E Pn-zl then for each j = 1 , . .. ,d, Xjf E P2-l, hence, by the symmetry of Xj and (6):
(XjA,f) = ( A , X j f )= 0 because by assumption (A) E V, I Pn-l.If f E Vk with k > n + 1, then f I X i ] A ) because X i IA) is a polynomial of degree n 1. This proves the statement. I
+
Denote P, the orthogonal projection on V,. The following is a multidimensional generalization of the Jacobi relations for 1-dimensional orthogonal polynomials. Corollary 1.7 (Recurrence relations) Let us fix,for any n E N,an orthonormaZbasis{JA)=Jnl, ...,n d ) ; Cd j n j = n } ofv,. F o r e a c h f i = ( n l , ...,nd) E N d with C: nj = n, and for each j = 1,.. .,d we have
XjPn = P,+lXjPn
+ P,XjP, + Pn_1XjP,.
(11)
PROOF.We know from Theorem 1.6 that for each j = 1,... ,d, n E N,
la) E v n , XjlA) E Vn+l@Vn @ Vn-1.
(12)
197
Since Pn+i + Pn + Pn-i is the orthogonal projection on Vn+i ® Vn ® Vn-\, (12) implies that Xj\n) = Pn+lXj\n) + PnXjln) + P^X^n), and, since \n) € Vn is arbitrary, this is equivalent to (11).
I
Now define the following operators: D+U) := P^XjPn k € B(Vn, Vn+1), D°n(j)
:= PnXjPn
k
(13)
€ B(Vn, Vn) = B(Vn),
(14)
D~(j) := Pn-iXjPn k 6 B(VntVn^).
(15)
Given any orthonormal basis (|n)) of Vn, we can write the finite dimensional operators (13), (14), (15) as matrices: ^n(j)\m)(n\,
(16)
|m|=|n|=n
E
DlA(j)\m)(n\,
(17)
0n,*0')|m>
(18)
|n|=n,|m|=n+l
: n, "»
E |n|=7i,|m|=n— 1
where, if n = (m, . . . ,n d ), we use the notation |n| = ni + • • • + n^. We note that -D+(j) is represented by an r^ x r*+l matrix; D°(j) is represented by an r^ x r^ matrix; D~(j) is represented by an r* x r^_ x matrix. Lemma 1.8 The operators (13), (14), (15) satisfy the following relations: (19)
i)
(20)
£»ti d)D- (j) + D°n(i) D°n(j) + D-+l(i) D+(j) = ^_iO')^ W + D°n(i)D°n(j) + D~+1(j)D+(i),
(21)
for i ^ j, 1 < i, j < d and n > 0, u/Aere £>li(i) = 0. PROOF.
We have Vm+1
Pn+1^Pn |V. Vn
.
(22)
198
Thus, exchanging i and j :
Iv,
Q+l(i)O,+(j)= Pn+2XiPn+lXjPn On the other hand, since X i X j = X j X i , it follows that
Pn+2XiXjPn = Pn+2XjXjPn. Using (11) we see that the left hand side of (24) is equal to
Pn+2 X i Pn+1 X j Pn + Pn+2X i Pn X j Pn + Pn+2X i Pn- 1 X j Pn and the right hand side to
Pn+2Xj Pn+1XiPn + Pn+2Xj PnXi Pn
+ Pn+2X j Pn-
1 X i Pn *
Since
Pn+2XiPnXjPn = Pn+,XjPn-,XiPn = Pn+2Xj P,Xi Pn = Pn+,XiPn-,XjPn this is equivalent to
= 0,
Pn+2 XiPn+1 X j Pn = Pn+2Xj Pn X i Pn . Therefore (22) and (23) are equal, which proves (19). In a similar way we observe that Pn+lXiXjPn = Pn+1XjXiPm is equivalent to
+
Pn+lXiPn+lxjPn + Pn+lXiPnXjPn Pn+1XiPn-lXjPn = Pn+1XjPn+1XiPn + Pn+1XjPnXiPn + Pn+lXjPn-1XiPn.
(25)
Since
Pn+1XiPn-1XjPn the relation (25) implies
Pn+lXjPn-1XiPn = O
OX+l(i)O,+(j)+ Wi)DX(j) = DX+l(j)O,+(i)+ q ( j > O : ( i ) which proves (20). To prove (21) we apply the recurrence relation to the two sides of the equality PnXiXjPn = PnXjXiPn and argue as above.
I
199
PROOF.
Lemma 1.10 If p is generic, then D,+(j) is injective f o r any n E N and j = 1,..., d. PROOF.If there exists A E V, such that Pn+lXjA = 0 then
X j A = PnXjA
+ Pn-IXjii. +
But the left hand side of this equality is a polynomial of degree n 1 and the right hand side is a polynomial of degree n. Because of the genericity of p they can be equal if and only if they are both zero. Also because of genericity X j is invertible in R,, hence xjfi=o
*
A=O.
It follows that Pn+lXjPn = D R ( j ) is injective.
I
Lemma 1.11 For each n E N, the famdy D$(l), . . . ,D$(d) : Vn + Vn+l is surjective, in the sense that the linear span of the ranges of D R ( j ) is the whole of Vn+l.
<
PROOF.Let 0 # E Vn+l be an element orthogonal to the range of all D$(j)’s ( j = 1,. ~.,d). Then for any fi E V, and for any j = 1, . . . ,d . 0 = (ti Dn+ (3)n) = ((9 Pn+lXjfi) = ( E l X j f i ) .
<
But, by definition of Vn,it also follows that IXjVk for any k 5 n - 1. By taking linear combinations we see that is orthogonal to all polynomials of the form Xjqn where qn E P,. But, taking linear combinations of these, we obtain all possible polynomials of degree n 1 without constant term. This contradicts the fact that 0 # E Vn+l. I
<
<
+
200
2
Interacting Fock Spaces
Let us recall the definition of interacting Fock space. We use here the definition introduced by [AcSk99] which, although equivalent to the original one of [3], [4], is more suitable for our purposes. Definition 2.1 Let 3-1: be a preHilbert space. An interacting Fock space based on 3-1: is an Hilbert space 3-1, with a gradation indexed by N
[email protected]=C.+ a nZ0
(27)
@7Cn n>l
and with the following property: For all v E X:, there exists a densely defined linear operator a+(v) on 3t such that: (i) The map v E 3cy I+ a+(v) is complex lineax. (ii) For all n E N the set {a+('&)
- - .a + ( Y ) @; 211,. . . ,V n E x!}
(28) is contained in the domain D(a*(v)) of a+(v). For fixed n E N, we will denote by Nn the linear subspace (algebraically) generated by the vectors (28)* (iii) The union of all the N, (n E N) is a dense subspace of 3-1. (iv) Each a+(.) has an adjoint a(v) defined on N := UnNn. Remark The direct sum in (27) is in the Hilbert space sense. We will denote algebraic 3-10
=
@ an,
3-10
=c-0
(29)
n20
the algebraic direct sum, i.e. the subspace of 3t of the vectors which have a non zero component only on a finite number of spaces Bn. Now let us fix the following choices: algebraic
3-1=3-1/&,
@ a/&,
3c?=Cd,
R,=V,,
n
where the operators a: are defined by the following: Proposition 2.2 In the notation (10) Iet, for n E N, lfi) = ln1, ...,nd) denote an arbitrary orthonormal basis of V, and let D&+(j) denote the matrices in this basis, of the operators D h ( j ) ( E = +, -, 0, n € N) as defined by (16) , (17) , (18). Then:
201
(1) For each j = 1 , . . ., d the operator a:, defined on the vectors lit) = Inl,...,nd> b y aj+ ~nl,..:,nd) = o , + ( A=I ~ ~ > ~ ; ~ + ( j ) I f i + ) (30)
C
Ifi+l=n+l
has an adjoint, also defined on the vectors In1 ,. . . ,nd) by
.;lo,.
. . ,0 ) = 0 ,
1
a;Iit) = ~ ; ( j ) i = t
D;,*-(j)17f~-).
\fi-(=n-l
(2) The operators a; ( j = 1 , . . . ,d) defined b y
ajOlit) =
C ~:,~(j>~fi)
(33)
are symmetric on HE. ( 3 ) The following identity holds on HE:
Xj
j = 1,. .. , d .
:= aj” + a ; + a ; ,
(34)
Remark Notice that the definition of the operators a: depends on the choice of the orthonormal basis (711,. .. ,nd).An intrinsic definition is possible, but makes the intuitive connection with ordinary multiplication operators more obscure.
PROOF.Since, for each it # 0, a:
lit)
is always in some V k with k 2 1,
one has
(0, aj” i i ) = 0. Moreover since, by definition,
a + ( i ) Iv,=
oR(j)= Pn+1XjPn
and
K+dA
a - w Ivn+l=
it follows that ( C ( j ) ) *= pnXjpn+l = a,+i(j)i
(35)
which means that a - ( j ) is the adjoint of u + ( j ) on HE. Similarly
=(~:(j))*. a o ( j ) Iv,= @ ( j ) = P , X ~ P , = (pnxjpn)*
(36)
202
Therefore the operator u: is symmetric on HE. For each n E N the identity
+ a; + u j ) P , follows from the definition of the D L ( j ) , E = +, -, 0, and this implies (34). XjP, = (UT
I Conversely, Theorem 2.3 Let be given a Halbert space 'El with a N-gradation
x = @vn, vo=c-9 nEN
such that, in the notation (9)
V,
= c ~, f t/n E N \ ( 0 )
and let be given: (i) d sequences of (finite dimensional) operators oR(j)E BWn, vn+l), o:(j)E
a(vn,K ) ,
o,(j) E
~(vn, Vn-1)
satisfying the conditions (19), (20), (21). (ii) For each n E N an orthonormal basis of V,, denoted d
I A ) = 1121,.
.. , n d ) ,
I A I :=
C nj = n.
j=1
Then, o n 31, there exists a structure of interacting Fock space over C d , with creation and annihilation operators given by (30), (31), (33), respectively. If, in addition the commutativity relations (19), (ZO), (21) are satisfied, then the operators xj
:= uj+ + a; + u;,
j = 1 , . . . ,cl,
(37)
are a commuting family of symmetric operators satisfying the relations (ll), Pn being the orthonogonal projection onto V,. Moreover 9 is a cyclic vector for the polynomial algebra generated by the X j and the identity.
PROOF. Let us fix arbitrarily, for each n
E N , an orthonormal basis V,. Define the operators u j ' , u j , u : by (30), (32), (33), ( j = 1,. .. , d ) , respectively. The symmetry of the uy comes from the fact that the D:(j) are symmetric and the fact that u; is the adjoint of u; comes from the relation D ; ( j ) = ( I A ) ) of
203
( D $ ( j ) ) *as in the proof of Proposition (1). This implies that the operators Xj, defined by (37) are symmetric on HE and the relation (11) holds. The commutativity of the family (XI, . . . , Xd) comes from the relations (19), (20), (21) as follows. For each i , j = 1 , . . . ,d and n E N we have
XiXjlfi) = (a+(i)+ aO(i) + a - ( i ) ) ( a + ( j )+ a y j ) + a - - ( j ) > l A ) = (a+(i)a+(i)+ a+(i>a+(O)+ a + ( i )u+(-))IA ) + (aO(i)a+(i)+ aO(i)a+(o)+ aO(i)a+(-))Ifi) + (a-(i)a+(i)+ a-(i)a+(o) + u - (i) u+ ( - ) ) lf i) = (D;t+l(i>DR(j) + D , f ( i ) D : ( j )+ (D;t-dW;(A)Ifi)
+ P:+dWR(A + ~X(i)DFi(A+ (DX-lw;(~Nlfi) + (K+l(i)D;t(j>+ D ; ( i ) D : ( j ) + (D;-1(W;WIfi) = D:+1 (iPf(j)lfi) + (D,+-l(iPR(j)+ 034 0%) + K+l(~>DR(j))lfi) + (D,(i)DX(j) + DX-lW;(j) + D;-l(W;(j>)lfi) + (DR+l(i)Wj)+ ~,+(WEl(j>>lfi). The relations (19), (20), (21) imply that the above expression is equal to
xj xi pi) which proves the commutativity. By assumption, each D $ ( j ) is injective. Therefore the operators a: are injective. By assumption, for each n E N the linear span of the ranges of the operators D:(j) ( j = 1,. . ., d ) is the whole of V,.Therefore the vectors of the form (28) are total in X.Finally denote PD the (polynomial) algebra generated by the Xj and the identity. Then 3 E PD . 3 and since, due to (31) (u;
+ u; + a ; ) @ = a+@ 3 + DOo(j)3.
Since @ ( j ) is a constant, also the vectors of the form Now suppose, by induction, that &)vk k=O
3 are in PD @.
c PD *@
Then, because of the relation (ll),for any IA) E V,, one has XjIfi) = Pn+lXjlfi)
+ PnXjIfi) + Pn-lXjIfi)
204
and, by the induction assumption n
+ p ~ - 1 x j lE~@ ) v k c PI3
pnxjlfi)
*
a.
k=O
It follows that also xjlfi)-
(PnXjIfi)+ Pn-lxjlfi)
a:lfi) = o,f(j)lfi) E PD 0 .
The susjectivity condition then implies that Vn+l thesis follows by induction.
c PD - @ and therefore the I
The interacting Fock space defined by Theorem 2.3 will be denoted by (38)
3cD
Theorem 2.4 Suppose all the conditions of Theorem 2.3 are fullfilled. Then there m.sts a state p on the *-algebra A of the polynomials in d variables such that, denoting L2(Rd,p ) the GNS space of the pair { d , p } , there exists an unitary isomorphism
u : L ~ ( R ~+ , ~ ) such that f o r any j = 1 , . .. ,d ,
UXjU* = at + a0 + a -3 , ~ u1=
j
(39)
a.
PROOF. We define p ( P ) = (0,P ( X 1 , ... ,Xd)lO) for each polynomial P E A. It is clear that p is a state on A. Define U : L2(Rd,p)+ 3 c by ~ U ( P ) = P(X1,.. . ,X d ) o for each polynomial P E A. It is easy to verify that U is an unitary isomorphism which satisfies (39). I References 1. L. Accardi and M. Bozejko: Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.1 (1998), 663-270. 2. L. Accardi and M. Skeide: Interacting Fock space versus full Fock module, Volterra Preprint N. 328 (1998), Math. Notes 86 N6 (2000), 803-818. 3. L. Accardi and Y. G. Lu: The Wigner semi-circle law in quantum electrodynamics, Commun. Math. Phys. 180 (1996), 605-632. 4. L. Accardi. Y. G. Lu and I. Volovich: “The QED Hilbert Module and Interacting Fock Spaces,’’ Publications IIAS, Kyoto, 1997.
205
5. N. Asai: Analytic characterization of one-mode interacting Fock space, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 4 (2001), 409-415. 6. P. K. Das: Coherent states and squeezed states in interacting Fock space, preprint, 2001. 7. P. K. Das: A phase distribution in interacting Fock space, preprint, 2001. 8. C. F. Dunk1 and Y. Xu: “Orthogonal Polynomials of Several Variables,” Cambridge University Press, 2001. 9. Y. Hashimoto: Deformations of the semi-circle law derived from random walks o n free groups, Prob. Math. Stat. 18 (1998), 399-410. 10. Y. Hashimoto, A. Hora A and N . Obata: Central limit theorems f o r large graphs: Method of quantum decomposition, J . Math. Phys. in press. 11. Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 45-57, World Scientific, 2001. 12. R. Koekoek and R. F. Swarttouw: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Technical Report 94-05 Technical University of Delft (1994) Preprint math.CA/9602214 also available from twi.tudelft .nl in directory /pub/publications/ tech-reports/(l994) 13. M. A. Kowalski: The recursion formulas for polynomials in n variables, SIAM J . Math. Anal. 13 (1982), 309-315. 14. M. A. Kowalski: Orthogonality and recursion formulas f o r polynomials in n variables, SIAM J . Math. Anal. 13 (1982), 316-323. 15. H. L. Krall, I. M. Sheffer: Orthogonal polynomials in two variables, Ann. Mat. Pure Appl. 76 (1967), 325-376. 16. T. H. Koornwinder: Orthogonal polynomials in connection with quantum groups, in “Orthogonal Polynomials (P. Nevai, Ed.),” pp. 257-292, Kluwer Academic Publ., 1990. 17. S. Sakai: “C*-Algebras and W*-algebras,” Springer, 1971. 18. H. Szego: “Orthogonal Polynomials (4th ed.),” Amer. Math. SOC. Colloq. Publ. Vol. 23, Providence, R.I. 19. Y. Xu: O n orthogonal polynomials in several variables, Fields Institute Commun. 14 (1997), 247-270. 20. Y. Xu: Unbounded commuting operators and multivariable orthogonal polynomials, Proc. Amer. Math. SOC. 119 (1993), 1223-1231.
CAN “QUANTUMNESS” BE AN ORIGIN OF DISSIPATION? TOSHIHICO ARIMITSU Institute of Physics University of Tsukuba Ibaraki, 305-8571 Japan E-mail: arimitsu Ocm.ph. tsukuba. ac.jp In their constructions of system of quantum stochastic differential equations, mathematicians and/or several physicists interpret that the function of random force operator is to preserve the canonical commutation relation in time, i.e., t o secure the unitarity of time evolution generator even for dissipative systems. If this is the case, it means physically that the origin of dissipation is attributed to quantum non-commutativity (quontumness). The mechanism that the mathematician’s approaches rest on will be investigated from the unified view point of Non-Equilibrium Thermo Field Dynamics (NETFD) which is a canonical operator formalism of quantum systems in far-from-equilibrium state including the system of quantum stochastic equations.
1
Introduction
There are arguments [1-6]that the function of random force operator is to preserve the canonical commutation relation in time. The contents of issue are the following. The time-evolution of free Boson operators is given by da(t)/dt = -iwa(t), dat(t)/dt = iwat(t). The canonical commutation relation [a,at] = 1,at time t = o preserves in time, i.e., [a(t),a t ( t ) ]= 1. If a relaxation is introduced simply by
the canonical commutation relation decays as [a(t),a t ( t ) ]= e-2nt. This inconvenience is secured by introducing random operators F ( t ) and F t ( t ) which are assumed to satisfy [a,Ft(t)]= 0, [ u t , F ( t ) ]= 0,etc. for t > 0. The solutions of Langevin equations1)
da(t)/dt = -iwa(t) - ~ a ( t+) F ( t ) , dat(t)/dt = iwat(t) - ~ ~ t + ( tF )t ( t ) , l ) These stochastic differential equations should be interpreted as those of the Stratonovich type [16],since we perform calculations as if they were ordinary differential equations.
206
207
are given by
Then, one knows that the canonical commutation relation preserves in time if the condition
1 = [a(t),a+(t)l
is satisfied. It is realized by the commutation relation among the random force operators:
[F(t),Fyt')] = 2 4 t - t').
(4)
The above argument seems to show us that the function of random force operators is to recover the unitarity of time-evolution generator rather than to represent dissipative thermal effects. If it is correct, doesn't it mean, physically, that the origin of dissipation can be attributed to quantum noncommutativity (quantumness)? There are, however, physical systems described by classical mechanics, or quantum systems with commutative random force operators. In this paper, with the help of the framework of Non-Equilibrium Thermo Field Dynamics (NETFD) [7-141, we will investigate the above argument in a systematic manner by means of martingale operator paying attention to the non-commutativity among random force operators. In Section 2, the structure of the system of stochastic differential equations in classical mechanics is reviewed. In Section 3, the system of quantum stochastic differential equations within NETFD is introduced. In Section 4, Boson system will be treated, which has a linear dissipative coupling with environment system within the rotating wave approximation. The mathematician's arguments will be studied from the unified viewpoint based on the canonical operator formalism of NETFD by changing the intensity of non-commutativity parameter X among a martingale operator. In Section 5 , Boson system having a linear dissipative interaction between environment without the rotating wave approximation will be investigated with the help of NETFD. This is the case where the system has commutative random force operators. Section 6 will be devoted to some remarks.
208 2
System of Stochastic DifFerential Equations in Classical Mechanics
2.1 Stochastic Liouville Equation
We will show the structure of the system of classical stochastic differential equations [15] starting with the stochastic Liouville equation
n(u,t ) d t = -
(E)
du
where the flow du in the velocity space is defined by du = - p d t
+ m-'dR(t).
Here, the circle o represents the Stratonovich stochastic product [16] which is defined in Appendix A together with the definition of the Ito stochastic product [17]. The increment of random force dR(t) is a Gaussian white stochastic process defined by the fluctuation-dissipation theorem of the second kind:
( d R ( t ) )= 0, (dR(t)dR(t))= 2myTdt, (6) where y ( > 0) is a relaxation constant, and T a temperature of the environment represented by the random force dR(t). Note that within the Stratonovich calculus dR(t) and f(u,t) are not stochastically independent: (dR(t)o f(u,t)) # 0. The average (- . -) is taken over all the possibility of the stochastic process {dR(t)}. By making use of the relation between the Ito and Stratonovich products (62), the stochastic Liouville equation (5) can be rewritten as the one of the It0 type:
where du is the flow in the Ito calculus given by
Note that there appears temperature T in the flow,and that within the It0 calculus dR(t) and f(u,t) are stochastically independent: (dR(t)f(u,t)) = 0.
209
The initial condition for the stochastic distribution function f(u,t ) is given by f(u,0) = P(u,0), where P(u,t ) is the velocity distribution function defined below in Subsection 2.3. Note that the stochastic distribution function conserves its probability within the relevant velocity phase-space: J duf(u,t ) = 1.
2.2 Langevin Equation The system described by the stochastic Liouville equation (5) can be treated by the Langevin equation: du(t) = -yu(t)dt
+ m-ldR(t).
(8)
This Stratonovich type stochastic differential equation does not contain the diffusion term, which is very much related to the way how physicists originally introduced the Langevin equation.2) It is worthwhile to note here that one could have introduced the Langevin equation within the It0 calculus of the form
du(t) = -y[u(t) + m-'T(b/bu(t))]dt+ m-ldR(t),
(9)
which has a term with the functional derivative operator b/bu(t). In the system of quantum stochastic differential equations within NETFD, the Langevin equation of the It0 type, such as (9),can be introduced on the equal footing as the one of the Stratonovich type (8), although this was not the original motivation for the invention of NETFD (see Section 3).
2.3 Fokker-Planck Equation
In precise, the stochastic distribution function is given by f(u,t) = f(n(u,t),P(u,O)).Taking the random average (---),we have an ordinary velocity distribution function P(u,t ) = ( f ( n ( ut,) ,P(u,0))) which satisfies the Fokker-Planck equation
+
dP(u,t ) / a t = (a/au)y[u m-1T(a/au)]P(u,t ) .
(10)
This can be derived most conveniently from the stochastic Liouville equation (7)of the It0 type because of the orthogonal property mentioned in Subsection 2.1. 2)The Langevin equation was introduced by adding a random force term, such as m -'dR (t)/dt, to a macroscopic phenomenological equation, for example, like d u ( t ) / d t =
-74t).
210
The fluctuation-dissipation theorem (6) of the second kind is introduced in order that the stochastic Liouville equation ( 5 ) and the Langevin equation (8) are consistent with the Fokker-Planck equation (10).
3
System of Quantum Stochastic Differential Equations
3.1 Non-Equilibrium Thermo Field Dynamics In order to treat dissipative quantum systems dynamically, we constructed the framework of NETFD [7-141. It is a canonical operator formalism of quantum systems in far-from-equilibrium state which enables us to treat dissipative quantum systems by a method similar to the usual quantum field theory that accommodates the concept of the dual structure in the interpretation of nature, i.e. in terms of the operator algebra and the representation space. In NETFD, the time evolution of the vacuum is realized by a condensation of y*;j.*-pairs into vacuum, and that the amount how many pairs are condensed is described by the one-particle distribution function n ( t ) whose timedependence is given by corresponding kinetic equation (see Appendix
B). We further succeeded to construct a unified framework of the canonical operator formalism for quantum stochastic differential equations with the help of NETFD. To the author's knowledge, it was not realized, until the formalism of NETFD had been constructed, that one can put all the stochastic differential equations for quantum systems into a unified method of canonical operator formalism; the stochastic Liouville equation [15] and the Langevin equation within NETFD are, respectively, equivalent to the Schrodinger equation and the Heisenberg equation in quantum mechanics. These stochastic equations axe consistent with the quantum master equation which can be derived by taking random average of the stochastic Liouville equation.
3.2
Quantum Stochastic Liouville Equation
Let us start the consideration with the stochastic Liouville equation of the Ito type:
dlOf(t))= -i'&f,tdt lOf(t)).
(11)
The generator v f ( t ) , defined by lOf(t)) = Qf(t)lO), satisfies &f(t) = -i%f,tdt v f ( t ) with pf(0)= 1. The stochastic hat-Hamiltonian %f,tdt is a tildian operator satisfying (i%f,tdt)" = i%f,tdt. Any operator A of NETFD
21 1
is accompanied by its partner (tilde) operator A, which enables us treat nonequilibrium and dissipative systems by the method similar to usual quantum mechanics and/or quantum field theory. Here, the tilde conjugation is defined by (A1A2)" = A1&, (CIA1 c2A2)" = cyA1 c;&, (A)" = A, and (At)" = At with A's and c's being operators and c-numbers, respectively. The thermal ket-vacuum is tilde invariant: lOf(t))" = lO,(t)). From the knowledge of the stochastic integral, we know that the required form of the hat-Hamiltonian should be
+
+
f i f , t d t = Hdt
-
+ : d< :
+
(12)
+
where H is given by H = Hs i f i with Hs = H s - Hs, and fi = f i ~f i where I?, and I?D axe, respectively, the relaxational and the diffusive parts of the damping operator The martingale d< is the term containing the operators representing the quantum Brownian motion d B t , dBf and their tilde conjugates, and satisfies (IdA&I) = 0. The symbol : d< : indicates to take the normal ordering with respect to the annihilation and the creation operators both in the relevant and the irrelevant systems (see (23)). The operators of the quantum Brownian motion are introduced in Appendix C, and satisfy the weak relations:
a.
dBi dBt = Adt,
dBt dBJ = ( A + 1) d t ,
dBt dBt = iidt,
dBf dBf = ( A + 1)dt,
(13)
(14) and their tilde conjugates, with A being the Planck distribution function given in Appendix B. (I and I) are the vacuum states representing the quantum Brownian motion. They are tilde invariant: (I" = (I, I)" = I). It is assumed that, at t = 0, a relevant system starts to contact with the irrelevant system representing the stochastic process included in the martingale d< .3)
3.3 Quantum Langevin Equation The dynamical quantity A ( t ) of the relevant system is defined by the operator in the Heisenberg representation:
A ( t ) = Pj-'(t) A P f ( t ) where vj-'(t) satisfies
d P y l ( t ) = Pj-'(t) i f i i t d t 3, Within the formalism, the random force operators dBt and dBf are assumed to commute with any relevant system operator A in the Schrodinger representation: [A,&] = [A,mfl= o for t 2 0.
~
212
with
%Xtdt = *f,tdt
+i d d t dMt.
In NETFD, the Heisenberg equation for A(t) within the Ito calculus is the quantum Langevin equation of the form
d A ( t ) = i [ f i f ( t ) d t ,A(t)]- d ' d ( t ) [ d ' a ( t ) , A(t)],
(15)
with % f ( t ) d t = PF1(t)%f,tdt P f ( t ) ,and
d ' G ( t ) = PT1(t)dX&
Pf(t).
(16) Since A(t) is an arbitrary observable operator in the relevant system, (15) can be the Ito's formula generalized to quantum systems. Applying the bra-vacuum ((11 = (l(11 to (15) from the left, we obtain the Langevin equation for the bra-vector ((lIA(t)in the form
d ( ( l l A ( t ) = i ( ( l J [ H s ( t )A, ( t ) J d t+ ( ( l I A ( t ) h ( t ) d t- i((llA(t)d ' d ( t ) .
(17)
In the derivation, use had been made of the properties (lIAt(t)= ( l ( A ( t ) , (Id'&(t) = (Id'B(t), and ( ( l l d ' d ( t ) = 0.
3.4
Quantum Master Equation
Taking the random average by applying the bra-vacuum (I of the irrelevant sub-system to the stochastic Liouville equation ( l l ) ,we can obtain the quantum master equation as
(a/at)lo(t))= -iBlO(t)),
(18)
with H d t = ( / % f , t d t ( )and l O ( t ) ) = ( l O f ( t ) ) .
3.5 Stratonovich- Type Stochastic Equations By making use of the relation between the It0 and Stratonovich stochastic calculuses, we can rewrite the It0 stochastic Liouville equation (11) and the It0 Langevin equation (15) into the Stratonovich ones, respectively, i.e.,
d A ( t ) = i[l?f(t)dt :A(t)], with
H f ( t ) d t = l?s(t)dt
+ i ( h ( t ) d t+ Z1d ' M ( t ) d ' h ( t ) ) + : d ' M ( t ) : .
213
3.6 Fluctuation-Dissipation Relation The fluctuation-dissipation theorem of the second kind for the multiple of martingales, d f i t d k t , is determined by the criterion that there is no diffusive term comes out in the terms f i d t + i d f i t d f i t appeared in Hf,tdt in Subsection 3.5:
d f i t dMt = - 2 f l o d t .
(21)
The origin of this criterion is attributed to the way how the Langevin equation was introduced in physics, as explained before, i.e., relaxation term and random force term were introduced in mechanical equation within the Stratonovich calculus. Therefore, there is no dissipative terms in stochastic equations of the Stratonovich type. We adopted this criterion in quantum cases. The operator relation ( 2 1 ) may be called a generalized fluctuationdissipation theorem of the second kind, which should be interpreted within the weak relation.
A System in the Rotating W a v e Approximation
4 4.1
Model
We will apply the above formalism to the model of a harmonic oscillator embedded in an environment with temperature T. The Hamiltonian H s of the relevant system is given by HS = wata where a, at and their tilde conjugates are stochastic operators of the relevant system satisfying the canonical commutation relation [u,at] = 1, and [ii, iit] = 1. The tilde and non-tilde operators are related with each other by the relation (llat = (116 where (11 is the thermal bra-vacuum of the relevant system. Since we are interested in the system in the rotating wave approximation, we will confine ourselves to the case where the stochastic hat-Hamiltonian '?it is bi-linear in a, at , dBt, dBi and their tilde conjugates, and is invariant under the phase transformation a + aeie, and dBt 4 dBt eie. This gives us the system of linear-dissipative coupling. Then, f i and ~ l ? consisting ~ of l? introduced in Subsection 3.2 become
respectively, where we introduced a set of canonical stochastic operators "/y = uGt, yp = at - ii with p u = 1, which satisfy the commutation relation
pa
+
+
214
[yvlyP]= 1. The parameter v (or p ) is closely related to the ordering of operators when they are mapped to c-number function space with the help of the coherent state representation [ll],i.e., v = 1 for the normal ordering, v = 0 for the anti-normal ordering, and v = 1/2 for the Weyl ordering. The new operators 7%and yP annihilate the relevant bra-vacuum:
(1lTP = 0.
(1lyP = 0,
4.2 Martingale Operator Let us adopt the martingale operator:
with
and : d&,(+) : = - i ( d W t y v
+ dW$yv).
Here, the annihilation and the creation random force operators d W t and dW$ are defined, respectively, by
d W t = 6( p d B t
+ v d B f ),
d W f = &(dBf
- dBt).
The latter annihilates the bra-vacuum
(IdW$ = 0 ,
(I of the irrelevant system: (IdIVf = 0.
Note that the normal ordering : .. . : is defined with respect to y’s and dW’s. The real parameter X measures the degree of non-commutativity among the martingale operators:
In deriving this, we used the facts that dWt dWt = d f i t dWt = 2 ~( f.i + v) d t , d W t d W t = dWt dW$ = 2ndt,
(24)
215
and that the other combinations are equal to zero. Note that [m/v,,dWt] = 2ndt should be compared with (4). There exist at least two physically attractive cases [11,14], i.e., one is the case for X = 0 giving non-Hermitian martingale:
d f i t = id'% [(at - 6 ) d B f
+ t . ~ . ,]
(25)
and the other for X = 1 giving Hermitian martingale:
[
d f i t = id% (atdBt - d B f u )
+ t.c.1 ,
(26)
where t.c. stands for tilde conjugation. The former follows the characteristics of the classical stochastic Liouville equation where the stochastic distribution function satisfies the conservation of probability within the phase-space of a relevant system (see Section 2). Whereas the latter employed the characteristics of the Schrodinger equation where the norm of the stochastic wave function preserves itself. In this case, the consistency with the structure of classical system is destroyed [11,14]. The fluctuation-dissipation theorem of the system is given by
: d f i t : : d f i t : = - 2 ( X f i ~4- f i o ) d t ,
(27)
where we used the relations :d f i j - ) : :d f i i - ) := - 2 f i ~ d t , :d M i - ) : :dM;+) := -
2h~dt
and
:dfi:+): : d f i i + ) :=:dfii+) : dfij-1 := 0, which can be derived by making use of (24). The hat-Hamiltonians of the model are given by
Hif,tdt = H s d t
+ i(l - X ) f i R d t + d f i t ,
~ fiD)dt + d k t , 7?Ztdt = H s d t -I-i ( ( 1 - 2 X ) f i f i i f ( t ) d t= f i s ( t ) d t
+i(l - x)fiR(t)dt+ :d'M(t) : .
(28) (29) (30)
4.3 Heisenberg Operators of the Quantum Brownian Motion The Heisenberg operators of the Quantum Brownian motion are defined by
B(t) = QF1(t)Bt Qif(t), Bt(t)= QF1(t)Bj Q'(t),
(31)
216
cf(t)),
and their tilde conjugates. Their derivatives dB#(t) = d(V;’(t) Bt# (# : nul, dagger and/or tilde) with respect to time in the Ito calculus are given, respectively, by
+ 6 [(1- A) v (iit(t) - ~ ( t )-)Aa(t)]dt, = dB,t - 6 [(l- A) p (a+@) - ?i(t))+ xa+(t)]dt,
dB(t) = dBt
@(t)
(32) (33)
and their tilde conjugates. Then, we have
dW(t) = dWt - XZny,(t)dt,
mqt)= mP , - 2Kyqt)dt.
(34)
Since, by making use of (34), we see that
dn;r(t) = d‘n;r(t) = i[yP(t)dWt+ ;i.+(t)dWt]- iA[dW,Py,(t) + dw:T”(t)],
(35)
we know that the martingale operator in the Heisenberg representation keeps the property: (Id&f(t)l) = 0.
4.4 Quantum Langevin Equations The quantum Langevin equation is given by
217
with
&(t) = Vil(t)BSVf(t) = H s ( t ) - as@). Note that the Langevin equation is written by means of the quantum Brownian motion in the Schrodinger (the interaction) representation (the input field [18]) in (36), and by means of that in the Heisenberg representation (the output field [IS]) in (37). The Langevin equation for the bra-vector state, ((lIA(t), reduces to
d((llA(t) = i((ll[Hs(t),A(t)ldt
- {((l"A(t),at(t)la(t)+ ((llat(t)[a(t),A(t)I}dt
+ 2/€fi((lI[a(t),"1,
at(t)lldt
+ ((lIIA(t),at(t)]&
dBt
+ ( ( 1 1 6 d B j [ a ( t ) ,A(t)]
(38)
= i((ll[Hs(t),A(t)ldt
- 4 1 - 2x1 { ((11[A(t),at(t)14t)+ ((ll,t(t)[a(t),A(t)l} d t
+ 2nfi((ll[a(t),[A(t), at(t>lldt
+ ((lI[A(t), at (t)]d% dB(t)+ ((lid% dBt ( t ) [ a ( t )A(t)]. ,
(39) The relation between the expression (38) and (39) can be interpreted as follows. Substituting the sohtion of the Heisenberg random force operators (32) and (33) for dB(t)and d B t ( t ) ,respectively, into (39), we obtain the quantum Langevin equation (38) which does not depend on the non-commutativity parameter A. The Langevin equations for a ( t ) and a t ( t ) of the system reduce to
+
da(t) = (-iw - /€)a(t)dt dWt - 2(1 - X)VK
[lit(t)- a(t)]d t - Xvdw:,
(40)
+
da+(t)= (iw - /€)at(t)dt dWt
+ 2(1 - X ) ~ K[at@)- TL(t)]dt + XpdW:.
(41) Note that the last two terms in the above equations disappear when one applies ((11 to them. For X = 0, (40) and (41) become, respectively, to
+ dat(t) = iwat(t)dt - /€ZL(t)dt+ d W t ,
da(t) = -iwa(t)dt - /€TLt(t)dt d W t ,
(42) (43)
where we put p = v = 1/2, for simplicity. For X = 1, we get da(t) = -iwa(t)dt
- /€a(t)dt+ &dBt,
dat(t) = iwa+(t)dt- d ( t ) d t + d%dB,t,
(44)
(45)
218
which may correspond to ( 2 ) . Applying ((11 = (11(1 to (40) and (41), we obtain, for any values of A, p and Y , the Langevin equations of the vectors ((lldt)and ( ( l l a t ( t ) in the forms
+
d((lla(t) = -iw((lla(t)dt - n((lla(t)dt &((11dBt, d ( ( l l a t ( t )= iw((llat(t)dt- tc((llat(t)dt+ d G ( ( 1 1 d B f .
(46) (47)
Note that these have the same structure as those in ( 2 ) . 5
A System with Commutative Random Force Operators
Let us investigate Boson system having x - X type interaction between environment, i.e., a system without the rotating wave approximation. The Hamiltonian of a harmonic oscillator can be written in the form
Hs
=: p 2 / ( 2 m )
+mw2x2/2:
with
x =J F ( a
+at),
p = -i&Gp(a
- a+>,
where x and p satisfy the canonical commutation relation [x,p] = i. The normal ordering : ... :, here, is taken with respect to a and a t . The relaxational and the diffusive parts in
i7 = i7R + fi, are given, respectively, as fifi
= - - i K ( x - z)(p +$),
f i D = -2Kmw(A
+ 1 / 2 ) ( x - z)2.
(48)
The martingale operator corresponding to x - X type interaction may have the form
d< = 2-(xdXt
- ZdXt),
(49)
with
dXt = (dBt
+d B i ) / h
where dBt and d B i are the quantum Brownian motion defined in Appendix C. Then, we have
dXt dXt = dXt dXt = ( A
+ 1/2)dt
which gives us the fluctuation-dissipation theorem
dMt dMt = - 2 f i ~ d t .
219
The form of the martingale (49) was adopted by following the structure of microscopic interaction Hamiltonian of the 2-X type. The stochastic hat-Hamiltonian 'fif,,dt for the stochastic Liouville equation (11) of the Ito type is given by
Ifif,tdt = Hs
+ i ( k+~f i ~+)d&.
(51)
Then, the stochastic hat-Hamiltonian of the Stratonovich type becomes
Hj,tdt = H s d t where one does not see f We can also check that
+ i f i R d t + diClt,
(52)
i thanks ~ to the fluctuation-dissipation theorem (50).
'fiXtdt = H s d t
+ i ( k-~k D ) d t + dA?t.
(53)
The Langevin equation has the forms
1
+K ( x ( t ) - Z(t))dt, d p ( t ) = - m w 2 z ( t ) d t - rc(p(t) + j ( t ) ) d t + 4i~mw(A+ 1 / 2 ) ( 2 ( t )- 5 ( t ) ) d t- 2
dz(t) = -p(t)dt
(54)
m
6 dXt.
(55)
Applying (11 to (54) and (55), we have the Langevin equation for (llz(t)and ( 1lp( t ) in the forms
d ( l ) z ( t )= m - l ( l l p ( t ) d t , d(llp(t)= -mw2(llz(t)dt
(56)
- 2 ~ ( l l p ( t ) d-t 2-
(lJdXt,
(57)
- i f i ( l [ ( d B t+d B J ) ,
(58)
respectively. They can be written in terms of a and ut as
d ( l ( a ( t )= - i w ( l ( a ( t ) d t - ~ ( l[a(t) l - at(t)]dt
d(llat(t)= iw(llat(t)dt - ~ ( 1 1[at@)- a ( t ) ] d t - i f i ( l l ( d B f + d B t ) .
(59)
If we take the rotating wave approximation at this stage the coefficients in front of the quantum Brownian motion are not equal to those appeared in (44) and (45). It may indicate that a naive procedure of taking the rotating wave approximation will not give us correct results. It might also be related to the renormalization procedure needed to derive stochastic differential equations for the system with z-X type interaction from a microscopic Heisenberg equation [19].
220
6
Concluding Remarks
We have revealed that the non-commutativity among d&li-) and d&li+) appeared in the martingale operator of the model within the rotating wave approximation affect the relaxation part of the stochastic hat-Hamiltonian. When the measure X of the non-commutativity has the value X = 1, the hatHamiltonian becomes Hermite, and therefore, it looks like being related to a microscopic description. On the other hand, for X = 0, the system of the quantum stochastic differential equations has the same structure as that of classical mechanics, and it is related to a semi-macroscopic description. As has been shown in this paper, Hermiticy of the hat-Hamiltonian is realized thanks to the non-commutativity between d M i - ) and d&li+). Does this mean that the system with commutative martingale does not have any microscopic realization? As an example of system with commutative martingale, we studied the system corresponding to the quantum Kramers equation which has z - X type interaction Hamiltonian between the relevant system and environment system. Since, there is no non-commutative parts in the martingale operator, this system cannot have an Hermitian stochastic hat-Hamiltonian. The non-commutative parts appears when one takes the rotating wave approximation to the interaction Hamiltonian. Does dissipation originate in the approximation causing non-commutative character in martingale? Can quantumness, appeared in this way, be the origin of dissipation? On the contrary, the following question arises naturally. Is it always possible to put all random force operators to be commutative? There are still a lot of problems to be resolved before we know the origin of dissipation. However, with the help of NETFD, we can see the problems from a unified viewpoint which may provide us with good prospects for further developments. Introducing the parameter X in the martingale term as given by (23), we can transform the equation to the non-Hermitian version by shifting X + 0 (see (37)). In other words, it seems that the non-commutativity is renormalized into the relaxational and diffusive terms. Substituting the solution of the random force operators (32) and (33) in the Heisenberg representation (the output field) into (37), we have the Langevin equation (36) expressed by means of those in the Schrodinger (or, more properly, the interaction) representation (the input field). Note that the Langevin equation (38) for the bra-vector state ((lIA(t)does not depend on X when it is represented by the random force operator in the Schrodinger representation (the input field). We are intensively investigating what is the physical meaning of the renormalization of non-commutativity by changing the parameter A. The relation
22 1
between the present argument and the procedure of the coarse graining is under investigation. Related to the system with commutative random force operators, a microscopic derivation of quantum stochastic equations corresponding to the quantum Kramers equation are in progress [19]. There, an appropriate renormalization is required in accordance with the separation of two time-scales, i.e., microscopic and macroscopic time-scales. Without the renormalization, one gets quantum stochastic equations in the rotating wave approximation, which do not correspond to the system described by the Kramers equation. Including these studies, the further progress will be reported elsewhere. Acknowledgments
Somewhat preliminary contents of the paper was presented at the International Workshop, New Developments in Statistical Physics, held in University of Tokyo in 1997, celebrating Prof. M. Suzuki’s sixtieth birthday at the occasion of his retirement from University of Tokyo. The present version was mostly developed at Prigogine Center for Statistical Mechanics and Complex Systems in the University of Texas at Austin in Autumn, 1999. The author would like to express his sincere thanks to all members of the center, especially to Prof. I. Prigogine and Dr. T. Petrosky, for their warm hospitality in the productive tense atmosphere. Appendix A
The definitions of the Ito [17] and the Stratonovich [16] stochastic products are given, respectively, by
and
xt 0 dYt = X t + d t2 +xt ( Y t + d t
- yt>>
for arbitrary stochastic operators X t and yt. From (60) and (61), we have the formulae which connect the Ito and the Stratonovich products in the
222
differential form
Appendix B
The time-evolution of the thermal vacuum lO(t)), satisfying the quantum master equation (18) with the hat-Hamiltonian for the semi-free system specified by Hs = wata and (22), is given by
{
lO(t)) = exp [n(t) - 4011
w} lo),
(63)
where the one-particle distribution function, n(t) = ((llat(t)a(t)lO)), satisfies the kinetic (Boltzmann) equation of the model: dt
= -2n[n(t) - A]
with the Planck distribution function A = (ew/= - 1)-l. temperature of environment system.
Here, T is the
Appendix C
Let us introduce the annihilation and creation operators bt, bf and their tilde conjugates satisfying the canonical commutation relation: [bt, bf,] = b(t - t'),
The vacuums (01 and 10) are defined by
[bt,
bl,] = b(t - t').
(64)
-
btlO) = 0,
btlO) = 0
(Olb,t = 0,
(OlQ = 0.
and
The subscript or the argument t represents time. Introducing the operators
l-dt 1 t
Bt =
dBtf =
dt' btt,
l-dt1 t
BJ =
dB:, =
and their tilde conjugates for t 2 0, we see that they satisfy B(0) = 0,
Bt(0) = 0,
[B,, Bj] = min(s,t),
dt' bi,
223
and their tilde conjugates, and that they annihilate the vacuums 10) and (01: dBt)O) = 0,
dBt10) = 0 ,
(O(dB! = 0 ,
(Old@ = 0.
These operators represent the quantum Brownian motion. Let us introduce a set of new operators by the relation d C c = B””dBy with the Bogoliubov transformation defined by BP”=
( -1 1 ) ’ l+ii-ii
where ii is the Planck distribution function. We introduced the thermal doublet: dBf=’ = d B t ,
dBr=2 = dB;,
dBf=’ = d B J , d B r E 2 = - d B t ,
(66)
and the similar doublet notations for d C r and d c c . The new operators annihilate the new vacuum (I and I): dCtl) = 0 , dC‘t1) = 0 ,
(IdC! = 0,
(IdC’J = 0.
We will use the representation space constructed on the vacuums Then, we have, for example,
(I
and
I).
(IdBtI) = (IdB,tI) = 0 , ( I d B J d B t I ) = fidt,
( I d B t d B J I ) = (fi
+ 1) dt.
References 1. R. F. Streater: J. Math. A: Math. Gen. 15 (1982), 1477. 2. H. Hasegawa, J. R. Klauder and M. Lakshmanan: J. Phys. A: Math. Gen. 18 (1985), L123. 3. R. L. Hudson and K. R. Parthasarathy: Math. Phys. 93 (1984), 301. 4. R. L. Hudson and J. M. Lindsay: Ann. Inst. H. Poincarh 43 (1985), 133. 5. K. R. Parthasarathy: Rev. Math. Phys. 1 (1989), 89. 6. K. R. Parthasarathy: “An Introduction to Quantum Stochastic Cdculus,” Monographs in Mathematics Vol. 85, Birkhauser Verlag, 1992. 7. T. Arimitsu and H. Umezawa: Prog. Theor. Phys. 74 (1985), 429. 8. T. Arimitsu: Phys. Lett. A153 (1991), 163. 9. T. Arimitsu: J. Phys. A: Math. Gen 24 (1991), L1415. 10. T. Arimitsu and N. Arimitsu: Phys. Rev. E 50 (1994), 121. 11. T. Arimitsu: Condensed Matter Physics (Lviv, Ukraine) 4 (1994), 26. Accessible at [http://www.px.tsukuba.ac.jp/home/tcm/arimitsu/cmp4. Pdfl.
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12. T. Saito and T. Arimitsu: J. Phys. A: Math. Gen. 30 (1997), 7573. 13. T. Imagire, T. Saito, K. Nemoto and T. Arimitsu: Physica A 256 (1998), 129. 14. T. Arimitsu: Quantum Stochastic Diflerential Equations in view of NonEquilibrium Thermo Field Dynamics (2002), submitted. 15. R. Kubo, M. Toda and N. Hashitsume: ‘‘ Statistical Physics 11,”Springer, Berlin 1985. 16. R. Stratonovich: J. SIAM Control 4 (1966), 362. 17. K. Ito: Proc. Imp. Acad. Tokyo 20 (1944), 519. 18. C. W. Gardiner and M. J. Collett: Phys. Rev. A 31 (1985), 3761. 19. T. Saito and T. Arimitsu: “Stochastic Processes and their Applications (A. Vijayakumar and M. Sreenivasan Eds.) ,” Narosa Publishing House, Madras, 1999, 323.
EVENTUM MECHANICS AS THE CROSSROAD OF PROBABILITY, INFINITE-DIMENSIONALITY AND NON-COMMUTATIVITY VIACHESLAV P. BELAVKIN Mathematics Department Unaversity of Nottingham NG7 2RD, UK E-Mail: vpb Omaths. not t. ac.uk Web Page: http://www.maths.nott.ac.uk/personal/upb/ It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements. Although they all have received an adequate mathematical treatment in the phenomenological information dynamics such as quantum stochastics of open systems with counting or continuous observations, the measurement problem in these models is simply lifted to the apparatus level. Recent development in infinite-dimensional analysis of boundary-value problems for quantum measurement theory has indicated a possibility for resolution of this interpretational crisis. It is done in the new framework of Eventum Mechanics, a time asymmetric event enhanced quantum mechanics, by divorcing the algebra of the dynamical generators and the algebra of the actually observable events. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the past events with the potential future. This gives a dynamical solution, in the form of a boundary value problem for an infinite number of particles, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schriidinger and Bohr. The information dynamics are obtained then by a conditioning as the dynamical filtering equations. We prove that the quantum stochastic model for the continuous-in-time measurements of the spontaneous events is equivalent to a Dirac type boundary-value problem for the second quantized input “offer waves from the future” in one extra dimension, and to a reduction of the algebra of the consistent histories of past events t o an Abelian subalgebra for the ‘Ltrajectories of the output particles.” This supports the wave-particle duality in the form of the thesis that everything in the future will be quantized waves, everything in the past will be the trajectories of recorded particles.
1
Introduction Have the ‘jump’ in the equations and not just the talk - J. Bell.
In 1929 Erwin Schrodinger complained “If I had known we were going to go on having all this damned quantum-jumping, I would never have got involved in the subject” [l:p. 3441.
225
226
After so many years quantum events, such as spontaneous jumps, which are the only experimental evidence in quantum physics, are not yet a part of the present formulation of quantum theory. There is no place for the causal events in quantum mechanics, apart of the trivial ones 0 and I, with respect to which any initial state could be conditioned. Moreover, it is now well understood that the events can not be a part of any closed, reversible quantum theory, quantum gravity say, in which the whole world is quantized as a Hamiltonian system, and there is no preferred arrow of time which is necessary for causality, or even no time. It is known since von Neumann that this greatest crisis of the dynamical interpretation of quantum measurement cannot be resolved within the purely quantum world based on the maximal non-commutativity, or even semi-quantum, based on a partial commutativity, if the place for the events and their probabilities is a finite-dimensional Hilbert space. This indicates the only one option left for resolving this crisis which is on the crossroad of non-commutativity, infinite-dimensionality and eventum probability. The quantum world should be considered as only an open, unstable part of a larger, semi-quantum time-asymmetric world. Moreover, according to the completeness of the quantum probabilistic description of the reality, it must have started from a pure state, which has to remain such if conditioned by all past events. In this extended world there should be a infinite-dimensional place for all observable events, the Bell’s beables [17], which may dynamically remain causal in only one, the chosen to be positive time arrow. We call such non-commutative infinite-dimensional eventenhanced quantum-probabilistic mechanics the Eventum Mechanics. As the matter wave mechanics it should be Hamiltonian in the Schrodinger picture in order to retain all essential features of quantum mechanics such as linearity and unitarity in the Hilbert space, but not time-symmetric in the Heisenberg picture in order to be compatible with quantum causality, see the review paper [2] for more ditails. In this paper we show that quantum jumps, or at least their Markovian stochastic model, can be made compatible with a special unitary dynamics of an extended quantum theory, and can be derived from a boundary value problem for a second quantized extended stable semi-quantum system. This model is sufficiently general to include any type of quantum system Hamiltonian and any type of Bell’s beables, and is exactly solvable, not perturbative which would require a renormalization. The generalized wave equation includes the Bamiltonian boundary interaction of an infinite number of incoming quantum and outgoing classical particles, with the unstable quantum system placed at the boundary of the whole system. The mathematical formulation of this dynamical problem is as continuous and reversible as Schrodinger could have
227
wished, and the jumps as the observable events appear only in the corresponding singular interaction picture in the positive time arrow. This reduces the whole quantum measurement problem in continuous time to the statistical inference by the simple conditioning (filtering) of a quantum unstable subsystem on the classical part of the system. Among the founders of quantum theory perhaps the closest to this truth was Bohr when he said that it ‘must be possible to describe the extraphysical process of the subjective perception as if it were in reality in the physical world’. He regarded the measurement apparatus, or meter, as a semiclassical object which interacts with the world in a quantum mechanical way but is essentially classical: it has only commuting observables - pointers. It relates the reality to a subjective observer as the classical part of the classical-quantum closed mechanical system. Thus Bohr accepted that not all the world is quanturn mechanical, there is a classical part of the physical world, and we belong partly to this classical world. Schrodinger himself tried unsuccessfully to derive the time continuous jumps from a boundary value problem between past and future for a more general formulation than his wave equation, which would be relativistic and with infinite degrees of freedom. Here we shall see that in order to realize this program one should indeed consider a quantum field Dirac type boundary value problem in a cylindrically-extended semi-classical world, and this boundary value problem can be obtained as an ultrarelativistic limit from any special relativiastic closed Hamiltonian system in such world [3]. However in order to solve the quantum jump as a time-continuous measurement problem we will need not only the new wave equation but also a new quantum causality principle in the form of a superselection rule. This rule will uniquely define the arrow of time in which the induced Heisenberg dynamics will be compatible with the observable events as quantum jumps which will satisfy our nondemolition causality principle, briefly formulated as ‘The future is quantum but past is classical’. It is well known that there is no nontrivial Hamiltonian interaction of classical and quantum systems which would lead to a reversible, authomorphic dynamics of the total system. The recent phenomenological models [4-101 for quantum spontaneous jumps in quantum optics, as well as the event enhanced phenomenological quantum mechanics of Blanchard and Jadczyk i11-131 are based on the non Hamiltonian quantum Markov Master equation. However they all can be derived from quantum stochastic Hamiltonian models using the quantum statistical filtering method [14-161. This indicates a possibility to construct an irreversible, but endomorphic nonphenomenological eventenhanced quantum theory, inducing it from quantum stochastic unitary evo-
228
lution in the corresponding Hilbert space. Here we shall treat these quantum stochastic models simply as the interaction representations of a Dirac type boundary value problems for an infinite number of the auxiliary systems, and explicitly construct the corresponding Heisenberg irreversible dynamics of the event enhanced extended system. In realizing this program we will start along the line suggested by John Bell [17] that not all quantum observables should be related to the actual events, but only those which he, failing mathematically to define them, called 'beables. 2
Quantum Events and SchrSdinger Cat
The matrix, the completely noncommutative form of quantum mechanics discovered by Heisenberg, and its statistical interpretation put forward by Bohr, is still a corner stone of all quantum physics. It abandons the traditional, deterministic causality, but taken in its orthodox form it is also inconsistent with the statistical causality which is based on the Bayes prediction formula for the conditional probabilities
P r ( F J E )= P r ( F A E ) / P r ( E ) . Indeed this formula assumes the consistency assumption that the probability of any future event F must be the statistical average
+
Pr ( F ) = Pr (FIE)Pr ( E ) Pr (FIEL)Pr (EL), of the conditional probabilities respectively to the actual event E and the alternative event E L = I - E . Representing the events as usual by the orthoprojectors in a Hilbert space W such that P r ( F ) = ( x J F x ) ,one can easily find that the consistency condition holds for every state-vector x E W if and only if
F A E + F A E ~= F , i.e. iff the largest event (orthoprojector) F A E majorized by both E and F is the relative orthogonal complement of F A E L in F . This is possible only under the condition of the compatibility of E and F when E F = F E is the orthoprojector F A E. The predictability of the future by the conditioning in the result of statistical inference is the heart of the weakest type of physical causality, and should be taken as the defining property for what should represent the events as the actualities. If we assume that any orthoprojector F is admissable in future, then there is no place for a nontrivial event E in W,
229
apart from E = 0 and E = I , which are the only orthoprojectors commuting with any such F . In 1935, Einstein launched a brilliant and subtle attack on quantum causality in a paper [18] with two young co-authors, Podolski and Rosen, which has become of major importance to the world view of physics. They showed that this inconsistency leads to nonlocality, “a spooky action at distance” which cannot be explained by the dynamical interactions of quantum mechanics. Thus, they derived, that quantum mechanics must be incomplete. However instead of completing it by adding a part which would be a home for the events, the necessary elements without which there is no interpretation of any physical theory, Einstein simply suggested replacing quantum mechanics by a classical hidden variable theory. Unfortunately, there is no such classical theory which would be fully compatible with quantum mechanics, and even the Bell’s famous arguments that von Neumann’s proof of the nonexistence of hidden variables was ‘wrong’ (see the review [30] for the detailed analysis of these arguments), didn’t add anything positive to the solution of this problem. Motivated by EPR paper, in 1935 Schrodinger published a three part essay [19] on ‘The Present Situation in Quantum Mechanics’. He turns to EPR paradox and analyses completeness of the description by the wave function for the entangled parts of the system. (The word entangled was introduced by Schrodinger for the description of nonseparable states.) He notes that if one has pure states 1c, ((T) and x (w) for each of two completely separated bodies, one has maximal knowledge, (a,w)= 1c, (o)x(w), for two taken together. But the converse is not true for the entangled bodies, described by a non-separable wave function $1 ((T,w ) # $J ((T)x (w): ‘Mazimal knowledge of a total system does not necessary amply maximal knowledge of all its parts, not even when these are completely separated one from another, and at the tame can not influence one another at all. ’ To make the absurdity of quantum causality in the EPR argument even more evident he constructed his famous burlesque example in quite a sardonic style. A cat is shut up in a steel chamber equipped with a camera, with an atomic mechanism in a pure state described by the superposition
1 1c, = -10)
a
1 + -11) &
of only two orthogonal states: (0) (atom disintegrates) and 11) (the single level atomic state). The mechanism triggers the release of a phial of cyanide if an atom disintegrates spontaneously, and this event is represented by a onedimensional projector FO = lO)(Ol. It is assumed that it might not disintegrate in the course of an hour t = 1 with probability Tr(F1P~) = 1/2, where
230
Fl = 11)(11and P+ = $$t. If the cyanide is released, the cat dies, if not, the cat lives. Because the entire system is regarded as quantum and closed, after one hour, without looking into the camera, one can say that the entire system is still in a pure state in which the living and the dead cat are smeared out in equal parts. Schrodinger resolves this paradox by noting that the cat is a macroscopic object, the states of which (alive or dead) could be distinguished by a macroscopic observation as distinct from other events, whether observed or not. He calls this ‘the principle of state distinction’ for macroscopic objects, which is in fact the postulate that the directly measurable system (consisting of the cat) must be classical: ‘It is typical in such a case that an uncertainty initially restricted to an atomic domain has become transformed into a macroscopic uncertainty which can be resolved through direct observation. ’ The dynamical problem of the transformation of the atomic, or “coherent” uncertainty, corresponding to a probability amplitude $ (a),into a macroscopic uncertainty, corresponding to a mixed state p, is called quantum decoherence problem. Thus he suggested that the solution of EPR paradox is in the non-equivalence of the two spins in this thought experiment, one is being observed and thus must be a macroscopic (i.e. classical) system, and the other (nonobserved, microscopic) should stay quantum but open subsystem. This was the true reason why he replaced the observed spin by the classical two state cat corresponding to the only two possible events, and the other by an unstable atom modelled as a quantum two level interacting subsystem. The only problem was to construct the corresponding classical-quantum interaction and corresponding state transformation in a consistent way. In order to make this idea clear, let us formulate the toy dynamical model of the Schrodinger’s cat problem in a purely mathematical way. For the notational simplicity instead of the values &1/2 for the spin-variables u and w we shall use the indexing values ( 0 , l ) describing the states of a “bit”, the “atomic” system of the classical information theory. These states are identified with the distinguishable events upon which the any mixed state can be conditioned by usual classical inference. Consider the atomic mechanism as a quantum “bit” with Hilbert space b = C2,the pure states of which are described by $-functions of the variable T E {O,l}, i.e. by 2-columns with scalar (complex) entries $(T) = defining the probabilities (7) l2 of the quantum elementary propositions corresponding to T = 0 , l . If the atom disintegrats, $ = 11) corresponding to T = 1, if not, = 10) corresponding to T = 0. However these states cannot be considered as directly observable events because any other, nonorthogonal atomic ‘event’ E # F,, cannot be conditioned by Fo = lO)(Ol or Fl = 11)(11 in
(TI$
23 1
+
a consistent way due to E # E A FO E A F1 = 0 . Indeed, because of zero infima EAF, = 0, v = 0 , l the probability of observing something compatible simultaneously with F and any of F, is always zero, but the left hand side P r ( E ) = TrEp is not zero for p = E , say. The observable events are only the Schrodinger’s cat states which can be conditioned as a classical bit with only two pure states v E (0,l). They are identified with the Kronicker delta probability distributions 60(w)when the cat is alive (v = 0) and 61 (v) when the cat is dead (v = 1). Moreover, as we shall see in the next section, every state of the combined semi-classical system can be conditioned upon the events of the classical part of the system, and hence can be predicted simply as the posterior state, i.e. as the result of the statistical inference based OR the measurement of these events. Such predictability for any prior state of the total system is the defining property for the events, and this is why the nontrivial events do not exist in the orthodox quantum theory without adding a classical part (the Schrodinger cat). By adding such a classical auxiliary system and considering appropriately correlated states one can derive the projection, or any other reduction postulate, simply as the statistical inference. The only remaining problem is to explain how these correlations can by dynamically achieved for noncorrelated states of initially independent classical and quantum systems. This was seen always as an unsolvable problem as there is no nontrivial Hamiltonian interaction of the classical and quantum systems corresponding to the reversible automorphic dynamics which would give such correlations. However by a further enlargement of this system we will construct the desired correlations by reducing them from a special Hamiltonian unitary evolution which induces an irreversible endomorphic dynamic of the total system only in the chosen positive direction of time. 3
The Dynamical Toy Model
Both pure states of the classical Shrodinger’s cat, and even any their mixture can be described by the complex amplitudes x (v) = (vlx of v = 0 , l as if the cat like the atom were a quantum bit. However the 2-columns x are not unique, they are defined by the probabilities Ix (.)I2 up to a phase function 4 of v, not just up to a phase constant as in the case of the atom (only constants commute with all atomic observables B E C on the Hilbert space $, but any phase multiplier 4 E C2, I+ (w)l = 1 commutes with all cat observables c (v)). Initially the cat is alive, so its amplitude xo (v) is defined up to a phase function 4 (v) by the probability &distribution S G 60 on ( 0 , l ) as 4 (v) SO (w), where 60 (w) = (wJ0) is equal 1 if v = 0, and 60 (w) = 0 if v = 1 such that
232
xo can be identified with the unit vector 60 10) in C2. The dynamical interaction in the semiclassical system “atom plus cat” can be described by the unitary transformation S = Fo 8 i + F1881 = 8,XBi
(1)
in b 8 g as it was a purely quantum composed system. Here 81 is the unitary flip-operator in g, (61x1 (v)=
( v h x = .( A 1lx = x (vA 1) 7
where v A T = 1v - 7 ) = T A v is the difference (mod2) on (0, l}, Fv = Iv)(vl, and X = OFo lFl is the orthoprojector Fl in b. This is the only meaningful interaction affecting the cat but not the atom during the hour in a way suggested by Schrijdinger,
+
s [$ 8 XI (7,211 := ( T , 4 S ($ 8 x>= $ (7)x.(
A TI,
where ( T , v ~= (71 @ (vl. Applied to the initial product-state $0 = $J 8 6 corresponding to x = 6 x o it has the resulting probability amplitude
$ ~ ( T , ~ ) = $ ( T ) ~ ( w =AOT ) if
~ f v .
(2)
Because the initial state xo = 10) is pure for the cat considered either as classical bit or quantum, the initial composed state $0 = $J@ 60 is also pure even if this system is considered as semiquantum, corresponding to the Cartesian product ($,O) of the initial pure classical v = 0 and quantum states $ E b. Despite this fact one can easily see that the unitary operator S induces in W = b @ g the mixed state for the quantum-classical system, although it is still described by the vector $1 = S$O E W as the wave function $1 ( ~ , vof ) the “atom+cat” corresponding to $0 = $ 8 6 . Indeed, the potential observables of such a system at the time of observation t = 1 are all operator-functions X of 0 with values X (w)in Hermitian 2 x 2-matrices, represented as block-diagonal (T,v)-matrices X = [ X (v) S,”,] of the multiplication X (v) $1 (., v) at each point ‘u E (0,l). This means that the amplitude $1 (and its compound density matrix P$,) induces the same expectations
( X )=
c
$1
(4+x >.(
$1
(v>=
v
as the block-diagonal density matrix
c
Trx (v)e (v)= nX4
(3)
V
4 = [e( v )a,”,] of the multiplication by
233
Here ~ ( v=) I $ ( v ) ~ ~ , F ( v ) = Iv)(vl is the projection operator Fo if v = 0 and F1 if v = 1 represented as the multiplication
[F (v)$1
(7)= 6 (v A
I.
$ (7)= $ (v)6,
(4)
(7) 7
by 6, (.) = 6 (. A v), and PF(,)+= F, is also this projector onto 6, (-) =. ) .1 The 4 x 4-matrix d is a mixture of two orthogonal projectors F, €3E,, v = 0,1, where E, = Pa,: 1
6 = [F (v)6,vtw (v)]=
X T(v)F, @ E,. v=o
The only remaining problem is to explain how the cat, initially interacting with atom as a quantum bit described by the algebra A = B ( g ) of all operators on g, after the measurement becomes classical, is described by the commutative subalgebra C = 23 (g) of all diagonal operators on g. As will be shown in the next section even, this can be done in purely dynamical terms if the system “atom plus cat” is extended to an infinite system by adding a quantum string of “incoming cats” and a classical string of “outgoing cats” with a potential interaction (1) with the atom at the boundary. The free dynamics in the strings is modelled by the simple shift which replaces the algebra A of the quantum cat at the boundary by the algebra C of the classical one, and the total discrete-time dynamics of this extended system is induced on the infinite semi-classical algebra of the “atom plus strings” observables by a unitary dynamics on the extended Hilbert space 7-l = W@Zo. Here W = g @ g, and ?lo is generated by the orthonormal infinite products IT^, v r ) = @1)~i, vi) for all the strings of quantum TO” = (71, ~ 2 ,...) and classical v r = ( q , v 2 , . . .) bits with almost all (but finite number) of T,, and v, being zero. In this space the total dynamics is described by the single-step unitary transformation
U
: 170,v)@
IT
u TO”, v r ) I+ I T O,
TO
+T)@
incorporating the shift and the scattering S, where (which coincides with T A v = I T - vl),and ITF,VUVr)=1Ti,T2
IT^, v LI vr), T
,...) @ I v , v 1 , 2 1 2 , ...),
(5)
+ v is the sum mod2 7,VE
{0,1}
are the shifted orthogonal vectors which span the whole infinite Hilbert product space 7-lo = @,>owr of W, = gr @ gT (the copies of the four-dimensional Hilbert space W). Thus the states ]TO”,vr) = 170”)@ Ivr) can be interpreted as the products of two discrete waves interacting only at the boundary via the atom. The incoming wave 170”) is the quantum probability amplitude wave describing the state of “input quantum cats”. The outgoing wave I v r ) is the
234
classical probability amplitude wave describing the states of “output classical cats.” Now consider the semi-classical string of “incoming and outgoing cats” as the quantum and classical bits move freely in opposite directions along the discrete coordinate r E N. The Hamiltonian interaction of the quantum (incoming) cats with the atom at the boundary T = 0 is described by the unitary scattering (1). The whole system is described by the unitary transformation (5) which induces an injective endomorphism 6 ( A ) = UtAU on the infinite of the atom-cat observables X E A at r = 0 product algebra U = A @ and other quantum-classical cats ‘2l0 = @,>OAT.Here A = a($)@ C is the block-diagonal algebra of operator-valued functions ( 0 , l ) 3 v I+ X (v) describing the observables of the string boundary on the Hilbert space b @ g7 where $ = (C2 = g, and A, = B (b,) @ C, are copies of represented on tensor products $, @ g, of the copies $, = (C2 = gr at r > 0. The input quantum probability waves 170”) = @,>o~T,) describe initially disentangled pure states on the noncommutative algebra l3 (3-10) = @,,oB (b,) of “incoming quantum cats” in 3-10 = @,>o$,, and the output classical probability waves I v r ) = @r>~(v,)describe initially pure states on the commutative algebra = @,>oC, of “outgoing classical cats” in GO = @,.>og,. At the boundary T = 0 there is a transmission of information from the quantum algebra B (3-1) on 3-1 = bB3-10to the classical one C = C@Gon Q = g@Go which is induced by the Heisenberg transformation 19 : Q Q. Note that although the Schrodinger transformation U is reversible on W = 3-1 @ Q, U-l = U t , and thus the Heisenberg endomorphism 19 is one-to-one on the semi-commutative algebra 0 = 23 (3-1) @C, it describes an irreversible dynamics because the image subalgebra 6 (U) = UtQU of the algebra Q does not coincide with 2l C B (W). The initially distinguishable pure states on B may become identical and mixed on the smaller algebra U t UU, and this explains the decoherence. Thus, this dynamical model explains how the actual events E, = Pa, on the part of the cats remain compatible with any future observable, despite their Hamiltonian interaction with the atom. They simply remain in the center of the total semiclassical system in the Heisenberg picture for the chosen arrow of time. This quantum causality is achieved only due to the admittance of the infinite degree of freedom for the auxiliary system (infinite number of incoming and outgoing cats) having the irreversible Heisenberg dynamics induced by the unitary shift and the reversible interaction. Because of the increase of the center, the pure initial states of the total dynamical system become mixed even though they are evolved by a unitary transformation, and the von Neumann irreversible decoherence u = P+ I+ p of the atomic state is due to the ignorance of the results of the measurements described by the f&
235
partial tracing over the cat’s Hilbert space g = C2 on each step:
c 1
p =n
g @
=
7l
(v)Fu = e ( 0 )+ e (1),
(6)
u=o
where e(v) = F,. It has entropy S ( p ) = -Trplogp of the compound state @ of the combined semi-classical system prepared for the indirect measurement of the disintegration of the atom by means of cat’s death:
v=o
It is the initial coherent uncertainty in the pure quantum state of the atom described by the wave-function II, which is equal to one bit in the case I$ (0)12 = 1/2 = I$ (l)12.Each step of the unitary dynamics adds this entropy to the total entropy of the state on (21 at the time t E N, so the total entropy produced by this dynamical decoherence model is equal to exactly t. The described dynamical model of the measurement interprets filtering p I+ u, simply as the conditioning
r, = e (v)./ (v)= F,
(7)
of the joint classical-quantum state e ( . ) with respect to the events E, = Iv)(vl = Ps, on the part of the cat by the Bayes formula which is applicable due to the commutativity of actually measured observables C E C generated by E, (the life observables of cat at the time t = l),with any other potential observable of the combined semi-classical system. Thus the atomic decoherence is derived from the unitary interaction of the quantum atom with the cat which should be treated as classical due to the projection superselection rule in the “Heisenberg” picture of von Neumann measurement. The spooky action at distance, affecting the atomic state by measuring v, is simply the result of the statistical inference (prediction after the measurement) of the atomic posterior state u, = F,: the atom disintegrates if and only if the cat is dead. 4
Stochastic Decoherence Equation
Quantum events are usually observed in the form of quantum jumps which occur spontaneously in the continuous time. As it was already mentioned in the introduction, there are many phenomenological theories of quantum jumps, but neither gives a dynamical explanation of these jumps. The time
236
which appears in these recent theories is not the time at which the experimentalist decides to make a measurement on the system, but the time at which the system does something for the experimenter to be observed. What it actually does and why, remains an unexplained mystery in these theories. In this section we shall make the first step towards such an explanation, following the line suggested by John Bell [17]; that the “development towards greater physical precision would be to have the ‘jump’ in the equations and not just the talk - so that it would come about as a dynamical process in dynamically defined conditions”. We shall use the same quantum causality idea and the quantum filtering (conditioning) method as for the toy Schrodinger’s cat model in the previous section. Other phenomenological continuous reduction and spontaneous localization models [20,21,22,23,24,25,26,27] of the individual continuous-in-time stochastic decoherence, lead to quantum state diffusions which we also derived by the filtering method [14,28,29] (see also the recent review [30]). It is also possible to treat these as appropriate Dirac type boundary value problems, and such treatment will be soon published elsewhere. The generalized, stochastic wave mechanics which enables us to treat the quantum spontaneous events such as quantum state jumps, and diffusions in the unstable systems and other stochastic processes of time-continuous observation, or in other words, quantum mechanics with stochastic trajectories w = (zt), was discovered relatively recently, in [14,15,28]. The basic idea of this theory is to replace the deterministic unitary Schrodinger propagation $ I+ $ ( t )by a linear causal stochastic one $ I+ $ (t,w ) which is not necessarily unitary for each history w, but is unitary in the mean square sense, M [ll$(t)112] = 1, with respect to a standard probability measure p (dw) for the measurable history subsets dw. The unstable quantum systems can also be treated in the stochastic formalism by relaxing this condition, allowing the decreasing survival probabilities M [ll$(t)112] 5 1. Due to this, the positive measures lim P ( t ,dw) P (t,dw) = 11t,h ( t ,w)l12p (dw) , ji (dw) = t-hw are normalized (if 11$1,11 = 1) for each t , and are interpreted as the probability measure for the histories wt = { (0,t] 3 T I+ z T }= of the output stochastic process xt with respect to the measure fi. In the same way as the abstract Schrodinger equation can be derived from only the unitarity of propagation, the abstract decoherence wave equation can be derived from the mean square unitarity in the form of a linear stochastic differential equation. The reason that Bohr and Schrodinger didn’t derive such an equation despite their firm belief that the measurement process can be described ‘asif it were in reality in
~9
237
the physical world’ is that the appropriate (stochastic and quantum stochastic) differential calculus had not been yet developed early in that century. As classical differential calculus has its origin in classical mechanics, quantum stochastic calculus has its origin in quantum stochastic mechanics. Assuming that the superposition principle also holds for the stochastic waves such that $ ( t , w ) is given by a linear stochastic propagator V ( t , w ) , let us derive the general linear stochastic wave equation which preserves the mean-square normalization of these waves. Note that the abstract Schrodinger equation = E$ can also be derived as the general linear deterministic equation which preserves the normalization in a Hilbert space b. For notational simplicity we shall consider here only the finite-dimensional, maybe complex trajectories xt = (xi), k = 1 , . . . ,d. The infinite-dimensional trajectories (fields) with even continuous index k can be found elsewhere (e.g. in [14,361). It is usually assumed that the these x i as input stochastic processes have stationary independent increment dxi = xL+dt - x i with given expectations M [dxi] = Akdt. The abstract linear stochastic decoherence wave equation is written then as
imt$
Here E is the system energy operator (the Hamiltonian of free evolution of the system), R = Rt is a selfadjoint operator describing a relaxation process in the system, Lk are any operators coupling the system to the trajectories xk, and we use the Einstein summation rule LkXk = C L k z k Lx, A2 = with Ak = x k . In order to derive the relations between these operators which will imply the mean-square normalization of $ ( t , ~ let ) , us rewrite this equation in the standard form
=
+
d$ ( t ) K$ ( t )dt
Lk$ ( t )dyi,
A2
K = -R 2
i + -E ti
- LA,
(8)
where y i = x i - t A k are input noises as zero mean value independent increment processes with respect to the input probability measure p. Note that these noises will become the output information processes which will have dependent increments and correlations with the system with respect to the output probability measure ji = P (00, dw). If the Hilbert space valued stochastic process $ ( t , w ) is normalized in the mean square sense for each t , it represents a stochastic probability amplitude $ ( t )as an element of an extended Hilbert space 3c0 = b @ L;. The stochastic process t I+ $ ( t )describes a process of continual decoherence of the initial pure state p (0) = P+ into the
238
mixture
of the posterior states corresponding to GU (t) = $ (t,w ) / 1111, (t,w)ll, where M denotes mean with respect to the measure p. Assuming that the conditional expectation (d&dy:), in (d(lllt$))t = (d$td$
+ $+d$ + Wt$)t
= $t (Lkt (dfjkdYk)tLk - (K
+ Kt) dt) $
is dt (as in the case of the standard independent increment processes with 5 = y and (dy)2 = dt+adyt), the mean square normalization in the differential form (d($t$))t = 0 (or (d($t$)), I 0 for the unstable systems) can be expressed [15,29] as K Kt 2 LtL. In the stable case this defines the selfadjoint part of K as half of LtL, i.e.
+
where H = Ht is the Schrodinger Hamiltonian in this equation when L = 0. One can also derive the corresponding Master equation
for mixing decoherence of the initially pure state p ( 0 ) = $$t, as well as a stochastic nonlinear wave equation for the dynamical prediction of the posterior state vector (t), the normalization of $ (t, w ) at each w . 5
Quantum Jumps as Unstabie Dynamics
Actually, there are two basic standard forms [28,16] of such stochastic wave equations, corresponding to two basic types of stochastic integrators with independent increments: the Brownian standard type xi = b i , and the Poisson standard type x: = n: with respect to the basic measure p. We shall consider only the Poisson case of the identical ni having all the expectations Mn: = ut and characterized by a very simple differential multiplication table dn: ( w ) dni ( w ) = $dn; ( w )
as it is for the only possible values dn: = 0 , l of the counting increments at = n:/u1f2 such that they have the expected each time t. By taking all rates x k = u1f2 we can get the standard Poisson noises y: = xi - v1f2tG mtk
239
with respect to the input Poisson probability measure p = P, the multiplication table dmkdml = &(dt
+ U-lI2drnk),
described by
dmkdt = 0 = dtdmk,
Let us set now in our basic equation (8) the Hamiltonian H = ti (K - Kt) /2i and the coupling operators Lk of the form Lk = X(Ck - I), H = E
+ i -U2( C k - Ck),
with the coupling constant A = u1/2 and Ck 2 C l given by the collapse operators Ck (e.g. orthoprojectors, or contractions, CiCk 5 I). This corresponds to the stochastic decoherence equation of the form
where R 2 CtC - I, or in the standard form (8) with y: = mi. In the stable case when R = CtC - I this was derived from a unitary quantum jump model for counting nondemolition observation in [15,39]. It correspond to the linear stochastic decoherence Master-equation de(t)
+ [Ge(t)+ e(t)Gt - w(t)]dt = [cke(t)ck- e(t>ldni, e(o>= P,
for the not normalized (but normalized in the mean) density matrix e ( t , w ) , where G = 5ckck $E (it has the form $ ( t , w ) $ (t,w)t in the case of a pure initial state p = $$+). The nonlinear filtering equation for the posterior state vector
+
$ld
( t )= $ (t,w ) / 1M (t,w>ll
has in this case the following form [16];
where 11$11 = ($I$J)~/~ (see also [38] for the infinite-dimensional case). It corresponds to the nonlinear stochastic Master-equation dp,
+ [Gp, + pwGt - ~p,TrC'p,Ck]dt
= [Ckpp,Ck/TrCkpwCk - p]dnt,'P,
240
for the posterior density matrix pw ( t ) which is the projector P., ( t ) = $J., ( t ) (t)t for the pure initial state pw (0) = P+. Here (t, w ) = are the output counting processes which are described by the history probability measure
+.,
ni
nk2t)
P (t,dw) = n ( t ,w ) p (dw) , n (t,w ) = Trp (t,w )
with the increment dni ( t )independents of
ni ( t )under the condition pw ( t )=
p and the conditional expectations
M [dni ( t )Ip., ( t )= p] = VTrCipCkdt which are vllCl+112dt for p = P+. The derivation and solution of this equation was also considered in [37], and its solution was applied in quantum optics in [31,32]. This nonlinear quantum jump equation can be written also in the quasilinear form [28,16]
W., ( t )+ e ( t )l/lw ( t )dt = Lk ( t )lcIw (t) d&'P,,
(11)
~522~'~)
where mi (t, w ) = are the innovating martingales with respect to the output measure which is described by the differential dfhi ( t )= v-1/211Cl~.,(t)ll-1dni(t)- v ~ ' ~ ~ ~ C ~dt $J.,(~)~~ with p = P+ for $J= $., ( t )and the initial similar to K has the form
fki (0) = 0, the operator
(t)
e ( t )= -fz (t)t z ( t )+ ifi ( t ), 2 ti and
6 ( t ),z(t) depend on t (and w ) through the dependence on -1c, = $.,
(t):
The latter form of the nonlinear filtering equation admits the central limit v -+ 00 corresponding to the standard Wiener case when yk = wk, dwkdwl = $dt,
dwkdt = 0 = dtdwk,
with respect to the limiting input Wiener measure p. If Lk and H do not depend on v, i.e. Ck and E depend on v as c k
=I+V-1/2Lk,
y1/2 E=H+-(Ll-Lk), 22
241
then 6;(t) + 6:, where the innovating diffusion process
defined as
d6: ( w ) = dw:(w) - 2Re ($w(t)lLL$u(t)) dt, are also standard Wiener processes but with respect to the output probability measure ji (dw) = p (d3) due to d'lZkd6l = 6idt,
d6kdt = 0 = dtd6k.
If (t)II = 1 (which follows from the initial condition II$II = l), the stochastic operator-functions ( t ) , (t) defining the nonlinear filtering equation have the limits
zk
-Lk = Lk - Re($ILk$),
=H
+ 2 (LL - Lk)Re ($ILk$).
The corresponding nonlinear stochastic diffusion equation d$w (t)
+ K (t)
$w
( t )dt = Zk (t) $w (t) d6:
was first derived in the general multi-dimensional density-matrix form dpw
+ [Kpw + pwKt - L'pwLk]
dt
+
+
= [Lkpp, pwLkt- pwTr(Lk Lkt)pw]d8:
for the renormalized density matrix pw = p ( w ) / T r p (w) in [14,?] from the microscopic reversible quantum stochastic unitary evolution models by the quantum filtering method. It was applied in quantum optics [23,25,24,26]for the description of counting, homodyne and heterodyne time-continuous measurements introduced in [39]. It has been shown in [21,27] that the nondemolition observation of such a particle is described by filtering of the quantum noise which results in the continual collapse of any initial wave packet to the Gaussian stationary one localized at the position posterior expectation. The connection between the above diffusive nonlinear filtering equation and our linear decoherence Master-equation de(t) + [Ke(t) + e(t)Kt - Lk@(t)Lk] dt = [Lke(t) + e(t)~"] dwi,
e(o>= P,
for the stochastic density operator e (t, w ) defining the output probability density Tre(t,w), was well understood and presented in [20,26]. However it has also found an incorrect mathematical treatment in recent Quantum State Diffusion theory [40] based on the case E = 0 of the filtering equation. This particular nonlinear filtering equation is empirically postulated as the 'primary quantum state diffusion', and its more fundamental linear version d$+K$dt = Lk$dwk is 'derived' in [40] simply by dropping the non-linear terms without
242
appropriate change of the probability measures for the processes @k = Iijk and Y k = Wk. 6
The Derivation of Jumps and Localizations
Here we give the solution of the quantum jump problem for the stochastic model described by the equation (9) in the case CtC 5 I, R = 0 which corresponds to the Hamiltonian evolution between the jumps with energy operator E, and the jumps are caused only by the spontaneous decays or measurements. The corresponding boundary value solution for the general stochastic decoherence equation will be published elsewhere. When CtC = I, the quantum system certainly decays at the random moment of the jump d n i = 1 to one of the m products ending in the state Ck$/llCk$lI from any state $ E b with the probability llCk$[12,or one of the measurement results k = 1 , . . . ,m localizing the product is gained at the random moment of the spontaneous disintegration. The spontaneous evolution and its unitary quantum stochastic dilation was studied in detail in [41]. When CtC < I, the unstable system does not decay to one of the measurable products with probability ll$1I2 - llC$112, or no result is gained at the jump. This unstable spontaneous evolution and its unitary quantum stochastic dilation was considered in details for one dimensional case m = 1 in [42]. First, we consider the operator C as a construction (or isometry if CtC = I) from b into b 8 f, where f = Cm. We dilate this C in the canonical way to the selfadjoint scattering operator
S=
[-(I -
CtC)1/2 Ct C (I 8 i - CCt)l/2]
where I 8 i is the identity operator in b 8 f, and Ct is the adjoint construction b 8 f + b and CCt is a positive construction (orthoprojector CtC = I) in this space. The unitarity St = S-' of the operator S = St in the space g = C @ f = C1+m is easily proved as (I - CtC)
s2=[
+ CtC
0 CC'+(I@i-CC')I
=
,:[ I S ]
by use of the identity (I 8 i - CCt)'/2C = C(I - CtC)ll2. The operators Ck = Sgk are obtained from S as the partial matrix elements (I @ (kl)S (I 8 10)) corresponding to the transition of the auxiliary system (pointer) form the initial state 10) = 1 @ 0 to one of the measured orthogonal states 0 @ Ik), Ic = 1 , . . . ,m in the extended space g.
243
Second, we consider two continuous seminfinite strings indexed by s = kr, where r > 0 is any real positive number (one can think of s as the coordinate on the right or left semistrings on the real line without the point s = 0). Let us denote by g@= g1 €3 g2 €3 .. . mg, the infinite tensor product generated by ax,,with almost all components X, E g, 3 g equal to cp = 10). We shall consider right-continuous amplitudes @ (w) with values in g@for all infinite increasing sequences w = { T I , r 2 , . . .}, rn-l < r, having a finite number nt (w) = Iw n [O,t)l of elements r E wt in the finite intervals [ O , t ) for all t > 0 such that f (T,) E g, for the generating products @ (v) = @f (r,). Let us also define the Hilbert space L; €3 g@of the square-integrable functions w I+ @ (w) in the sense
=
ll@1I2= /(@
).(
I@ (.I)
dP:
= M(11@(*)112)<
-
with respect to the standard Poisson measure p: = P, with the constant intensity u on R+. In other words we consider a countable number of the similar auxiliary systems (Schrodinger cats, bubbles or other pointers) described by the identical state spaces g, as independent and randomly distributed on R+, with the average number u on any unit interval of the string. Let Q* = L; €3 g z be two copies of such space, one for the past, another for the future, and let 0 = L; €3 G@,with G = g- €3 g+, be canonically identified with the space 8- €3 Q+ of square-integrable functions 9 (w-, w+) with respect to pE €3 p:, having the values in g? €3 .g: In particular, the ground (v) = @4:, where 4 O = cp €3 cp, state described by the constant function is identified with @."_ €3 @$,where @% (v) = @cp,. In order to maintain the quantum causality we shall select a decomposable algebra 24- @J 2l+ of the string with left string being classical, described by the commutative algebra Q- = V (Q-), and the right string being quantum, described by the commutant I#+ of the Abelian algebra LF of random scalar functions f : w+ H C represented by multiplication operators f on Q+. More precisely, are the von Neumann algebras generated respectively on Q* by the functions w* I+ A(w*) with operator values A ( v + ) E d? and A ( v - ) E d?,where d? = @B (g,) is the algebra of all bounded operators B (g@) corresponding to A+ = B (C1+m),and d? = €323 (g,) is its diagonal subalgebra 2)(g@) corresponding to A- = V (C1+,). The canonical triple (0,2l, @ O ) is the appropriate candidate for the dynamical dilation of quantum jumps in the unstable system described by our spontaneous localization equation. Third, we construct the time-continuous unitary group evolution which will dynamically induce the spontaneous jumps in the interaction representation at the boundary of the string. Let Tt be the one parametric con-
244
tinuous unitary group on 9 = 9- 8 G+ describing the free evolution by right shifts at (w)= 9 (w- t ) when 9 (w-, v+) is represented as 9 (w) with (-we) U (+v+) c R for the two sided string parametrized by R 3 w . The corresponding Hilbert space 9 for such 9 will be denote as Go] 8 90 with Go = 9+ and Go] obtained by the reflection of 9-. The free dynamics Tt in 9- 8 E+ induced by the shift in Go] 8 Go can be written as
Tt9 (w-,v+)= 9 (v”,.:)
,
where vi = f[([(-v-)U (+v+)] - t ) n R,]. Let us denote the selfadjoint generator of this free evolution on 6 by P such that Tt = e--iPt/tr. This P is the first order operator
on R+ which is well defined on the differentiable functions Qr (v) with respect to each r E v which are not constants only for a finite number of r E w. It is selfadjoint on a natural domain V O in 9 corresponding to the boundary condition 9 (0 U v-,v+) = (v-,0 U w+) for all w, > 0, where Qr (0 U v-,v+) = lim Qr
( r U v-,v+),
rl0
9(v-,OUv+) = l i m 9 ( v - , r U v + ) rl0
as induced by the continuity condition at s = 0 on the whole R. Here r U v is adding a point r 4 v to the ordered sequence = (0,r1, . . .}, and 0 U vf can be formally treated as adding two zeroth s = f O to { f r l , f r 2 , . . . } such that --T < -0 < +O < +r for all r > 0. Note that the Hamiltonian -P, not +P corresponds to the right free evolution in the positive arrow of time in which the states 9 (w+) describe the incoming “from the future” quantized waves, and the states 9 (v-)describe the outgoing “to the past” classical particles. This is a relativistic many particle Dirac type Hamiltonian on the half of the line Iw corresponding to zero particle mass and the orientation of spin along R. Note that this Hamiltonian has unbounded from below spectrum in the Hilbert space of not restricted momenta. However as we showed in [3,43], the Heisenberg free evolution corresponding to this Hamiltonian can be obtained as a WKB approximation in the ultra-relativistic limit (p) + 00 of any free evolution with bounded from below Hamiltonian, e.g. with a positive single particle Hamiltonian E (p) = IpI. The unitary group evolution Utcorresponding to the scattering interaction at the boundary with the unstable system
245
which has its own free evolution described by the energy operator E can be obtained by resolving the following generalized Schrodinger equation
a
ifi-Qt
at
(v-,w+)+ PQt (v-,w+)= (E €3 I ) Qt ( t ,v-, v+)
(14)
in the Hilbert space 'fl = $ €3 G, with the following boundary condition Qt (0u 21- ,v+) = SOP (v- ,0 u v+) ,
vt > 0, w& > 0.
(15)
Here SOis the boundary action of S defined by SOQ(0 U v) = S [$ @ cp] €3 CP (v) on the products Q (0 U v) = $ @ cp €3 @ (w).Due to the unitarity of the scattering matrix S this simply means IIQ (0u v-,v+)II = IIQ (v-,O u w+)II, that is Dirac current has zero value at the boundary T = 0. Note that this natural Dirac boundary condition corresponds to an unphysical discontinuity condition Q (-0 U w ) = (S €3 I) Q (+0 U w ) at the origin s = 0 when the doubled semi-string is represented as the two-sided string on P. Fourth, we have to solve this equation, or at least to prove that it has a unitary solution which induces the injective Heisenberg dynamics on the algebra 23 = B (g) 8 U of the combined system with the required properties U-t%Ut 23. The latter can be done by proof that the Hamiltonian in the equation (14) is selfadjoint on a natural domain corresponding to the perturbed boundary condition (15). It has been done by finding the appropriate domain for the perturbed Hamiltonian in the recent paper [44].However we need a more explicit construction of the time continuous resolving operators U t : Q I-+ Qt for the equation (14). To obtain this we note that this equation, apart from the boundary condition, coincides with the free evolution equation given by the Hamiltonian -P up to the free unitary transformation e-aEtlK in the initial space $. This implies that, apart from the boundary, the evolution U t coincides with e-iEtlfi @ T-t such that the interaction evolution U ( t )= TtW, where Tt is the shift extended trivially onto the component of 'fl = Go] @ g €3 Go, is adapted in the sense
fil)
Gil
fil
where 4(t) = W i ( $ €3 E $ €3 for all $ E and E 3;l. Here we use the tensor product decomposition GO = F,'@ Gt, where F,]is Fock (w)= BTEVf( T ) with finite v C [0, t ) and space generated by the products f ( T ) E g (we use the representation of 'fl as the Hilbert space Go] €3 8 €3 GO for the two sided string on R with the measured system inserted at the origin s = 0, and the notations & , = G+, Go] = G-, identifying s = +r with z = Is1
fil
246
for all T > 0, and s = -T with z = - Is1 including T = 0). Moreover, in this interaction picture U ( t )is decomposable,
[ U ( t ) P ] ( w= ) u ( t , w ) s ( w ) , U ( t , w ) = I01 8 w2(w;l) 8 It as each W: is decomposable into the unitary operators W$ (u) in 8 Fil (v) with F,,’ (u) = 8ILlgn for any finite v C [ O , t ) . The dynamical invariance U-tBUt E 23 of the algebra 23 = B (b) 8 B for any positive t > 0 under the Heisenberg transformations of the operators B E 23 induced by U t = T-JJ ( t ) simply follows from the right shift invariance Ttf?T-t E ‘15 of this algebra and U (t,w ) t 23 ( w ) U ( t , w ) = 23 ( w ) for each w due to unitarity of W2 (u) and the corresponding simplicity of the local algebras of future Bil ( u ) = B (b) 8 to A+ = B ( 0 ) . And finally we can find the nondemolition processes NL, k = 1 , . . . ,m which count the spontaneous disintegrations of the unstable system to one of the measured products k = 1,.. . ,m, and with N& corresponding to the unobserved jumps. These are given on the space 8- as the sums of the orthoprojectors Ik)(kl E g corresponding to the arriving particles at the times rn E u- up to the time t : n’(v-)
C
Ni(~-,v+)=18 1~-”8(~k)(k~81+)8In. n= 1
(16)
Due to commutativity of this compatible family with all operators B 8 I of the unstable system, the Heisenberg processes Nk ( t ) = U-tNLUt satisfies the nondemolition causality condition. The independent increment quantum nondemolition process with zero initial expectations corresponding to y i = mk in the standard jump-decoherence equation (8) then is given by
Y i = v-l12 (NL - vt) = Xi - At, where Xi = X-ll2NL with the coupling constant X = d12. Hence the quantum jumps, decoherence and spontaneous localization are simply derived from this dynamical model as the results of inference, or quantum filtering without the projection postulate by simple conditioning. The statistical equivalence of the nondemolition countings NL in the Schrodinger picture Pt = Ut (Ic, 8 9”) of this continuous unitary evolution model with fixed initial ground state 9” = 9 : @I 9 : and the stochastic quantum reduction model based on the spontaneous jump equation (9) for an unstable system corresponding to R = 0 and CtC 5 I was proved in the interaction representation picture in [41,38].
247
Note that before the interaction the probability to measure any of these products is zero in the initial states @' (u-, u+) = @& as all 4; = 10) @ 10). The maximal decreasing orthoprojector
Et(u-,v+) = I@&(O) @'..@&~(,-)(O)
En = lO)(Ol
@L+-),
@ I+,
which is orthogonal to all product countings NL, k = 1 , . .. ,m, is called the survival process for the unstable system. Thus, the quantum jumps problem has been solved as the time-continuous quantum boundary-value problem formulated as Let V ( t , w ) = V ( t , m $ ) ,t E R+ be a reduction family of isometries o n 9 into f~ 8 Lz resolving the quantum j u m p equation (9) with respect t o the input probability measure p = P , for the standard Poisson noises defining the classical means
mi
M [ g V ( t ) ' B V ( t ) ] = / g ( Y : ) V ( t , Y o t1 ) ' B V ( t , y t l ) dP,. Find a triple (G,%, @) consisting of a Hilbert space G = [email protected]+ embedding the Poisson Hilbert space L; by a n isometry into G+, a n algebra M = 2- @%+ o n li with a n Abelian subalgebra U- generated by a compatible continuous family Y!', = {Yi, k = 1 , . . . ,d , s 5 0) of observables (beables) o n G-, and a statevector Qo = @: @ E Q such that there exist a t i m e continuous unitary group Ut o n N =B @ Q, inducing the dynamical semigroup of endomorphisms Q 3 B e UdtBUt E !B, which represents this reduction o n the product algebra Q = B (9) 8 B as
+;
M[gV(t)+BV(t)= ] n'(4-t @ B ) . Here
7rt
is the quantum conditional expectation
nt ( i - t @ B) = (I €Q a')' U-t (I3 8 g-t (Y!],)) Ut (I @ * O )
,
which provides the dynamical realization of the reduction as statistically causal inference about any B E B ( f ~ ) with respect t o the algebra 2L of the functionals i-t = g - t ( ~ -01, ) of operators ~1.1 = { Y S : s E (-t, 01) , all commuting o n G, representing the shifted measurable functionals g-t(g"]) = g ( y i l ) of gotl = {y: : r E (0, t ] }for each t > 0 in the center C of the algebra 2 ' 3. Note that despite strong continuity of the unitary group evolutions Tt and Ut , the interaction evolution U (t,w ) is time-discontinuous for each w . It
248
is defined as U ( t )= Iol8 W ( t )for any positive t by the stochastic evolution W ( t )= Wil 8 It resolving the Schrodinger unitary jump equation [41,38] d90 (t,v)
+
(E8 I) 90(t,v)dt = (S - I ) t (v)90(t,v)dnt (v)
(17)
on ?l =o 0 8 GO as q0 (t,v) = W (t,v)90.Here Lt (v)= L,t(,) 8 It is the adapted generator L = S - I which is applied only to the system and the n-th particle with the number n = nt (v)on the right semistring as the operator L, = T:It (It-'] 8 LO)T:] obtained from Lo = TLTt by the transposition operator T ( $ 8x) = x 8 $ generating T]: ($ 8 xi') = xi18 II, on b @ g': by the recurrency
Ti] =
(.
)(
@ T T:-l]
@I),
TE1=I.
The operator W: can be explicitly found from the equivalent stochastic integral equation
with WE1 = I. Indeed, the solution to this integral equation can be written for each v in terms of the finite chronological product of unitary operators as in the discrete time case iterating the following recurrency equation
W: (v)= e-iEt/hS (tn>(w:]
(v)8 I) ,
w:]
(v)= I,
where S(t,) = eiEt-/"Sne-aEtnIh with n = n t ( w ) . From this we can also obtain the corresponding explicit formula [41]
v (t,v.)= e--iEt/fiCk*(t,) (V (tn,v)8 I), v (v.)= I, resolving the reduced stochastic equation (9) as $ ( t ) = V ( t )1c, for R = 0 and any sequence v.of pairs ( r n ,k,) with increasing { r , } , where n = nt (v.) is the maximal number in {T,} n [0, t).
7 Conclusion: A Quantum Message from the Future Although the conventional formulation of quantum mechanics and quantum field theory is inadequate for the temporal treatment of modern experiments with the individual quantum system in real time, it has been shown that the latest developments in quantum probability, stochastics and in quantum information theory made it possible to reconcile the dynamical and statistical aspects of its interpretation. All these observable phenomena, such as quantum
249
events and quantum jumps, which do not exist in usual quantum mechanical formalism but do exist in the modern experimental quantum physics, can be interpreted in the modern mathematical framework of quantum stochastic processes in terms of the results of the generalized quantum measurements. The problem of quantum measurement which has always been the greatest problem of interpretation of the mathematical formalism of quantum mechanics, is unsolvable in the orthodox formulation of quantum theory. However it has been recently resolved in a more general framework of the algebraic theory of quantum systems which admits the superselection rules for the admissible sets of observables defining the physical systems. The new superselection rule, which we call quantum causality, or nondemolition principle, can be formulated in short as the following resolution of the corpuscular-wave dualism: the past is classical (encoded into the trajectories of the particles), and the future is quantum (encoded into the propensity waues for these particles). This principle does not apply (it simply does not exist) in the usual quantum theory with finite degrees of freedom. And there are no events, jumps and trajectories and other physics in this theory if it is not supplemented with the additional phenomenological interface rules such as the projection postulate, permanent reduction or a spontaneous localization theory. This is why it is not applicable for our description of the open quantum world from inside of this world as we are a part of this world, but only for the external description of the whole of a closed physical system as we are outside of this world. However the external description doesn’t allow to have a look inside the quantum system as any flow of information from the quantum world (which can be obtained only by performing a measurement) will require an external measurement apparatus, and it will inevitably open the system. This is why there is no solution of quantum measurement and all paradoxes of quantum theory in the conventional, external description. Recent dynamical models of the phenomenological theories for quantum jumps and spontaneous localizations, although they all pretend to have a primary value, extend in fact the instantaneous projection postulate to a certain, counting class of the continuous in time measurements. As was shown in the paper, there is no need to supplement the usual quantum mechanics with any such mysterious quantum spontaneous localization principles, even if they are formulated in continuous time. They all have been derived from the time continuous unitary evolution for a generalized Dirac type Schrodinger equation, and ‘that something’ that the system does to be spontaneously observed, is simply caused by a singular scattering interaction at the boundary of our Hamiltonian model. The quantum causality principle provides a time continuous nondemolition counting measurement in the extended system which
250
enables to obtain ‘all this damned quantum-jumping’ simply by time continuous conditioning called quantum jump filtering. Our mathematical formulation of the extended quantum mechanics equipped with the quantum causality to allow events and trajectories in the theory, is just as continuous as Schrodinger could have wished. However it doesn’t exclude the jumps which only appear in the singular interaction picture, as they are a part of the theory but not only of its interpretation. Although Schrodinger himself didn’t believe in quantum jumps, he tried several times, although unsuccessfully, a possible way of obtaining the continuous reduction from a generalized, relativistic, “true Schrodinger”. He envisaged that ‘if one introduces two symmetric systems of waves, which are traveling in opposite directions; one of them presumably has something to do with the known (or supposed to be known) state of the system at a later point in time’ [45], then it would be possible to derive the ‘verdammte Quantenspringerei’ for the opposite wave as a solution of the future-past boundary value problem. This desire coincides with the “transactional” attempt of interpretation of quantum mechanics suggested in 1461 on the basis that the relativistic wave equation yields in the nonrelativistic limit two Schrodinger type equations, one of which is the time reversed version of the usual equation: ‘The state vector $J of the quantum mechanical formalism is a real physical wave with spatial extension and it is identical to the initial “offer wave” of the transaction. The particle (photon, electron, etc.) and the collapsed state vector are identical to the completed transaction.’ There was no proof of this conjecture, and now we know that it is not even possible to derive the quantum state diffusions, spontaneous jumps and single reductions from such models involving only a finite particle state vectors $J ( t ) satisfying the conventional Schrodinger equation. The nondemolition principle defines what is actual in the reality and what is only possible, what are the events and what are just the questions, and selects from the possible observables the actual ones as the candidates for Bell’s beables. It was unknown to Bell who wrote that “There is nothing in the mathematics to tell what is ‘system’ and what is ‘apparatus’, . . .” , in 117: p. 1741. The mathematics of quantum open systems and quantum stochastics defines the extended system by the product of the commutative algebra of the output trajectories, the measured system, and the noncommutative algebra of the input quantum waves. All output processes in the apparatus are the beables which “live” in the center of the algebra, and all other observables which are not in the system algebra, are the input quantum noises of the measurement apparatus whose quantum states are represented by the offer waves. These are the only possible conditions when the posterior states exist
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as the results of inference (filtering and prediction) of future quantum states upon the measurement results of the classical past as beables. The act of measurement transforms quantum propensities into classical realities, and our model explains this as a result of the dynamical propagation from quantum future through the present as the boundary into the statistical inference from the classical past. References
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16. V. P. Belavkin: A stochastic posterior Schrodinger equation for counting nondemolition measurement, Lett. Math. Phys. 20 (1990), 85-89. 17. J. S. Bell: “Speakable and Unspeakable in Quantum Mechanics,” Cambridge UP, 1987. 18. A. Einstein, B. Podolski and N. Rosen: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 (1935), 777-800. 19. E. Schrodinger: Naturwis. 23 (1935), 807-812, 823-828, 844-849. 20. G. C. Ghirardi, P. Pearl and A. Rimini: Markov processes an Hilbert space and continuous spontaneous localization of systems of identical particles, Phys. Rev. A 42 (1990), 78-89. 21. D. Chruscinski and P. Staszewski: O n the asymptotic solutions of the Belavkin’s stochastic wawe equation, Physica Scripta 45 (1992), 193-199. 22. N. Gisin and I. C. Percival: The quantum state d i f i s i o n model applied to open systems, J. Phys. A: Math. Gen. 25 (1992), 5677-5691. 23. H. M. Wiseman and G. J. Milburn: Phys. Rev. A 47 (1993), 642. 24. H. M. Wiseman and G. J. Milburn: Phys. Rev. A 49 (1994), 1350. 25. P. Goetsch and R. Graham: Quantum trajectories for nonlinear optical processes, Ann. Physik 2 (1993), 708-719. 26. P. Goetsch and R. Graham: Linear stochastic wave equation for continuously measurement quantum systems, Phys. Rev. A 50 (1994), 52425255. 27. V. N. Kolokoltsov: Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate, J. Math. Phys. 36 (1995), 2741-2760. 28. V. P. Belavkin: A new wave equation for a continuous nondemolition measurement, Phys. Lett. A, 140 (1989), 355-358. 29. V. P. Belavkin: In “Stochastic Methods in Experimental Sciences (W. Kasprzak and A. Weron Eds.),” pp. 26-42, World Scientific, 1990; A posterior Schrodinger equation for continuous nondemolition measurement, J. Math. Phys. 31 (1990), 2930-2934. 30. V. P. Belavkin: Quantum noise, bits and jumps, Progress in Quantum Electronics, 25 (2001), 1-53. 31. H. J. Carmichael: “An Open System Approach to Quantum Optics,” Lect. Notes in Phys. Vol. 18, Springer-Verlag, Berlin, 1993. 32. H. J. Carrnichael: In “Quantum Optics VI (J. D. Harvey and D. F. Walls, Eds.) ,” Springer-Verlag, 1994. 33. G. C. Wick, A. S. Wightman and E. P. Wigner: The intrinsic parity of elementary particles, Phys. Rev. 88 (1952), 101-105. 34. R. L. Stratonovich and V. P. Belavkin: Dynamical interpretation f o r the
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WHAT IS STOCHASTIC INDEPENDENCE? W E FRANZ Institut f i r Mathematik und Informatik Ernst- Moritz-Arndt- Universitat Greifswald fiedrich-Ludwig- Jahn-Str. 15 a D-17487Greifswald, Germany E-mail: [email protected] http://hyperwaue.math-inf. uni-greifswald. de/algebra/franz The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections. In this context, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Schiirmann become classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. The notion of a reduction of one independence to another is also introduced. As examples the reductions of Fermi independence and boolean, monotone, and anti-monotone independence to tensor independence are presented.
1
Introduction
In this paper we will deal with the question, what stochastic independence is. Since the work of Speicher [13] and Schiirmann [1,2,12] we know that a ‘universal’ notion of independence should come with a product that allows to construct the joint distribution of two independent random variables from their marginal distributions. It turned out that in classical probability there exists only one such product satisfying a natural set of axioms. But there are several different good notions of independence in non-commutative probability. The most important ones were classified in the work of Speicher [13], Ben Ghorbal and Schiirmann [1,2], and Muraki [9,10], they are tensor independence, free independence, boolean independence, monotone independence and anti-monotone independence. We present and motivate here the axiomatic framework used in these articles. We show that the classical notion of stochastic independence is based on a kind of product in the category of probability spaces, which is intermediate to the notion of a (universal) product in category theory - which does not exist in this category - and the notion of a tensor product. Furthermore, we show that the classification of stochastic independence by Ben Ghorbal and Schiirmann [1,2,12] and by Muraki [9,10] is also based on such a product,
254
255
which we call a tensor product with projections or inclusions, cf. Definition 3.3. This notion allows to define independence for arbitrary categories, see Definition 3.4. If independence is something that depends on a tensor product and projections or inclusions between the original objects and their tensor product, then it is clear that a map betwe! en categories that preserves independence should be tensor functor with an additional structure that takes care of the projections or inclusions. This is formalized in Definition 4.2. We show in several examples that these notions are really the correct ones, see Subsections 3.1, 3.2, 3.3, and 4.1, and Section 7. But let us first look at the notion of independence in classical probability. 2
Independence for Classical Random Variables
Two random variables X I : ( R , F , P ) -+ (E1,&1) and X2 : ( R , F , P ) + (E2,€2), defined on the same probability space ( R , F , P ) and with values in two possibly distinct measurable spaces ( & , & I ) and (E2,&2), are called stochastically independent (or simply independent) w.r.t. P , if the o-algebras X,'(&1) and XF1(€2)are independent w.r.t. P , i.e. if
P ( ( X 1 " M l ) n XF'(M2)) = p ( ( x , 1 ( M 1 ) ) p ( x , - 1 ( M 2 ) ) holds for all M1 E € 1 , M2 E € 2 . If there is no danger of confusion, then the reference to the measure P is often omitted. This definition can easily be extended to arbitrary families of random variables. A family ( X j : ( R , F , P ) + (Ej,&j))jE~, indexed by some set J, is called independent, if / n
\
n
holds for all n E N and all choices of indices j 1 , . . . ,j , E J with j k # j , for # C, and all choices of measurable sets Mj, E &jk. There are many equivalent formulations for independence, consider, e.g. , the following proposition. Proposition 2.1 Let X1 and X2 be two real-valued random variables. The following are equivalent. j
(i) X1 and X2 are independent. (ii) For all bounded measurable functions
fi,f2
on R we have
lE(f1 ( X d f 2( X 2 ) ) = q f l (Xl,) lE( f 2 ( X 2 ) ) .
256
(iii) The probability space (R2, B(R2),P(x,,xZ)) is the product ofthe probability spaces (R, B(R),Px,)and (R, B(R), Px,),i.e.
q x ,,X2)= px, 63 pxz . We see that stochastic independence can be reinterpreted as a rule to compute the joint distribution of two random variables from their marginal distribution. More precisely, their joint distribution can be computed as a product of their marginal distributions. This product is associative and can also be iterated to compute the joint distribution of more than two independent random variables. The classifications of independence for non-commutative probability [1,2,9,10,13] that we are interested in are based on redefining independence as a product satisfying certain natural axioms. 3
Tensor Categories and Independence
We will now define the notion of independence in the language of category theory. The usual notion of independence for classical probability theory and the independences classified in [1,2,9,10,13] will then be instances of this general notion obtained by considering the category of classical probability spaces or the category of algebraic probability spaces. First we recall the definitions of a product, coproduct and a tensor product, see also MacLane [7] for a more detailed introduction. Then we introduce tensor categories with inclusions or projections. This notion is weaker than that of a product or coproduct, but stronger than that of a tensor category. It is exactly what we need to get an interesting notion of independence. Definition 3.1 (See, e.g., Maclane [7]) A tuple (B1rIB2,nl,n2)is called a product or universal product of the objects B1 and B2 in the category C, if for any object A E ObC and any morphisms f 1 : A + B1 and f 2 : A + B2 there exists a unique morphism h such that the following diagram commutes,
B l ~ B l n B 2 ~ B 2 . An object K is called terminal, if for all objects A E ObC there exists exactly one morphism from A to K . The product of two objects is unique up to isomorphism, if it exists. Furthermore, the operation of taking products is commutative and associative
257
up to isomorphism and therefore, if a category has a terminal object and a product for any two objects, then one can also define a product for any finite set of objects. The notion of coproduct is dual to that of a product, i.e., its defining property can be obtained from that of the product by 'reverting the arrows'. The notion dual to terminal object is an initial object, i.e. an object K such that for any object A of C there exists a unique morphism from K to A . Let us now recall the definition of a tensor category. Definition 3.2 A category (C, 0) equipped with a bifunctor 0 : C x C + C, called tensor product, that is associative up to a natural isomorphism QA,B,C : AO(B0C)
5 (ADB)OC,
for all A, B , C E ObC,
and an element E that is, up to isomorphisms AA : EOA 5 A ,
and
P A : AOE
7 A,
for all A E ObC,
a unit for 0 , is called a tensor category or monoidal category, if the pentagon axiom
(AOB)O(COD)
A O ( B O ( C0 D ) ) idn
A . B.C
((A0B)UC)OD
t
i
AO((B0C)OD)
QA,BOC.D
a A . B . C OidD
* (AO(B0C))OD
and the triangle axiom
A0 (EOC)
~A.E.C
* (AOE)C
AOC are satisfied for all objects A , B , C,D of C. If a category has products or coproducts for all finite sets of objects, then the universal property guarantees the existence of the isomorphisms a,A, and p that turn it into a tensor category. In order to define a notion of independence we need less than a (co-) product, but a little bit more than a tensor product. What we need are inclusions or projections that allow us to view the objects A , B its subsystems of their product AOB.
258
Definition 3.3 A tensor category with projections (C, 0 ,T) is a tensor category (C,O) equipped with two natural transformations T I : 0 + PI and + P2, where the bifunctors Pl,P2 : C x C + C axe defined by 7r2 : Pl(B1,Ba) = B1, Pz(B1,Bz) = B2, on pairs of objects B1,B2 of C, and similarly on pairs of morphisms. In other words, for any pair of objects B1, B2 there exist two morphisms B B , : B1OB2 -+ B I , T B :~B1OB2 + B2, such that for any pair of morphisms f 1 : A1 + B1, f 2 : A2 + B2, the following diagram commutes,
A1 fll
- A - 1. “A1
AlOA2 fl
B1-
“El
J.
“A2
A2
f2
B1OB2
“%
B2.
Similarly, a tensor product with inclusions ( C , 0 ,i ) is a tensor category (C,O) equipped with two natural transformations il : PI + 0 and i 2 : P2 + 0 , i.e. for any pair of objects B1, B2 there exist two morphisms i~~ : B1 3
BlOB2, ig, : B2 + BlOB2, such that for any pair of morphisms B1, f 2 : A2 + B2, the following diagram commutes, Ai
- -
fl/
B 1
AiOA-2
2A1
iB
flAf2
fi
: A1
+
A2
2A2
+ is, 1 B1UB2
f--
if2
B2.
In a tensor category with projections or with inclusions we can define a notion of independence for morphisms. Definition 3.4 Let (C, 0,T) be a tensor category with projections. Two morphisms fi : A + B1 and fi : A + B2 with the same source A are called independent (with respect to 0),if there exists a morphism h : A + BlOB2 such that the diagram
commutes. In a tensor category with inclusions (C, 0 ,i ) ,two morphisms f 1 : A1 + B and f 2 : A2 + B with the same target B are called independent, if there
259
exists a morphism h : AlOA2
+ B such that the diagram
commutes. This definition can be extended in the obvious way to arbitrary sets of morphisms. If 0 is actually a product (or coproduct, resp.), then the universal property in Definition 3.1 implies that for all pairs of morphisms with the same source (or target, resp.) there exists even a unique morphism that makes diagram (1) (or (2), resp.) commuting. Therefore in that case all pairs of morphisms with the same source (or target, resp.) are independent. We will now consider several examples. We will show that for the category of classical probability spaces we recover usual stochastic independence, if we take the product of probability spaces, cf. Proposition 3.5.
3.1 Example: Independence in the Category of Classical Probability Spaces The category meas of measurable spaces consists of pairs (0,F),where R is a set and T P ( R ) a a-algebra. The morphisms are the measurable maps. This category has a product, ( & , ~ l ) ~ ( R 2 , 3 . 2= ) (01
x
0 2 , F l €472)
where R1 x 0 2 is the Cartesian product of R1 and 522, and TI €4 T 2 is the smallest a-algebra on R1 x R2 such that the canonical projections p l : R 1 x Rz + R1 and p 2 : R1 x 0 2 + 0 2 are measurable. The category of probability spaces p r o 6 has as objects triples (0,F,P ) where (52,T)is a measurable space and P a probability measure on ( R , F ) . A morphism X : (01,T I ,PI) + (R1, 3 2 , P 2 ) is a measurable map X : (al,T I )+ (Ql, F 2 ) such that P1 0
x-l = P 2 .
This means that a random variable X : ( 0 , F ) + ( E , € ) automatically becomes a morphism, if we equip ( E ,€) with the measure
Px = p induced by X .
o x - 1
260
This category does not have universal products. But one can check that the product of measures turns Prob into a tensor category,
(%,FI,Pl)8 (n2,F2,P2)= (01x Q2,Fl 8F2,PI 8 P 2 ) , where PI 8 P2 is determined by
(Pl 8 P2)(Ml x
M2)
= S(Ml)P2(M2),
for all M I E Fl,Mz E F2. It is even a tensor category with projections in the sense of Definition 3.3 with the canonical projections pl : ( 0 1 x R2,Fl 8 . T 2 , 9 ~ P z ) - , ( R l , F l , P l ) , p z : ( R l xn2,318F2,P18'2)-,(Rz,.T2,P2) g i v e n b y p l ( ( w 1 , ~ 2 )= ) 4 , ~ 2 ( ( w l , w 2 )=w2forw1 ) Ef%,w2En2. The notion of independence associated to this tensor product with projections is exactly the one used in probability. Proposition 3.5 Two random variables X I : ( R , F , P ) + (E1,El) and X2 : (0,F,P ) + (E2,€ 2 ) , defined o n the same probability space (0,F,P ) and with values in measurable spaces (El,&) and (E2,€2),are stochastically independent, i f and only if they are independent in the sense of Definition 3.4 as morphismsX1 : ( R , F , P )3 (El,€l,Px,) andX2: ( R , F , P )+ (E2,€2,PxZ) of the tensor category with projections (ptob, 8 , p ) .
PROOF.Assume that X1 and X Z are stochastically independent. We have to find a morphism h : ( R , F , P ) 3 (El x & , € I @ € ~ , P x8 , Px,) such that the diagram
(a,3,PI (E1,E1,PX1)%(El
x
E2,El@62,[email protected],)
pEz- (E2,E2,Px2)
commutes. The only possible candidate is h ( w ) = ( X l ( u ) , X ~ ( u )for ) all w E R, the unique map that completes this diagram in the category of measurable spaces and that exists due to the universal property of the product of measurable spaces. This is a morphism in vrob, because we have
P(h-l(M1 x M2)) = P ( X i l ( M 1 )n X z ' ( M 2 ) ) = P(X,1(M1))P(X,1(M2)) = PX,(Ml)PX,(MZ)= (Px, @PX,)(Ml x Mz) for all M I E
€ 1 , M2
E
€2,
and therefore
P o h-' = Px, 8 Px,.
26 1
Converselx, if XI and X2 are independent in the sense of Definition 3.4, then the morphism that makes the diagram commuting has to be again h : LJ c) ( X l ( w ) ,X2(w)). This implies
P(xl,x,)= P 0 h-' = Px, @ PxZ and therefore
P(xF1(Ml)n x ; ~ ( M ~ ) )= P(X;~(M~))P(X;~(M~)) for all M I E €1, M2 E €2.
I
3.2 Example: Tensor Independence in the Category of Algebraic Probability Spaces By the category of algebraic probability UIgProb spaces we denote the category of associative unital algebras over C equipped with a unital linear functional. A morphism j : (d1, cpl) + (d2, cp2) is a quantum random variable, i.e. an algebra homomorphism j : d1 + d2that preserves the unit and the functional, i.e. j ( l d l ) = Id2 and cp2 0 j = cp1. The tensor product we will consider on this category is just the usual tensor product (dl @ dz,(PI @ p2), i.e. the algebra structure of d1 8 d 2 is defined by 1d1@d2 = Id1 @ l d z , (a1 8 az)(bi 8 b2) = aibi @ a2b2, and the new functional is defined by
(cpl @ cP2)(a1 @ a2) = cpl(al)cpz(a2),
for all al, b l E d1, a2, b2 E d 2 . This becomes a tensor category with inclusions with the inclusions defined bY idI(a1) = a 1 @Id,, idz(a2) = Id1 @ a27 for a1 E d1, a2 E d2. One gets the category of *-algebraic probability spaces, if one assumes that the underlying algebras have an involution and the functional are states, i.e. are also positive. Then an involution is defined on dl @ d 2 by (a1@a2)*= a; @ a$ and 9 1 @ q.72 is again a state. The notion of independence associated to this tensor product with inclusions by Definition 3.4 is the usual notion of Bose or tensor independence used in quantum probability, e.g., by Hudson and Parthasarathy.
262
Proposition 3.6 Two quantum random variables j1 : (Bl,$1) -+ (A, cp) and : (232, $2) + (A,cp), defined on algebraic probability spaces ( B l , $ I ) , (B2, $2) and with values in the same algebraic probability space (A,cp) are independent i f and only if the following two conditions are satisfied.
j2
(i) The images of jl and j2 commute, i.e. [j,(a1),j2(a2)]
= 0,
f o r all a1 E d1,a2 E d 2 . (ii) cp satisfies the factorization property
cp(j1(a1) j 2 ( a 2 ) > = cp(j1 (a1) ) c p ( j 2 (a2)) for all a1 E A1, a2 E d 2 .
7
We will not prove this Proposition since it can be obtained as a special case of Proposition 3.7, if we equip the algebras with the trivial iZ2-grading A(’) = A, = (0).
3.3 Example: Fermi Independence Let us now consider the category of &-graded algebraic probability spaces Z2-%Ig!JJtob. The objects are pairs (d,cp)consisting of a &-graded unital and an even unital functional cp, i.e. (pIA(1) = 0. algebra A = A(’) @ The morphisms are random variables that don’t change the degree, i.e., for j : (d1,cpd-+ (d2,cp2), we have j ( d ? ) )C dp)
and
j ( A y ) )E dc).
The tensor product (dl @zZdz,cp1 @ 92) = (d1,cpl) @.az (d2, cp2) is defined as follows. The algebra A1 @zz A2 is the graded tensor product of A1 and A2, i.e. (A1 @zzA2)(’) = A?) 8 df)@ A?) @ dc),(dl@zZ d2)(l)= A?) @ A?) @ A?) @ A:), with the algebra structure given by 1dI@-,dz
= Id1 @ 1 d z ,
. (a2 @ b2) = ( - l ) d e g b l degaz albl 8 a2b2, for all homogeneous elements a1 ,bl E d1,a2, b E d2.The functional (PI 18972 (a1 @ b l )
is simply the tensor product, i.e. (cpl 8 cpz)(al @ a2) = cpl(a1) @ cpz(a2) for all al E d1,a2 E d2. It is easy to see that 91 @ 9 2 is again even, if and 9 2 are even. The inclusions il : (A1,cpl) -+ (dl@zz d2,cp1 @ 9 2 ) and i 2 : (-42, cp2) -+ (dl@zz d 2 , (PI @ 972) are defined by ii(ai) = a1 @ Idz
and
h(a2)
=
@ U2,
263
for a1 E d l , a2 E d 2 . If the underlying algebras are assumed to have an involution and the functionals to be states, then the involution on the &-graded tensor product is defined by (a1 @ a2)* = ( - l ) d e g a l degazaT @ u;, this gives the category of &-graded *-algebraic probability spaces. The notion of independence associated to this tensor category with inclusions is called F e m i independence or anti-symmetric independence. Proposition 3.7 Two random variables j l : ( & , $ I ) + (d,cp) and j 2 : (B2,$2) -+ ( A ,cp), defined on two &-graded algebraic probabi2ity spaces (B1,$ I ) , ( B 2 , $ 2 ) and with values in the same &-algebraic probability space (d,cp)are independent if and only i f the following two conditions are satisfied. (i) The images of j1 and j2 satisfy the commutation relations
j2(a2)j1(a1) = (-1)degal for all homogeneous elements a1 E
d1,
dega2
a2 E
j l (a1)j2 (a21
d2.
(ii) cp satisfies the factorization property
f o r all a1 E d1,a2 E
d2.
PROOF. The proof is similar to that of Proposition 3.5, we will only outline it. It is clear that the morphism h : ( & , $ I ) @z2 ( a s , & ) + (d,cp) that makes the diagram in Definition 3.4 commuting, has to act on elements of B 1 @ l a 2 and l a l @ B2 a~ h(b1 @ la,) = jl(b1)
and
h ( b , @ b2) = h ( b 2 ) .
This extends to a homomorphism from ( B l ,$ 1 ) @z2(f32, $ 2 ) to (A,cp), if and only if the commutation relations are satisfied. And the resulting homomorphism is a quantum random variable, i.e. satisfies cp o h = $1 @ $9, if and only if the factorization property is satisfied. I
4
Reduction of Independences
In this Section we will study the relations between different notions of independence. Let us first recall the definition of a tensor functor.
264
Definition 4.1 (see, e.g., Section XI.2 in MacLane [7]) Let (C, 0)and (C', 0') be two tensor categories. A cotensor functor or comonoidal functor F : (C, 0 ) + (C', 0') is an ordinary functor F : C + C' equipped with a morphism FO : F(Ec) + E p and a natural transforrnation F2 : F ( - 0 + F ( .)O'F( .), i.e. morphisms F z ( A , B ) : F ( A 0 B ) + F(A)O'F(B)for all A , B E ObC that axe natural in A and B , such that the diagrams a )
F (AO(B0C))
F(~A.B.c)
* F ((AOB)OC)
(3)
FoO'idB
F(B)
XF(B)
Ect O'F ( B )
commute for all A, B , C E ObC. We have reversed the direction of Fo and F2 in our definition. In the case of a strong tensor functor, i.e. when all the morphisms are isomorphisms, our definition of a cotensor functor is equivalent to the usual definition of a tensor functor as, e.g., in MacLane [7]. The conditions are exactly what we need to get morphisms Fn(Al,... , A , ) : F ( A 1 O . - . O A , )+ F ( A I ) O ' . . . O ' F ( A , ) for all finite sets { A l , . . . , A , } of objects of C such that, up to these morphisms, the functor F : (C, 0 ) (C', 0') is a homomorphism. For a reduction of independences we need a little bit more than a cotensor functor.
265
Definition 4.2 Let (C, 0 ,i) and (C’, O’, i‘) be two tensor categories with inclusions and assume that C is a subcategory of C’. A reduction ( F , J ) of the tensor product 0 to the tensor product 0’ is a cotensor functor F : (C,O) + (C’,O’) and a natural transformation J : idc + F , i.e. morphisms j A : A + F(A) in C’ for all objects A E ObC such that the diagram
A 3F(A)
commutes for all morphisms f : A + B in C. Such a reduction provides us with a system of inclusions Jn(A1,. . . ,A,) = F,(A1,. . . ,A,) 0 J A ~ ~ . . . ~ A : , A10 * . .OA, + F(A1)O‘ * * O‘F(A,) with &(A) = JA that satisfies, e.g., J,+,(Al ,... ,A,+,) = F2(F(A1)0‘--*O’F(A,),F(A,+1)0‘..-O’F(A,+,)) 0 (J,(Al,. . . ,A,) OJ,(A,+1,. . . ,A,+,)) for all n , m E N and Al,. . . ,A,+, E ObC. A reduction between two tensor categories with projections would consist of a tensor functor F and a natural transformation P : F + id. We have to extend our definition slightly. In our applications C will often not be a subcategory of C’, but we have, e.g., a forgetful functor U from C to C’ that “forgets” an additional structure that C has. An example for this situation is the reduction of Fermi independence to tensor independence in following subsection. Here we have to forget the &-grading of the objects of Z2-UIglprob to get objects of UIglprob. In this situation a reduction of the tensor product with inclusions I3 to the tensor product with inclusions 0‘ is a tensor function F from (C, 0) to (C‘, 0’) and a natural transformation J:U+F.
-
4.1
Example: Bosonazation of Fermi Independence
We will now define the bosonization of Fermi independence as a reduction from (Wglptob, @,i) to (Z2-U[glprob, @z2,i). We will need the group algebra CZ2 of Z2 and the linear functional E : CZ2 + C that arises as the linear extension of the trivial representation of 252, i.e. &(1)= & ( g ) = 1,
if we denote the even element of
22
by 1 and the odd element by g .
266
The underlying functor F : Z2-UKg'33rob
F:
Ob Z2-UKgPtob 3 (A,cp) Mor Z2-%Ig!$hob 3 f
I+ I+
+ UIgPro b is given by
( A @z2CZ2, cp @ E ) E ObU[@J3rOb, f @ id@zzE Mor UKg!J3rob.
The unit element in both tensor categories is the one-dimensional unital algebra C1 with the unique unital functional on it. Therefore FO has to be a morphism from F(C1) E CZ2 to C l . It is defined by Fo(1) = Fo(g) = 1. The morphism F2 (A1,d2) has to go from F(d@zz B) = ( A@z2B)63(CZ2 to F(A)@ F ( B ) = ( A@z2CZz)@ (B@z2CZ2). It is defined by
a@b@l*
{
(a @ 1) @ ( b @ 1) if b is even, (a @ 9) @ ( b CQ 1) if b is odd,
a@b@g*
{
(a 18 g ) 18 ( b @ 9) if b is even,
and (a @
1) @ ( b CQ 9) if b is odd,
for a E A and homogeneous b E B. Finally, the inclusion JA : A + A @zz CZ2 is defined by
=U@ 1
JA(U)
for all a E A. In this way we get inclusions Jn = Jn(dl,.. . ,A,) = Fn(A1,. . . ,An) 0 J A 1 ~ , z z . . . ~ 3 2of A the , graded tensor product d1 @zz. . .@zz An into the U S U ~ tensor product (d1@zzCZ2) @ . . . @ (A, @zZCZ2) which respect the states and allow to reduce all calculations involving the graded tensor product to calculations involving the usual tensor product on the bigger algebras F(d1) = dl @z2CZ2, . . . ,F ( A n )= An @z2CZ2. These inclusions are determined by
JnQ. @ k
-1
@ :@a@? @ .-; times
for odd a E Ak, 1 5 k
n
1) = ? @
times
k
-
*;-@?18ii 18 i @I.-.. @i
- 1 times
n
-k
times
5 n, where we used the abbreviations
ij=l@g, 5
-k
@
ii=a@l,
i=l@l.
Forgetful Functors, Coproducts, and Semi-universal Products
We are mainly interested in different categories of algebraic probability spaces. There objects are pairs consisting of an algebra A and a linear functional cp on A. Typically, the algebra has some additional structure, e.g., an involution, a unit, a grading, or a topology (it can be, e.g., a von Neumann algebra or
267
a C*-algebra), and the functional behaves nicely with respect t o this additional structure, i.e., it is positive, unital, respects the grading, continuous, or normal. The morphisms are algebra homomorphisms, which leave the linear functional invariant, i.e., j : ( A ,cp) + (B,$) satisfies cp=$oj
and behave also nicely w.r.t. to additional structure, i.e., they can be required to be *-algebra homomorphisms, map the unit of A to the unit of B,respect the grading, etc. We have already seen one example in Subsection 3.3. The tensor product then has to specify a new algebra with a linear functional and inclusions for every pair of of algebraic probability spaces. If the category of algebras obtained from our algebraic probability space by forgetting the linear functional has a coproduct, then it is sufficient to consider the case where the new algebra is the coproduct of the two algebras. Proposition 5.1 Let ( C , 0, i) be a tensor category with inclusions and F : C + D a functor from C into another category 2, which has a coproduct LI and an initial object ED. Then F is a tensor functor. The morphisms F2(A,B ) : F ( A ) LI F ( B ) + F ( A 0 B ) and FO : ED + F ( E ) are those guaranteed by the universal property of the coproduct and the initial object, i.e. FO : ED + F ( E ) is the unique morphism from ED to F ( E ) and F2(A,B ) is the unique morphism that makes the diagram
F(A)
- F (i .4)
F(A0B)
F(~B)
F(B)
commuting.
PROOF.Using the universal property of the coproduct and the definition of F2, one shows that the triangles containing the F ( A ) in the center of the diagram
268
commute (where the morphism from F ( A ) to F ( A O B ) U F ( C ) is F ( ~ A ) idF(C)), and therefore that the morphisms corresponding to all the different paths form F ( A ) to F ( ( A U B ) O C )coincide. Since we can get similar diagrams with F ( B ) and F ( C ) ,it follows from the universal property of the triple coproduct F ( A )U ( F ( B ) F ( C ) ) that there exists only a unique morphism from F ( A ) ( F ( B ) F(C))to F ( ( A 0 B ) O C )and therefore that the whole diagram commutes. The commutativity of the two diagrams involving the unit elements can I be shown similarly.
IJ
U
IJ
U
Let C now be a category of algebraic probability spaces and F the functor that maps a pair (A,cp) to the algebra A, i.e., that “forgets” the linear functional cp. Suppose that C is equipped with a tensor product 0 with inclusions Let (d,cp),(a,+) be two algebraic probaand that F(C) has a coproduct bility spaces in C, we will denote the pair (A,cp)U(B,+) also by (AUB,&I+). By Proposition 5.1 we have morphisms F2(A,f?) : A u B + AUB that define a natural transformation from the bifunctor to the bifunctor 0. With these morphisms we can define a new tensor product fi with inclusions by
u.
The inclusions are those defined by the coproduct. Proposition 5.2 If two random variables f1 : (d1,cpl)+ (a,+) and f i : (A1,cpl) + (f?,+) are independent with respect to U, then they are also independent with respect to fi.
PROOF.If f1 and f2 are independent with respect to 0 , then there exists a random variable h : (AlOA2, cp1Ucp2) + (23, +) that makes diagram (2) in ~ (a, 2 ) $) Definition 3.4 commuting. Then h 0 F2(A1,A2) : (A1 A2, ~ ~ 1 0 + makes the corresponding diagram for 0 commuting. 1
u
The converse is not true. Consider the category of algebraic probability spaces with the tensor product, see Subsection 3.2, and take B = A1 A2 and II, = ( 9 1 63 9 2 ) o Fz(A1,Az). The canonical inclusions id1 : (A1,cpl)+ (a,+) and id2 : (A2, cp2) + (B, $) are independent w.r.t. 6, but not with respect to the tensor product itself, because their images do not commute in B = A1 A2. We will call a tensor product with inclusions in a category of quantum probability spaces semi-universal, if it is equal to the coproduct of the corresponding category of algebras on the algebras. The preceding discussion shows that every tensor product on the category of algebraic quantum probability spaces BIgpro b has a quasi-universal version.
IJ
269
6
The Classification of Independences in the Category of Algebraic Probability Spaces
We will now reformulate the classification by Muraki [lo] and by Ben Ghorbal and Schurmann [1,2] in terms of semi-universal tensor products with inclusions for the category of algebraic probability spaces UIgprob. In order to define a semi-universal tensor product with inclusions on UIgprob one needs a map that associates to a pair of unital functionals (cpl, cp2) on two algebras A1 and A2 a unital functional cp1 . cp2 on the free product A1 u A 2 (with identification of the units) of A1 and A2 in such a way that the bifunctor 0 : (Ai,cpi) x
(A2,cpi) r-) ( A i U A 2 , c p i *cp2)
satisfies all the necessary axioms. Since 0 is equal to the coproduct on the algebras, we don't have a choice for the isomorphisms Q, A, p implementing the associativity and the left and right unit property, we have to take the ones following from the universal property of the coproduct. The inclusions and the action of 0 on the morphisms also have to be the ones given by the coproduct. The associativity gives us the condition
((vl
*
(P2)
. (P3)
Qd1,dz,ds = cp1
'
(92
. (P3),
(6)
for all (d1,cpl), (A2, p2), (A3,cp3) in UKgprob. Denote the unique unital functional on Cl by 6,then the unit properties are equivalent to
(p'6)"pd=cp and
(6.(P)OAd='P,
for all (A,cp) in Q[gprob. The inclusions are random variables, if and only if (Vl . P2) 0 id1
91
a d
(91
. cP2) id2 = 9 2
(7)
for all (A1,cpl),(A2,cp2)in Q[gprob. Finally, from the functoriality of 0 we get the condition
-
(cpl (472) 0 (jl J&2)
= (cpl 0 jl)*
($72
0
j2)
(8)
for all pairs of morphisms j 1 : (a1,$1) + (A1,cpl), j 2 : (a2,$2) -+ (A2, cp2) in 31Igqrob. Our Conditions ( 6 ) , (7), and (8) are exactly the axioms (P2), (P3), and (P4)in Ben Ghorbal and Schurmann [l],or the axioms (U2), the first part of (U4), and (U3) in Muraki [lo].
270
Theorem 6.1 (Muraki [lo], B e n Ghorbal and SchGrmann [1,2]) There exast exactly two semi-universal tensor products with inclusions o n the categoy of algebraic probability spaces UIg?J?tob,namely the semi-universal version 6 of the tensor product defined in Section 3.2 and the one associated to the free product * of states. Voiculescu’s [14] free product cp1 * cp2 of two unital functionals can be defined recursively by
for a typical element (1102 .. . a, E dl d2,with ak E d,, €1 # €2 # . . . # em, i.e. neighboring a’s don’t belong to the same algebra. #I denotes the number of elements of I and ak means that the a’s are to be multiplied in the same order in which they appear on the left-hand-side. We use the convention cp2) kc!? ak) = 1. Ben Ghorbal and Schurmann [1,2] and Muraki [lo] have also considered the category of non-unital algebraic probability nuMIg?J?to b consisting of pairs (d,cp) of a not necessarily unital algebra d and a linear functional cp. The morphisms in this category are algebra homomorphisms that leave the functional invariant. On this category we can define three more tensor products with inclusions corresponding to the boolean product 0 , the monotone product D and the anti-monotone product a of states. They can be defined by
nZI
(n’
m
for (1102 .. .a, E dl d2 ak E A,, €1 # €2 # . . . # em, i.e. neighboring a’s don’t belong to the same algebra. Note that denotes here the free product without units, the coproduct in the category of not necessarily unital algebras. For the classification in the non-unital case, Muraki imposes the additional condition
IJ
(91 . cps)(a1a2)= (Pel (al>cpc,(a21 for all (€1,€2) E { (1,2), (2, I)} a1 E dcl , a2 E d,,.
(9)
27 1
Theorem 6.2 (Muraki [lo]) There exist exactly five semi-universal tensor products with inclusions satisfying (9) o n the category of non-unital algebraic probability spaces nuUKgyro6, namely the semi-universal version 6 of the tensor product defined in Section 3.2 and the ones associated to the free product *, the boolean product 0, the monotone product D and the anti-monotone product 4.
The monotone and the anti-monotone are not symmetric, i.e. (A1 U A2, 9 1 D cp2) and (A2 U A2, c p D ~ cpl) are not isomorphic in general. Actually, the antimonotone product is simply the mirror image of the monotone product,
for all (A1, cpl), (A2,92) in the category of non-unital algebraic probability spaces. The other three products are symmetric. At least in the symmetric setting of Ben Ghorbal and Schiirmann, Condition (9) is not essential. If one drops it and adds symmetry, one finds in addition the degenerate product
and a whole family 'p1 *Q cp2
= q((q-lcp1)
(q-lcpz)),
parametrized by a complex number q E C\{O}, for each of the three symmetric products, 0 E {6, *, o}.
7 The Reduction of Boolean, Monotone, and Anti-Monotone Independence to Tensor Independence We will now present the unification of tensor, monotone, anti-monotone, and boolean independence of F'ranz [5] in our category theoretical framework. It resembles closely the bosonization of Fermi independence in Subsection 4.1, but the group 2 2 has to be replaced by the semigroup M = { l , p } with two elements, 1 . l = 1, 1 . p = p . 1 = pop = p. We will need the linear functional E : C M + C with ~ ( 1= ) ~ ( p= ) 1. The underlying functor and the inclusions are the same for the reduction of the boolean, the monotone and the anti-monotone product. They map the algebra A of (A,'p) to the free product F(A) = A U C M of the unitization A of A and the group algebra (CM of M . For the unital functional F(cp) we
272
+
take the boolean product $ o E of the unital extension of cp with E . The elements of F ( A ) can be written as linear combinations of terms of the form P " ~ I P .. * WmP"
with m E N,a , w E (0,l},al, . . . .am E A, and F(cp) acts on them as m
F(cp)b"alP.-.pamP")= r]: c p ( 4 . k=l
The inclusion is simply Jd : A 3 a
I-)
a E F(A).
The morphism FO : F(C1) = (CM + C l is given by the trivial representation of M , Fo(1) = F o b ) = 1. The only part of the reduction that is different for the three cases are the morphisms
We set
for the boolean case,
for the monotone case, and
for the anti-monotone case. For the higher order inclusions J,?, = F,'(A1,. {B, M, AM}, one gets
. . ,A,)
u...
o Jdl
A,,
0
E
= p@(k-l) @ a @ p ( n - - k ) J,"(a) = l@(k-l) @ a @p@(,-k) J,AM (a) = p @ ( k - 1 ) 8 a @ I@(,-&), J,"(.)
if a E d k . One can verify that this indeed defines reductions (FB,J), (F', J), and (FAM, J) from the categories (nu%lgVtob,0 , i), (nuQIg!$Jrob, D, i), and (nuUlg!prob, a, i) to (%lgprob, @, i). The functor U : nu%[gyrob+ MIgTrob
273
mentioned at the end of Section 4 is the unitization of the algebra and the unital extension of the functional and the morphisms. This reduces all calculations involving the boolean, monotone or antimonotone product to the tensor product. These constructions can also be applied to reduce the quantum stochastic calculus on the boolean, monotone, and anti-monotone Fock space to the boson Fock space. Furthermore, they allow to reduce the theories of boolean, monotone, and anti-monotone L6vy processes to Schiirmann’s [ll]theory of LBvy processes on involutive bialgebras, see Franz [5].
8
Conclusion
We have seen that the notion of independence in classical and in quantum probability depends on a product structure which is weaker than a universal product and stronger than a tensor product. We gave an abstract definition of this kind of product, which we named tensor product with projections or inclusions, and defined the notion of reduction between these products. We showed how the bosonization of Fermi independence and the reduction of the boolean, monotone, and anti-monotone independence to tensor independence fit into this framework. We also recalled the classifications of independence by Ben Ghorbal and Schiirmann [1,2] and Muraki [lo] and showed that their results classify in a sense all tensor products with inclusions on the categories of algebraic probability spaces and non-unital algebraic probability spaces, or at least their semi-universal versions. There axe two ways to get more than the five universal independences. Either one can consider categories of algebraic probability spaces with additional structure, like for Fermi independence, cf. Subsection 3.3, and braided independence, cf. Franz, Schott, and Schiirmann [3], or one can weaken the assumptions, drop, e.g., associativity, see Mlotkowski [8] and the references therein. Romuald Lenczewski [S] has given a tensor construction for a family of new products called m-free that are not associative, see also Franz and Lenczewski [4]. His construction is particularly interesting, because in the limit m + 00 it approximates the free product. But it is not known, if a reduction of the free product to the tensor product in the sense of Definition 4.2 exists.
274
References
1. A. Ben Ghorbal and M. Schurmann: O n the algebraic foundations of a non-commutative probability theory, Prkpublication 99/17, Institut E. Cartan, Nancy, 1999, to appear in Math. Proc. Cambridge Philos. SOC. 2. A. Ben Ghorbal: “Fondements algkbrique des probabilitks quantiques et calcul stochastique sur l’espace de Fock boolCen,” PhD thesis, Universitk Henri PoincarbNancy 1, 2001. 3. U. Franz, R. Schott, and M. Schurmann: “Braided independence and LCvy processes on braided spaces,” Prkpublication Institut Elie Cartan 98/n 32, 1998. 4. U. Franz, and R. Lenczewski: Limit theorems f o r the hierarchy of freeness, Probab. Math. Stat. 19 (1999), 2 3 4 1 . 5. U. Franz: Unification of boolean, monotone, anti-monotone, and tensor independence and Lkvy processes, EMAU Greifswald Preprint-Reihe Mathematik 4/2’001, 2001, to appear in Math. 2. 6. R. Lenczewski: Unification of independence in quantum probability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), No. 3, 383-405. 7. S. MacLane: “Categories for the Working Mathematician (2nd edition),” Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, Berlin, 1998. 8. W. Mlotkowski: Free probability o n algebras with infinitely many states, Probab. Th. Rel. Fields 115 (1999), 579-596. 9. N. Muraki: The five independences as quasi-universal products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 113-134. 10. N. Muraki: The five independences as natural products, EMAU Greifswald Preprint-Reihe Mathematik 3/2002, 2002. 11. M. Schiirmann: “White Noise on Bialgebras,” Lect. Notes in Math. Vol. 1544, Springer, Heidelberg, 1993. 12. M. Schurmann: Direct sums of tensor products and non-commutative independence, J. Funct. Anal. 133 (1995), No. 1, 1-9. Universal products, in “Free Probability Theory 13. R. Speicher: (D. Voiculescu Ed.),” Papers from a workshop on random matrices and operator algebra free products, Toronto, Canada, Mars 1995, Fields Inst. Commun., Vol. 12, pp. 257-266. American Mathematical Society, 1997. 14. D. Voiculescu, K. Dykema, and A. Nica: “Free Random Variables,” American Mathematical Society, Providence, RI, 1992.
CREATION-ANNIHILATION PROCESSES ON CELLAR COMPLECIES WKIHIRO HASHIMOTO Chzid-dai 6-8-3, Kasugai, Aichi, 487-0011, JAPAN E-mail: coy0 @sun-inet.or.jp We study creation-annihilation processes associated with face maps on cellar complecies and develop a quantum probabilistic approach to the category of cellar complecies and cellar maps. We show that the join operation of cellar complecies induces a process on the symmetric toy Fock space, due to its commutativity. We investigate a chain complex defined by a set of binary trees, of which face maps define a large graph with multiple edges in contrast to the case of Cayley and distance regular graphs. As a result we obtain a one-mode interacting Fock space with Jacobi parameters wk = 2(2k l)(k + 2).
+
1
An Introductory Example
In algebraic probability theory, spectral analysis of adjacency matrices of discrete graphs has been studied as a discrete-time analogy of quantum stochastic process, where a pair of operators, the creation and annihilation, plays most fundamental roles. Let us consider a discrete graph (V,E ) with a set V of vertices and a set E of edges. Here we allow multiple edges (or parallel arcs) and then E @ V 2 . Suppose that the vertex set V is graded as a disjoint union V = v k with VO = {xo}, and for each k, the numbers of the edges w + ( x ) = { e E E I e joins x and y E V k - 1 ) and w - ( x ) = { e E E I e joins x and y E V k + l } are independent of the choice of z E V k respectively. This homogeneous property allows us to define the “creation” and “annihilation” operators on the Hilbert space 12(V),and their behavior, the creation-annihilation process ( CA-process for short), is described in terms of the one-mode interacting Fock space [l]. In the previous studies [4-61 we have introduced the quantum decomposition method to the spectral analysis of adjacency matrices of Cayley graphs associated with discrete groups and distance regular graphs, where we have seen that the homogeneous property of w+ and w - sets is satisfied asymptotically, and we have developed a quantum probabilistic approach to the subject. In this paper, we extend such a quantum probabilistic approach to the category of cellar complecies and cellar maps. We study two cases. In section 2, we consider a CA-process associated with the join of simplical complecies. Since the join operation is commutative, the limit process on the join
JJEo
275
276
of infinitely many complecies is described on the symmetric toy Fock space illustrated in Meyer's book [lo] (Theorem 2.1). In section 3, we consider a chain complex defined by the set of binary trees [3]. In contrast to the case of simple graphs such as Cayley or distance regular ones, the set of binary trees and face maps define a graph with multiple edges. The associated CA-process is described on the one-mode interacting Fock space with Jacobi parameters W k = 2(2k l)(k 2) (Theorem 4.1). Let us start with an example of a CA-process on an infinite dimensional Euclidean simplex. An n-dimensional Euclidean simplex A(n) or n-simplex is a convex hull of independent n 1 vertices V, = {wo, . , . ,wn} in RN ( N > n). The power set C, = 2vn gives a simplical complex structure A(n) = (Vn, En), that is,
+
+
+
a) u E C, and u' C
(T
then u' E C,, and
b) {w} E C, for any w E V,. By definition, the convex hull of any proper subset s C V, defines a simplex u,called a face of A(n). Ak(n) stands for the set of all k-simplecies in A(n), that is, simplecies in A(n) consist of k 1 vertices. The join of any face (T and any vertex w @ u,u * {w}, i.e., the convex hull of the set s U {w}, is again a face of A(n). Conversely, for any face u and vertex w E u there exists a unique face u' such that u = u' * {w}, denoted by u' = u/{w}. Then we have locally defined simplical maps
+
s, : Ak(n) 3 u H u * {w} E
Ak+'(n),if w @ u,
d, : Ak(n) 3 u H u/{w} E A"l(n),
if w E u.
From a combinatorial point of view, these maps are described as sv : s + s U {w} if v @ s and d, : s + s \ {w} if w E s for any s c V,. Let C(Ak(n)) be a C-vector space freely generated by k-faces Ak(n) with a canonical inner product (ulIUZ) = S,, . The simplical maps s,'s and dv's are extended to Clinear maps s, : C(Ak(n))+ C(Ak+'(n)) and d, : C(Ak(n)) + C(Ak-l(n)>, given by sdu) =
{
u * {w},
if w @ u,
0,
otherwise ,
dv(u)
=
{
u/{w},
if w E u,
0,
otherwise .
Our interest is in the limit behavior of two linear operators 1 1 d, a: = s, and a, = -
C
f i VEV,
C
f i u€",
277
as n + 00. One sees that for any a E Ak(n),
and
#{u' E Ak+l(n)I a C a'} =
r-!-') ri')
#{d E A"-'(n) 1 a' C a} =
=n-k-l,
=k+l.
Define vectors in C(Ak-l(n)),
for k 2 0, where we put A-l(n) = {@}, then we have
and, putting
@p)= &Qp), we have
One sees that as n 4 00, the CA-process associated with a: and an on the space r(A(n)) = @F==,C@p) converges to the CA-process on the one-mode interacting Fock space with Jacobi parameters {Wk = k, a k = 0). This system is identified with the toy Fock space illustrated in Meyer's book [lo]. 2
Join of Simplical Complecies and a CA-Process
The observation in the previous section is easily extended to the join of finite simplical complecies. A simplical complex K = (V,C) consists of a set of vertices V and a set C c 2v of subsets of vertices with the properties a) and b) in the previous section. If V is finite, we call K a finite simplical
278
complex. For finite simplical complecies Ki = join K1 * K2 = (V,C) is defined by
(K, Ci) with V1 n v2 = 8, the
V = V1 u V 2 , C = { a c V 1 a n V i E Ci, i = 1,2}. By definition, one sees the followings. Lemma 2.1 For finite simplical complecies Kj disjoint each other, one has ( 1 ) K1* K2 = K2 * K1 and (K1* K2) * K3 = K1* (K2 * K3) (=
K1* K2
* K3
for short). (2) For any simplex a E Kl * K2 and any vertex v E K3, a * {w} is a simplex in Kl * K 2 * K3.
Let (Ki= (K, &))El be a sequence of finite simplical complecies with vertices, i.e., 0-faces, = {wil,. . . ,wimi}. For a join K ( n ) = ( V K ( n ) C , K ( n ) )= K1 * ... * K,, K k ( n )denotes the set of all k-simplecies in K ( n ) . We have locally defined simplical maps sv and dv for any vertex o E V K ( n ) . Their domains are given by dom(s,) = {a E C K ( n ) I w @ a and a * {w} E C K ( n ) } , dom(d,) = { a E C K ( n ) I v E a}. One sees that by the definition of a simplical complex, for any simplex a E C K ( n ) and any vertex w E V K ( n )with w @ a there exists a unique simplex a/{.} that is a face of a. Note that dom(s,) is identified with a link complex Link({v}, K ( n ) )in the combinatorial topology. The simplical maps s, and d, are then given by sv : dom(sv)n K k ( n )3 a
I-+
a * {w} E Kk+'(n),
d, : dom(sv) n K k ( n )3 a I+ a/{w}E K k - l ( n ) . Let C ( K k ( n ) be ) a C-vector space freely generated by k-simplecies K'"(n) with a canonical inner product (a11 0 2 ) = d,,,,. We extend the maps to linear ) ) , by operators on a Hilbert space C ( K ( n ) )= @ ~ o C ( K k ( ngiven
{ *oT}, a
=
if a E dom(s,), otherwise ,
dv(a)
=
{
a/:;}, if a E dom(d,),
otherwise .
Let us consider a CA-process associated with operators 1 1 =
JmvE and an~ = (n)sv vEVK(n)
279
One sees that for any u E K k ( n ) ,
We assume a bounded condition for the vertices V , of the sequence (Ki = (K, Ci))go,l, SUP#V~= M
< 00.
(1)
i
Lemma 2.2 F o r k 2 1 and n 2 0, we have lim #(*?=I K ) = 1, #Kk(n)
n+w
where (*?=lK)k stands for a set of k-simplecies in K ( n ) given by ( * ~ = ~ ~ ) k = { { 2 ) o } * . ' . * { 2 ) k } 12)iEFi, l
<jO<jl<*.*<jk
PROOF. By definition of K ( n ) ,one sees that any simplex u E K k ( n ) has a unique expression u=uo**.-*G~, u ~ E K ~ ,
+
-(2)
+
with 1 5 j o < ... < j , 5 n, i = 0,...,I and C(ki 1 ) = k 1. Here K! denotes a set of all k-simplecies in Kj. Fix the sequence (ko,. . . ,ki) and assume that kp > 0 and then 1 < k . It follows from the assumption (1) that the number of k-simplecies in K ( n ) of the form (2) is at most
The number of choice of (ko, . . . ,a) with c ( k i 2k - 1. One sees then
+ 1) = k + 1 and 1 < k is
# ( K k ( n )\ ( * Z ~ K )<~ ) - 1)nkMk+l, while
Hence
280
I Lemma 2.3 For any simplex u E K k ( n ) ,
#VK(n)- M ( k
+ 1) 5 #{a' E K"'(n) I #{a' E K"'(n)
0
C a'} 5 # V K ( n ) - ( k
+ l),
I a' c a } = k + 1.
PROOF.For a E K k ( n )and a' E K"'(n), a C a' means there is a vertex w E V K ( n )such that a' \ a = {w}. Then for any 0 E K k ( n ) ,there exist at most # ( V K ( n )\ a ) = #VK(n) - ( k 1) simplecies a' E Kk+'(n), and hence the second inequality of the first statement. Since every k-simplex 0 E K k(n) has a unique form (2), u' = a * {w} is a (Ic 1)-simplex in Kk+' (n) for any vertex w E V K ( n )\ (KO U ... U q,). Hence the first inequality of the first statement. The second statement follows from the definition of a simplical complex. I
+
+
Lemma 2.4 We have for any k 2 0,
PROOF.By Lemma 2.2, we have # K k ( n )M #(*y=lVJk. the assumption ( 1 ) that for any 1 5 il < . < ik 5 n, # V K ( n ) - kM 5
c
i E ( 1 , ...,n } \ { i l , ...,i k }
It follows from
#vi I #VK(n).
By definition, we then have
#(*;yqk = l
# v i O * . . # K k
I n
Hence the assertion follows.
I
28 1
We define vectors in C(K"-'(n)),
for k 2 0, where we put K-'(n) = (0). By Lemma 2.3 and 2.4, we have
(3)
M
and
Putting
=
fi!J!F),
we have
@El
an M and a n @ g lM (k + l)@C). (5) Let us consider a one-mode interacting Fock space r ( K )which is a completion of an orthogonal sum @ g o { C @ k( ,' l ' ) k } where the scalar product is given by (z@klW@k)k
= xkzw,
z,w E
c, x k = k!.
The creation at and annihilation a are respectively defined by at@k = @ k + l ,
a@k+l = W k + l @ k ,
where W k + l is a Szegij-Jacobi parameter [l]given by come to a conclusion.
&+l/Xk
=k
+ 1.
We
Theorem 2.1 Given a sequence (Ki = (K, C i ) ) z l of finite simplical complecies disjoint each other which satisfies the assumptions ( 1 ) . Then we have a limit for any k,2 2 0 and any finite sequence ( ~ E i {k}), lim
n+w
( @ k la:
. . aim@ l ) r ( q n )=) ( @ k la'' . . .a', *
@l)r(K),
282
where we write a: = a t , a; - an, a+ = at and a- = a, that is, the CAprocess o n r ( K ( n ) )associated with a: and a, converges in moments to the CA-process o n r ( K ) with the Szego-Jacobi parameters (wk = k), which is identified with a symmetric toy Fock space [lo]. PROOF.It follows from Lemma 2.3 that for w = CuEKk-l(n) w,u (v, E
C),
where 11 . 11 stands for the canonical norm (-1.). We see llanw112 5 (k by a similar way. The assertion then follows from (3), (4) and ( 5 ) .
+ l)llw1/2 I
3 The Set of Binary Trees and its Almost-Simplical Structure
A tree is a connected acyclic graph. A binary tree is a tree consists of external vertices i.e., vertices of degree 1, and internal vertices of degree 3. We fix an external vertex and call it the root. The remaining external vertices are called leaves. Y, denotes the set of binary trees with n + 1 leaves, which are labeled as 0,1,. .. ,n from left to right, and we call its element n-tree. Let (0) and (01) stand for the unique 0-tree and 1-tree respectively. 2-trees are obtained by substitution and re-labeling of leaves so : (01) e ((01)l) e ((01)2),
s1 : (01) e (O(12)).
We define inductively substitution and re-labeling maps si : Y k + Y k + l (i = 0, ..., k). For y = (--.i...) E Y k , we substitute i I+ (ii 1) and re-label j e j 1 ( j > i), what we call the i - t h degeneracy m a p [3,9]. One sees that every k-tree is expressed by parenthesis and labels (0,. . . ,k}: each label i
+
+
283
in the expression corresponds to the i-th leaf and each pair of parentheses ”(” and ”)” corresponds to a internal vertex (excepting 0-tree (0)). It is then shown that the number of k-trees is given by the Catalan number [7] l)!. The degeneracy map is one of the grafting operations on (2k)!/k!(k binary trees. For yl E Yp, y2 E Yq and 0 5 i 5 p , we define (p q)-tree y = si(y1; y2) by substituting y2 for the i-th leaf of 91,
+
+
y l = ( ...i...)++ ( . . . y 2 . . . ) =Y: and re-labeling the leaves of yi as 0,1, . . . , p q from left to right. One then sees si(y) = Si(y;(Ol)). Let (z) be a parenthesized block, i.e., an internal vertex, of a k-tree y. One sees then there exists a unique p&r ( i , z ) of an integer i and a binary tree z such that y = si(z;(z)). The left grafter Z(z) and the right grafter T ( + ) are grafting operations Yk + Y&+1defined by
+
Z(=)(Y) = si(z;sl((Ol); (.)I) and T(+)(Y) = si(z;so((Ol); (.))I respectively (see F’rabetti [3]). For instance, the 3-tree y = ((0(12))3) is expressed as y = s1(((01)2); (12)) and then 412) : T(12)
Y = ((0(12))3) * ((0(0(12)))3) * ((0(1(23)))4),
: Y = ((0(12))3)
* ((0((12)1))3)* ((0((12)3))4).
The i-th face map [3,9] is the map di : Yk i-th leaf and re-labeling. For instance,
*
+ Yk-1
*
which is a deletion of the
*
do : ((0(12))3) (((12))3) ( W 3 ) ((OW), di : ((0(12))3) I+ ((0(2))3) ((0213) ((01)2) and
*
*
d3 : ((0(12))3) I+ ((O(12))) I+ (O(12)). Remark 3.1 In the study of the set of planar binary trees by F’rabetti [3], it is shown that the set of binary trees (Yk,di,si) has the almost-simplical properties:
< j, Sj-ldi, for i < j ,
(d) didj = dj-ldi for i (ds) disj =
{
id,
+ 1, + 1,
for i = j , j
sjdi-1, for i
(s) sisj = sj+lsi for i
>j
< j.
Hence, for any field k, the sequence k[Yo] t k[Y1] e . . . k[Y,] t . . is a chain complex with a boundary operator d = xr=o(-l)idi. It is shown that the chain complex of binary trees is acyclic. We note that the set of binary trees occupies an important role in the studies of dialgebra homology theory [3,9].
284 4
A CA-Process on the Set of Binary Trees
We define the set
for each k-tree y E Yk. Proposition 4.1 For any y E Yk, we have #w-(g) = 2(2k
+ 1). +
PROOF.We note that any k-tree y E Yk has k 1 leaves and k internal vertices. For 0 5 i 5 k, one sees that di(si(y)) = di+l(si(y)) = y by the almost-simplical properties of (Yk, d i , si), and hence (i, si(y)), (i 1,si(y)) E u-(y). Let (2) be a parenthesized block in y. max(z) (resp. min(z)) stands for the largest (resp. smallest) label appears in the block (2). By definition, we see that drnin(z)(Z(=)b)) = ~rnax(z)+1(T(z)b)) = Y, and hence (min(z>,z(z)(~>>, (max(.) + 17(=)(9))E u-(Y>Let min(z) = i for a block (5) in y. By definition, si(y) contains the block (ii 1) while Z(,)(y) contains the block ( i ( z ) )which , implies si(y) # Z(z)(y). Similarly, we see si(y) # Z[,)(y) for the case min(z) = i 1. Hence (i, si(y)) and (i l , s i ( y ) ) are different from (rnin(z),Z(=)(y))for any choice of labels i and blocks (z). By the similar manner, we see that (i, si(y)) and (i 1,si(y)) are different from (max(z) l,r(=)(y)) for any choice of labels i and blocks (x).Thus, we have proved #u-(y) 2 2(2k 1). Suppose that there exists a k-tree y such that #w-(y) > 2(2k + 1). It follows from the disjoint union
+
+
+
+
+
+
( 0 , . ..,k
+
+ 1) x Yk+l =
u-(Y) Ycyk
that we have
(k 2)#yk+l =
#u-(Y)
> 2(2k 4-I)#&,
YEYk
while we have the equality
(k + 2)#&+1 = 2(2k + I)#& since #Yk is given by the Catalan number (2k)!/k!(k+l)!. shows that #u-(y) = 2(2k 1) holds for all y E Yk.
+
This contradiction
I
285
For a k-tree y , I(y) stands for the set of all parenthesized blocks i.e., the set of internal vertices of y. We define the sets for each (k 1)-tree y E Yk+1,
+
= { ( i , g f ) E { o , . - - , k }x yk I Si(y’> =y), uF(y) = {((zC),yf)1 y f E yk, (2)E I(y), l ( z ) b f ) = Y> wz(y)
and “J:(y>
= { ( ( z ) , g f )I yf E yk, (2)E
I(y), T ( z ) b f )=
Proposition 4.2 For any y E Yk+l, we have
# w f ( d . 2 + #wI+(y)
+ #W,+(Y)
= k + 2.
PROOF.We note that for each i-th leaf of (k + 1)-tree y, only the one of the three cases occurs: there exists a k-tree y f and 1) either s i ( y ’ ) = y or s i - l ( y f ) = y holds, 2) there is a block (z) E I(y) such that l ( z ) ( y f )= y and min(z) = i, 3) there is a block (z) E I(y) such that r(%)(y‘)= y and ma(.) 1 = i.
+
By definition, one sees that y f E Yk and (z) E I(y) are uniquely determined in each case. The case of 2) and 3) give an injection L
: w;t(y)
u u,+(y) + (0,. . .,k + 1>
and the case of 1) gives a 2-to-1 correspondence
( 0 , . . . ,k
+ 1) \ Range(L) + wz(y). I
Hence the assertion follows.
Let us construct a CA-process on the set of binary trees. C(Yk) denotes a C-linear space freely generated by k-trees with a canonical inner product (zly) = bzy for z,y E Yk. Let us put a vector
The face map di and degeneracy map si are linearly extended to the maps
di : c(&)
c(Yk-1)
and
Si : c(Yk)
c(Yk+l)
by a canonical way. For a parenthesized block (5)E UyEykI(y),the left and right grafters are extended to the linear maps c(Yk)+ c(Yk+l) given by
286 We then define the creation a+ : C(Y,) + C(Y*+l),
i=O
(*)EI(Y)
and the annihilation a : C(Y*)+ C(Y*-l), m ._
a = Cdi.
(7)
Here I ( Y )stands for the set of all parenthesized blocks which appear in binary trees, I ( Y ) = Ur=.=, UyEykI(y). It follows from Proposition 4.1 and 4.2 that we have
and
Put
@k
=J
m
\
E
k
,
then we have
Summing up, we have the following result. Theorem 4.1 The CA-process on the set of binary trees {Yk} associated with the creation a+ of ( 6 ) and the annihilation a of ( 7 ) is described on the one-mode interacting Fock space r ( Y ) with Szego-Jacobi parameters W k + l = 2 ( 2 k + l ) ( k + 2 ) , which is a completion ofthe orthogonal sum @ r = o { ( c @ k , ( * \ ' ) k } where the scalar product is given b y
287
Acknowledgments
I would like to thank Professor N. Obata for the invitation to contribute t o this book. References 1. L. Accardi and M. Boiejko: Interacting Fock spaces and Gaussianization of probability measures, Infinite Dimen. Anal. Quantum Prob. 1 (1998), 663-670. 2. S. Ali, J.-P. Antonie and J.-P. Gazeau: “Coherent States, Wavelets and Their Generalizations,” Graduate Texts in Contemporary Physics, Spriger-Verlag, 2000. 3. A. Frabetti: Simplical properties of the set of planar binary trees, J . Algebraic Combinatorics 13 (2001), 41-65. 4. Y. Hashimoto: Quantum decomposition in discrete groups and interacting Fock spaces, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 4 (2001), 277-287. 5. Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 45-57, World Scientific, 2001. 6. Y. Hashimoto, A. Hora and N. Obata: Central limit theorems f o r large graphs: A method of quantum decomposition, to appear in J . Math. Phys. 7. P. Hilton and J. Pedersen: Catalan numbers, their generalization, and their uses, Math. Intelligencer 13-2 (1991), 64-75. 8. A. Hora: Central limit theorems and asymptotic spectral analysis o n large graphs, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), 22 1-246. 9. J.-L. Loday, A. Frabetti, F. Chapoton and F. Goichot: “Dialgebras and Related Operads,” Lect. Notes in Math. Vol. 1763, Springer-Verlag, 2001. 10. P. A. Meyer: “Quantum Probability for Probabilists,” Lect. Notes in Math. Vol. 1538, Springer-Verlag, 1993. 11. W. Schoutens: “Stochastic Processes and Orthogonal Polynomials,” Lect. Notes in Stat. Vol. 146, Springer-Verlag, 2000.
WHITE NOISE ANALYSIS
- A MORE GENERAL APPROACH TAKEYUKI HIDA Meijo University Nagoya 468-8502 Japan E-mail: [email protected] White noise theory has well developed in stochastic analysis, and it is now the suitable time to expect further developments in line with the original idea for the study of stochastic processes. To be more concrete, we take random complex systems and wish to analyze them by following the steps: Reduction & Synthesis Analysis. To establish such a way of research systematically, we need to introduce new classes of random functions different from the Hilbert space which is familier for us. By doing so, it is illustrated that interesting questions can be discussed within our new setup. Mathematics Subject Classification (2000): 60H40
1 Foundation of a Stochastic Process, Revisited 1 . 1 Innovation
Concerning the foundation of stochastic process theory, we should like to be all the way back to the book
J. Bernoulli: Ars Conjectandi, published 1713, after his death. We may understand that conjecture or prediction for random events would be suggested to the readers in the study of stochastic processes as well as limit theorems. Recurrence to the idea, that is seen in this old literature, should be emphasized and the basic idea has been well illustrated by many probabilists, so far. Among others, E. Slutsky, in his 1928 C. R. Acad. Sc. paper, discussed a theory of stochastic integral. Then, S. Bernstein proposed, in 1938, stochastic differential equations. Having skipped many other approaches, we come to the innovation approach (in our terminology) proposed by P. LBvy in his 1937 book (also see [4],Chapitre 11). Later more concerete formula, called stochastic infinitesimal equation was proposed in 1953 Berkeley publication [5]. It would be fine if the variation of stochastic process X ( t ) over an infinitesimal time interval [t,t d t ) could be expressed in the form
+
6 X ( t ) = @ ( X ( s )s, 5 t , Y ( t ) ,t , d t ) , where { Y ( t ) } is the innovation, which will be prescribed later. Surely, the equation has only a formal significance, however it tells us the idea how to
288
289
approach the investigation of the X ( t ) . Based on the systematic research on white noise analysis, we shall further come to the study of general random complex systems, which develop as time or space-time varies. For this purpose the innovation approach is just fitting. In the case of a stochastic process X ( t ) , the Y ( t )in the above stochastic infinitesimaln equation stands for, intuitively speaking, a random variable which is a function of the X ( s ) ,s 5 t + d t , being independent of the X ( s ) , s 5 t , and the collection { Y ( s ) s, 5 t } contains the same information as that gained by the X ( s ) , s 5 t . In terms of the sigma-fields, we can say as follows. Let B t ( X ) be the o-field generated by X ( s ) , s 5 t. Then, Y ( t )is such a random variable that it is independent of B t ( X ) , and B,(X) v B(K) = ne,oBt+e(X). As a result, it can be proved that the innovation is a generalized process with independent values at every instant t under minor assumption. Because of the relationship of the sigma-fields, construction of the innovation can be done as the step of reduction. Once the innovation of the given system is obtained, the random complex system in question may be expressed as a functional of the innovation Y ( t ) . This leads to the next step synthesis. Since the innovation can be viewed as the set of variables of the functional just constructed to represent the complex system, so that we are ready to analyze it by using those variables. Thus, the three steps are now in order: Reduction
3
Synthesis
Analysis.
Needless to say, applications can follow after analysis in many cases. Our innovation approach is certainly beyond the so-called L2-theory. In fact, sample paths or trajectories of stochastic process are explicitly and heavily involved in the expression and in the analysis. With this fact in mind, suitable spaces formed by functionals of paths will be introduced. Remark The term reductionism first appeared in 1948 in the OED Dictionary. P. W. Anderson used this word together with emergence. J. Wakefield suggests reserchers to move beyond the reductionist approach and tackle "the inverse problem" of putting the pieces back together to look at the complex system (Scientific American Jan. 2001, 24-25). This may come thanks to the computational power now available.
1.8 Examples There are various problems to which the innovation theory can be used efficiently and the classical L2 theory can not be applied.
290
a) Nonlinear Prediction Theory. N. Wiener’s theory ([8],[9]) of linear and nonlinear prediction as well as the coding and decoding problem could be one of the motivations of white noise theory, although it has not so many results in the same line as us. We shall not remain within the too beautiful and rather established results on L2-linear prediction theory, but we should construct a nonlinear theory for random complex systems more systematically, starting afresh from necessary background. b) Le‘vy’s Brownian Motion and General Gaussian Random Fields. It is interesting to find characteristics (or characterizations) of Brownian motion, in particular the motion with multi-dimensional time parameter. It is necessary to have a slight generalization of the concept of innovation. Namely, we take additional notion of multiplicity into account. We refer to the representation theory of Gaussian processes in [l] Part I1 no.8, and to McKean’s paper [6], 1963. The higher multiplicity, the more complexity is there. From each space with unit multiplicity exists an innovation. LBvy’s Brownina motion is a typical example of random field which enjoys complex and profound probabilistic structure with higher multiplicity, in fact, infinite multiplicity. c) Subordination by Random Time. Let X(t) be a given stochastic proces. Take an increasing LCvy process L ( t ) , t 2 0, which is independent of X ( t ) , and form a new stochastic process Z(t) = ( X 0 L)(t) = X(L(t)).
Such an operation defining a new process by the product of two stochastic processes is called subordination. This operation o can be considered as that defined by the sample function-wise product, and never be done within the L2-theory. d) Linear Process. Take a LBvy process L(t) without Gaussian component. To fix the idea, we assume that L(t) has only finitely many components of Poisson processes with different quantity of jumps. Then, two different kind of stochastic integrals can be defined. One is a stochastic integral based on the random measure dL(t), which is well defined, since the L(t) has independent increments and the existence of the second order moments is guaranteed. Another one is an integral defined by sample function-wise integral. This is possible, since the paths are bounded variation over a finite time interval. It is interesting to note that the probabilistic properties of two integrals are very much different, as we shall see later. Here is a short remark. Suppose we are given an integral in the second sense, it is an interesting question how to find the jumps (jump points and
29 1
heights) by some analytic or probabilistic method. Such a question suggests a connection with computability problem. e) Sample Function Properties. Analytic properties or regularity of a sample function of the innovation should be investigated for our calculus. Also, we often meet a question, when some applications are considered, asking if individual sample function enjoys any optimal properties. Such questions have been discussed in many places, but it should be noted that there are many significant questions are still left unsolved. 2
Spaces of Functionals of Innovation Process
We are now in a position to define suitable spaces of random functions, where our analysis, including the treatments of the examples listed above, can be carried out within those spaces. There, addition, multiplication, differentiation in the time parameter and other usual operations from analysis can, of course, be done without any difficulty.
2.1
The Hilbert Space ( L 2 )
First of all we must have a review of the basic space for Gaussian white noise. Let p be the satandard Gaussian measure on the space E* of generalized functions on Rd. Namely, p is the probability distribution of Gaussian white noise. The complex Hilbert space L 2 ( E * , p ) = ( L 2 ) can be formed in the usual manner. A member of ( L 2 )is called a white noise functional. The space (S)*of generalized white noise functionals can be defined by a Gel’fant triple
( S ) c ( L 2 )c (S)*, which may be viewed as an infinite dimensional analogue of the triple involving the Schwartz space in the left and the space of tempered distributions in the right. 2.2
Spaces of Poisson Functionals ( L 2 ) p and (P)
Next, take the Poisson noise with d-dimensional parameter space. In exactly the same manner as in the case of ( L 2 ) ,we can form a Hilbert space ( L 2 ) pof square integrable functionals of Poisson noise. We remind the characteristic functional of Poisson noise is of the form
292
We then come to another important spaces. For convenience, consider a Levy process L ( t ) , t >_ 0, without Gaussian part; namely it is a compound Poisson process. It is known that its elemental component, denoted by PU(t)is a Poisson process with fixed jump u, where u runs through a Bore1 set U. Let (L)obe a vector space spanned by all the random variables {P,(t),u E U , t >_ 0). Let the space be topologized by the convergence in probability, or equivalently by the norm
The closure of (L)o with respect to this norm is denoted by (L). The following assetion is rather triviality. Proposition 2.1 The Ldvy process L ( t ) forms a continuous curve living in the space ( L ) . The next space is generated by nonlinear functionals of members in (L), which is closed under the norm I( . 11. It is denoted by (P). Proposition 2.2 Let L ( t ) be a Lkvy process given in d) in the last section. Then, we have (L2>P
c
(PI
and the injection is continuous. 3
Analysis
3.1 Linear Processes Again we take the LBvy process L ( t ) introduced in d) of $1.2. Let F ( t ,u ) be a kernel function such that it is continuous in ( t ,u ) and that F ( t , t ) # 0. Define a stochastic process
X ( t ) = J, F ( t , u ) d L ( u ) . Remind what we briefly mentioned in d) of 51.2. The integral can be defined in two ways; one is a stochastic integral, since L ( t ) has independent increments and dL(u) has finite variance; while, another way is to define a sample function-wise integral, noting that dL(u) defines a locally finite measure almost surely. Fkom the.information theoretical viewpoint, we may say that
293
x(t)
Proposition 3.1 If the kernel F ( t , u ) is not a canonical kernel, the defined by stochastic integral has less information than that of L ( t ) over a time interval [0,T ] for any f i e d T . While, in the sense of the second integral (path-wise antegral) X ( t ) has the same information as that of L(t). The first assertion can be proved in the similar manner to the Gaussian case (see [l]Part I1 no.8), and the second assertion comes from the result by Hida-Ikeda [2]. 3.2 Nonlinear Functaonals with Finite Variance
As was introduced before, the Hilbert space ( L 2 ) p is taken to be the basic space. In order to have good representation in terms of classical functionals, we introduce an infinite dimensional analogue of the Laplace transform, where arises some difference between Gaussian case and that of Poisson type. In order to show this significant difference, two extremal cases will be shown. a) Gaussian Case, that is White Noise. The Hiibert space is just the (L2).The transform in question is the so-called S-transform:
where C(t) is the charcateristic functional of white noise with the expression and where t is a test functional in a nuclear space E . The e~p[-+1<1~], representation of cp in terms of a non-random functional of 5 is well known, together with the calculus base on this representation theory. b) Poisson Process and Poisson Noise. The case of functionals of a single Poisson process P ( t ) , we introduce the U-transform following the paper by K. Sait8 et al [7] to have an analytic representation. It is of the form
WCp)(t)= C P W
1
exp[(z, s)ICp(z)dPP(4.
Fact Under the U-transform, a discrete chaos (in Wiener’s sense) cp of degree n has a representation of the form
where u = ( u i , - - .,u,.,). Let it be denoted by Un(t). Since exp[it(t)] corresponds to P(t),the F’rkchet derivative &j plies the partial derivative acting on ( L 2 ) p . ap(t)
im-
294
c) The Discrete Chaos. Let Un(()defined above be depending on the time parameter t , in particular, assume that the kernel function involves t such that F = F ( t ,u). And the given Poisson functional X n ( t ) represents an evolutional phenomenon. Such a process is easier to deal with. As for a more general evolutional nonlinear Poisson functional, in particular for a random field X ( C ) formed by discrete chaos and parameterized by a smooth ovaloid C,we can construct a generalized innovation, which enjoys the orthogonality instead of independence at every u. Before the theorem is strated, a notation is provided. The symbol (C) denotes the domain enclosed by C. Theorem 3.2 If a random field X ( C ) formed b y the discrete chaos of degree n given above is causal, that is F ( t ,u) = 0 for u $ ( C ) , and if F ( t , u) # 0 , for u E ( C ) , then we can find a generalized innovation : P ( u j ) :. The proof can be given by using the variational calculus as was done in the Gaussian case (see Hida-Si Si [3]), although some modification is necessary. d) Multi-Dimensional Parameter Case. There are many interesting questions in this case and important applications can be seen in quantum field theory. However, there arises a difficulty when restriction of the parameter to a hyper surface is requested. In fact, such a restriction is necessary when variational calculus for X ( C ) is applied. This might seem to be a simple problem, but it is not quite. Some results concerning this problem have been reported, as a joint paper with Si Si, in the Proceedings of the Tunis Conference on Stochastic Analysis and Applications, to appear.
nj
Concluding Remark When we have discussed compound Poisson process, in fact, too strict condition has been assumed, namely the L6vy process in question involves only finite number of elementar Poisson processes. We may construct a compound Poisson process with countably many components by applying the quasi-convergence in probability. So far as we stick to the sample function-wise calculus in the space (P),it is natural to consider the computability of the limiting procedure, and conversely that of jump finding. The author hopes to discuss this question later. References 1. L. Accardi et a1 (Eds.): “Selected Papers of Takeyuki Hida,” World Scientific, 2001. 2. T. Hida and N. Ikeda: Note on linear processes, J. Math. Kyoto Univ. 1 (1961), 75-86.
295
3. T. Hida and Si Si: Innovations for random jelds, Infinite Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998), 499-509. 4. P. LCvy: “Processus Stochastiques et Mouvement Brownien,” GauthierVillars, 1948. 5. P. Lhy: Random ficnctions: General t h e o y with special reference to Laplacian random functions, Univ. of California Pub. in Statistics, 1 (1953), 331-388. 6. H. P. McKean: Brownian motion with a several-dimensional time, Theory Probab. Appl. 8 (1963), 335-354. 7. K. Sait6 et al: Poisson noise analysis base on the Lkvy Laplacian, in “Quantum Information IV (T. Hida and K. Sait6, Eds.),” pp. 103-114, World Scientific, 2002. 8. N. Wiener: “Exprapolation, Interpolation, and Smoothing of Stationary Time Series,” M.I.T. Press, 1949. 9. N. Wiener: ‘‘NonlinearProblems in Random Theory,” M.I.T. Press, 1958.
CHARACTERS FOR THE INFINITE WEYL GROUPS OF TYPE B,/C, AND D,, AND FOR ANALOGOUS GROUPS TAKESHI HIRAI’ 22-8 Nakazaichi-Cho, Iwakura, Sakyo, 606-0027 Kyoto, Japan
E-mail: hirai.takeshiOmath.mbox.media.kyoto-u.ac..jp
ETSUKO HIRAI Department of Mathematics, Faculty of Sciences, Kyoto Sangyo University, Kita-Ku, 603-8555 Kyoto, Japan All the characters of the infinite Weyl groups W B , of type B , f C , and W D , of type D , are given. Also all the characters of wreath product groups of arbitrary finite abelian groups with the infinite symmetric group are given. These groups contain inductive limits limn-,, G(T,1,n) of complex reflection groups. The proofs are outlined.
1
Preliminaries for Characters of Infinite Discrete Groups
We study in this paper, characters of the infinite Weyl groups, 6, of type A,, Wg, of type B,/C, and WD, of type D,. Furthermore we study characters of analogous groups given as wreath products of finite abelian groups T with the infinite symmetric group B , , which we denote by e , ( T ) (cf. ~41). Let us begin with some preliminaries on characters of infinite discrete groups. Let G be such a group. Denote by Cc(G) the *-algebra of all compactly supported functions on G with the operations ($1
* $2)(9)
:=
‘$‘l(gh-1)$2(h),
$ *(9)= $(g-’),
(1)
hEG
for $1,$2,$ E Cc(G), and g E G. Then it has a basis ( 6 , ; g E G), where d, denotes a function on G having the value 1 at g, and zero elsewhere. The identity element of Cc(G) is given by 6, with e the identity element of G. The completion of Cc(G) with respect to a certain special norm is the C*-algebra C*(G) of G. *SUPPORTED BY JSPS-PAN JOINT RESEARCH PROJECT “INFINITE DIMENSIONAL HARMONIC ANALYSIS.”
296
297
A unitary representation T of G corresponds bijectively to a nondegenerate representation of Cc(G)and that of C*(G) through
We refer [3, $61 for the theory of traces and characters for representations of C*-algebras, and for our special case of discrete groups, we refer also [13]. For a C*-algebra, a character t is, by definition, a trace which is semifinite, semicontinuous from below, and such that any such trace majorized by t is proportional to t . We translate this notion for a C*-algebra, on the level of Cc(G)in our case. A trace t on Cc(G)is a positive definite functional satisfying
t($i
* $2)
= t($2
* $1)
($1,$2
E Cc(G)1-
(2)
It is determined by a positive definite function f as
’
cC(G)
3
c--)
f($) := C g E G f(g)$(g)
E
c,
(3)
and the property (2) implies
f(g9’) = f(g‘9) or f(gg’g-9 = fb’) (979’E GI. This means that f is an invariant (or class) function on G. In the case of a discrete group, a trace t is a character in the sense of C*-algebras if and only if the corresponding f is indecomposable or extremal [13, Korollar 2 to Lemma 21. Let us explain the situation a little more in detail. Let L+(G) be the set of all positive definite class functions on G, and introduce in it an order f1 2 f2 by f1- f2 E L+(G). The set of all f E L+(G) normalized as f(e) = 1, forms a compact convex set P(G) under pointwise convergence. Denote by E(G) the set of extremal points and by F ( G ) its closure in P(G). Then F ( G ) is compact and E ( G ) is measurable. Any f E P(G) is expressed as an integral on F ( G ) against a unit measure p supported on E ( G ) ,that is, p ( F ( G )\ E ( G ) )= 0. Our present problem is to determine explicitly all elements of E(G) for the groups listed above. For the infinite symmetric group 6,, the problem has been worked out by E.Thoma [14], and we reexamined it in [6,7] by a completely different method. For a non-zero positive definite function f on G, we can associate, by socalled GNS construction, a cyclic representation rf as follows. Put U = Cc(G) for simplicity. Introduce in U a positive semidefinite inner product as ($17$2)f
:=
f(h-lg)$l(g)m g,hEG
($1,$2
E
a).
298
This inner product is invariant under the left translation (90,g E G,$ E U>. (L(go)$,)(g) := $(go-lg) Therefore the kernel J f of (. , . )f is L(G)-invariant. Taking a quotient 2lf = U/Jf and then its completion, we get a Hilbert space fjf on which a unitary representation 7 r f is induced from L(g0) (go E G). Let 210 E V(7rf)= fjf be the image of the delta-function Se E M, then it is a cyclic vector for 7rf and the original f is recovered as a matrix element of 7 r f as f(g) = (7rf(g)wo,vo). When f is invariant in addition, it has a certain special structure as explained below. For an f E P(G), the kernel J f is a two-sided *-ideal and Uf is a *-algebra. Furthermore, on the Hilbert space Bf,the left and right multiplications of 8, generate respectively representations 7 r f ($), pf ($) of Uf and accordingly of Q. They generate von Neumann algebras Uf := 7rf(U)", ?Z?f := pp(U)", and they are mutually commutants of the other. Denote by M+(f) the set of h E L+(G)such that h 5 cf for some c > 0, and by 3+(f)the set of all positive operators A in the common center 3(f) := Uf r l Bf. Then, there exists a bijective order-preserving mapping h H A between these two sets such that h ( $ l , $ z ) = ( A $ [ , $ ; ) for all E M, where $f := $ Jf E Qf C 4jf. Thus the common center 3(f) reduces to C .I if and only if f is extremal, where I denotes the identity operator [13, cited above]. When 7 r f is a factor representation, the character assosiated to it is t : C,(G) 3 $ t-)f($) E C , or on the level of von Neumann algebras
+
-
c
($ E CJG) 1This character has a finite value 1 for the identity operator I,since np(6,) = I,f(&) = f(e) = 1. Therefore, when dim7rf = co,the factor is of type 111, and when dim7rf < o, it is of type I, with n such that n 5 dimnf 5 n2 because 7rf is cyclic.
Uf 3 T ( $ ) f($) E
2
2.1
Infinite Weyl Groups of Type B,/C, Analogous Groups
and D,, and
Wreath Product Groups 6,(T) and 6&,(T)
The infinite Weyl groups of type B,/C, and D, are defined as infinite Coxeter groups, or as the inductive limits of finite Weyl groups of type B,/C, and D, when n + 00. We define them and analogous groups as follows. For a set I , denote by 6 1the group of all finite permutations on I , where a permutation u is called finite if the support of u,supp(u) := { i E I ; u(i) # i } ,is finite. We put 6 , = 6~ for the set of natural numbers N .
299
Take a finite group T , and define wreath product groups of T with symmetric groups as follows. First define a restricted direct product
D r ( T ) := n i E I T i , T; = T (i E I), and, for d = (t&r
(4)
E Dr(T), put
where the product is taken in a fixed order, and eT denotes the unit element of T . In the case where T is abelian, a subgroup of D r ( T ) is defined by
The group 6 1acts on D I ( T ) and D i ( T ) as a(d):= d' = (tL);€r with t; = t , , - ~ ( ~ (i)E I ) ,
(7)
and the semidirect products B r ( T ) := D { ( T ) x 61and, in case T is abelian, 6 ; ( T ) := D;(T) x 6 1are defined with this action. Identifying D r ( T ) and 61 with their images in these semidirect products, we have adu-l = ~ ( d and ) ( d , g ) ( d ' , d ) = ( d ( ~ d ' ~ - ' ) , a d ) (d,d' E D r ( T ) , g , d E 61).
When I = N , we replace frequently the suffix N by 00. The Weyl group of type A, is naturally equal to 6 ., Those of type B,/C, and D,, denoted respectively by WB- and WD-, can be realized
as
with T = 22 := 2/22. Moreover, the wreath product groups 6,(2,) = D , ( Z , ) >a 6, for T = 2, the cyclic group of order r , is the inductive limit of finite wreath product groups 61, ( Z , ) , denoted also by G(T,1, n). The latter is a series of complex reflexion groups and has Hecke algebras with good properties, called ArikiKoike algebras, and accordingly has recently attracted many mathematicians' interests and there appear many works on the representation theory, from which we refer here [ll]and [8]. The fact 6,(2,) = limn+, G(r,1,n) is used to establish the necessity condition for a positive definite class function to be extremal.
300
2.2 Finite-Dimensional Irreducible Representations Let us study finite-dimensional irreducible representations (= I&) of 6,(T) and 6 & ( T ) . Let r be such an IR of G := 6,(T). Consider a series of subgroups G, := 61,, ( T ) with I , := { 1 , 2 , . . . ,n } . Then G, 2 G as n + 00. Since d i m r < 00, there exists an n such that the restriction rl~,,of r on G, is already irreducible. Then, x(Gn) generates the full operator algebra 23 of V ( r ) .Take the commutant ZG(G,) of G, in G. Then for any h E Zc(G,), the operator r ( g ) commutes with every element in Q, and so is a scalar operator. On the other hand, any g E G is conjugate under G to an element h E Zc(G,). Therefore r ( g ) is a scalar operator together with r ( h ) . This means that d i m r = 1. Thus we get the following result. Lemma 2.1 (i) A finite-dimensional irreducible representation r of 6,(T) is a one-dimensional character, and is given in the form
r ( g ) = C ( P ( d ) ) (sgnJ
(a) for 9 = (d,a) E s,(T)
= & o ( T ))Q 6 ,
7
where 5 is a one-dimensional character of T , sgne(a) denotes the usual sign of a and E = 0 , l . (ii) W h e n T is abelian, a finite-dimensional irreducible representation T of G&(T) is a character given by r ( g ) = (sgn6)" (a) for g = (d,a)E 6 & ( T )= D&(T) x 6., As a consequence of this lemma, when T is akelian, one-dimensional characters of 6,(T) are parametrized by (<, E ) € T x { 0, 1 }, where r?; denotes the dual group of T . Cases of Infinite Weyl Groups. A one-dimensional character of W g , is given as follows. For d E D,(Z2), put sgnD(d) := sgn(P(d)). Then, for 9 = (4a) E D m ( Z 2 ) 6, , xa,b(g) := (%nD)"(d) * (sgn6)b(a) ( a , b E { 0,1})-
(9)
On the other hand, that of WD- is given by
Xb(g) := ( s g n ~ ) ~ ( c )( b E { (),I}) for 9 = (d,a) E D k ( Z 2 ) X
2.3 Standard Decomposition of Elements and Conjugate Classes Let G be 6,(T) or 6 & ( T ) .An element g = ( d ,a ) E G is called basic in the following two cases: CASE1: a is cyclic and supp(d) c supp(c);
301
CASE2: u = 1 and for d = ( t i ) i E ~t,, # eT only for one q E N . The element ( d , 1) in Case 2 is denoted by <, and put supp(f,) := supp(d) = Cq).
For a cyclic permutation u = (il, ia, . . . ,it) of l integers, we define its length as l ( u ) = C, and for the identity permutation 1, put C ( l ) = 1 for convenience. In this connection, 5, is also denoted by (t,, ( q ) ) with a trivial cyclic permutation ( q ) of length 1. In Cases 1 and 2, put C(g) = l ( a ) for g = ( d , o ) , and l(<,)= 1. An arbitrary element g = ( d , u ) E G , is expressed as a product of basic elements as
with g j = ( d j , u j ) in Case 1, in such a way that the supports of these components, q1,42,. . . , q,., and supp(gj) = supp(uj) (1 5 j 5 m ) , are mutually disjoint. This expression of g is unique and is called standard decomposition of 9. Note that C(<,,) = 1 for 1 5 k 5 r and C(gj) = l ( u j ) 2 2 for 1 _< j 5 m, and that, for 6,-components, u = u1uz . . . urngives the cycle decomposition of 0. To write down conjugate class of g = ( d , ~ )there , appear products of components ti of d = (ti),and then the orders of taking products become crucial when the finite group T is not abelian. For the present short note, we avoid this complication and restrict ourselves from now on to the case where T is abelian, except the case where the contrary is explicitly announced, for instance, as in 59.1. Assuming T be abelian, let us give a parameter of the conjugate class of g in (10). For gj = ( d j , u j ) , put 9; := (d>,uj)with d> = ( t l ) i E N with t:, = P ( d j ) = t i ,K j := supp(uj), for some io E Kj and t: = eT elsewhere.
niEKj
Lemma 2.2 Let T be abelian. For a g = (d,o) E G,(T), let its standard decomposition be g = Eq2 . .<,,glg2 . . .gm an (10). Define gi (1 5 j 5 m) as above and p u t 9' = cqz. . . <,,gig; ..g;, Then, g and g' are conjugate in 6,(T). Since T is abelian, the set of its conjugate classes is equal to T itself. Therefore, the conjugate class of gj and gi is characterized by the pair of P ( d j ) = P(d[i) E T and Cj = l(0.j)>_ 2. Thus we get the following corollary. Corollary 2.3 A complete set of parameters of the conjugate classes of non-
302
trivial elements g E B,(T)
is given by
{ti,tL,...,t i } and
{ ( u j , L j ) ;l < j < _ r n } ,
(11)
where ti = t,, E T* := T \ { e T } , uj = P(dj) E T and L j 2 2, and r or m should be > 0. Cases of Infinite Weyl Groups WB- and WD-. Let G be WB, or WD,. Let g = (d,a) E G be a basic element in Case 1, and let sgnD(d) = €1 ( E = *). In case E = +, g is conjugate to g' = ( e o , g ) ,and of the order L, where eD = (1,1,1, . . . ) denotes the identity element of D,(Z2). In case E = -, g is conjugate to &,g' = (d',a),d' = (1,. . . ,1, -1,1, . . .), for any p E supp(a), and of order 2C, where -1 in d' is placed at p . Their conjugate classes are parametrized by ( E , L ) . For a basic element En in Case 2, we take ( ~ , e=) (-, 1) as the parameter of its conjugate class. An arbitrary element g E G has a standard decomposition into a product of basic elements as g = -..tq,g1g2 . - . g r n , gj = ( d j , u j ) in Case 1 with L ( g j ) 2 2 for 1 5 j 5 m. Let the parameter of conjugate class of gj be ( ~ jL ,j ) , and denote by n,,e(g) the multiplicity of the parameter ( E , L ) , then
Then this system of parameters determines the conjugate class of g.
3
Characters for the Infinite Weyl Groups
The infinite symmetric group 6, is the Weyl group of type A,, and the symmetric group 6,,which is the Weyl group of type A,, is imbedded in 6 , as GI,, with I , := { 1 , 2 , .. . ,n } C N . Take a # 1 from B,, and decompose it into a product of mutually disjoint cycles (= cyclic permutations) as (7
= a1a2 . . . u r n , aj = (ijJ ij,2
.. .
(13)
ij,!,).
I{
-!}I
By definition, L j is the length of the cycle aj,and put ne(a) = j ; L j = the number of cycles ai with length C. The set of multiplicities { ne(o); L >_ 2 } determines the conjugacy class of a. In [14], Thoma has determined all the characters of the Weyl group WA- = 6,, and we review it a little later for the reference to our present results for WB, and WD,. For the infinite Weyl group G = WB, of type B,/C,, all the indecomposable (or extremal) positive definite class functions, which are also called
303
characters (of factor representations) of G, will be given explicitly here. Recall that G is naturally realized as a semidirect product group as with
G = WB-= 6,(T) = Dm(T)X 6 , T = Z 2 ,Dm(T) = &NT;, Ti = T (i E N ) .
( 14)
A one-dimensional character of G is given as a tensor product of such ones (sgnD)a of D = D w(T ) and (sgne)b of 6,: for g = ( d , a ) E G = D >a G W , xa,b(g) := (sgnD>"(d)' (s@e)b('J)( a ,b E { 0 , 1 )I,
niEN
(15)
with sgnD(d) = t; for d = (ti);cN E D. By our result, all the characters of G are written as follows. First let us prepare a set of parameters as
llall =
IIPII = l
l
Take a g E G and let
be a standard decomposition in (lo), i.e., a decomposition into mutually disjoint basic elements. Recall that for the basic component gj = ( d j , a j ) of g = ( d , ~ with ) d = (ti)iEN E Djv(T), we have supp(dj) C supp(aj) =: Kj (put). Then, dj
=(
t i ) i E ~ DK, ~ ~ (T)
D N ( T ) , sgn,(dj) = J'K~
(dj),
304
and the length ((a,) of the cycle aj is also denoted by ((gj). By definition, and in general
l(tq) = 1 for Cq = ( t q , ( q ) ) . We have xo,o(gj) = l,xO,l(gj) = (-l)'(gJ)-'
xo,o(tp) = 1, Xl,O(Sp)
= 1 = (-l)l(tp)-ll Xl,l(Cp) = -1.
XO,l(tPP)
= -1,
Theorem 1 Let G = WB,,, = 6,(T) with T = 22 be the infinite Weyl group of type B,. For a character of G there correseponds uniquely a parameter ( Q , ~ , ~ , ~ given , I C ) in (16)-(17), and it is expressed as fm,p,y,6,K in the following formula: f o r a g E G express it as in (19), then
s1
= s1;o + K ,
s1;o
:= IIQII+ IlPll - (Ilrll+ Il~ll).
(21)
The case where a1 = 1 (resp. = 1,yl = 1, and 61 = 1) corresponds to one-dimensional character XO,O = 1~ (resp. X O , ~= sgng,Xl,o = sgnD, and x1,1 = sgnD 8 sgnG) of G. The case ' a = P = y = 6 = 0 and K. = 0 ' corresponds to the regular representation XG of G. The case where 'y = 6 = 0 and n = 1 - (Ilall ~ ~ / 3 ~whence ~ ) ' , s1 = s1;o K. = 1, corresponds to a factor representation of the quotient group 6, S G / D of G = 6,(T) by D = D,(T), and in particular the case of K. = 1 (hence Q = P = y = 6 = 0 ) corresponds to the regular representation of the quotient group 6, or the representation of G induced from the trivial representation 10 of D. The case where 'a = /3 = 0 and K = -1 (Ilyll Il6ll) 5 O', whence s1 = s1;o K = -1, corresponds to a factor representation of G for which any d E D is represented by a scalar operator sgn,(d)I, where I denotes the identity operator on the representation space, and in particular the case of K = -1 (hence a = /3 = y = 6 = 0 ) corresponds to a representation of G induced from the one-dimensional character sgnD of D. (Note that the character sgnD is invariant under the action of the subgroup S := 6, of the semidirect product G = D >a S.)
+
+
+
+
+
305
Except the above mentioned 4 cases of one-dimensional representations of G, a character given above corresponds to a 111 type factor representation of G (cf. [13]). These factor representations can be decomposed into irreducible representations. (For explicit forms in certain cases of B,, see [lo].) Another Expression of Characters. We can rewrite the formula (20) in more compact form. For a, b E { 0 , 1}, put Qa,b;k 1 0 as a0,O;k
=a
k
,
a0,l;k
=Pk
,
a1,O;k
,
= yk
al,l;k
= 66 (k 2 1). (22)
For a basic element h = (&a) E G with L = C(h) := C(a),sgn,(d) = €1,we have xa,b(h) = ( ~ 1 .)(-l)b(e-l), ~ and put
=
c
a,&{ O J }
{a:
Ilk<,
+ (-l)'-'O;
+ ( ~ l ) y L+ (-l)e-l(~l)d;}.
(23)
l
In case C = 1, we have C(gP) = 1,sgnD(cp)= -1, and ~ - ; I ( < P ):= C a , b ~ { o ,1i Xa,b(h) ( C l < k < m aa,b;k)
= IlQll+ IlPll - IlYll - lldll = s1;o We define s-;1 adding some deviation s-;1 := &;l([p)
K.
to
+ - ; I ( < ~ ) as
+ K. = s1;o + K. = s1 .
Then, the formula (20) of positive definite function fcr,p,7,6,n(g)= (s-;1lr x
n
(24)
*
4ej;e, (gj),
Lj
(25)
fa,o,y,6,nis
= C(gj),
Ejl=
written as xl,o(gj)- (26)
1<j<m
For L 2 2, denote the number of gj = (dj,aj) with L(aj) = L and = +) by
~ g n D ( d j= ) €1( E
ne;e(g) =
i { j ; lbj) = 4 <1,o(gj)= sgnD(dj) = €1> ( ,
and put n+(g) = T = the number of again rewritten as fcr,P,7,a,n(g) = (S-;i)
<jS in
n-;l(g)
x
(27)
(19). Then, the formula (26) is
n
(S,;e)ne;c(g). €=i=,e22
(28)
Review of the Case of Infinite Symmetric Group 6,. Let us rewrite here the formula of characters of WA- 2 B,, given in [14], into
306
another form which suggests our method of calculation in the present paper. For such a character, Thoma takes a paprameter (Y = ( ( Y k ) k ? l , P = ( P k ) k > l of decreasing sequences of non-negative real numbers satisfying
l l 4 + IlPll 5 1 . We newly put x(k)
= le,,
x(-k)
= sgnGm ,
a-k
=Pk
,
for k = 1 , 2 , . . . . Take a LT E 6, and let its cycle decomposition be and put t?, = l ( r j ) .We have formula in [14] is rewritten as
(-1)'j-l
=
We expand this product into a sum of monomial products as follows. Let K+ = max{k; (Yk > O}, K- = min{k; (Yk > O}, and let Z , , p be the intersection of the interval [K-,K+] c 2 with Z*.Then the sum over k E Z * in (29) is actually over k E Z,,p. Thus we get
4
Method of Proving Theorem 1
Let us explain our method of proving Theorem 1. Our proof consists of two parts. The first part is to prepare seemingly sufficiently big family of positive definite class functions on WB, = 6,(22). The second part is to guarantee that actually all extremal positive definite class functions or characters have been already obtained in the first part. 4.1
The First Part of the Proof
The first part of our proof has two important ingredients. One is a method of taking limits of centralizations of positive definite functions. This method, which will be explained in the next section, has been applied in [6,7] to the case of 6, and reestablished the results in [14].
307
The other is inducing up positive definite functions from subgroups. After choosing appropriate subgroups H and their representations T ,we use their matrix elements fx as positive definite functions on H to be induced up to G, and then to be centralized. Following our method in [4] of constructing a huge family of IURS of a wreath product group G = 6,(T) of any finite group T with 6,, we take so-called wreath product type subgroups H and their URS T of certain simple forms to get p = Indgr. This ingredient will be explained in the succeeding section. 4.2
T h e Second Part of the Proof
The second part contains also two ingredients. The first one is to generalize Thoma’s criterion, Satz 1 in [14], for that a positive definite class function is extremal or indecomposable. The second one is to determine the range of parameters appearing for extremal positive definite class functions. Actually in Theorem 1, the range of (a, p, y, 6, K ) should be specified. To do so, we apply in part Korollar 1 to Satz 2 in [14]. 5
Centralizations of Positive Definite Functions
Let us explain our method of taking limits of centralizations of positive definite functions. For a function f on a countable discrete group G and a finite subgroup G’ c G, we define a centralization of f with respect to G’as
c
1 f G ’ ( g ) := 7
IG I uEG’
f (ago-1).
Taking an increasing sequence of finite subgroups GN 7G, we consider a series f G N of centralizations of f with respect to G N and study its pointwise convergence limit, limN+., f G N , which depends heavily on the choice of the series GN 7 G. In our previous papers [6,7], we studied positive definite functions f ( a ) on G = 6, of three different types given in [1,2]: for a E G, T14 (-1
5 T 5 1);
qll‘ll
(0
5 q 5 1); sgn(a)qll‘ll
(0
5 q 5 11,
where r and q are constants. Here 101 denotes the usual length of a permutation a coming from its reduced expressions by simple transpositions, and llall denotes the block length of a, which is by definition the number of different simple transpositions appearing in a reduced expression of a. Then we have proved the following.
308
Theorem 2 Let f be one of the above positive definite finctions, and GN = G N ( N 2 1). Assume (TI < 1 or 0 < q < 1 correspondingly. T h e n the series of centralizations f G N of f converges pointwise to the delta function 6, o n G = 6, as N tends to 00:
f G N ( e )= 1; f G N ( a + ) 0 for (T # e ( N 3 00). (32) The delta function 6, is the character of the regular representation XG of G which is known to be a factor representation of type 111, and moreover it is a matrix element of XG corresponding to a cyclic vector vo = 6, E L2(G) : 6, (0)= (XG (0)vo 110). In the topology of weak containment of unitary representations, we can translate this convergence as follows. Each of the representations rf contains weakly the regular representation XG of G = 6,. We have also calculated various limits of centralizations of positive definite matrix elements of irreducible or non-irreducible representations which are induced from subgroups of wreath product type. Especially we observed the following fact in [6,7],which suggests strongly our present method of getting all the characters for the Weyl group WB- and so on. For a certain irreducible or non-irreducible unitary representation, the family of limits of centralizations of its matrix elements covers all the char., acters of the infinite symmetric group (5 6
Inducing up of Positive Definite Functions
In a general setting, let G be a discrete group, and H its subgroup. Take a unitary representation T of H on a Hilbert space V ( n ) ,and consider an induced representation p = Indgn. The representation space V(p) of p is given as follows. For a vector v E V ( r ) ,and a representative go of a right coset Hgo E H\G, put
Let V be a linear span of these V(T)-valued functions on G, and define an inner product on it as
(r(h)w,v')
if hgo = gb (3h E H ) , if Hgo # Hg;.
The space V ( p )is nothing but the completion of It.
(34)
309
The representation p is given as p(gl)E(g) = E(gg1) ( g 1 , g E G, E E V(P)1. Now take a non-zero vector v E V ( K )and put E = EV,=E V(p). Consider a positive definite function on H associated to T as
f7T(h)= ( T ( h ) V , 4
( hE HI,
(35)
(9 E GI.
(36)
and also such a one on G associated to p as
F(g) = (p(g)E,E)
Then, we can easily prove the following lemma. Lemma 6.1 The positive definite function F on G associated to p = Indgx is equal to the inducing up of the positive definite function fir on H associated to K : F = Indz fir , which is, by definition, equal to f i r on H and to zero outside of H .
Centralizations of F = IndZfr. Let G N 7 G be an increasing sequence of subgroups going up to G, and consider a series of centralizations FGN of F . Since F is zero outside of H , the value of centralization FGN(g) is # 0 only for elements g which are conjugate under G N to some h E H . Moreover, for h E H , we get
The condition aha-l E H for (T E G N ,is translated into certain combinatorial conditions, and to get the limit as N + 00, we have to calculate asymtotic behavior of several ratios of combinatorial numbers. The details in the case of G = 6, are given in [6,7]. For our present case, we give an explicit formula for all the characters of the infinite Weyl group G = WD- in Theorem 3 in the next section, 57, and that for all the characters of the wreath product groups G = 6,(T) with T any finite abelian groups, in Theorem 4 in 58. After giving these general formulas, we will explain about three important points of our proofs: (1) subgroups H , (2) their representations K , and (3) increasing sequences of subgroups G N i” G, in the last section, 59.
7 Characters for the Infinite Weyl Group of Type D, For the infinite Weyl group G o := WD- of type D,, all the extremal positive definite class functions, or characters (of factor representations) of G, are given
310
explicitly here. Recall that GD is realized as a semidirect product group as
GD = WD, = e&(T) = Dk((T)>a 6, with T = 2 2 , D&(T) = { d = (ti)icN E o,(T), s @ D ( d ) = eT 1, D m ( T ) = n&Ti, Ti = T (i E N ) , sgnD(d) = P(d) := f l i E N t i .
c N , and also sgnD(d)
Put Pr(d) = n i E r t i for a subset I
(38)
= Pl(d)for
d E Dr(Z 2) 9&0(Z2).
A one-dimensional character of G o is given as (sgng)", b = 0 , l . However we need one-dimensional characters of so-called wreath product type subgroups H of G D , and so we keep notations in the case of the Weyl group GB := WB,. Quite similarly as in the case of WB, , we prepare a set of parameters as
Here a , p , ~6 ,and
K:
satisfy the condition
Take a g E GD and let 9 =
..
'
. .gm '
be a standard decomposition, i.e., a decomposition into mutually disjoint basic elements. Then, for a basic component gj = ( d j , aj) of g = (d, a) with d = (ti)iEN E D m ( T ) ,we have supp(dj) C supp(aj) =: Kj , dj
= (ti)i€Kj E DKj ( T )9Dm(T), s @ D ( d j ) = PKj ( d j ) .
Note that each component
gj
=
(dj,aj)
does not necessarily belong to
G o , because, for d j , it may naturally happen s g n D ( d j ) = -1, whence gj E GB \ G o , and that each component Cqbedoes not belong to GD but to GB. After careful discussions on the relation between GD and G B , we obtain the following result for the infinite Weyl group GD from the result for GB. Theorem 3 Let GD = W D , = G&(T) with T = 2 2 be the infinite Weyl group of type D,. For a character of GD there corresponds a parameter
31 1
(a, 0,y,6, K ) given in (39)-(40), and it is expressed as fa,p,y,6,n in the following formula: for a g € Go take its standard decomposition as above, then
+ xi,o(gj)
r;"') + x1,1(gj) l_
6i(g'))
(41)
l_
with s1 = s1;o + 4
s1;o := 1.
+ IlPll - (Ilrll + IlW
(42)
T w o parameters ( a ,0, y, 6, K ) and (a',p', y',IS,K ' ) determine the same character of GD if and only if they coincide with each other or (Q', P') 8
(7, 6>, (Y',6') = (a,P),
K'
= --Ice
(43)
Characters of Wreath Product Group 6,(T)
Here we give our general results on characters of a wreath product group G = 6,(T) for any finite abelian group T . First let us introduce several notations. Let T^ be the dual group consisting of all one-dimensional characters of T . Denote by 1~ the identity representation of T , and put T^* := T^ \ { 1~ }, T* := T \ { eT }. Then
Cc,
I T Is, =
as functions on T ,
(44)
c&
C C,
O=CC,
1 ~ = -
onT*.
(45)
C€?*
CErT
Take an element g E G = 6,(T) and let its standard decomposition into basic components be as in (10)
. . *tqvg1g2. . 'grn (46) where the supports of components ql,4 2 , . . . ,qr and supp ( g j ) = supp ( u j ) (1 5 j 5 m) are mutually disjoint. Furthermore, uj is a cycle of length e(uj) 2 2 and suPP(dj) c suPP(uj) =: Kj and t q k = ( t q r , (qk)),tqr # eT, =tqltq,
7
7
with t(&J= 1 for 1 5 k 5 r. For 6,-components, Q = 01u2..-orngives the cycle decomposition of u. For dj = ( t i ) i E K j E D K (~T )c) D,(T), put PKj
(4) = n i E K j ti
7
W j )
:= W K j
(4)) = H i E K j <(ti).
312
For one-dimensional charcters of 6,, we introduce simple notation as x E ( o ):= sgnG((cr)€
(CJ
E t?jm; e = 0 ,l).
(47)
As a parameter for characters of G = 6,(T), we prepare a set QC,E
(C E T^, & E { 0,1)>, and P = (11&f
,
(48)
of decreasing sequences of non-negative real numbers "C,E
and a set of non-negative
with
Il%Ell
,
1 q , , , 2 L q , , , 3 2 ... 2 0 ; p~ 2 0 (C E ?), which satisfies the condition
= bC,E,i)i€N
=
q , E J
c
CrC,€,i 3
(EN
IlPll =
c
PC
.
C€F
Then we have the following result. Theorem 4 Let G = 6 , ( T ) be a wreath product group of a finite abelian group T with 6,. Then, for a parameter
A := ( ( ~ c ~ E ) ( C s ) E ~ x { o , l }; 11) ' in (48)-(49), the following formula determines a character element g E G , let (46) be its standard decomposition, then
(50) fA
of G: f o r an
Conversely any character of G is given an the form of f A . The parameter A of character is not necessarily unique because of the linear dependence (45) on T"of functions C , C E T^. To establish uniqueness of parameter, we transfer from the parameter A, to another parameter B = 4(A) given by
313
with K
= (.c)c@.
7
“c
(C
= Pc - P1T
E
m.
Then, because of the linear relation (45), we have
cc&
Pc
c
. C(tq,) =
Kc
.C(tqd7
CE?’
and the uniqueness of parameter is established. However the inequality (49) for the range of parameter A containing p cannot be translated in a compact form in another parameter 4 ( A ) containing n in place of p. Note that the factor for tq,= ( t q k(,q k ) ) in the formula is rewritten as
c (ll%oll + II“c,1II + c(II’yc,oll + Il%lll> Pc)
.C(tq,)
CE?
=
.C ( t Q k ) +
C€?
9
c
nc . C ( t Q L ) .
CE?*
Subgroups, Their Representations for Wreath Product Group 6,(T)
9.1 IURs of 6,(T), T a Finite Group, as Induced Representations
In the previous paper [4], we have constructed a big family of IURs by the method of inducing up from wreath product type subgroups. Let us review it briefly. Take a subgroup H of G = 6,(T) of the form
H = Ho x n;,,Hp Hp
1
Ho = 6 r , ( T ) ,
= 6 1 p ( T p ) = Dr,(Tp)
(53)
61, 7
where I0 is a finite subset (we admits empty set), and Ip’s are infinite subsets of N all mutually disjoint, and Tp’sare subgroups of T. Thus H is determined by the datum c := P o ,
(4,TPIP€P)
and is denoted also by H C .We assume that H is “saturated” in G in the sense that N = I0 u ( U p E p I p is ) a partition of N . As an IUR of H , we take so-called factorizable one: n = no 8 ( @ : G p n p ) for
HOx
niEP Hp .
(54)
314
Here b = (b,),~p,b, E V ( T , ) , ~=~ 1, ~ ,is~a~reference vector to take tensor product of wp's, when P is infinite, and IURs TO and T, are given as follows. First choose an IUR CP E (resp.
F.
I ( a ) : v 3 v = @iEI,Wi (Vi E
v: = ".-'(i) Take a one-dimensional character 61,(T,) by the formula: T P ( ( d 9 4 ) := T,D(d)I
x:
-
<,
@iE.r,v: E
v,
V(C,,i),i E I,).
of 61,, then we get an IUR
(4XF(4
(
E DIp(Tp)>a
T,
of H p =
6IP),
cp
and similarly for HO = 6,(T). In case is one-dimensional or P is finite, the reference vector a, or b is not necessary. Thus the IUR T of H = H Cis determined by the datum (c, 3) with 0 := ( ( C 0 , X f )
>
(C,,a,,XpG)pEP; b ) ,
and is denoted also by n(c,a). We know in [4] that, under the saturation condition: N = I. U ( U p E p I p ) , the induced representation p(c,P) = Indg?r(c,3) is irreducible, and equivalence relations among these IURs are also clarified there. On this occasion, we can give a conjecture to generalize this method of constructing IURs as follows. Conjecture 2002-5 To have the irreducibility f o r induced representation p = Indgn, it can be generalized in two points: (i) f o r TO of Ho in (54), take any IUR of HO = 6,(T);
(ii) for H p and rP in (54), take the full group T as Tp and a cyclic representation of T as C,, then for a, = ( ~ , , i ) i ~take ~ ~ a, cyclic vector as ap,i E V(<,,i)= V(C,).
315
A Support for (ii): For the original setting as H p = 6 r p ( T p )and mp coming from E Tp, consider the induced representation Ind(7rp;H p f Hk) from H p to Hh = el,(T). Then, we get a representation n; of HL coming from a cyclic representation := Ind(cp;Tp f T ) of T , similarly as for mp from (see Theorem 3.12 in $3.6 in [4]). Usually 7 r L l D , D =
cp
h
cp
c;
@i:rpcL,i (L,i
DIP( T ) ,which is nothing but , = ci, with a certain reference vector a;, is far from irreducible and decomposed into a continuous integral of irreducible representations of D. In particular, if Tp = { e T } , we get as ci the regular representation AT of T . In these cases, the fact that we have still the irreducibility of 7rL for D M GI,,means that the symmetric group 6 r P G m acts 'ergodically', in a certain sense, on the tensor product space a'
4;v (c; ) . 1,
9.2 Subgroups and Their Representations for Matrix Elements
fn
In our present case where T is assumed to be abelian, T^ is nothing but the dual group of T and the situation is simpler at the point that no reference vectors are necessary. In place of the purpose in [4]of getting IURs, our present purpose is to get all the characters of G = e,(T) as limits of centralizations G where fn itself is a positive of matrix elements F = Indgfn of p = IndH7r, definite matrix element of a UR 7r of H . So we looked for a better choice of subgroups H and their (mainly non-irreducible) representations 7r. We follow principally the case of [4], but change slightly the choice of H and 7r to be well fit to our purpose. To give subgroups H , we take first a partion of N as
where each P c , ~ is an infinite index set, and all the subsets I, are infinite. Corresponding to this partition, we define a subgroup
with
Hp = elP(T), Hc = D r C ( T ) , He = { e } . Here e is the identity element of G, and we consider He as a trivial subgroup of el.(TI. For a representation m of H to be induced up to G, we take =
(@(<,E)&x{o,I)
( @ P € ~ c , ~ ~ P@) )(
@ c ~ i ' ~ C@) l H e
7
(57)
,
316
xc(dc) =
n
~ ( t c , i=) <(dc) for dc = (tc,i)i€I, E Dr,(T)
i€Ic
where we employed the abbreviated notation in Theorem 4 as ((4) := C(%(dP)) and C(dc) := C(Pq(dc))respectively for characters of the groups H p = 61,( T )= DI, ( T ) A 61,and Hc = DI, (T).
9.3 Increasing Sequence of Subgroups GN /' G = 6,(T) Depending on the choice of increasing series G N /' G of subgroups, we get various positive definite class functions of G as limits of centralizations FGN for F = Indz fT , which turn out to be characters. We choose a series GN as G N = ~ J , ( T JN ) , 7 N , and demand an asymptotic condition as
where P := U ( e , e ) E ~ x { O , I is ) Pthe c , Eunion of index sets. Then, Cp,EPAP
+ CCETPC 5 1.
(59)
For each (C, E ) E ?x { 0, 1}, let reorder the numbers { A, ; p E P c , }~in the N , also put p := ( p ~ ) ~ ~ ? . decreasing order and put it as ac,€= ( c Y c , , , ; ) ~ ~and Then we have
c
IIQc,€ll + IIPII I
1,
(C,E)€~X{O,l)
which is nothing but the condition (49). As a pointwise limit of the series of centralizations FGN, we obtain the character f A with
A = (("c,')(c,€)ETx(0,l} ; P ) in Theorem 4. The detailed calculation will be given elsewhere and it is also explained in [6,7] in the case of 6,. Finally we remark that, to obtain all the characters of G, it is sufficient for us to use only one set of H and A above, and this means that the induced representation p = Indgx contains weakly all the factor representation of type 111 of G.
317
References
1. M. Bozejko: Positive definite kernels, length functions o n groups and a non commutative von Neumann inequality, Studia Math. 95 (1989), 107-118. 2. M. Bozejko and R. Speicher: Completely positive maps o n Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), 97-120. 3. J. Dixmier: “Les C*-alghbres et Leurs Repr&entations,” GauthierVillars, Paris, 1964. 4. T. Hirai: Some aspects in the theory of representations of discrete groups, Japan. J. Math. 16 (1990), 197-268. 5. T. Hirai: Construction of irreducible unitary representations of the infinite symmetric group 6,, J. Math. Kyoto Univ. 31 (1991), 495-541. 6. T. Hirai: Centralization of positive definite functions, Thoma characters, weak containment topology f o r the infinite symmetric group, RIMS K6kyiiroku 1278 (2002), 48-74. 7. T. Hirai: Centralization of positive definite functions, weak containment of representations and Thoma characters f o r the infinite symmetric group, submitted. 8. N. Kawanaka. A q-Cauchy identity f o r Schur functions and imprimitive complex reflexion groups, Osaka J. Math. 38 (2001), 775-810. 9. N. Obata: Certain unitary representations of the infinite symmetric group,I, Nagoya Math. J. 105 (1987), 104-119; 11, ibid. 106 (1987), 143-162. 10. N. Obata: Integral expression of some indecomposable characters of the infinite symmetric group in terms of irreducible representations, Math. Ann. 287 (1990), 369-375. 11. T. Shoji: A Frobenills formula f o r the characters of Ariki-Koike algebras, J. Algebra, 226 (2000), 818-856. 12. H.-L. Skudlarek: Die unzerlegbaren Charactere einiger diskreter Gruppen, Math. Ann. 223 (1976), 213-231. 13. E. Thoma: h e r unitare Darstellungewn abzahlbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111-138. 14. E. Thoma: Die unzerlegbaren positiv-definiten Klassenfunktionen der abzihlbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 4061.
NONCOMMUTATIVE ASPECT OF CENTRAL LIMIT THEOREM FOR THE IRREDUCIBLE CHARACTERS OF THE SYMMETRIC GROUPS AKIHITO HORA Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology Okayama University Okayama 700-8530, Japan E-mail: horaaems. okayama-u. ac.jp We report our recent study on noncommutative central limit theorem related to the symmetric groups and Young diagrams, which includes an extension of Kerov’s Gaussian limit for the Plancherel measure of the symmetric group [S. Kerov: C. R. Acad. Sci. Paris 316 (1993)]. Applying this result, we further investigate the central limit theorem for the adjacency operators on the symmetric groups [A. Hora: Commun. Math. Phys. 195 (1998)] from the viewpoint of convergence of mixed matrix elements of relevant operators.
1
Introduction
It is a substantial problem in asymptotic combinatorics to clarify behavior of the irreducible characters and the conjugacy classes of the symmetric groups and structure of the ensembles of Young diagrams in the large volume limit. We begin with recalling a result due to Kerov on Gaussian limit of the irreducible characters. The symmetric group of degree n is denoted by S(n). The Young diagrams with n cells parametrize both the equivalence classes of the irreducible representations and the conjugacy classes of S(n). Let us assemble some notations in Young diagrams. Y,, denotes the set of Young diagrams with n cells. Set Y = Ur=oY,,. Here Yo = {0} (empty diagram). Y odenotes the set of Young diagrams which contain no rows consisting of a single cell. 0 is regarded as an element of Yo. The number of cells in v E is denoted by Ivl. We use the cycle expression v = ( 1 k 1 ( ” ) 2 k z ( v ) . . - j k J ( ”.). .) where k j ( v ) is the number of rows of length j in v E y . Note lvl = Cg,j l c j ( v ) . The union of two diagrams v1 and v2 is a new one V I U v2 in which kj (v1 U vz) = k j (VI) kj (v2) for V j E N. According to these notations, if p E Y o and IpI 5 n, we have p U ( l n - l p l ) E Y,,. For p E Y , and X E Y,, the conjugacy class of S(n) corresponding to p and the irreducible character of S(n) corresponding to X are denoted by C, and xx respectively. The value of xx at an element of Cp is denoted by x:. In particular, = dim X is the dimension of X E Y,,.
+
318
319
One can equip the family of Young diagrams with various structures of statistical ensembles. The most important one is induced by the Plancherel measure defined as M,,(X) = dim2 X/n! (A E Y,). M , is a probability on Y,. Given k E N, k 2 2, we consider a function 4 k on Y,, for n 2 k defined as
The mean and the variance of a normalized irreducible character with respect to the Plancherel measure are given by X X(k)U(ln--k)
X€Yn
dimX
dim2 --
n!
-0
and
respectively, where nLk = n(n- 1). . . (n- k + 1). Hence (1) is a normalization according to the order of the srandard deviation. Kerov [ll]showed the following central limit theorem (CLT in short) which describes asymptotic joint distribution of 4 k 7 S with respect to M,. Theorem 1.1 (Kerov [ll])For Vm E N, rn _> 2 and V z 2 , . . . ,x, E IR,
holds. Equation (2) implies that f#Jk'S corresponding to cycles of distinct length k are probabilistically independent in asymptotics, each obeying a Gaussian distribution. Such an asymptotic independence of cycles of different lengths suggests that they can play a role of building blocks of probabilistic models for the symmetric groups. Ivanov-Olshanski [lo] made much of this observation and developed CLT for irreducible characters of the symmetric groups, Young diagrams and their transition measures by succeeding Kerov's works. Also related to Kerov's works, remarkable results were obtained by Biane [2,3] on asymptotic behavior of irreducible characters of the symmetric groups under a suitable scaling limit. In $2, we will discuss a noncommutative version of the above Kerov's theorem. We assign an adjacency operator on S ( n ) to p E Y o and n 2 Ip( by
320
where L, denotes the left regular representation of S(n). Apu(ln-lpl) is a self-adjoint operator acting on 12(S(n)).Apu(ln-lpl) has the spectral decomposition
where {ExlX E Y,,} is a complete system of orthogonal projectors. Regarded as a multiplication operator, appropriately normalized x;u(ln-lPl~ (a function on Y,) can be identified with adjacency operator ApU(ln-IPI). For p1,. . . ,pl E Y o and n L max(lpl1,. . . , IplI), we get from (3) (bey AplU(ln-IP1l) . . .
ApIu(l~-~~i~)~e)~~(s(,))
The Plancherel measure thus corresponds to the vacuum state (bey * b e ) ~ ( ~ ( r z ) ) * Our noncommutative CLT is as follows. We decompose adjacency operator A ( k ) u ( l n - b ) (2 5 k 5 n) corresponding to a cycle into a sum of two noncommuting operators. Those noncommuting components are normalized according to the scale of CLT. Then we show that they converge to certain operators as n + 00 (in the sense that the matrix elements of an arbitrary mixed product of them converge). See §2 for a precise formulation. The limit operators act like creation and annihilation on the Hilbert space equipped with an orthonormal basis parametrized by Y o . Such a noncommutative (or quantum) CLT summarized as (i) quantum decomposition of observables (ii) construction of a certain Fock space and creation/annihilation operators acting on it (iii) convergence of arbitrary mixed products of quantum components in (i) to the same type of products of creation operators and annihilation operators in (ii) (in the sense of convergence of matrix elements) goes beyond adjacency operators on the symmetric groups to those on various discrete groups and regular graphs. We refer to Hashimoto [4],HashimotoObata-Tabei [6],HashimoteHora-Obata [ 5 ] ,Hora [9]. As a general merit we
32 1
mention that combinatorial argument and limiting operation are much easier after quantum decomposition than before and that combinatorial structure of limit operators get transparent (for example, some commutation relations satisfied by them). As for graphs enjoying certain symmetries, the limit operators are described by using orthogonal polynomials in one variable. This description is based on the treatment of one-mode interacting Fock spaces due to Accardi-Bozej ko [11. Applying the noncommutative CLT given in §21we reinvestigate in $3 the result of extension of Theorem 1.1 to the adjacency operators corresponding to arbitrary Young diagrams in Y o (beyond cycles) which was given by Hora [7]. Ivanov-Olshanski [lo] gave another proof of it. We discuss also connection with their work. 2
Noncommutative Extension of Kerov's Theorem
Equation (4)shows that (2) is equivalent to
(Vm E N,Vpz,.. . lpm E N). Following Hora [81, we first introduce quantum decomposition of adjacency operator A ( k ) U ( l n - k ) and then a Fock space, creation operators and annihilation operators which describe the limit picture as
n + 00.
Let n E N. On the Cayley graph of S ( n ) in which the generators consist of all transpositions {(i j ) } , the length of a geodesic joining z E S(n)with the unit element e is denoted by In(z). Then n-ln(z) is the minimal number of the cycles expressing z as their product. For j E { 2 , , . . ,n} and s E C(j)u(Ia-j), we define operators s+ and s- on Z2(S(n))by if Z,(sz) > Zn(z) if zn(sz) = ln(z) S-S, = if Zn(sz) < In(x), respectively. Set
if l n ( s z )< Zn(z)
322
Then A(jI"(1n-j) is decomposed into a sum of mutually adjoint operators A(j)u(ln-,l + and A(JTu(ln-j). We consider matrix elements of these operators on a subspace of Z2(S(n)). For p E Y o and n 2 IpI, set
Note that this coincides with 6, when p = 0. { @ ( p U(l"-If'I))Jp E y o ,JpJ5 n} forms an orthonormal system in Z2(S(n)). Their linear hull
r(qn))=
@
q p u (in-+"))
pEYO,lplln
is an invariant subspace for A&u(ln-jl ( j E ( 2 , . . . , n } ) which plays a role of a finite-dimensional Fock space (see Hora [ S ] ) . We define length function Z on yo by m
I(P) = \PI - tt(rows of P) =
C Cj- l)kj(p) j=2
(P E y o ) -
This definition is consistent with Zn(z)for z E S(n) because p E Y o ,IpI 5 n and 2 E Cpu(ln-IpI) (C S(n))imply Z(p) = Zn(z). The nontrivial conjugacy classes of the infinite symmetric group S(o0) = h q n S ( n ) are parametrized by yo. Let j3 denote the diagram obtained by adding a copy of the longest column of p E Y as a new column to p. Namely, kj+l(p) = k j ( p ) for V j 2 1. We have jj E Y o and Z(p) = I p ( . In particular, = 0. Under the operation p E y H jj E Y o ,each stratum Ynof the Young graph Y is tranformed into a stratum { p E y"lZ(p) = n} of y oinduced by the length function 1. (Compared to the notation IpI J = IpI CjE k j ( p ) in Ivanov-Olshanski [lo], l p l ~= lpl holds for p E Y.) As an oo-version of r ( S ( n ) ) we , define Hilbert space r(S(o0))to be the completion of
+
00
ro= @ Q ( p ) = @ @ p€YO
*(p)
(orthogonal sum)
r-0 p : l ( p ) = r
where Q(p)'s are all unit vectors. For j E {2,3,. . . }, we define creation operator BT and annihilation operator BJ: acting on the Fock space r(S(o0)) by
323
Here it is adopted that !O(p \ (j))= 0 if k j ( p ) = 0. B: and BjT are mutually adjoint operators which satisfy CCR on ro:
[BJ7,Bj+] = I, [Bf,BJ = [B',Bj+] = [B,,Bj+] = 0 (i # j ) . Theorem 2.1 (Hora [ 8 ] ) For V p , a E y o ,Vm E I+?, k 1 , . . . ,em E {+, -} and V j l , . .. ,j, E {2,3,. .. }, lim @(a u ( I ~ - I ~ I ) ) ,
n-+w
(
A(ii)U(ln-Jl ) &jl)u(l"-Jl)
...
A(l,")U(ln-Jm)
+(P
u (ln-lpl)))lz(s(n))
&%$Jp-Jm)
= (Q(a),B;:. . .B;z Q(p))r(s(w))
holds. Theorem 2.2 (Hora [8]) For Vm E N and Vp2,. . . , p , E A(2)U(1"-2)
??+A (J-) im 6,,
pz
... (
A(+J(l--m) J#c(m)u(l~-m)
)
N,
Pm 6e ) P (S(n))
= (*W, (B,++ B,)P1*(0))r(~(m)) * . . (*(@),(G+ K-JPm*(0))r(qw)) holds. Restricted on rj = @n>oQ((jn)), BTl,, and BJ:(rJ are a creation operator and an annihilation operator on the one-mode Boson Fock space respectively. Hence field operator (B; BJ:)(rJobeys the standard normal distribution with respect to the vacuum state. Theorem 2.2 thus yields ( 5 ) (equivalent to Theorem 1.1). In order to prove Theorem 2.1 we need to observe the action of on the basis vectors of r ( S ( n ) )and to analyze asymptotic branching behavior. {ApU(ln-IPI)IP E Y o ,[PI 5 n} generates a C-algebra which is called the BoseMesner algebra of (the group association scheme of) S(n). These generators satisfy a linearizing formula
+
The structure constants, called the intersection numbers, are given by
for any choice of 2,y such that z-'y E C T U ( l n - l r l ) . The following estimate is crucial for our purpose.
324
Proposition 2.3 (Hora [S]) For V j E {2,3,. . . } and Vm, 7 E
Yo,
holds unless T = D U ( j ) nor 7 = D \ ( j ) . Following Kerov-Olshanski [12] and Ivanov-Olshanski [lo], let tion on Y , be defined for p E Y by
8, a func-
We refer to Ivanov-Olshanski [lo] also for computing 9;. Let us see the connection of g& with the intersection numbers through the spectral decomposition of adjacency operators. Since
+
+
implies g;:('J) = 0. Equation j > IpI + if lpl 1 . 1 5 n. Note that 1 . 1 (10) leads us to an alternative proof of Proposition 2.3.
325
3 Central Limit Theorem for Arbitrary Adjacency Operators We extended Theorem 1.1 to adjacency operators corresponding to arbitrary conjugacy classes. Let H k ( z ) be the monic Hermite polynomial of degree k obeying
Theorem 3.1 (Hora [7])For Vm
E
N, Vpl, ... ,pm
E
Y o and
Vrjp'l,... , T , E N ,
holds. We showed Theorem 3.1 by purely combinatorial argument without using representation theory of the symmetric group. (Hermite polynomials appeared as matching polynomials of complete graphs in that argument.) Basic observations in the proof were:
(Obl) Rows of different lengths in Young diagrams behave like statistically independent in asymptotics. (Ob2) k multiplicity (or interaction) of rows of the same length is described by the Hermite polynomial of degree k. After observing that {p!,)JkE N} in (7) generates the algebra A of polynomial functions on the Young diagrams (see Kerov-Olshanski [12]) and that {ptflp E Y } forms a basis of A, Ivanov-Olshanski [lo] analyzed asymptotic behavior of p$'s by introducing appropriate filtrations in A. Their result provides us with an alternative proof of Theorem 3.1 through (9). Ivanov-Olshanski [lo] discussed the properties of (Obl) and (Ob2) in a more systematic way. From the viewpoint of noncommutative CLT in this note, we obtain the following.
326
holds. Applying the similar method to (Obl) to the left hand side of (ll),we can separate asymptotically the interaction between rows of different lengths. Essentailly we have only to compare (@(T
Apl"p2U(1"-IPlUPz1)
u
and
~ # ~ ~ l u p z u ( l ~ - I ~ l u P ~ ~ )
in the case where p1,p2 E Y o do not share any rows of the same length. Combinatorial argument as in Hora [7] works well for this aim. Alternatively, we can apply the estimate of pplupz n - &p:, obtained by Ivanov-Olshanski [lo] together with spectral decomposition (9). Note that, in this assumption, +lUPZ
= ZPl%
and
+
= #CpIu(l~-~P1i)#Cpzu(l"-iPzI)(l 4 1 ) ) . We can see how the Hermite polynomials appear as stated in (Ob2). UCplup2u(ln-IP1UPzi)
Essentially we have only to show
+ o(1). Equation (12) is seen by induction on k. Applying Proposition 2.3 to
(12)
327
we have
+o(l) . On the other hand, the recurrence formula for the Hermite polynomials yields
Combining these, we see that the induction proceeds to obtain (12). Finally we note that the right hand side of (11) is a nice expression. In fact,
holds on r0 for Qk E N and Qj E {2,3,. . .}. Since BT and BJ: satisfy CCR, we can rewrite inductively to be the normal order. The action of the operators having the expression (13) to Q(o)'s is clear. References
1. L. Accardi and M. Bozejko: Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), 663-670. 2. P.Biane: Advances in Math. 138 (1998), 126-181. 3. P.Biane: Int. Math. Res. Notices (2001), 179-192. 4. Y. Hashimoto: Infin. Dimen. Anal. Quantum Probab. Relat. Top. 4 (2001), 277-287.
328
5. Y. Hashimoto, A. Hora and N. Obata: Central limit theorems for large graphs: method of quantum decomposition, J. Math. Phys. in press. 6. Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 45-57, World Scientific, 2001. 7. A. Hora: Commun. Math. Phys. 195 (1998), 405-416. 8. A. Hora: A noncommutatiwe version of Kerov’s Gaussian limit for the Plancherel measure of the symmetric group, preprint, 2001. 9. A. Hora: Scaling limit for Gibbs states of the Johnson graphs, preprint, 2002. 10. V. Ivanov and G. Olshanski: Kerov’s centrd limit theorem for the Plancherel measure on Young diagrams, preprint, 2001. 11. S. Kerov: C. R. Acad. Sci. Paris, Skrie I316 (1993), 303-308. 12. S. Kerov and G. Olshanski: C. R. Acad. Sci. Paris, S&ie I 319 (1994), 121-126.
BROWNIAN MOTION AND CLASSIFYING SPACES REMI LEANDRE Institut Elie Cartan, Universite‘ de Nancy I Vandoeuvre-les-Nancy 54000 fiance E-mail: 1eandreOiecn.u-nancy.fr
1
Introduction
Let us consider a compact manifold. Associated to it, there are various cohomology theories: -) The Cech cohomology. -) The singular cohomology. -) The de Rham cohomology. -) The complex K-theory or the real K-theory. If we consider values in R, the first three cohomology theories are equal. The K-theory, by the theory of characteristic classes, gives a refinment of these three cohomology theories. In particular, by the Chern-Weil isomorphism, the complex K-theory tensorized by R of a compact manifold is equal to the even de Rham cohomology of the compact manifold. In order to understand characteristic classes, one can give explicit expressions in term of the curvature associated to a connection on the bundle. The second approach is to use the universal bundle associated to the infinite Grassmanian, called the classifying space, and the classifying map from the manifold M into the classifying space, whose pullback bundle gives the original bundle on M . We refer to the book of Milnor-Stasheff [56] about characteristic classes. The deep relationship between the theory of bundles and characteristic classes arises in the index theory (see [5,11,20]). Physicists (see [25,62]) replace the finite dimensional manifold by the loop space of it, and consider “Dirac” type operators over it. J&e-Lesniewski-Osterwalder [24] tensorized the “Dirad’ type operator by a finite dimensional bundle, given by an idempotent over an algebra of functions over the loop space (see [13,19]) too. Witten [62] claims that the K-theory of the free loop space is related to the elliptic cohomology (see [28] for details). The K-theory of the free loop space in the sense of Witten [62] is largely conjectural. The algebraic K-theory of the loop space in the sense of Jaffe-Lesniewski-Osterwalder [24] is not conjectural, but it is difficult to find a non-trivial example of finite dimensional bundle over the loop space which is a subbundle of a finite dimensional trivial bundle over
329
330
the loop space. Namely, if we consider for instance a loop group, it is not clear that the line bundle associated to the Kac-Moody cocycle satisfies this assumption. Let Bn(S1) the Grassmannian U ( n l)/U(n) x S1. The theory of Jaffe-Lesniewski-Osterwalder relies to the study of the injective limit of the set of maps from the loop space into B,(S1).We are motivated by this work in the set of maps from the loop space into the injective limit of B,(S1) (which is too the infinite complex projective space). The measures of physicists over the loop space are formal. A natural applicant is the Brownian bridge measure. Some stochastic regularizations of “Dirac” type operators over the loop space are studied in [27,31,43,44,53]. Some stochastic bundles were constructed in [35,36,38] by using a system of transition maps. In particular, it is shown by using a cohomological argument ([42]), that a stochastic line bundle (with fiber almost surely defined), is isomorphic to a true line bundle over the Hoelder loop space, if l 1 2 ( M )= 0 because in such a case a line bundle is determined by its curvature. This motivates in part the study of various stochastic cohomology of LQandre (see [39,46] for surveys). LQandre [38] gives the general definition of a finite dimensional bundle in the Malliavin sense over the loop space, by using transition functionals. This means that the transition functionals belong to all the Sobolev spaces over the loop space defined in [30,34]. There is a smaller class of bundles, which are given by an idempotent smooth in the Malliavin sense (see [50]). We can define a stochastic Chern character associated to it. We consider in [50] functionals smooth in the Malliavin sense in a finite dimensional Grassmannian. In the first part of this work, we define a richer class of bundles in the Malliavin sense by looking at maps which are smooth in the Malliavin sense into the infinite dimensional complex Grassmannian. We avoid the NualartPardoux Calculus in order to define a stochastic Chern Character, by pulling back as in [56] the characteristic classes over the infinite dimensional Grassmannian. The stochastic Chern Character belongs to the algebraic stochastic de Rham complex over the Brownian bridge of [51]. But Malliavin Calculus is inefficient to define stochastic cohomology classes with values in 2 or 2/22. Therefore, by Malliavin Calculus, we cannot reach stochastic Stiefel-Whitney classes, if we consider stochastic real bundles. There is another Calculus than Malliavin Calculus or its refined version the Nualart-Pardoux Calculus, which was inspired by the works in the deterministic case of Chen and Souriau (see [12,18,22,60]for related topics). The stochastic Chen-Souriau Calculus ([37,41,42]) is much more flexible than the Malliavin Calculus. It allows to define stochastic line bundle, because it allows to define stochastic Z-valued forms. We show in the second part
+
331
that the stochastic Chen-Souriau Calculus allows to define stochastic bundles, by using a system of transition functionals smooth in the Chen-Souriau sense, which coincides with the homotopy classes of functioanlas smooth in the Chen-Souriau sense in some classifying spaces, as it is traditional in the deterministic category (see [56]). Moreover, we can define a stochastic 2 / 2 2 singular cohomology on the Chen-Souriau sense, and, therefore by pullbacking via the stochastic classifying map the Stiefel-Whitney classes on the infinite real Grassmannian, we can define stochastic Stiefel-Whitney classes in the stochastic Chen-Souriau sense associated to a real bundle in the ChenSourian sense over the stochastic loop space. 2
Malliavin Calculus and Stochastic Chern Character
Let us recall some elements of the Malliavin Calculus on the loop space (see [30,34]). We consider a compact Riemannian manifold M . Let 2 be a based point in M . Let A be the Laplace-Beltrami operator over M . Associated to it, there is a semi-group exp[-tA] as well as a heat kernel pt(z,y). Over L,(M), the based loop space of continuous applications y from the circle into M starting from z,we consider the Brownian bridge measure. If F is a smooth functional from M' into R and if 0 < s1 < .. < sr < 1 are some deterministic times, we get:
where d~ is the Riemannian measure on M . s + -ys is a semi-martingale with respect to its natural filtration (see [7,15,23]) and we can consider the stochastic paralllel transport T~ associated to the random loop for the LeviCivita connection. A tangent vector field is given by
X , = rsHs.
(2)
where H. is of finite energy and Ho = H I = 0 (see [7,26] for a preliminary Id/dsHSl2ds. We get Bismut's form). The Hilbert norm of X. is IIX.112 = type integration by part formulas (see [7,14,30]):
Ji
E[(dF,X)]= E[Fdiv X ]
(3)
332
if in (2) H. is deterministic and F a cylindrical functional of the type considered in (1). These integration by parts formulas allow us to define first order Sobolev spaces
llF1112,P = E
Ji
[
]
PI2 1IP
( J 10
IWW)
(4)
Ji
if (dF,X ) = k(s)d/dsH,ds with k(s)ds = 0. We introduce a connection in order to get higher order Sobolev spaces
V X . = r.VH.
(5)
which allows us to iterate the operation of stochastic gradient
(C& F, X 1 ,. . . ,X') =
J
k(s1, -. sr)d/dSH:, . . . d/dsHlrdS1 .. .ds,,
(6)
[OJI'
where
Jik ( s 1 , ...,s,)dsi = 0, see [30,34]. We get the notion of Sobolev space:
We consider the injective limit M , ( C ) of the subset M , ( C ) of linear applications from C" into C". We consider the Hilbert norm llA1I2 = C IAe,12 where e; is the canonical basis of C". This norm is compatible with the canonical injection from M , ( C ) into M n + l ( C ) . We get a structure of prehilbert space over M,(C) (it is not complete). To take derivative, we consider its natural completion. B,(U(n)) can be considered as the set of orthogonal projectors of rank n belonging to M , ( C ) . But instead of taking the inductive topology as usual in B,(U(n)), we consider the topology which is issued of the prehilbert norm which is considered (see [56]). We can define maps from L , ( M ) into M,(C) which belong to all the Sobolev spaces by the following: Definition 2.1 F from L,(M) into M,(C) belong to all the Sobolev spaces if the kernel of its derivatives d'-,F(sl,..s,) satisfy to the property
for the prehilbert norm considered before. We approach the functional F by cylindrical functionals with values in M,(C) in the previous definition.
333
Definition 2.2 A Sobolev bundle of rank n over L,(M) is given by map p from L,(M) into B,(U(n)) which belongs to all the Sobolev spaces for the collection of norms (8). Let us provide an example of such map p if in (1) we replace p s ( z ,y) by p,,(x,y) for E small enough. That is we consider small random loops. We suppose that lll(M) = I12(M) = 0. The construction is taken from [56] and from [35,38,39]. We consider a set of N points xi on M , and the set of polygonal loops joining the xi associated to the subdivision t k = k / N of [0,1]. Let us consider such polygonal curve y+,. There are at most C N such polygonal curves. We can suppose we can find 6 small enough and the set of xi enough rich such that the open balls €or the uniform distance B(-yi,~;d) constitutes a cover of L,(M). Let us proceed as in [38] Chapter 11. We consider a representative w of an element of H 3 ( M ;2). There exists a 1 stochastic line bundle with curvature ~ ( w=) 27ri J, w(dy,, ., .) (see [38]). We can construct some transition functional p i , ~ ; jover , ~ B ( y i , ~S); n B ( y j , ~S); with values in U(1) = S1restriction to some maps over L,(M) which belongs to all the Sobolev spaces with Sobolev norms smaller than C(N+l)*(M+l)" ( a depends only on the chosen Sobolev norm). Moreover, associated to the cover B ( T ~ , N there ; ~ ) , exist a partition of unity Fi+ such that the Sobolev norms of F ~ , Nare smaller than (N 1)" exp[-CN] for a big C if E is small enough (see [39] pp. 127-128). Moreover, 1 < C F&. Over B ( y i , ~S), ; the bundle is trivial by a unitary trivialmization h(i,N). We associate the injective map C F ; , ~ h ( N) i , (see [56]) and the projector p ( . ) over its image. Since C I I F ~ , N<~ CCI ~ ~and , ~ by the estimates of the Sobolev norms of the transition functionals, we see that y(.) + p ( y ( . ) ) = C F i , ~ p ( y ( . )is) a map which belongs to all the Sobolev spaces into P,(C), the classifying space of U1,but here not endowed with the inductive topology. In order to state an homotopy theory, instead of taking the norm C lpei12 over our random projectors, we take over M,(C) the operator norm such that llABll 5 llAllllBll. Let us recall that we choose orthogonal projectors such that the canonical bundle E , (U( n ) )over B , (U( n ) )is hermitian and is endowed with the canonical Hermitian connection pdp. We can consider the same Sobolev norms as in (8), but with the operator norm over d';F(sl,. ..,s~). We say that (y(.), t) + p ( t , y(.)) with values in B,(U(n)) belongs to all the Sobolev spaces over [0,1] x L,(M) if for all ( a ) and for all T :
+
where we take the operator norm (see [61] for analoguous considerations). By
334
Kolmogorov lemma, we deduce if p(t, y( .)) belongs to all the Sobolev spaces in t E [0,1] and y(.), that t + p(t,y(.)) as almost surely a version which is smooth for the operator norm. Therefore, we can solve the linear equation:
dUt = -Utp(t, y(.)dtp(t, Y(.)))
(10)
because we consider the operator norm. U1 realizes, since the canonical connection pdp over Em (U(n))is unitary a random isometry from the fiber of the pullback bundle over the Brownian bridge associated to the stochastic classifying map y(.) + p(O,y(.)) to the fiber of the pullback bundle associated to the stochastic classifying map y + p(1, y(.)). Moreover, by developping UIin series of iterated integrals, because it is a solution of a linear equation, we see that U1 belong to all the Sobolev spaces, because we use the operator norm and because the parallel transport along the path t + (t,y( .)) is an isometry. So, the choice of the operator norm over M m ( C ) allows to state a deformation theory of the stochastic classifying map, corresponding for the Malliavin Calculus, to the deformation theory of deterministic classifying maps over a paracompact topological space (see [56]). This will allow to define a stochastic Chern Character in Malliavin sense, which depend in cohomology only of the homotopy class of the stochastic classifying map (see [50,54] for similar considerations). Let us clarify this point: let p be the stochastic classifying map. Since Itr(pb)( 5 Kllbll for the operator norm, tr(pdpApdp)"k) defines a smooth form over L,(M) which belongs to all the Sobolev spaces (see [32,45])for analogous statements, but in the refinment of Malliavin Calculus which is constituted of the Nualart-Pardoux Calculus). The big remark is that d2p = 0 such that tr((pdpApdp)Ak)belongs to the analogue in Malliavin sense of the algebraic de Rham complex defined in [51] for the Nualart-Pardoux Calculus. Therefore, by using the cyclicity of the trace: dtr((pdp A p d ~ ) " ~=)Ic(tr(dp A dp A pdp A (pdp A pdp)^("'))
- tr(pdp A dp A dp A (pdp A pdp)^("'))
(11)
By dp A dp is an even form. By cyclicity of the trace tr(pdp A dp A dp A (pdp A p d ~ ) ^ ( ~ - ' ) ) = tr(dp A dp A (pdp A pdp)"("')p
A dp)
= tr(dp A d p A (pdp) A (pdp A pdp)"(k-l)).
(12)
Therefore the stochastic Chern character related to the stochastic classifying map p which belongs to all the Sobolev spces for the operator norm over
335
M , (C) associated to the curvature R = p d p A p d p Ch(p) = Trexp[R]
(13)
is closed. That is all its components are closed (see [50,54] for analoguous conditions). We get: Theorem 2.3 In cohomology, ch(p) depends only on the homotopy class of P given by (9).
PROOF. ( t , y ( . ) )+ p ( t , y ( . ) ) defines a bundle over [0,1] x L,(M) in the Malliavin-Taubes sense. Its stochastic Chern character is closed over [0,1] x L,(M). We deduce as in [50] that:
a
-n-exP[Rtl
at
= d-f(.)%
(14)
where crt is a form over L,(M) whose components of order Ic are bounded in t E [0,1] in all the Sobolev spaces.
Rt = P(t, T(.)d7(.)P(GY(.) A P(t, 7 ( W 7 ( . ) P ( t r(.)i > is the curvature of the bundle over L,(M) by taking the stochastic classifying map over L,(M) y(.) -+ p(t,y(.)). Therefore the result, by operating component by component in the stochastic Chern character. I
3
Stochastic Chen-Souriau Calculus and Stiefel-Whitney Classes
Let (52, F,, P ) be a filtered probability space. Let M be a compact manifold. Following the terminology of [8,9],we consider the strong Hoelder based loop space L1/2--s,*,,(M)of M of maps y from [0,1] into M such that:
and such that yo = Yl = 2,
(16)
where d is the Riemannian over M supposed Riemannian. L1/2--E,+,~ ( m ) is a Banach manifold (see [8,9]). Let us recall (see [42] Definition 2.1), after imbedding M isometrically into a linear space RJ:
336
Definition 3.1 An admissible process over Rd is a process
where almost surely s,' IC,lds < 00 and sup, E[IA,IP]'lP < 00. Here 6 denotes the It8 integral and A, and C, are predictables. The Brownian motion starting from x is the solution of the Stratonovitch differential equation: dyt = r(yt)dBt,
(18)
where r ( y ) is the orthogonal projection from Rdinto T t ( M ) ,the tangent space of A4 in y, and Bt a Brownian motion with values in Rifixed definitively in all this part. yt has an heat kernel pt(x,y). If we put Xi(y) = r ( y ) e ; where ei is the canonical basis of Rd, the Brownian bridge is the solution of the Stratonovitch differential equation:
drt = 4 r t ) d B t
+ C x i ( r t ) ( X i ( y t ) , F a d logPl-t(yt,z))dt.
(19)
The finite variational term Ct in (19) checks 1 ; IC,lds < 00 almost surely and therefore the Brownian bridge is an admissible process. The sequel in this part is an adaptation of the considerations of Souriau and Iglesias ([22,60]). The reader can see the works of Chen [12] too for the study of cohomology. Definition 3.2 A stochastic plot of dimension n of L 1 / 2 - e , * , z ( Mis) given by a countable family (U,&, Ri) where U is an open subset of R" such that: i) The Ri constitute a measurable partition of R. ii) 4i(u)is an admissible process over Rd such that u + A,(u) is smooth is almost surely for the family of norms sup, E[IA,IP]llpand u -+ Cs(u) : IC,(u)lds is for all u almost surely finite. (The measurable smooth and J set of probability 1 where & IC,(u)lds l is finite does not depend on u.) iii) Over Ri, s + &(u) is almost surely an element of L1/2--E,*,r(M).
"3)
We identify two stochastic plots (U,&, R t ) and (V,45, if 4: = 4; almost surely over R t n Let us recall (see [41] Lemma 4.1): over Ri, there exists a version of 4i which is almost surely smooth for the 1/2 - E strong Hoelder topology over the loop space of M .
"5.
337
In the sequel, we will define a GL,(R) bundle in the Chen-Souriau sense over the based loop space, but we could do the same for a complex bundle. Let us recall first of all what is a functional (or a form) related to the ChenSouriau Calculus: Definition 3.3 Let 0 be an open subset of the strong Hoelder based loop M ) the strong Hoelder topology. A smooth stochastic space L 1 ~ 2 - e , * , x ( for k-form uO+t over 0 is given by the following data: let q5st = (U,q5i,Ri) be a stochastic plot. Let Ui = q5;'O on Ri. It is a random subset of U . Over Vi, we associate a random smooth form q5:tu0,st.It checks the following properties: i) If j is a deterministic map from V into U and if we consider the composite plot $st = q5st o j , then almost surely as smooth forms K p 0 , s t
(20)
= j*q5:tuo,st.
ii) If dst = (U,q5i,Ri) and $st = (Ulq5j,RS) are two stochastic plots such that there exists an Ri and anhn R[i and a measurable transformation H defined over a set of probability strictly larger than 0 over Ri into 0; such that $ j o H = 4i, then q5:tuo,,t = $:tuO,st
0
H
(21)
on this set. This allows us to define a G L , ( R ) bundle in the stochastic Chen-Souriau sense over the loop space: Definition 3.4 A stochastic G L , ( R ) bundle in the Chen-Souriau sense is given by the following data: i) A countable open cover Oi of L l p - e , * , z ( M ) . ii) Over Oi n O j , a functional g i , j smooth in the Chen-Souriau sense with values in GL,(R) which satisfies to the following requirements: g z., J.g , ,.z . - I d over Oi n O j (22) gi,jgj,kgk,i
= Id
over oi n oj n o k .
(23)
Let us recall (see [8,9]) that there exists smooth partition of unity gi over
L ~ I ~ - ~ , * ,associated ~ ( M ) to this cover. We can define the set of sections of the bundle given by Definition 3.4: Definition 3.5 A section $ of the stochastic GL,(R) bundle given by the data ( O i , g i , j ) is given by a family of functionals Fi smooth in the ChenSouriau sense with values in R" submitted to the relation Fj = gj,iFi over
<
oi n oj.
338
We call Ec the set of sections smooth in the Chen-Souriau sense of the stochastic bundle <. We take the tensor product Ec @ Ast of the space of these sections by the stochastic forms in the Chen-Souriau sense Ast. We can define an element $ of this tensor product as in Definition 3.5, but Fi is a collection of random smooth forms in the Chen-Souriau sense over Oi. Let us recall that we can define the stochastic exterior derivative on A,t (see [41,42]). A connection V is an application from Ec @ Ast which satisfies to the following request:
v($A a ) = v$A a +
A da.
(24)
Since there are partition of unity, and since over Oi the bundle is trivial, a connection is given over Oi by a stochastic 1-form which takes its values in the Lie algebra of GL,(R). Reciprocally, starting from a collection of stochastic 1-forms with values in the Lie algebra of GL,(R), we get connections V ; over 0; and a global connection, by putting V = CgiVi. So we can construct a stochastic connection in the Chen-Souriau sense associated to the bundle <. Let us consider the set 0; of open balls for the uniform distance B ( y j ;6 ) for 6 small enough for a countable set of smooth loops ~j such that the set of B ( y j ;6 ) constitutes an open cover of the strong Hoelder based loop space. If y E B(7j;6 ) for 6 small enough, we have a distinguished path joining y to yj by taking Fj(t)(-Y)(S) = exP,(s)[t(y(s)
- Yj(S))l,
(25)
where exp is the Riemannian exponential starting from yj(s) and y(s) -yj(s) is the vector of the unique geodesics joining yj(s) to y(s). Let us suppose II1(M) = 0. There exists a curve joining yj to the constant loop z.. If y E B(yj;d), let l j be the distinguished curve joining the loop y to x, by concatenating the two previous curves. Let ai be the parallel transport along this curve. Ify E B(y;;G)nB(yj;a), we put g;,j = aia;'. It is clearly smooth in the Chen-Souriau sense and check the properties (22) and (23). So we can come back to the system of balls for the uniform distance B(yj;6 ) in order to trivialize our stochastic bundle. Let B,(GL,(R)) be the classifying space of the Lie group. It is the injective limit of the finite dimensional real Grassmannian GL,+,(R)/GL,(R) x GL,(R) (see [10,56]). It can be seen as the set of symmetric projectors from R" into R" of rank n where R" is the set of (21, ..,zk, 0 , ..,O..) with xj = 0 for the integer j larger than some k endowed with the inductive limit topology (see [56]). R" is endowed too of a real prehilbert structure. (It is not complete.) These projectors p are of rank n. To each projector, we associate its image, and we get a bundle E,(GL,(R)) over B,(GL,(R)). Moreover,
339
B,(GL,(R)) is a manifold (see [56]). Unlike the previous part, we will consider the inductive topology over the classifying space (see [56]). We can define a stochastic diffeology over S, the "total" space of the stochastic bundle 5. A stochastic plot gst = (U,$i, ni) on 3 is given by the following data: -) A stochastic plot rGst = dst on the strong Hoelder based loop space. -) Over q5z10i a family of smooth random functions Fj,1 = 1 , . . . ,n satisfying the following requirement:
(F.") = g i , A w
(26)
over 0: n 0;.
If $it and $:t are two stochastic plots in E such that r$it = r$$, we can add $it and $,", and multiply them by a random functional g. We can define a smooth functional from E into R" through a system of plots, and we can define a functional smooth in the Chen-Souriau sense from the strong Hoelder based loop space into B,(GL,(R)) through a system of plots. It is a functional p with values into the the set of symmetric projectors of rank n from R" into R" which depends smoothly on the Chen-Souriau sense of the random path. This means, if we consider a stochastic plot dst = (U,&, ni) with values in the strong Hoelder based loop space, p o &t is almost surely smooth in the finite dimensional parameter u E U for the considered inductive topology over B, (GL,(R)) modulo some consistency relations we don't write. We can define a smooth homotopy pt between p o and p1 in the sense that if we consider a plot dst = (U,q5i, ni) on the strong Hoelder loop space, pt o dst is almost surely smooth in U x [O,11. Theorem 3.6 A bundle in the stochastic Chen-Souriau sense 5 over the strong Hoelder based loop space determines a map smooth in the stochastic Chen-Souriau sense from the strong Hoelder based loop space into the classifying space B,(GL,(R)) such that this bundle is isomorphic to the pullback bundle by this map of the canonical bundle over the classifying space B,(GL,(R)). Another map p' is necessarily homotopic to p . PROOF. It is the same as the proof in the deterministic case for a paracompact manifold of [56] pp. 67-68. We use here that there are partition of unity smooth in the traditional sense, therefore in the Chen-Souriau sense, of the strong Hoelder based loop space for the cover 0;. I
Bm(GL,(R)) coincides with the set of n-planes in R". Its 2 / 2 2 singular cohomology groups coincides with the polynomial algebra in the StiefelWhitney classes on the infinite dimensional real Grassmannian (see [56]). An
340
equivalence class of GLn(R)bundle in Chen-Souriau sense over the strong Hoelder based loop space is given by the previous theorem by an homotopy class of a map smooth in the Chen-Souriau sense pform the strong Hoelder based loop space into the real classifying space B,(GL,(R)). In order to define the stochastic Stiefel-Whitney classes associated to the stochastic bundle <,we will pullback through the stochastic classifying map p as it is traditional in algebraic topology (see [56]) the (deterministic) Stiefel-Whitney class associated to the canonical bundle E,(GL,(R)) over the classifying space B,(GL,(R)). For that, we have first of all to define stochastic singular cohomology groups with values in Z/2Z. Let A" be the canonical oriented simplex of R" included in some open subset U of R". We say that (An,&) is an oriented stochastic simplex with values in the strong Hoelder based loop space if it is the restriction for +it ni) to A". We can add and subtract some U to some stochastic plot (U, oriented stochastic simplices. The boundary of a stochastic simplex (A", q5st) is a union or a difference of oriented stochastic simplices. We can define the boundary of a difference or a union of n-dimensional stochastic simplices, say a n-dimensional surface C, considering the stochastic simplices which constitutes the boundaries of each stochastic simplices which constitutes C,. We get a (n - 1)-dimensional surface dC,. We say that an n-dimensional surface is a random cycle if its random boundary vanishes. The boundary of a random surface is a random cycle. In the sequel, we will consider components in 2/22. Definition 3.7 a E Hz(Ll,2-c,*,z(M);Z/2Z) is defined by the following data: to each stochastic cycle C of dimension n, we associate a Z/2Z-valued random variable (&a) which is 2 / 2 2 linear in C. If C = dC', (C, a) = 0. If there exists a measurable transformation H on the underlying probability space given on a set of probability strictly larger than 0 such that C o H = C', then almost surely on this set
(C, a)0 H = (C', a)
(27)
as random variables. Let us consider a stochastic bundle over the strong Hoelder based loop space and associated to it some stochastic classifying map p from the strong Hoelder based loop space into the classifying space B,(GL,(R)). Let wk the k-th order Stiefel-Whitney class on the classifying space B, (GLn(R)). If we consider a random k-dimensional cycle C over the strong Hoelder based loop space, p ( C ) constitutes a random cycle over the classifying space. By definition, (C,p*wk)= (pC,wk). If we consider another classifying map p l , it is homotopic by pt to p = p o , and ptC realizes a k + 1 dimensional surface
<
34 1
whose boundary is p l C - poxc.Therefore,
<
-) The stochastic Stiefel-Whitney classes Wk(<)depend only of the isomorphism class in Chen-Souriau senes of the bundle <. (It is possible to define what are bundle isomorphic in the Chen-Souriau sense. Equivalently, their classifying maps are homotopics).
-) Let us consider a smooth map f from ( M , z ) into (N,y) such that f(z) = y. We deduce an infinite dimensional map fm : y. + f(7.) which is compatible with the considered stochastic diffeologies. (It transforms a stochastic plot into a stochastic plot.) Let be a stochastic real bundle over L1/2--c,*,y(N)given by a stochastic classifying map p . We can construct the pullback bundle f&
<
-) If o and o' are two elements of H:t(L1/2--s,*,r(M),2 / 2 2 ) , we can define their cup product CT U o' as in [56] pp. 263-264, by taking random cycles instead of deterministic cycles as in pp. 263-264 of [56]. (27) is still clearly checked for u UCT'.We can define the sum [@ <'of two stochastic bundles [ and ['. We get: k
w i ( < )u wk--i(<')
wk(< @ 5') =
(29)
i=O
For that, we consider the two stochastic classifying maps p and p' from the and B,(GL,(R)) strong Hoelder based loop space into B,(GL,(R)) associated to the stochastic bundles and 5'. The classifying map of [@[' is the sum p e p ' . The result comes from the fact if we consider the bundle E,(GL,(R)) e E m ( G L ( R ) ) over Bm(GLn(R)) x %(GL,(R)), its deterministic Stiefel-Whitney classes satisfy to (29) by [56] p. 38.
<
342
Acknowledgements
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FOCK SPACE AND REPRESENTATION OF SOME INFINTE DIMENSIONAL GROUPS TAKU MATSUI AND YOSHIHITO SHIMADA Graduate School of Mathematics Kyushu University 1-10-6 Hakozaki, Higashi-ku Fukuoka, 812-8581, Japan E-mail: matsuiOmath.kyushu-u.ac.jp We explain here relationship between quasi-equivalenceof (projective) unitary r e p resentations for infinite dimensional groups U ( w ) , SO(oo), and Sp(oo,R), and that of quasifree states of CAR and CCR algebras.
1
Introduction
The theory of unitary representations for infinite dimensional groups is one of typical examples where the analytical tools of infinite dimensional space show its power and weakness at the same time. People have noticed that unitary representations of infinite dimensional groups appear naturally in the context of quantum field theory, integrable models of statistical mechanics and nonlinear equations. So far two classes of infinite dimensional groups are considered. (1) groups whose matrix elements are functions: Examples are loop groups and the diffeomorphism group of the circle (See Segal-Pressley [13], Carey-Ruijsenaars [6] and the references therein), and their higher dimensional analogue. (2) inductive limit of classical groups, SO(o0) or U(o0). See Borodin-Olshanski [5], Pickrell [12] and StrtitilkVoiculescu [14]. Usually, the construction of unitary representations has been carried out in the following ways. The first method is to construct measures on an infinite dimensional space quasi-invariant under the group in question (c.f. OlshanskiVershik [ll]). Another method is to use Fock spaces of the quantum field theory and unitary implementors of Bogoliubov automorphisms. (c.f. CareyRuijsenaars [6] and Striitilti-Voiculescu [14]). Asymptotics of combinatorics of representation of classical groups is another interesting aspect of the theory. In what follows, we consider projective unitary representations of the inductive limit of classical groups, U(oo),SO(oo), and Sp(o0,R)constructed in Fock spaces. Due to infinite dimensionality, we can implement a large class of representations of these groups with the help of embedding into the group of Bogoliubov automorphisms. The representation of CAR and CCR
346
347
algebras (the algebra of creation and annihilation operators) is highly nonunique, and we obtain a huge class of projective unitary representations on the representation spaces of CAR and CCR algebras and the above mentioned infinte dimensional groups. In case of SO(o0), our representation of Fock spaces is an infinite dimensional analogue of the spin representation while the symplectic case corresponds to the metaplectic (Weil) representation (c.f. Lion-Vergne [7]). In our construction, the gauge invariant part of the algebra of creation and annihilation operators (CAR and CCR algebras) plays an role of group algebra. For the SO(o0) case, we have an inner action of SO(o0) on CAR algebra (the algebra of Fermion creation annihilation operators) and the linear span of these representatives is dense in the Z2 gauge invariant part of the CAR algebra (the even part of the CAR algebra). Thus any representation of the latter yields a projective unitary representation of SO(o0). The Fock representation of the the gauge invariant part of the CAR algebra contains two sectors (two irreducible components) which is labeled by the Z2 Fredfolm index and associated projective unitary representations is classified with the equivalence classes of Fock representations of the CAR algebra and this Z2 Fredfolm index. In the same fashion, the ordinary Fredfolm index appears in this classification problem of irreducible representations U (00) on Fock spaces. In Section 2, we explain the results for sO(00) and the cases of U(o0) and Sp(o0) are mentioned in Section 3. 2
SO(o0) and its Spin Representations
First of all, we realize the inductive limit SO(o0) of the classical group S O ( N ) as a group of invertible operators on a Hilbert space. Set K: be the Hilbert space of square summable sequences:
The inner product of K: will be denoted by (h,f)c and we employ the convection that the second variables of the inner product is linear.
j=1
Let en be the standard basis of
K: determined by
en = {en(j)}>
e n ( j ) = aj,n
348
and let K N be the finite dimensional subspace of K spanned by {e1,e2,.. . , e ~ } By . PN we denote the orthogonal projection to K N . The special orthogonal group S O ( N ) acts on K N and any element g of S O ( N ) is extendible naturally to a group of invertible operators A on K such that ( 1 - PN)g(l - PN) = 1 - PN.
In this realization of S O ( N ) as operators on in SO(N + 1) and set SO(o0) =
K ,we have inclusion of S O ( N )
u S O ( N ) c B(K).
N>2
Next we introduce infinite dimensional Clifford algebra We follow the notation of H. Araki [ l ]as it can be employed for Sp(o0) and O(o0) in a unified way. Let J be the anti-unitary involution on K defined by the complex conjugation:
(Jt)(A = m. Then SO(o0) is real in the sense that JgJ = g for any g in SO(o0). Let d(K,J ) be the C*-algebra generated by B(h) satisfying
( B ( h l ) , B ( h 2 ) *= } (h2,hl)Kl’ B(h)*= B ( J h ) (1) where h2 and hl are vectors in ic. d ( K , J ) will be referred to as the selfdual CAR algebra over K where ‘CAR’ is the abbreviation of Canonical AntiCommutation Relations. The equation ( 1 ) contains the relation of Clifford algebra and *-structure. Definition 2.1 A projection E on K satisfying JEJ=l-E will be called a basis projection. Given a basis projection, we set
(2)
a*(h)= B ( E h ) , a(h) = a ( J h ) for any h in EK. Then,
{a(hl),a(h2))= 0, {a*(hl),a*(h2)) = 0, {a(hl),a*(h2)) = (hl,h2)K1This is the relation of CAR. The basis projection specifies the creation operators in our Clifford algebra. Suppose that u is a unitary satisfying the reality condition J u J = u. Then, we can introduce the Bogoliubov automorphism ^lu determined by ^ l u ( W ) ) = B(uh).
(3)
349
The equation (3) gives rise to a *automorphism of the selfdual CAR algebra W,J). In particular, we can introduce an involutive automorphism 0 determined by 0 = 7-1. By definition, 0 is an automorphism of d characterized by the equation:
O ( B ( h ) )= -B(h)
(4)
for any h in K. 0 gives rise to a Z2 grading of the selfdual CAR algebra d ( K , J). We denote the fixed point algebra under 0 by d ( K , J)+:
4 G J)+
= { Q E d ( K , J)
I @(Q)= Q> .
Next we introduce an action of SO(m) on d ( K , J ) via the Bogoliubov automorphism ^/s (9 € SO(m)). This action is inner. To see this, recall that any element of S O ( N ) is a composition of reflections with respect to lines in RN. The reflection with respect to the one dimensional space C f (f = J f E K , = 1) is implemented by the adjoint action A d ( B ( f ) )of the selfadjoint unitary B(f):
A W ( f ) ) ( Q= ) W)QB(f)*. The number of reflection appearing in the element of S O ( N ) is even so that the representative T satisfying Ad(T) = rg is a product of an even number of selfadjoint unitaries B ( f ) ,thus we conclude that there exists r(g)in d ( K , J)+ such that
Arn(9) = 7 9 . We can describe the explicit form of r(g),however we will not use it in what follows. The representative r(g)is unique up to a phase factor, we obtain a projective representation r(g)of SO(m) in d ( K , J)+. In the finite dimensional case, the cocycle of the projective representation r(g)for S O ( N ) is non-trivial and we obtained the unitary representation of the double cover of S O ( N ) . It is often referred to as the spin group Spin(N). Note that the linear hull of r(g)is norm dense in d ( K , J)+. We summarize the these remarks as below. Lemma 2.2 Given a representation T of d(K, J ) + on a Halbert space, we obtain a projective representation T ( r ( g ) ) . The projective representation Q'(g)) is irreducible (resp. factor) if and only if the representation T of d ( K , J ) + is irreducible (resp. factor). Next we introduce the Fock representation of the CAR algebra and an infinite dimensional analogue of spin representation of S O ( N ) . Fix a basis
350
projection E on
X. Consider the exterior algebra 00
/ \ ( E X )= @ Ak(EX), k=O
where & ( E X ) is the anti-symmetric part of the k hold tensor of the Hilbert space EX. In what follows we consider the completion / \ ( E X ) as a Hilbert space and we will use the same notation A ( E X ) for this Hilbert space. Set 00
00
A(EX)+ = @ A z k ( E X ) , A ( E X ) - = @ h k + i (EX).
(5)
k=O
k=O
Given h in E X , define the bounded operator a * ( h )on A ( E X )by the following equation:
a* (h)hl A h2 * * * A h k = h A hi
*
-
*
A
hk
E Ak+l
(EX).
Let a(h)be the adjoint (a*(h))* of a*(h). The operators a*(h)and a (h ) satisfy the canonical anticommutation relations. Set
n E ( B ( f ) )= a*(Ef)-k a(EJf)-
(6) It is straightforward to verify that R E gives rise to the representation n,y(d(IC,J)) of d(IC,J) on the Hilbert space & ( E X ) . The representation RE restricted to the even part d ( X , J)+ is decomposed into two irreducible on A(EX)*. As a consequence of previous lemmas, we obcomponents tain two projective irreducible representation rL*)(I'(SO(m)) of the group SO(w) on /\(EX)*. We call &*Ithe spin representation for SO(o0). L e m m a 2.3 The spin representations and are not unitarily equiv-
RL*)
RF) RL-)
alent. The construction of spin representations rises a question. When are these spin representations equivalent for different choice of the basis projection? This means that even though the algebraic construction is same the representations may not be unitarily equivalent in infinite dimensional groups. We will present our answer to this question now. Let El and E2 be basis projections such that El - E2 is compact. By El A (1- E z ) , we denote the projection to the intersection of the range of El and that of E2. Due to compactness of El - Ez, the range of El A (1 - E2) is finite dimensional. Moreover the parity (even-oddness ) of the range of El A (1 - E2) is continuos (constant) in norm topology for El and E2. Then, combined with what we have explained so far, a result of Araki-Evans [4] leads to the following conclusion.
351
Theorem 2.4 Let El and E2 be basis projections on K. Consider spin representations and of SO(o0). (1) and are unitarily equivalent if and only if El - E2 is of Halbert Schmidt class and the dimension of the range El A ( 1 - E2) is even. ( 2 ) 7rLT) and are unitarily equivalent if and only if El - E2 as of Hilbert Schmidt class and the dimension of the range El A ( 1 - E2) is odd. The state of the selfdual CAR algebra A(& J ) is the Fock state in the sense of H. Araki [l]. Namely, consider the unit vector RE in the one di~ d(& J ) specified by the mensional space A,,(EIC). RE yields the state c p of following identities:
TK) TE’ TLT) TE) TL~)
--
~ ~ ( B ( h i ) B ( h *B(h2n+1)) z) = 0,
(7)
n
~ E ( ~ ( h l ) ~ . (- h . ~2( )h 2 n = ))
C ~ign(p>~ ( ~ h , ( z j - l ) , ~ h , ( z j ) ) r c(8), j=1
where the sum is over all permutations p satisfying P(1)
< P(3) < . .. < p(2n -
P(2j - 1) < P ( 2 j ) ,
111
and sign(p) is the signature of the permutation p . The above equation (7) and (8) defines a state if we replace the basis projection E with a positive operator S satisfying 0 5 S 5 1 JSJ = 1- S. In this way, we can introduce a class of states, quasifree states for the selfdual CAR algebra d ( K , J ) . Definition 2.5 The state cp is called a quasifree state if and only if
~ ( B ( h l ) B ( h 2- *)B. ( h n + l ) )= 0, cp(B(hl)B(h2).* . B(h2n)) =
c
(9)
n
sign@) ]II(P(B(h,(2j-l))B(h,(2j)), j=1
(10)
where the sum is over all permutations p satisfying P(1) < P(3) < . . . < p(2n - 11, P(2j - 1) < P(23.1,
and sign@) is the signature of the permutation p . Suppose a quasifree state cp is given. Due to canonical anticommutation relation, there exits a bounded operator S such that
0 5 S 5 1, JSJ = 1- S, cp(B(hl)B(hz))= (Jh1,Shz)rc. We denote cps by the quasifree state determined by S via the above equation. A quasifree state cps restricted to d ( K , J ) + is pure if and only if S is
352
a basis projection. We can construct (not necessarily irreducible) projective representations of SO(o0) starting from any quasifree state cps. The conditions for factoriality and quasi-equivalence of representations for d(K, J)+ are obtained by T. Matsui [8,9]. 3
U(o0) and Sp(o0)
In the previous section, we show how the infinite spin representations are related to the representation of the CAR algebra. The same construction can be carried out for the infinite unitary group U(o0). If a group G is an inductive limit of compact groups, we obtain a AF algebra B attached with the inductive limit group algebra and the set of primitive ideals J of d parametrizes the set of quasi equivalence classes of type I1 factor representations 'of G . The detail of this correspondence is examined for U(o0)in Str2itilbVoiculescu [14]. The gauge invariant CAR algebra dV(l) is the quotient of the above mentioned algebra 23 by an primitive ideal 3.The construction of representations is similar and we sketch the results here. Now we set
K = 12(N)@ Zz(N).
(11)
Let en be the same standard basis of 12(N)of previous section. Let 12(N), be the finite dimensional subspace of 12(N) spanned by {el, e 2 1 . .. en}. By Pn we denote the orthogonal projection to KN. Consider the group of unitaries u on /2(N) satisfying [u,Pn] = 0, (I. - Pn)~(1-Pn) = 1 - Pn.
This group is naturally identified with the group of U ( n ) . Then, U(o0)is the union of U(n):
=
u
U ( n )c
B@2",
rill
where 17(12(N)) is the set of all bounded operators on 12(N). Let JO be the anti-unitary involution on 12(N) defined by the complex conjugation: (JO" = ad. Set J ( h l @h2) = (JOh2 @ JOhl).
and for u in U(o0)
t = Zd @ JouJo E B ( K ) .
353
Then
JEJ = E. As in the previous section d ( K , J ) be the C*-algebra generated by B(h) ( h E lc) satisfying (1) and the Bogoliubov automorphism % can be introduced as before by the following identity: We consider a U(1) action
%(B(h)) = B(;Tih). determined by
Pe(B((fi @ f 2 ) ) ) = B((e"fi
e-"fd).
We denote the fixed point algebra under p,g by dU(')(lc, J):
d'(l)(lc,J ) = { Q E d ( K , J) I Po(&) = Q (0 < B < 2 ~ ) ) . (12) As before we introduce an action of U(o0) on d(lc,J) via the Bogoliubov automorphism % ( u E U(o0)). This action % is inner and this time, we have the unitary representation r(u)of U(o0) into d U ( l ) ( K ,J) such that
A Q ( r ( 4 )= %The triviality of the cocycle of r(u)follows from the fact that U ( N ) is simply connected. Namely we have only to construct representation ctr(t) of the Lie . we represent here the explicit form of ctr(t). algebra ~ ( c o )Next The Lie algebra u(o0)is generated by the diagonal elements ti and off diagonal elements sij, t i j where these are bounded selfadjoint operators on /2(N) described in terms the basis vectors {en I n = 1 , 2 , .. . } as follows:
tie, = &,,en
Then, set a t = B((en @ 0 ) ) , an = B((OCB en)). and a,,,satisfy CAR. We introduce the representation ctr via the following equations: a:
dr(tn)= ata, c t r ( s i j ) = araj d ~ ' ( t i j )=
+ a;ai
& (.;ai i - aiaj)
(14) It is easy to,see that the algebra generated by dI'(tn) dI'(sij) and dI'(tij) is
dU(l)(lc, J).
354
Lemma 3.1 The equations (14) give rise to the representation & of the Lie algebra u(m) and the unitary representation I? of U(o0) in d U ( l ) ( K J, ) . In particular, i f a representation A of dU(l)(K:, J ) o n a Hilbert space is given, the associated representation a ( r ( u ) )of U(o0) is irreducible (resp. factor) i f and only if the representation a of d U ( l ) ( K J, ) is irreducible (resp. factor). We turn to the Fock representation of d L T ( l ) ( KJ). , Let P+ be the projection on K: to 12(N)@ 0 and set P- = 1- P+ We consider a basis projection E on K commuting with P*:
P+E = EP+, P-E = EP-. (15) Lemma 3.2 Suppose that the basis projection E commutes with P*. Consider the exterior algebra m
A(EK:)= @ Ak(EK). k=O Then, the irreducible representation r E ( d ( K , J ) ) of d ( K , J ) restricted to d U ( l( ) K , J ) is decomposed to countably many irreducible representations of d U ( l ) ( K J, ) which are mutually disjoint. Note that if the basis projection E does not commute with P*,the representation AU(l)(IC, J) can be non type I. Definition 3.3 We assume that the basis projection E commutes with P&. Let 0~ be the vector unit vector in Ao(EK) The irreducible representation of U ( m )on the cyclic component a E ( d ( L ,J ) ) ~ isE called the Fock representation for U(o0). We denote it by A E ( U ( W ) ) . If the difference El - E2 of two projections El and E2 is compact, E2E1 defined on the range of El is a Fkedholm operator and we denote its index by ids,E2 El : indE, E2 El = dim kerE2 El - dim kerE1E2 = dim{< 1 E2E1< = 0, El< = <} - dim{< I E1E2<= 0, E2< = <}. The following is due to Str5tilii-Voiculescu [14]. Theorem 3.4 Let El and Ez be basis projections on K which commute with P&. Consider the Fock representations A E ~(U(o0)) and AE~(U(.CO)) of U(o0). AE,(U(W))and A E ~ ( U ( of ~ ~U(o0) ) ) are unitarily equivalent i f and only i f El - E2 is of Hilbert Schmidt class and the Fredholm index indp+E1P+E2E1 vanishes: indp,~,P+ El E2 = 0.
355
Suppose that S is a bounded operator on 05S
K
satisfying
5 1, J S J = 1 - S, [P*,S] = 0.
(16)
We can introduce the U(1) invariant (70 invariant) quasifree state cps of d ( K , J ) via (9) and (10). The state cps for d ( K , J ) and its restriction to du(l)(KI J) is factor. Theorem 3.5 Suppose that Sl and S2 are strictly positive operators on K satisfying (16). Consider unitary representations xsl and nsz of U(o0) associated with states (ps, and cps2. as1 and xs, are quasi-equivalent if and only is of Halbert Schmidt Class. if f i Next we consider representations of the infinite (real) symplectic group Sp(o0,R). Handling the symplectic group case is some what delicate, for the creation and annihilation operators in canonical commutation relations axe unbounded and we have problem of the domain and the selfadjoint extension of polynomial of the creation and annihilation operators. In particular, it is not clear that an analogue of Lemmas 2.2 and 3.1 is valid. Nevertheless we obtain similar results for the symplectic case Sp(o0, R). Let Co(N) be the set of all complex finite sequences:
Co(N) = {( = {(i (i = 1 , 2 , . . .)}
I & =0
( i >> 1 ) ) .
Set
K: = Co(N) @ Co(N).
(17)
We introduce a non-degenerate symplectic form on K. For h = (h(l)@ h ( 2 ) ) and f = (f(') @ f ( 2 ) ) ,
7 ( h ,f ) = @ ( I ) ,
f(2))lz(N)
- ( h ( 2 )f,( l ) ) l z ( N )
We also introduce the complex conjugation involution J via the following equations: ( J O W )
=V ( 8 ,
JO on Co(N) as before and an
J(h1 @ h2) = (Job
Joh2).
Then we obtain
J2 = 1, y ( J h , J f ) = -7(f,h). Let en be the same standard basis as before and Co(N), be the finite dimensional subspace Co(N) of spanned by { e l , e2,. . . ,e n } . By Pn we denote the
356
orthogonal projection to K , = Co(N), @ Co(N),. Consider the group S p ( K ) of invertible operators g on K satisfying Y(Sh gf) = Y(h
f),
JgJ = 9.
(19) The real symplectic group S p ( n , R ) is identified with the subgroup of S p ( K ) whose elements g satisfy
[g,P,] = 0,
(1 - P,)u(l- P,) = 1.
Then, we define Sp(00, R) as the union of Sp(n,R):
u 00
Sp(n,R) =
SP(%R).
n=l
Next we consider the CCR algebra (the algebra of Boson creation annihilation operators) on which S p ( K ) acts as a group of *automorphisms. Definition 3.6 By B(K,J) we denote the *algebra with unit generated by B(f ) (f E K) satisfying the following equations:
B(f)*= B(Jf)l B(fl)B(f2)*- B(f2)*B(fl)= Y(f2, fill.
(20)
B(K,J) will be called the self dual CCR algebra. Next we introduce the action rg of Sp(00, R) on ubov automorphisms:
B(K,J ) via the Bogoli-
rg(B(h))= B(gh). Unlike the case of SO(o0) or V(m), rg is not inner, however the action ~ of the Lie algebra sp(00,R) is bilinear in B ( f ) . Namely, for any element u in the Lie algebra sp(00,R) of Sp(00, R) we introduce a derivation d l ? ~ on B(K,J ) via the following equation: &H((B(f)) = B ( H f ) . Then there exist complex coefficients t i j and vectors bilinear elements ( B ,H B ) in B(K,J) such that
c
fj
in
K
such that the
M
( B ,NB)=
{tijB(fi>B(fj)* + w w j ) B ( f i ) * }1
j,i=l
&H(Q)
= [ ( BH , B ) ,Ql
for any Q in B(K,J ) . It is a non trivial matter whether a representation of B ( K , J ) gives rise to a representation of Sp(00, R) as the essentially selfadjointness of ( B ,HB)
357
must be proved. In case of the GNS representation associated with quasifree states, we can construct unitary representations of the double cover $p(oo, R) of Sp(cm, R). gp(00, R) is called the metaplectic group. Now, we introduce quasifree states for the self dual CCR algebra B ( K , J) in the same manner as in the CAR case. Definition 3.7 A state cp of B(K,J) is called a quasifree state if and only if
cp(B(hl)B(h2).. .B(h2n+l))= 0,
cn
(21)
n
cp(B(hdB(h2).. .B(hzn))=
cp(B(h,(j,>~(h,(j+,,)), ( 2 2 )
j=1
where the sum is over all permutations p satisfying
P(1) < P ( 2 ) < * - . < p ( n ) , P ( j ) < P ( j + n). Fix a state cp of following equation:
B(K,J).
Define an inner product (f,g), on
(9,f), = cp(B(f)B(g)*)+ cp(B(g)*B(f)).
K: via the (23)
Let K be the completion of Ic by the topology induced by (f,g),.There exists a bounded linear operator S, on E such that
(f,S,d,
= cp(B(f)*B(g))
for any f and g in K. Then if S, is a projection, the GNS representation associated with 'p is the representation on a Fock space for our CCR algebra
m,J).
Suppose that S, = E is a projection. Consider the standard Boson Fock space &(L) and the vacuum vector R. 00
c = EE,
RE
o:c
= c,
&(L) = @@y.
(24)
n=O
The annihilation (resp. creation) operator a(f) is identified with B((1- E)f) (resp. B ( E f ) * )on the Boson Fock space &(L) so the representation 7r of B ( K , J) is realized via the following equations:
4 f ) = T(B((1- E M ,
a*(!)
= T(B(Ef)*).
(25)
If Jf = f ?r(B(f)) is essentially selfadjoint on the finite particle states and the unitary e z p ( i r ( B ( f ) ) generated ) by n ( B ( f ) )satisfies the Weyl form of the canonical commutation relations. The algebra exp(in(B(f))) is weakly dense in B(Fb(L)).In this sense the above Fock representation of B(Ic, J) is irreducible.
358
By use of (25) we can introduce the representation of the Lie algebra sp(00,R) of Sp(00, R):
n(H)=4 m ' H ) for H in sp(00,R). The vectors with finite particle number in the Boson Fock space &(L) are analytic for n ( H ) and we obtain the projective unitary representation of n ~ ( S p ( 0 0 R)) , on the Boson Fock space &(l). Lemma 3.8 The projective unitary representation of ?r~(Sp(oo,R)) is decomposed into mutually disjoint irreducible representations nEf(Sp(00,R ) )on (L)where
3bf
00
00
C ( L )= @ €g"L,
q - - ( C ) = @ €gn+lL.
n=O
n=O
Note that this representation r ~ ( S p ( 0 0R)) , is the infinite dimensional extension of the Weil representation for the double cover Sp(oo,R)) of Sp(oo,R)). Unlike the finite dimensional case, we have a huge number of mutually non-equivalent representation associated with Fock states and the classification up to unitary equivalence was obtained H. Araki-Shiraishi [2] and H. Araki [3]. The following theorem reduces the classification (up to unitary equivalence) of infinite dimensional Weil representations to that of representations of the self dual CCR algebra B(K,J) on Fock states. Theorem 3.9 Let E and F be the basis projections associated with states c p ~and ( P F . The irreducible unitary representations nz ( g p (00, R)) and rFf(gp(00,R ) )are unitarily equivalent i f and only if the GNS representations ~ B(K,J ) are unitarily equivalent. associated with quasifree states (PEand c p of We can also generalize the construction of the unitary representation r,+,of gp(00,R)) on the general quasifree states for B(K,J). ~,(,!&(oo, R)) can be non type I in the sense that the von Neumann algebra generated by r,+,((S;(m,R)) is non type I. We say that two unitary representations nl and 7r2 Sp(00, R)) if there exists an isomorphism 9 ' of the von Neumann algebras rl (Sp(00,R))" and n~(,!?p(00,R))" such that WTl(9)) = T 2 b )
for any g in gp(oo,R). Theorem 3.10 Let cp1 and cp2 be quasifree states of B(K,J ) . The projective unitary representations nV1(Sp(00, R)) and xvPz(Sp(00,R ) ) of Sp(00, R) are quasi-equivalent if and only if quasifree states cp1 and cp2 of B(K,J ) are quasiequivalent.
359
The proof of the symplectic case is obtained by Taku Matsui and Yoshihito Shimada [lo]. References 1. H. Araki: O n quasifree states of CAR and Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci. 6 (1970/71), 385-442. 2. H. Araki and M. Shiraishi: On quasifree states of the canonical commutation relations. I, Publ. Res. Inst. Math. Sci. 7 (1971/72), 105-120. 3. H. Araki: O n quasifree states of the canonical commutation relations. 11, Publ. Res. Inst. Math. Sci. 7 (1971/72), 121-152. 4. H. Araki and D. E. Evans: A C*-algebra approach to phase transition in the two-dimensional Ising model, Commun. Math. Phys. 91 (1983), 489-503. 5. A. Borodin and G. Olshanski: Infinite random matrices and ergodic measures, Commun. Math. Phys. 223 (2001), 87-123. 6. A. L. Carey and S. N . M. Ruijsenaars: On fermion gauge groups, current algebras and Kac-Moody algebras, Acta Appl. Math. 10 (1987), 1-86. 7. G. Lion and M. Vergne: “The Weil Representation, Maslov Index and Theta Series,” Progress in Mathematics, 6. Birkhauser, 1980. 8. T. Matsui: O n quasi-equivalence of quasifree states of the gauge invariant CAR algebras, J.Operator Theory 17 (1987), 281-290. 9. T. Matsui: Factoriality and quasi-equivalence of quasifree states for 2 2 and U(1) invariant CAR algebras, Rev. Roumaine Math. Pures Appl. 32 (1987), 693-700. 10. T. Matsui and Y. Shimada: On quasifree representations of infinite dimensional symplectac group, preprint , 2002. 11. G. Olshanski and A. Vershik: Ergodic unitarily invariant measures on the space of infinite Hermitaan matrices, Contemporary Mathematical Physics, pp. 137-175, Amer. Math. SOC. Transl. Ser. 2, 175 Amer. Math. SOC.,Providence, RI, 1996. 12. D. Pickrell: Separable representations f o r automorphism groups of infinite symmetric spaces, J. F’unct. Anal. 90 (1990), 1-26. 13. A. Pressley and G. Segal: “Loop Groups,” Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. 14. 5. Str2itilii and D. Voiculescu: On a class of K M S states for the unitary group U(oo),Math. Ann. 235 (1978), 87-110.
CAUCHY PROCESSES AND THE LEVY LAPLACIAN NOBUAKI OBATA Graduate School of Information Sciences Tohoku University Sendai 980-8579, Japan E-mad: obatabmath. is.tohoku.ac.jp KIMIAKI SAITO Department of Information Sciences Meijo University Nagoya 468-8502, Japan E-mail: ksaitobccmfs.meijo-u.cac.jp The LBvy Laplacian is formulated as an operator acting in a direct integral space of white noise functions. From Cauchy processes an infinite dimensional stochastic process is constructed, of which the generator is the LBvy Laplacian.
Introduction An infinite dimensional Laplacian was introduced by P. E v y in his famous book [17]. Since then this exotic Laplacian has been studied by many authors from various aspects see [1-6,18,20,23]and references cited therein. In this paper, generalizing the methods developed in the former works [16,19,28,29], we construct a new domain of the LBvy Laplacian and associated infinite dimensional stochastic processes. Our new domain is obtained in two steps: we first find suitable spaces of white noise distributions on which the L6vy Laplacian acts in a nice manner; then we form a direct integral of these spaces as its domain. This formulation is slightly outside the usual white noise distribution theory, while the L6vy Laplacian has been discussed within the framework of white noise analysis in [4,7,8,10,13,14].Another construction of stochastic processes associated with the LBvy Laplacian is found in [1,3,6]. This paper is organized as follows. In Section 1 we summarize some notions in white noise theory following Kubo-Taken& [12]. This particular framework is enough for our discussion though the white noise theory has been considerably generalized in recent years. In Section 2, following the recent works Kuo-Obata-Saitb [16], Saitb [28] and Saitir-Tsoi [29], we formulate the Ldvy Laplacian acting on a direct sum Hilbert space consisting of white noise generated distributions and give an equi-continuous semigroup of class (CO) by the Laplacian. This situation is further generalized in Section 3 by means
360
36 1
of a direct integral of Hilbert spaces. In Section 4, based on infinitely many Cauchy processes, we give an infinite dimensional stochastic process generated by the LQvyLaplacian. 1
Standard S e t u p of White Noise Calculus
Following Kubo-Takenaka [12] we assemble some basic notations of white noise analysis, for related discussion see also [7,9,14,21]. Let E = S(R) be the Schwartz space of rapidly decreasing R-valued functions on R. There exists an orthonormal basis { e v } v r o of L2(R) contained in E such that
Ae, = 2(v+ l)e,)
8
v = 0 , 1 , 2)...,
A = --
du2
If/,
+u2
+ 1.
For p E R define a norm 1 .,1 by = I A p f l p ( ~for ) f E E and let E, be the completion of E with respect to the norm I . , 1 Then E, becomes a real separable Hilbert space with the norm I .,1 and the dual space EL is identified with E-, by extending the inner product (., of L2(R) to a bilinear form on E-, x E,. It is known that
-
0 )
E = proj lim E,,
E* = indlim E-,,
P-==
The canonical bilinear form on E* x E is also denoted by -). We denote the complexifications of L2(R), E and E, by L&(R),E c and Ec,,, respectively. The standard Gaussian measure p on E* is defined by the characteristic function: ( 0 ,
where I 10 is the norm of L2(R). Let L2(E*,p)be the Hilbert space of C-valued square-integrable functions on E*. The famous Wiener-It6 decomposition theorem says that: m
n=O
where 3, is the space of multiple Wiener integrals of order n E N and 30 = C by definition. According to (1) each YJ E L2( E * ,p ) is represented as m
362
where L&(R)6n denotes the n-fold symmetric tensor power of L$(R) (in the sense of a Hilbert space). Then the norm Ilcpllo is given by
n=O
where 1.10 stands for the norm of L&(R)6n. Given cp in the form (2), we introduce another norm llcpllp for p E the formula:
R by
m n=O
where I . I p is the norm of EEB. For p 2 0 let be the space of all cp E L 2 ( E * , p )such that IIcpllp < 00 and (E)-p the completion of L 2 ( E * , p ) Then, for each p E R, equipped with the with respect to the norm 11 . norm 11 . lip, ( E ) pbecomes a Hilbert space. The dual space ( E ) ; is identified with ( E ) - pthrough the inner product of L2(E*, p ) . Moreover, the Wiener-It6 decomposition (1) is valid for all p E R
We set
( E )= proj lim(E), = P+m
n( E ) ~ , P 3
which is a countably Hilbert nuclear space. The strong dual space (E)* is identified with the inductive limit space (E)* = indlimp+m(E)-p. An element of (E)* is called a white noise distribution. We denote by ((-, .)) the canonical bilinear form on (E)*x (E). Then, for @ E (E)* and cp E ( E ) we have m
m
where the canonical bilinear form on ( E p ) *x (EEn)is denoted also by (., .). The S-transform of 9 E (E)*is defined by S@(<)= ((@,cpO), where c p ~= exp{
(a,
< E Ec,
<) - (<,<)/2} E ( E )is an exponential vector.
363
Theorem 1.1 [24] (see also [9,14,21]) A C-valued function F on E c is the S-transform of an element in (E)* if and only if for e v e y <,q E E c , the function z I+F(< + zq), z E C , is an entire function of z and there exist constants K 2 0, a 5 0 and p 2 0 such that
lF(t)II Kexp (alrr;} 2
5 E Ec.
7
The LQvy Laplacian Acting on a Hilbert Space of White Noise Distributions
Consider F = Sip with ip E (E)*. By Theorem 1.1, for any < , q E Ec the function z I+ F(< zq) admits a Taylor series expansion:
+
-
where F(")(<): E c x . x E c + C is a continuous n-linear functional. Fixing a finite interval T of R, we take an orthonormal basis { c n } ~ =C o E for L2(T) which is equally dense and uniformly bounded (see e.g. [14,15]). Let DL denote the set of all ip E (E)* such that the limit
-
N-1
exists for any S E E c and is in S[(E)*].The Levy Laplacian AL is defined by
A ~ i p= S - l A ~ S i p ,
ip
E VL-
Given X E R, n E N and f E Eg", we consider f ( u 1 , ..
@ = S , n
ip
E (E)* of the form:
. ,un) : eiA\r(ul)...eiAz(ern): d u l . . .dun,
(3)
where : . : is the Wick ordering. In fact, the above @ is defined through the S-transform given by Sip(() =
Ln
f ( u 1 , .. . ,un)eiAc(ul) . ..eiXE(un)dul . . .dun,
< E Ec.
For any pair n E N and X E R let Dn,xdenote the space of ip which admits an expression as in (3), where f belongs to Egn and suppf c T". Put Do,x = C . It is known [16]that D,J is a linear subspace of whenever p > 5/12. Then . Dn,x itself becomes a Let D,J be the closure of Dn,x in ( ~ 3 ) - ~ Hilbert space with norm 11 . ll-p. Using a similar method as in [29], we get the following
364
Theorem 2.1 [29] (see also [16,27]) For each pair n E NU (0) and X E R the Lkvy Laplacian AL becomes a scalar operator on Dn,x such that
In particular, A= is a self-adjoint operator on Dn,x. Proposition 2.2 [29] Let X E R be jixed. Consider two generalized white noise functions of the form:
If
m
m
n=O
n=O
= q in ( E ) * ,then an = qnfor all n E N U (0). Taking (4)into account, we put
For N E N,p > 5/12 and X E R let E?P,Nbe the space of admits an expression
E (E)*which
m
such that m
n=l
By the Schwaxtz inequality we see that E?p,N is a subspace of ( E ) - p and becomes a Hilbert space equipped with the new norm 11) . I I I - p , ~ , ~ defined in (5). Moreover, in view of the inclusion relations:
we define m ~~
x E-p,OO = proj lim E xe p , = ~ N+w
Note that for any X E R we have
N=l
365
The operator AL becomes a continuous linear operator defined on E!p,N+l into E1p,Nsatisfying ~ ~ ~ A L @ ~ ~5 ~ -III@III-p,N+1,X, P,NJ
@
E
x
E-p,m,
N E N.
(6)
Summing up, we have the following Theorem 2.3 [16,29] The operator AL is a self-adjoint operator densely defor each N E N , p > 5/12 and X E R. fined in It follows from (6) that A, is a continuous linear operator on E-p,,. In view of the action of (4), for each t 2 0 and X E R we consider an operator G: on EtP,, defined by 00
00
n=l
n= 1
Theorem 2.4 [16,28] Let X E R a n d p > 5/12. Then the family of operators of which { G t ; t 2 0) on EtP,, is an equi-continuous semigroup of class (CO) the infinitesimal generator is AL. 3 The LQvyLaplacian Acting on a Direct Integral Hilbert Space of White Noise Distributions Let &(A)
be a finite Bore1 measure on R satisfying
Fix p > 5/12 and N E N. Let & - p , ~ be the space of (equivalent classes of) measurable vector functions (@A) with @A E D n , x , where n E N and X E R, such that
Then & - p , ~ becomes a Hilbert space with the norm given in (7). In other words, & - p , ~ is a direct integral of Hilbert spaces:
where dP(n, A) = a&(.) x (counting measure) x &(A).
366
Proposition 3.1 The map
(9:) e 9 = C / R 9 : d v ( X ) n=l
is a continuous linear map from
& - p , ~ into
(E)-p.
PROOF.By the Schwartz inequality we have
Since ah(n)2 (nA2/IT1)2for N 2 1, we have
which is finite by assumption.
I
For (9;)E L p we , ~define W
n=l
Then
E E!p,N for v-a.e. X E
R. In view of (7), we see that
As for the uniqueness of (8) we only mention the following Proposition 3.2 Let N E N andp > 5/12. If dv(X) is absolutely continuous with respect to the Lebesgue measure on R, then El’p,N n El’p,N = (0) holds for v 8 v-almost all (A1, X2) in R2.
PROOF.Take an element 9 in E?p,N has two expressions:
n= 1
n El’p,N. Then the functional 9
n= 1
For each n E N, 9, and Q n are expressed in the forms n
367
and
fiN1 +
E E$" and giN1E E$". Take CT E Ec with CT = 1 respectively, where on T . Put = (Y
<
Let Q denote the rational numbers. Since UTE^{ (A, r X ) ; X E R} is a null set in R2with respect to the Lebesgue measure, this implies @ , = Q n = 0 for v @ v-almost all (Al, Xa) in R2and for each n E N. Consequently we obtain @ = 0 for v @ v-almost all (XI, X2) in R2. I We go back to the Hilbert space & - p , ~ . In view of the natural inclusion: c N E N, which is obvious from construction, we define
& - p , ~ + l & - p , ~ for
n m
LP,= proj lim & - p , ~= N+w
&-P,N*
N= 1
The Levy Laplacian A t is defined on the space &-p,m by
At@ = (At*;),
@
= ( a nx ) E LP,-.
The operator A, becomes a continuous linear operator from &-p,m into itself. Similarly, for t 2 0 we define
Then we come to the following:
Theorem 3.3 Let p > 5/12. The family { G t ; t 2 0 } is an equi-continuous semigroup of class (Co)on &-p,w whose generator is given b y A,.
368 PROOF.
For any t 2 0 and N E N, the norm IIIGt+lIl-p,N for 9 =
(at) E E - p , ~can be estimated as follows:
m
-
Hence the family { G t ; t 2 0) is equi-continuous in t . It is obvious from Theorem 3.4that GtGs9 = Gt+s@and Go9 = 9 for t , s 2 0 and 9 E E - p , ~ . For { G t } being of (CO) it is sufficient to show that
9E L p , ~ .
lim IIIGta - Gt,@III-p,N = 0,
t+to
This follows from
Ill(G?
x
-Gs)
+A
2
5 4111a
A
2
E R, and the Lebesgue convergence theorem together with Theorem 3.4. I~I-pJVJ
III-p,N,X,
To complete the proof we prove that the infinitesimal generator of the semigroup { G t } is A L . By definition we have
and
where we used a simple inequality:
Since
369
and lim
t+O
4
1
t
+-IT1 =o,
An Infinite Dimensional Stochastic Process Generated by the LQvyLaplacian
For p E R let EE,p be the linear space of all functions X I+ E EC,~, X E R, which are strongly measurable. An element of EE,p is denoted by = (&)x€R. Equipped with the metric given by
<
the space EE,p becomes a complete metric space. Similarly, let CR denote the linear space of all measurable function X I+ . z ~E C equipped with the metric defined by
Then CR is also a complete metric space. In view of dp 5 dq for p 3 q, we introduce the projective limit space EE = projlim,,, EE,p. The S-transform can be extended to a continuous linear operator on EP,+, by
< = (<X)X€R s@(<) = (s@x(b))XCR,
E
Eg,
for any 8 = ( a X ) x EE~~5-~,+,. The space S[E-,,,] is endowed with the topology induced from LP,+, by the S-transform. Then the S-transform becomes a homeomorphism from &-p,m onto S[E-,,,]. The transform S 8 of 8 E €-p,+, is a continuous operator from EE into CR.We denote the operator by the sa_me notation S@. Let Gt be an operator defined on S[E-,,,] by Gt = SGtS-l,
{at;
t 2 0.
Then by Theorem 3.6, t 2 0) is an equi-continuous semigroup of class (CO) generated by the operator EL.
370
Let { X i } and {X,"}be independent Cauchy processes with t running over [0,co),of which the characteristic functions are given by
E[eizx:]= e-tlzl,
z E R,
j = 1,2.
Set
Take a smooth function qT E E with q~ = 1/ITI on T. Define an infinite dimensional stochastic process {Yt;t 2 0) starting at 6 = ( < x ) x € RE EE by yt
= (6
+Y$vT)~€R,
t 2 0.
Then this is an Eg-valued stochastic process and we have the following Theorem 4.1 If F i s the 5'-transform of an element in & - p , w , we have
G F ( 6 ) = E [ F ( Y t ) l Y o= <],
t 2 0.
(10)
R the PROOF.We first consider the case when F ( 6 ) = ( F ' ( & , ) ) X ~has form:
Hence (10) holds. Next let F = (C,"==, Fi)xERE S[&-,,,]. Then for any n E N and for almost all X E R, F; is expressed in the following form:
371
where
(fi:)~ is a sequence of functions in Ec(R)@".Hence, for almost all
X E R, we have W
n= 1
W
n=l
For almost all X and for each n, since F: E S[E-,,,], there exists some E E-p,w such that F,"= S[@A].By the Schwarz inequality, we see that W
W
n=l
n= 1
for almost all X E R and for each N E N. Therefore by the continuity of @ j we obtain E[F(Yt)lYo = 4 = (E[Fx(<x K A 7 ~ ) ] ) x where E ~ , the Xcomponent is given by
+
W
E[FX(
+ q x v T ) ]=
E[F,
(
-k Kx?7T)]
n=l
n=l
This proves the assertion.
n=l
I
References 1. L. Accardi and V. Bogachev: The Ornstein-Uhlenbeck process associated with the Ldvy Laplacian and its Dirichlet f o r m , Prob. Math. Stat. 17 (1997), 95-114.
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2. L. Accardi, P. Gibilisco and I. V. Volovich: Yang-Mills gauge fields as harmonic functions f o r the Lkvy Laplacian, Russ. J. Math. Phys. 2 (1994), 235-250. 3. L. Accardi and 0. G. Smolyanov: Dace formulae f o r Levy-Gaussian measures and their application, in “Mathematical Approach to Fluctuations Vol. I1 (T. Hida, Ed.),” pp. 31-47, World Scientific, 1995. 4. D. M. Chung, U. C. Ji and K. Sait6: Cauchy problems associated with the Lkvy Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 5. M. N. Feller: Infinite-dimensional elliptic equations and operators of Ldvy type, Russ. Math. Surveys 41 (1986), 119-170. 6. K. Hasegawa: Ldvy’s functional analysis in t e r n s of a n infinite dimensional Brownian motion I, Osaka J. Math. 19 (1982), 405-428. 7. T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, No. 13, Carleton University, Ottawa, 1975. 8. T. Hida: A role of the Lkvy Laplacian in the causal calculus of generalized white noise functionals, in “Stochastic Processes,” pp. 131-139, SpringerVerlag, 1993. 9. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise: An Infinite Dimensional Calculus,” Kluwer Academic, 1993. 10. T. Hida and K. SaitB: White noise analysis and the Lkvy Laplacian, in “Stochastic Processes in Physics and Engineering (S. Albeverio et al. Eds.),” pp. 177-184, 1988. 11. E. Hille and R. S . Phillips: “Functional Analysis and Semi-Groups,” AMS Colloq. Publ. Vol. 31, Amer. Math. SOC,1957. 12. I. Kubo and S. Takenaka: Calculus o n Gaussian white noise I-IV, Proc. Japan Acad. 5 6 A (1980) 376-380; 5 6 A (1980) 411416; 5 7 A (1981) 433436; 5 8 A (1982) 186-189. 13. H.-H. Kuo: O n Laplacian operators of generalized Brownian functionals, Lect. Notes in Math. Vol. 1203, pp. 119-128, Springer-Verlag, 1986. 14. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 15. H.-H. Kuo, N. Obata and K . Sait6: Lkvy Laplacian of generalized functions o n a nuclear space, J. Funct. Anal. 94 (1990), 74-92. 16. H.-H. Kuo, N. Obata and K. Sait6: Diagonalization of the Lkvy Laplacian and related stable processes, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.5 (2002), 317-331. 17. P. LCvy: “Lesons d’Analyse Fonctionnelle,” Gauthier-Villars, Paris, 1922. 18. R. Lkandre and 1. A. Volovich: The stochastic Lkvy Laplacian and YangMills equation on manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.4 (2001) 161-172.
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19. K. Nishi, K. SaitB and A. H. Tsoi: A stochastic expression of a semigroup generated by the Ldvy Laplacian, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 105-117, World Scientific; 2000. 20. N. Obata: A characterization of the Ldvy Laplacian in t e r n s of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 21. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. 22. N. Obata: Quadratic quantum white noises and Ldvy Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 23. E. M. Polishchuk: “Continual Means and Boundary Value Problems in Function Spaces,” Birkhauser, Basel/Boston/Berlin, 1988. 24. J. Potthoff and L. Streit: A characterization of Hida distributions, J. Funct. Anal. 101 (1991), 212-229. 25. K. SaitB: It6’s formula and Ldvy’s Laplacian I, Nagoya Math. J. 108 (1987), 67-76; II, ibid. 123 (1991), 153-169. 26. K. SaitB: A (Co)-group generated by the Ldvy Laplacian II,Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 27. K. Sait6: A stochastic process generated by the Ldvy Laplacian, Acta Appl. Math. 63 (2000), 363-373. 28. K. Sait6: The Ldvy Laplacian and stable processes, Chaos, Solitons and Fractals 12 (2001), 2865-2872. 29. K. Saitd and A. H. Tsoi: The Ldvy Laplacian as a self-adjoint operator, in “Quantum Information (T. Hida and K. SaitB, Eds.),” pp. 159-171, World Scientific, 1999. 30. K. SaitB and A. H. Tsoi: The Ldvy Laplacian acting o n Poisson noise functionals, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 503-5 10. 31. K. Sait6 and A. H. Tsoi: Stochastic processes generated by functions of the Lkvy Laplacian, in “Quantum Information I1 (T. Hida and K. Sait6, Eds.) ,” pp. 183-194, World Scientific, 2000. 32. K. Yosida: “Functional Analysis (3rd Edition),” Springer-Verlag, 1971.
SEPARATION OF NON-COMMUTATIVE PROCEDURES - EXPONETIAL PRODUCT FORMULAS AND QUANTUM ANALYSIS MASUO SUZUKI Department of Applied Physics Science University of Tokyo 1-3,Kagurazaka, Shinjuku-ku Tokyo 162-8601 Japan E-mail: msuzukiOap.kagu.sut.ac.jp Systematic methods to treat mathematically and numerically non-commutative operator funcions are discussed with focus on exponential operators. Quantum analysis is also briefly reviewed to explain a general strategy for constructing such systematic methods.
1
Introduction
The present paper reviews the general strategy [l-31 of treating noncommutative mathematical and physical objects, particularly exponential operators composed of the sum of non-commutative operators. The higherorder exponential product formulas obtained by the present author [4-181 are very useful in computational physics [19-1261. More mathematically, quantum analysis [127-1381 is also reviewed. This is a general formulation on differential calculus of an operator-valued function f(A)with respect to the operator A. 2
Separation of Non-Commutative Procedures
According to quantum mechanics, physical phenomena or processes are described in terms of non-commutative operators. This non-commutativity causes the uncertaintity principle in measurements. Thus, the noncommutativity of operators are essential in studying quantum systems, particularly quantum effects. Even in classical problems such as a chess game, the order of processes is very important. In a chess game, for example, to move a queen first and then move a rook (or castle), or to move them in the opposite order will change the game sometimes seriously. In quantum mechanics, the relevant Hamiltonian 31. is, sometimes, composed of the two or three non-commutative operators 31.1 and 31.2 (and 31.3) which are of the same order in magnitude. The synergism of quantum effects
374
375
caused by these operators is vital in many interesting quantum phenomena such as high-Tc superconductivity. In such a situation, a perturbational approach is useless. A strategy to solve such a problem is to make use of the concept [l]of separation of procedures. It is mathematically realized by using, for example, the following Trotter-like formula
(
e+(A+B)= n-+w lim e$+AeFXB)”,
(1)
or more generally [l] ex(Ai+Az+...+Aq)= lim n-bm
(e $ s A l e $ r A z
. . .e;;~Aq .
1”
(2)
The merit of this decomposition is to preserve [l]the symmetry of the original exponential operator such as the unitarity and symplectic property. The above formulas for finite n give approximations which can be made explicitly when A1 ,A2, . . ,A , can be diagonalized analytically. The above formulas have been effectively used, for example, to transform [139] quantum systems to classical systems. This transformation (secalled Suzuki-Trotter transformation) yields the quantum Monte carlo method [1391431.
-
3
Exponential Product Formulas
In practical applications, a smaller number n is more convenient in the formulas (2). However, the correction of it is of the order of z2/n,which is inversely proportial to n. If the correction of a product formula is of higher order of l / n , then such a formula will be very useful. In fact, the following symmetrized product formula
~
~= e (+ ~~~ l 1e $.e~ +~~2~. q. - l e ~ ~ q.e. .$e ~+~+q ~- lz e j ~(3) ~l
is of second order and has been used frequently. In the present section, we explain how to construct arbitrarily higher-order product formulas. The following recursive scheme [4-181 is very convenient. When we know the rn-th order product formula Qm(z),the (m+l)-th order formula Qm+l(z)is constructed in the form
with
376
By definition, we have
+
eZ(A1+A2+...+Aq)= Qm(z) zm+'R,
+ O(S"+~).
(6)
Here, R, is an operator-valued function of { A j } . Thus, we obtain eZ(Al+Az+...+Aq)- Qrn(Plz)Qm(fiz) *.*Qmbrz) r
+ zm+lX p T + l R , + O(Z"+~).
(7)
J=1
Therefore, for Q,+l(z) in (4) to be of the order of m+ 1, the parameters { p j } have to satisfy the condition
+ p y + l + . .. + p y + l
p;n+l
= 0.
(8)
Thus, we can construct any higher-order product formula recursively using only the two conditions (5) and (8). However, if m is odd, the solutions of (8) are all complex (not real). In order to construct higher-order product formulas with real decomposition parameters, we have only to make use of symmetrized product formulas in each step. When a symmetric product formula S2,(2) is known, S2,+2(z) can be constructed [4,5] recursively as 5'2m+2(2) = S2mb1z)S2rn(P2z) ...&rn(prz)
(9)
with Pr-j+l = Pj,
pl
+p2
+ ... +pr = 1
and p;m+l
+ p;m+l + . . . + p
+ l
= 0.
(10)
It is easily seen from (10) that the at least one of { p j } has to be negative. There are many different sets of solutions on { p j } . The following recursive scheme is now standard:
s;&> = [s~,-2(P,z)I2s;,-2((1
- 4P&wZ*m-2(P,z)l2
(11)
with the parameter given by
p , = (4 - 4 w - y .
(12)
In particular, the fourth-order product formula Sz(z) has been used very frequently [19-1261, because it is stable (0 < p , < 1 and 11 - 4pm1 < 1).
377
4
Quantum Analysis
In the previous section, we have explained only the recursive scheme to construct higher-order exponential product formulas. However, there are many other methods for it. In order to devise such methods, the following quantum analysis was formulated by the present author [127-1381. We start from the following Ggteaux differential # ( A ) of f ( A ) :
Clearly, df(A) is a functional linear in dA, namely we have
d f ( A )= f i ( A ,6 A ) . dA
(14)
with the inner derivation 64 defined by
~ A =B [A,B ] = AB - BA.
(15)
Here, A in f l ( A , b ~should ) be interpreted as a hyper-operator LA defined by L A B = A B . Then, two hyper-operators A and 6~ commute with each other. Thus, f1 ( A ,SA) is a functional convenient in obtaining higher-order exponential product formulas and in other operator manupulations, such as operator Taylor expansions. Therefore, we introduce the concept [127] of "quantum derivative" of f ( A ) with respect to A and introduce [127] the notation d f ( A ) / d Ato express it: -@ ( A )- f i ( A , 6 ~ ) .
dA
That is, the above quantum derivative d f ( A ) / d Ais a hyper-operator to map dA to @ ( A ) . However, the above notation d f ( A ) / d Ais very convenient in many applications. It has a great merit to emphasize the similarity of it to a classical derivative df(z)/dzfor an ordinary (real or complex) number z. Of course, there exists a big difference between the two derivatives. However, no confusion appears in explicit applications of quantum derivatives. The first derivative d f ( A ) / d Ais expressed as [127-1381
where f(")(z)denotes the nth derivative of f(z)with respect to an ordinary number I. Similarly the nth quantum derivative d"f(A)/dA" is expressed by
378
[127-1381
dtnf(")(A-t161 -t262*.*-tndn),
(18)
where 6, is a hyper-operator defined as
6,
:
B . . .. . B = Bj-1(hAB)Bn-j.
(19) The following general operator Taylor expansion formula has been derived [127- 1381 f ( A + ZB)=
c-+c 1 O0
n=O
X"d"f(A) n ! dAn
1
00
= f(A)
X"
n=l
I
B"
0
dtl
f 1
dt2.. .
0
tn-1
dt,,f(")(A - t161 - t262." - tn&) : B".
(20)
From this general formula, we can easily obtain the wellknown Feynman expansion formula on et(A+zB)as
where
B(t) = e-taAB - e-tABetA.
(22) Many other explicit operator expansion formulas have been derived [129-1381 and been applied to physical systems [130-1381. 5
Future Problems
The present formulations of exponential product formulas and quantum analysis will be applied to explicit calculations of the Hida calculus [2,3] and its extensions by Obata et al[144]. These applications will be reported elsewhere.
379
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Theory of quantum tunneling of the magnetic particles, Phys. Rev. B 56 (1997), 11761. 116. A. N. Drozdov and J. J. Brey: Accurate path-integral representations of the Fokker-Planck equation with a linear reference system Comparative-study of current theories, Phys. Rev. E 57 (1998), 146. 117. A. N. Drozdov and J. J. Brey: Operator expansions in stochastic dynamics, Phys. Rev. E 57 (1998), 1284. 118. A. N. Drozdov: Improved Feynman path-integral method with a large time-step - Formalism and applications, J. Chem. 108 (1998), 6580. 119. G. Dattoli, L. Giannessi, M. Quattromini and A. Torre: Exponential operators, operational rules and evolution problems, Nuovo Cimento B 113 (1998), 699. 120. C. Grosche and F. Steiner: Handbook of Feynman path-integrals - Introduction, Spr. Trac. Math. 145 (1998), 1. 121. H. Ishida and A. Kidera: Constant-temperature molecular-dynamics of a protein in water by high-order decomposition of the Liouville operator, J. Chem. Phys. 109 (1998), 3276. 122. K. Michielsen, H. Deraedt, J. Przeslawski and N. Garcia: Computersimulation of time-resolved optical imaging of objects hidden in turbid media, Phys. Report 304 (1998), 90. 123. A. N. Drozdov and J. J. Brey: Accurate quantum and statisticalmechanics from system-specific operator expansions, Phys. Rev. E 58 (1998), 2859. 124. H. W. Streitwolf Dynamics of a bond-disordered Peierls chain and mized gap states, Phys. Rev. B 58 (1998), 14356. 125. I. Saitoh, Y. Suzuki and N. Takahashi: The symplectic finite difference time domain method, IEEE Trans. Magn. 37 (2001), 3251; I. Saitoh and N. Takahashi: Stability of symplectic finite-difference time-domain methods, IEEE Trans. Magn. 38 (2002), 665. 126. Y. Hatsugai and A. Sugi: Effects of interaction f o r the quantum diffusion in coupled chains, Int. J. Mod. Phys. B 15 (2001), 2045. 127. M. Suzuki: Quantum analysis - Non-commutative differential and integral calculi, Commun. Math. Phys. 183 (1997), 339. 128. M. Suzuki: Quantum analysis, nonequilibrium density matrix and entropy operator, Int. J. Mod. Phys. B 10 (1996), 1637. 129. M. Suzuki: Compact exponential product formulas and operator functional derivative, J. Math. Phys. 38 (1997), 1183. 130. M. Suzuki: Algebraic formulation of quantum analysis and its applications to Laurent series operator functions, Phys. Lett. A 224 (1997), 337.
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FREE PRODUCT ACTIONS AND THEIR APPLICATIONS YOSHIMICHI UEDA Graduate School of Mathematics Kyushu University fikuoka, 810-8560 Japan E-mail: uedaamath.kyushu-u.ac.jp A quick review of a series of our recent works on free product actions, partly in collaborations with Dimitri Shlyakhtenko and with Fumio Hiai, is given. We also work out type I11 theoretic subfactor analysis on subfactors associated with free product actions of SU,(2). This part can be read as a supplementary appendix to those works.
1 Introduction Free product actions mean group or group-like symmetries on operator algebras (C*-algebras or von Neumann algebras), which are naturally arisen from the free product construction. In the early 80s, D. Voiculescu started his original study on the 111-factors associated with free groups IFn (so-called free group factors denoted by L(F,)) by the use of idea of probability. This line of study is now called free probability theory (see [42,43]), which has been widely accepted as one of the major (non-commutative) probability theories, and the free product construction plays particular r61e in the theory, in particular, naturally produces many explicit examples of non-commutative probability spaces where the theory works quite effectively. In [34], we began to use free product actions to deal with purely operator algebraic questions by the aid of free probability theory, and then, with D. Shlyakhtenko [31] (also see [36]) we showed that any positive real number greater than 4 can be realized as the Jones index of an irreducible inclusion of II1-factors both isomorphic to L(F,). We also use, with F. Hiai [12], free product actions to construct variety of examples of (aperiodic) automorphisms on interpolated free group factors L(F,), 1 < r 5 00, (independently introduced by K. Dykema [6], F. Rgdulescu [24]). One of the purposes of this paper is to give a quick review of the abovementioned progress and the other is to give supplementary results to the works [31,36], part of which was already commented in Remark 3.5 of [31]. The organization is as follows. The $ $ 2 4 are all preliminaries; namely we recall the compact quantum group SU,(2) in $2; the notion of free product actions in $3; and Wassermann’s construction for minimal actions of compact
388
389
quantum groups in 54. The 55 is quick review of [12,31], while in 56 we give type I11 theoretic subfactor analysis on the subfactors constructed in [36] (also see [31]). Finally, in 57 we collect some questions and comments related to the subject matter treated in this paper. 2
The Quantum Group SU,(2).
The von Neumann algebraic approach to compact quantum groups is needed for our aim, and we would like to employ the following as the (rather working) definition of compact quantum groups, which is sufficient for the later use. Let A be a von Neumann algebra and 6 : A + A @A be a (normal) *-isomorphism (called a co-multiplication) satisfying the co-associativity: (6 €9 IdA) o 6 = (IdA €9 S) o 6. The pair ( A ,6) is usually called a Hopf - von Neumann algebra. A Haar state h for ( A , 6 ) is a faithful normal state on A satisfying the left and right-invariance property: ( h €9 IdA) o 6(a) = h ( a ) l = ~ (IdA €9 h) o 6 ( a ) for a E A. We will here call a Hopf - von Neumann algebra equipped with a Haar state a (von Neumann algebraic) compact quantum group. All the usual notions in the group representation theory can be used even in this setting. We briefly explain those for the later use. (See [46] for details, also see [2].) A corepresentation x on a finite dimensional Hilbert space V, means a linear map x : V, + V, €9 A with the property:
(n €9 IdA) o K = (Idv- €36) o T . For a basis
{(j}
of V, one finds u(7r)ijin A with
a
Via the natural identification MdimV, ( A ) = B(V,) €9 A (with respect to the basis {&}), the operator matrix u(x):= [ u ( x ) i j ]can be regarded as an element in B(V,) €9 A , and the u(x)ij’sare called the matrix elements with respect to the basis (0). A corepresentation 7r is said to be unitary when the operator u(x) defines a unitary element. An intertwiner of two corepresentations T I , x2 means a linear map T E B(V,, , V,,) satisfying
(T €9 l)u(x1) = u(r2)(T€9 11, and the set of those intertwiners is denoted by End(xl,x2). When End(x1, x2) = {0}, we say that x1 and x2 are nonequivalent. A corepresentation x is said to be irreducible when the self-intertwiners End(x) = End(x, x) is 1-dimensional.
390
Since we consider only examples of compact-type, any finite dimensional corepresentation is reducible, that is, it is equivalent to a direct sum of irreducible ones, and also is equivalent to, or can be (by replacing the original inner product by a suitable equivalent one), a unitary one. Thus, in any case, for a given finite dimensional corepresentation T , we may and do assume that any given 7r is a direct sum of finitely many corepresentations: T
= m l T l @ .. . @ mnxn
with mutually nonequivalent irreducible unitary corepresentations T I , . . . ,xn and their multiplicities ml, . . . ,m,. In this case, one easily describe the selfintertwiners End(7r) as
(Mm1(C)€4C1vr1) @ - . . @ (Mmn(C)€4C1vr,). The quantum group SU,(2) (q 6 (0,l)) is a typical example of compact quantum group, which is given in the following way. Let us assume the indeteminates x,u, v, y satisfy the relations:
ux = qxu, '112)
vx = qxv, yu = guy,
= V U , x* = y,
yv = guy,
u* = -q-lv,
and consider the universal C*-algebra A generated by x,u,v, y. One can then construct a co-multiplication 6 : A + A @'min A by the matrix-product rule (with replacing the usual scalar multiplication by the operator tensor product):
+ u €3 21, S(V) = v €4 2 + y €4 v, 6(2)= x €4 2
6(u) = x €4 u 6(y) = v €4 u
+ u €4 y, + y €4 y.
It is known (see [45,46]) that there is a unique (faithful) Haar state (i.e., it state with the left and right-invariance property as above) for the pair (A,6 ) , and let A be the von Neumann algebra obtained from A via the GNS representation of the Haar state. It is straightforward to see that the co-multiplication has the unique extension to A , and we still denote it by 6 : A + A €4 A . By the construction, the GNS-vector gives, as a vector state, a Haar state h for the pair ( A ,6 ) . This compact quantum group ( A ,6, h) is called the quantum group SUq(2), and sometimes denote A = L"(SUq(2)) (and A = C(SU,(P))). 3
Wee Product Actions
Let N1, N2 be (a-finite) von Neumann algebras with faithful normal states cp1, cp2, respectively. We can then construct, from the given pairs ( N ~ , c p l ) ,
391
(N2,( ~ 2 ) ,a new pair ( M ,(P) of a von Neumann algebra and a faithful state in such a way that (i) there are two embeddings Nl, N 2 C ) M whose ranges generate M , i.e., M = N1 V N2; (ii) the restrictions of cp to N1, N 2 (via the embeddings above) coincide with cpl, ( ~ 2 respectively; , (iii) (the free independence) the expectation value of any alternating word in N;" := Kercpl, N," := kercpz is 0. The pair (M,cp) is called the (reduced) free product of and written as
(N1,cpl) and
(N2, p 2 ) ,
(M,(P)= (N1,Cpl)* ( N 2 , ( P 2 ) . The algebra M has many automorphisms naturally induced from automorphisms on N1, N 2 preserving the states (PI, ' p 2 , respectively: ( c Y ~ , QE~Aut,,(Ni) )
x Aut,,(N2)
*
++(Y := ( ~ 1 ( ~ E 2
AUtv(M)
determined (thanks to the characterization of ( M ,(P) explained just above) by the relations (YIN~= ( ~ 1 ( ,Y ~ I N ~ = ( ~ 2 .We would like to refer to group or group-like symmetries of this type as free product actions, and thus the most typical example is the modular action of R associated with the free state 'p:
t E R H 0: = 0 : '
* of2 E Aut,(M).
This is, of course, a special example, but the reader should pay their attention to it because many observations and/or motivations of [12,31,34,36] came in part from study on modular actions of this type so that many discussions there and/or here are closely related to the others [35,37-391. For example, the attempt of showing the minimality of free product actions is essentially same game as that of finding the IIIx-types of (type 111) free product von Neumann algebras. This is quite similar to the infinite tensor product situation (see p.212 of [44]for product-type actions of compact Lie groups and p.6546 of Takesaki's monograph [33] for the IIIx-classification of Powers factors). In the quantum group case, an action does no longer mean an automorphic one, that is, an action of a Hopf - von Neumann algebra ( A , 6 ) on a von Neumann algebra N is formulated as a normal *-isomorphism 9 : N + N 63 A satisfying (!I!63 IdA) o 4! = (IdM 63 6 ) o !I!. Assume that two actions rl : N 1 + N1 63 A , I'2 : N 2 -+ N 2 63 A of a Hopf - von Neumann algebra ( A ,6 ) on von Neumann algebras N1, N 2 equipped with faithful normal states (PI, ( ~ 2 respectively, , satisfy the following invariance condition:
(Pi63 IdA) O ri (z) = (Pi (XI,
((P2
€3IdA) 0 r
2
(Y) = cpi (Y)
392
for x E N l , y E N 2 . Then one can, even in this case, construct the free product action I?:d?l * r 2 : M
-+ M @ A
of ( A ,6 ) on the free product
( M ,9)= V
l , 91)
* (N2,9 2 )
(by the essentially same way as in the group case) in such a way that
riN1= rl,
rlN2
= r2.
See [34,36] for details. 4
Wassermann’s Construction for Minimal Actions of Compact Quantum Groups
Wassermann’s construction (see [44]) is a way of constructing subfactors of finite-index from (minimal) actions of compact groups together with their (finite dimensional unitary) representations. In this section, we will explain how it can be generalized to the quantum group case, which was done in [36]. (See also [4], where the only Kac case is treated.) For notational simplicity, we will deal with only the SUq(2) (i.e., ( A := L”(SUq(2)), 6, h),just explained in §2), but this is not essential; in fact we will not use any special characteristic of SUq(2). Let I‘ : M + M @ A be an action of SU,(2) on a factor M . Let us assume that (a) the fixed-point algebra Mr := {x E M : r ( m ) = m €31) has the trivial relative commutant in M ; (b) for any (irreducible) unitary corepresentation (T,Vm)of SUq(2) one can find a Hilbert space 3c in M in such a way that (r(N,R)is unitarily equivalent to See [27] for the terminology of Hilbert spaces in von Neurnann algebras. The actions given in [31,34,36] clearly satisfy these conditions by the constructions themselves, but in general the latter (b) should be replaced by the faithfulness in the sense of Izumi-Longc-Popa [13] with the aid of some (standard) techniques along the exactly same line as in [1,27]. The reader is encouraged to consult with Nakagami-Takesaki’s monograph [18] although only group actions are treated there. In fact, it is quite a straightforward work
393
to generalize the necessary parts given there to the quantum group case; see [5,13,48]for example. Let r be a finite dimensional corepresentation on V, of SU,(2). We then consider the following inclusion:
M := (B(V,) 8 M)Ad"@r2 N where Adr @ I? : B(V,) @ M
+ B(V,) @ M
:= Clv,, 8 Mr,
@A
is the action defined by
Adr 8 I? := Adu(r)lS 0 (IdB(v-1 @ I?). See $2 for the definition of u(r). Thanks to the assumption (b) we can construct an explicit isomorphism between M and N , and hence, in particular, M is a factor (see Remark 10 of [36] or Theorem 3.1 of [31]). When replacing the (b) by the faithfulness, one should do the argument after taking tensor product with B ( R ) , and then remove it so that the consequence is slightly changed, i.e., M and N are stably isomorphic. The first non-trivial problem is whether there is a (finite-index) conditional expectation for M 2 N . This is somewhat non-trivial because of the definition of the action Adr 8 I? and the non-commutativity of the algebra A. However, we can indeed overcome this by showing the following: Lemma 4.1 (see pp. 44-45 of [36]) Let { f z } z E ~ be the Woronowicz characters 0fSUq(2) (see [46]), and consider the element F, E B(V,) constructed as F," := (IdB(v-) 63 fz)(u(r)), z E C. Let us consider the quantum dimension and the positive/negative q-traces dim,(r) := Tr(F,) = T r ( F i l ) ,
where Tr means the non-normalized usual trace o n B(V,). Then we have (T:"'-'
63 IdM) o E A ~ , B=~Er o (T;,'+) @ IdM),
where Er, E~d",g,rdenote the usual conditional expectations onto the fixedpoint algebras constructed by using the Haar state h. Therefore, the restriction of the F'ubini-map T;,'-' 8 IdM to M gives rise to a (faithful normal) conditional expectation E : M -+ N . Unlike in the usual group case, we cannot provide the invariance principle of the exactly same form as in [44] because of the same reason as in the trouble treated above, and hence we start with constructing a candidate of the triple of basic extension, dual conditional expectation and Jones projection, and then show that they satisfy the so-called Pimsner-Popa characterization.
394
Consider the bigger algebra
Mi := (B(V+@ VT) M)Ad("@")@r inclusion M 2 N is embedded with x-e lv,
into which the @ x, where ii means the canonical dual corepresentation on V+ := V , of ?r (see [3]). By the same reason as above, it can be checked that M I is a factor, and also then that the restriction of the Fubini-map T;"-) @ IdB(v*) @ IdM to M1 gives rise to a conditional expectation El : M I + M thanks to the equation Ad(+ @ x) @ r = Adii @ (Adx @ I?). Looking at the basic extension of the inclusion
B(V,) 2
c1,
Ti">-'(
. ) : B(V,)
+ c1,
one can easily show that
satisfies one variation (see Theorem 8 of [ll])of the Pimsner-Popa characterization of basic extensions so that this tower is the Jones tower up to the 1st step. As a consequence one sees that the index IndE is given by the square of the quantum dimension Summing up the above discussions we have the following: Proposition 4.2 ([36]) W e have (1) M S N , and in particular, both being factors; (2) IndE = (dim,x)2 := Tr(F,)2 = T ~ ( F c ~ ) ~ . If one uses the faithfulness together with the usual technique of taking tensor product with B(7-t)instead of the assumption (A'), the first assertion should be changed into (1') M is stably isomorphic to N . (Ofcourse, this change is not needed when the fied-point algebra is properly infinite.) Clearly, the above procedure of finding the basic extension can be iterated, and the n-th basic extension M , has the simple description:
M , = (B(Vp,)@ M)Adp"@r with the embedding x e 1 @ x, where pn is determined by the recursive relations: ~2m+1=
+@~ 2 m ,
=
@ P2m-1,
P2m
po = x.
395
We should point out here that the conditional expectation En : M n + Mn-1 is of course described by the same way as above based on the q-traces. 5
5.1
Review on the Works with D. Shlyakhtenko and with F. Hiai Irreducible Subfactors of L(F,)
- Joint
Work with D. Shlyakhtenko
Based on [36] with the aid of free Araki-Woods factors introduced by D. Shlyakhtenko [28,29] (also see [30]), we showed with D. Shlyakhtenko the following theorem: Theorem 5.1 ([31]) A n y r e d number > 4 can be realized as the Jones index [M : Nl of a n irreducible inclusion M 2 N in such a way that M 2 N S L(F,). This together with his pioneering work of F. RSdulescu [24] in the subject matter implies Corollary 5.2 The possible Jones indices of irreducible inclusions consisting of L(F,) is the so-called Jones spectrum
In the proof, we first construct the free product action of SU,(2) by (MY ‘P) := ( L ( F N ) ,m
r :M + M
@A
~* ) ( A ,h ) , r := ( I ~ L ( F N@ ) 1 A ) * 6,
and then consider the Wassermann-type inclusion associated with an irreducible corepresentation. This inclusion consists of type IIIq2 factors so that we should show that it is of essentially type 11, and then get the desired type 111 subfactor via the structure theorem of type 111 factors. What we have to do is to investigate the fixed-point algebra M r , or more precisely its discrete core. In the process of identifying with L(F,), one of the free Araki-Woods factors (see [28]) naturally arises in the following way: Our type I11 subfactor M r can be described, via the Takesaki duality (for compact quantum group action), as the amalgamated free product
where A^ := Cm(SUq(2)) means the dual Hopf-von Neumann algebra of SUq(2). Since A^ is an infinite direct sum of matrix algebras, an argument similar to Theorem 5.1 of [29] works for identifying Mr (or the amalgamated free product of the above form) with the unique free Araki-Woods factor of
396
type 111,~.The free Araki-Woods factor of type IIIx (A # 0 , l ) is known to have the discrete core isomorphic to L(F,) €3 B(31), and hence we observe an exact relationship between our subfactor and the free group factor L(F,). Therefore, one of the key-points is “passing once through infinite (type 111) factors t o get a result on type 111 factors.” This should be much emphasized. In fact, S. Popa and D. Shlyakhtenko [23] generalized this theorem further to the final form, where the use of type 11, factors is one of the most essential points although the counter-part of the arguments explained above is much more complicated in their case.
5.2 Automorphisms of L(F,) with r > 1 - Joint Work with F. Hiai Using free product actions, we exhibited, with F. Hiai, several automorphisms with special characteristics on each (interpolated) free group factor L(F,) with r > 1 ([6,24]). Theorem 5.3 ([12]) There are continuously many non-outer conjugate aperiodic automorphisms o n the interpolated free group factor L(F,) with any f i e d r > 1, all of whose crossed-products are isomorphic. This theorem is a supplementary result to a famous one of J. Phillips (see [19,20]) obtained in the 70s. We also worked on this isomorphic crossedproduct algebra; it has the property but not McDuff or not strongly stable, and its quotient group m/Int is isomorphic to the 1st cohomology group of the unique amenable type 111 equivalence relation. Those proofs are essentially same as those given in [39]. Let P , Q be von Neumann algebras equipped with faithful normal states cpp, (PQ, respectively. Let G be a discrete (countable) group, and the compact can act on the free co-commutative Kac algebra & = (L(G),~G,KG,TG) product algebra
( M ,(PI := ( p€3 L(G),(PP €376)* (Q, (PQ) by the free product action
rG= (Idp €3 BG) * (IdQ €3 ~ L ( G ) ) . This action is related to the following action cr of G on the other free product algebra
defined by G ag := Idp * rg
397
*
with the G-free shift action yG on
(Q, 9 ~ )In~fact, . the following duality
gEG
holds: Theorem 5.4 ([12]) W e have
-
( M , r G )% ( N >arr G,G) with the dual action 8 (= Idp * r G ) (see [18]). This theorem explains (rather philosophically) that the minimal actions of SU,(N) constructed in [34,36] can be regarded as the dual actions of “free Bernoulli shift actions” of the dual P‘(SU,(N)). We will return to this point of view in near future. This duality can be used for example to investigate the crossed-products by free Bernoulli shifts on the free group factor L(IF,), see [12] for details. 6
Type I11 Theoretic Subfactor Analysis on Wassermann-Type Subfactors Associated with SU,(2)
Throughout this section, we keep and use freely the notations in $2, $3, $4. Let us assume that : M + M €3 A is a minimal action of SU,(2) equipped with an invariant state cp satisfying (i) the fixed-point algebra Mr is of type 111,~; 2. (ii) the modular automorphism or has the period To := --. log q 2 ’
(iii) there is a von Neumann subalgebra Q of Mr n Mu’ with Q’ n M = C1. These conditions are typical characteristic that the actions given in [31,34,36] (one of which is explained in $55.1briefly) has. Let us next choose an arbitrary finite dimensional unitary corepresentation 7r : V , + V, €3 A, and consider the Wassermann-type inclusion consisting of (isomorphic) type 111,~factors Ad@r Er
M ( T ):= (B(V,) €3 M )
2 N(.)
:= Clv, €3 Mr
Since we do not assume that ?r is irreducible, the inclusion may not be irreducible so that it is unclear whether or not the inclusion M ( T ) 2 N ( T ) is of essentially type 11. In what follows, we will discuss this aspect completely. Namely, following Hamachi-Kosaki’s analysis on type I11 subfactors ([9,10,14-16]), we investigate the factor map structure and the three-step inclusions associated with the type 111inclusion M (T) 2 n/(.), and give, as one consequence, a necessary and sufficient condition, in terms of fusion rule of K
398
decomposed into irreducible corepresentations, when the inclusion in question is of essentially type 11, i.e., it can be described by an inclusion of type 111 factors (together with a certain %action on its amplification) in principle. We encourage the reader to consult with [15] as a brief survey of HamachiKosaki's analysis. As explained in 52, we may and do assume that the n is the direct sum of (unitary) irreducible corepresentations: ?r
= mlnl @ . . . @ mnrn,
and hence
Therefore, one has:
N(x)'n M ( n ) = B(Vx)Adnc3 c
1 2 ~B(Vx)Ad"= End(x)
(since (Adr)' n M = c l ~so)that the inclusion M ( n ) 2 N(n) is irreducible if and only if so is K. Let us start with looking at the inclusion: E,
2
where
+ is the restriction of cp to Mr = N(n).Here Ex is defined by
(1:
1
W
m(t)h(i)dt):=
E,(m(t))X(t)dt. -W Ex The following simple application of the Fourier transform is essential in our computations: Lemma 6.1 (e.g. Proposition 5.6 of [8]) There is an isomorphism
a : Mr
x u R + (Mr x + T)63 LWIO,- logq2)
determined b y 9 ( ~ , ( m )= ) ~ b ( m€3) 1,
a (A'(t))
= ~ " ( [ tB]meit( ) .
).
Here, we set D := cr+ and u is the natural torus action of T := RJToZ anduced by D (cr has the period TO by the assumption (ii)) with the natural quotient mapping t E R e [t]E T. This lemma enables us to compute the relative commutant
399
(One should here note that this relative commutant involves continuous crossed-products so that the "Fourier series expansion" cannot be used to deal with generic elements in any sense!) We have (ivr X, R)' n (cix R)
T)@ L"[o, =" ((ivr = c1 @ L"[O, - logq2)
- l0gq2))' n ((CI >a T)CZI L"[o, - logq2))
(the first isomorphism given by $), and thus
(ivr M, ~ ) ' (ci n M R) = { x U ( q , )= 1@ xT0)" since
{$ (A"(T0)))" = (1 @ rne'To(
.
)}I!
= c1 €4 P [ O , -log&.
Therefore, we obtain
(G)' n ~7) E (C1@Q @ C1)' n (B(V,) ~3M)Ad"@r@ B(L2(R)) AdrW = 8 (9' n M ) ) @ B ( L 2 ( R ) ) (& sits in Mr) = B(V,)Ad" @ C1@B ( L 2 ( R ) ) (by the assumption (iii))
= End(r) @ C1@B ( L 2 ( R ) ) .
Since 0, tlo& = (AdF? @ 0:) l~(,) and F, E End(r)' (see Eq.(3.9) in p.47, lines 3-6 from the top of p.49 of [36]), we have
r,(z
@
1) = z @ 1@ 1 for all 2 E End(r),
and hence
(G)' nM x )
End(r) @ C1@{A, : t E R}"
because {At : t E R}" is a MASA in B ( L 2 ( R ) ) .Here, At means the (left) regular representation of R. Therefore, we get (%)I
n M T ) = (;v?;;j)' n (End(r) @ C1 @I {A, : t E R}") R))' n (End(.rr) @ (C >a R)) = End(n) @ C1@ AT^}" (2 End(r) 8 L"[O, - logq2)) . = (C1@(Mr
>a,
400
Set 2 := 2
((;IJ?;;i)' n M T ) ) = 2 (End(r)) @ C1@{ X T ~ } ' ' .
We have
OF(*)(z @ 1 8) ,A
= z @ 1 8 (e-itToATo), z E 2 (End(r)) ,
and hence, via the identification due to Lemma 1 (or a),
ey(*): meiTo(.
)
H m,;T0(.
-t)
= m( e ( i T o (
'
where Ft is the translation by t (mod:- logq2). Thus, we get the isomorphism !lj :
(2,BF(*)Iz)z (Z(End(x))@L"[0,-logq2),Id@FF,*).
What we want to do at first is to describe the relative position of the two subalgebras:
Z ( M T ) ) and 2 inside 2 = 2
(x)
((x)'
n M T ) ) by giving an explicit expression of the
factor maps
with the point-realizations:
The lemma below seems to be quite standard, but we will give the proof for the sake of completeness. Lemma 6.2 We have, via the \E, the following description: 2
= 2 (End(r)) @ L"[O, -logq2),
2 (MT)) = 2
(;IJ?;;i)
=
{F;iTo 8 m e i ~ o. () }" , C1@L"[O, -logq2).
PROOF. Only the non-trivial part is to compute 2 ( M T ) ) . Since cTo IltoEr = Ad (FiTo@ l),the F:To @ 1 is in 2 ( M ( T ) + ~ Eso , ) that we can find an invertible element h E 2 ( M ( K ) * ~ Ewith = ) FiTO @ 1 = hiTo. (Note that h
401
and Fx '8 1 are different and agree with only in the iTo-power!) Perturbing $ o Ex by h-l, i.e., 1/1 o E,(h-' . ) = 1c, 0 E,(h-'I2 1 5 - ' / ~ )we , get the new state 4 on M(7r) with the property gg0 = Id. The unitary operator U h on L2 (R,L2 (M(7r)))defined by
(Uh<)( s ) := h-Z"<(s),
< E L2 (R,L2(M(7r)))
gives us the isomorphism AdUh : M(7r) xu+ R satisfies that
E
M(7r) M ~ + ~ E ,R, , which
AdUh (~,m(m)) = n u + o ~ , ( m ) , AdUh ( X u m ( t ) )= (h-it '8 1) Xu'oEn ( t )= T , , + ~ E , (h-it) X u c l o E n ( t ) . Hence, we have
Z
(x))
= Z (M(7r)X U + o ~ , R) = AdUh (2( M ( T )~~4 R))
(since ng0 = Id, see around Lemma 6.1) =
{
7ru+oEn
(F;~TO'8 1)~
o
+
(T ~ ~~) }* " (since F ~ T O '8 1 = h i ~ o )
= { F l Z T o'8 1 @ A,}''. Therefore, we get the desired identification via the isomorphism q.
I
It is known that the irreducible corepresentations are indexed by the spins, i.e., the half-integers so that we can associate the half-integer t ( 7 r j ) E ;NO to any 7 r j , and we easily observe ~ , - i T o = FiTo =
c(
-1)2e(xj)pjE C p , @ . . . @ Cp, = 2 (End(7r))
based on the explicit computation of the matrix elements of the spin f2 corepresentations, see [17]. The standard identification z(End(7r))@Lm[0,-logq2)Z Loo ({1, ...,n } x [O,-logq')) sends the generators
FtiTo'8 mesTo( . ) ,
1'8 meiTo( .
)
402
of 2 ( M T ) ) ,2
(G) to the functions := (-1) Z l ( 7 T3. ) eiTos = eiTo(s-l(7rj) log$)
u'(")(j,s)
&(*I
>
( j ,s) := eiTos,
respectively. The mapping : ( j ,s) H ( j , s - C(.rrj) log q 2 )
defined on { 1,. . . ,n } x [0,- log q 2 ) satisfies uM(m) 0
= u N ( r ) or
-
z (NC.)),
so that the induced automorphism
aO( 2 ( M W ) ) Since both M ( T ) and the same space;
=
a0
uM(n)=~ N ( A 0 ro )
of 2 of
TO
satisfies
(etlz) o a 0 = a. 0 (etlZ>.
N ( r ) are factors of type IIIqz, the XM,,), XN(") are x M ( ~ )= xN(7~) =
[o,-logq2),
and the identification here enables us to describe the factor map 7 $ / ( " ) : ~ : = { 1,...,n } x [ ~ , - l o g q ~ ) + ~ N ( =[0,-logq2) ,) by the standard one ( j , s) H s while the factor map
)"('r
:
x := (1,...,n } x [o,-1ogq2) + x M ( ~ )= [0,-logq2)
= rN(") o T O , that is, by the deformation of the standard one by TO;)"('r ( j , s ) H s - C(rj)logq2 (mod: -logq2). Hence, we conclude E,
Theorem 6.3 T h e factor map structure attached to M ( T ) 2 ,p(-)
[O,-logq ) t {I,...,n } x [O,-logq )
N ( r ) is
#(-)
+ [0,-logq2)
with x'(")(j,
s) = s - C(.rrj) log q2 (mod: - log q 2 ) ,
7$/(7T) ( j ,s) = s.
Therefore, the inclusion is of essentially type I1 i f and only i f the set
{C : the spin C corepresentation occurs in I T } consists entirely of integers or consists entirely of half-integers.
403
The next question from the view-point of Hamachi-Kosaki's analysis is to describe the following inclusion of von Neumann algebras (not necessary being factors) inside M ( K )2 N ( K ) Letting :
-
-
% := M ( A )n 2' 2 5 := N ( A ) v 2, we set
-
U := U >a@ R 2 B := % ' M'
R
with 9 := O M ( " ) . We first easily observe that
2(%) = Z ( 5 ) = 2, 2 (U) = 2 (B) = 2' = 2 (End(r)) 63 C1 €3 C1 since Ft is a transitive flow on [0,- log q 2 ) . What we will do is to give a complete description of this inclusion U 2 B in terms of the irreducible decomposition of A. Consider the conditional mpectation
z M ( n )-+ I# :
-
= M ( A )n ( A , ( 2 (End(7r)) 8 Cl))'
given by
r,(u 63 l)Xn,(u €3 l)*du, X
6 ( X ) :=
E
-
M(A),
LZ(End(4))
-
with the modular automorphism ut := rfoE*.For any finite sum X = CjA , ( r n ( t j ) ) X u ( t j ) E M ( n ) we have
=
A,
((u€3 l)rn(tj)(U63 1)*)X"(tj)dU
l(Z(End(n)))
=
(u63 l)rn(tj)(U63 1)*du 'r
(k(.%'(Encl(~)))
Here the second equality comes from the fact that the modular automorphism ut acts on 2 ( E n d ( ~ )€3 ) C1 trivially. Note that all the
(u€3 l)rn(tj)(U 63 l)*du
404
are in the relative commutant
(B(V,) 8 M)AdK@r n ( 2(End(7r)) 8 Cl)‘ = ( 2(End(w))’ 8 M )
Adn@r
Ada@r
. Thus, by the
Here we used that 2 (End(w)) 8 C1 sits in (B(V,) €3 M ) appropriate continuity of 6,
5
% = M ( w ) n (7ru =
( 2(End(w)) 8 Cl))’
( M ( n ) )g ( 2(End(7r))’ 8 M )
Ada@r >QU
R,
and hence
% = (2(End(w))’ 8 M )A d r @ r x u R.
It is clear that
-
6 = N(7r)v 2 = ( C 1 8 (Mr x u R)) V ( 2(End(w)) €3 C1 €3 {XT~}’’) = 2 (End(n)) 8 (Mr x u R) .
Via the Takesaki duality, we get U = (2(End(w))’8 M )
Ad?r@r
, 23 = 2 (End(w)) €3 Mr.
We will next give the description of the decomposition of E, : M ( w ) +
N(7r)into
4
M ( w ) 3 U 5 23 N(7r) with E, = F o H o G. Define the conditional expectations F : 23 = 2 (End(w)) 8 Mr
+ N(7r)= C 1 8 Mr;
H : U = (2(End(7r))’ 8 M )
Ad?r@r
G : M ( n ) = (B(V,) 8
+ 23 = 2 (End(7r)) €3 Mr; + U = ( 2(End(w))’ €3 M )A d r @ r
by
F(X):= (T;,~-) 8 Id) (X), X E 23;
G(X):=
(u €3 1)X(u*8 1)du n
= C”(Pj 8 l)X(Pj €3 l), j=1
x E 23,
405
where the pj's form the partition of unity consisting of minimal projections in 2 (End(7r)). Here, the last expression of G is due to the simple fact that
with the Haar probability measure p ~ Since . g t = ofoEff acts on 2 (End(7r))m C1 trivially as remarked before, we have, by a direct computation, .1c, 0 F
0
H
0
G(X)= .1c, 0 E,(X), X
EM
(T),
and hence E, = F o H o G. One should note that G agrees with previously. Therefore, putting the detailed data of 7r
introduced
= m17r1@.. . @ mnnn
into the above computations we finally get the following descriptions:
(22 B)
=
1
@
(Mm3(C) €3 M(7rj) 2 C1m1
€3.(Tj))
,
j=1
where the (M(7rj)2 N(7rj))'smean the Wassermann-type inclusions associated with the 7rj's. The conditional expectation H : !2l -+ !B can be decomposed into
and we have Hj( . ) = -
-
406
where we denote the non-normalized usual trace by Tr and the normalized one by r. Thus the conditional expectation H : U + 23 has, via the above identification, the following central decomposition: n
j=1
where the Erj 's are the conditional expectations of the Wassermann-type inclusions ( M ( r j )2 N(rj))'s.Summing up the discussions, we conclude Theorem 6.4 The three-step inclusions
have the following description:
and the conditional expectations G , F are given by
c n
G(
=
"(Pj @
1) (
*
1 (Pj @ 1)
7
F = E7r
lB
f
j=1
where the p j 's mean the partition of unity consisting of minimal projections in 2 (End(n)). One should note that the factors M ( T ~ N ) , ( T ~ etc., ) , appearing in the above expression are all isomorphic to Mr by the construction itself. 7
Comments and Questions
(1) The existence question of AFD-minimal actions: We provide, in [34,36], examples of minimal actions of compact non-Kac quantum groups. However, their constructions involve the free product construction so that the factors on which the quantum groups in question act do never be AFD. Hence, the most important question in the subject matter is: Does a compact non-Kac quantum group possess minimal actions on AFD-factors ? If it was affirmative, we would like to ask further: What kind of compact non-Kac quantum groups does allow minimal actions on AFD factors ? (2) Bernoulli shift actions of discrete quantum groups: In a more or less connection to the above-mentioned questions, we would like to pose the research project to find true analog(s?) of (non-commutative) Bernoulli shift actions of
407
discrete quantum groups (or equivalently of the duals of compact (quantum) groups, e.g. SU,(2)). The duality given in Theorem 5.4 might be useful to try this research project. Indeed, as was commented after Theorem 5.4 the theorem explains philosophically that our examples of minimal actions of SU, ( N ) are thought of as the dual actions of “free Bernoulli shift actions” of the dual SU,(N). Thus, the first attempt seems to try to find an explicit construction of these “free Bernoulli shifts” out of the dual quantum group SU,(N) directly. However, the most interesting question on this topic is what should be the discrete quantum group version of non-commutative Bernoulli G-shifts (with discrete groups G) based on the infinite tensor product construction, which might help to solve the questions in (1).
(3) Around free Bernoulli schemes: Let p = (PI, . . . ? p n )be a probability vector, and there are two kinds of free analogs of the classical Bernoulli scheme associated with p - the “free commutative Bernoulli scheme” FCBS(p) = (L(F,), y p ) and “free non-commutative Bernoulli scheme” FNBS(p) = (L(F,),op), where y p and up are both aperiodic, ergodic automorphisms on L(lF,). See [12] for their constructions. These free Bernoulli schemes need new dynamical invariants like entropies. Indeed, any kind of noncommutative dynamical entropy cannot be used to distinguish them. In fact, it is known that they always have Connes-St~rmer’sentropy (see St~rmer[32]) H z t ( y p )= (up)= 0 (for any p ) , and also we can easily show that the perturbation theoretic dynamical entropy (see [40]) H p ( y p ) = Hp(crp) = 00 (for any p ) . However, we have very small facts on their crossed-products (see Proposition 14, 15 of [12]), which in particular say that, if all the interpolated free group factors were mutually non-isomorphic, then we would have some cocycle conjugacy classification results for free Bernoulli schemes. Moreover, K. Dykema [7] pointed out us that the questions of cocycle conjugacy and of conjugacy for %free shifts considered in Proposition 12 of [12] are both equivalent to the isomorphism question of their crossed-products, which can be easily derived from Theorem 5.4. Thus, from the view-point of the famous isomorphism question of free group factors, it might be very interesting to seek for a new kind of non-commutative dynamical entropy fit for a suitable class of 111-factors consisting of all free group factors, and the dynamical free entropy dimension (see 587.2 of [41]) seems to work for those Zfree shifts. (Unfortunately, its computation is probably quite difficult because of the socalled “semi-continuity problem” for free entropy!) (4) Some brief comments to $4 and 56: (41) It is possible to consider other quantum groups instead of SU,(2). Here, we will briefly explain what phe-
408
nomenon occurs in the SOq(3)-case. The quantum S0,(3) (0 < q < 1) is defined as the subalgebra of Lw(SUq(2)) generated by the unitary u(n1) of the spin 1 (irreducible, unitary) corepresentation n1 of SUq(2) with the same Hopf algebra structure, and it is known that the set of all nonequivalent irreducible corepresentations can be chosen as { n l } l E ~ , ,See . [21] for details. In this case, we can still prove that the fixed-point algebra Mr (with minimal action defined in the same manner as in Theorem 5.1) is of type IIIq2 (one of its proofs uses the explicit description of the fixed-point algebra by creation operators given in the Appendix I of [31]), and thus all the conditions given in the beginning of $6 are still satisfied, and all the results there are of course valid in this case. The most important thing in this case is the fact that the resulting inclusions always become of essentially type I1 because of all the irreducible corepresentations are indexed by only the integers. (4-2) Soon after the appearance of our work [31], F. Riidulescu [26] gave another proof (or construction) to Theorem 5.1. His subfactors are constructed by a similar (but not the exactly same) way as in [22]. However, he also used a (minimal) action of SUq(2) (or we should say a minimal action of S0,(2)) to identify with L(F,). What we would like here to point out is the rather simple fact that his subfactors are also Wassermann-type. In fact, one can easily see that the subfactors are of the form N 2 N-l, where N-1 is the downward basic extension of the Wassermann-type inclusion M 2 N constructed from the free product action r112: M + M @ A: r1/2 := (IdQ @ 1 ~* Adnip ) with M := Q * B(Vnl,2).
Acknowledgements The author would like to express his sincere gratitude to Fumio Hiai and Dima Shlyakhtenko for their friendship, many discussions and their collaborations with the author. The author also thanks Nobuaki Obata, Akito Hora and Taku Matsui for giving him this opportunity.
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of type Ill, Duke Math. J. 43 (1976), NO. 2, 375-385. 20. J. Phillips: Automorphisms of full 111 factors. II, Canad. Math. Bull., 21 (1978), NO. 3, 325-328. 21. P. PodleB: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1995), NO, 1, 1-20. 22. S. Popa: Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), No. 2, 375-405. 23. S. Popa and D. Shlyakhtenko: Universal properties of L(F,) in subfactor theory, preprint, 2000, (MSRI-Preprint, No. 2000-032). 24. F. Mdulescu: Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), No. 2, 347-389. 25. F. Mdulescu: A type IIIx factor with core isomorphic to the von Neum a n n algebra of a free group, tensor B(H), Recent Advances in operator algebras (OrlCans, 1992), AstCrisque 232 (1995), 203-209. 26. F. Riidulescu: Irreducible subfactors derived from Popa ’s construction for non-tracial states, preprint, 2000, (math.OA/0011084). 27. J. E. Roberts: Cross products of von Neumann algebras by group duals, in “Symposia Mathematica,” Vol. XX (Convegno sulle Algebre C* e lor0 Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria K , INDAm, Rome, 1974), pp. 335-363, Academic Press, London, 1976. 28. D. Shlyakhtenko: Free quasi-free states, Pacific J. Math. 177 (1997), No. 2, 324-368. 29. D. Shlyakhtenko: Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), 191-212. 30. D. Shlyalkhtenko: A-valued semicircular systems, J. Funct. Anal. 166 (1999), NO. 1, 1-47. 31. D. Shlyakhtenko and Y. Ueda: Irreducible subfactors of L(F,) of indez X > 4, J. Reine Angew. Math. 548 (2002), 149-166. 32. E. Stmmer: States and shifts o n infinite free products of C*-algebras, in ‘Tree probability theory,” Fields Inst. Commun. Vol. 12, pp. 281-291, Amer. Math. SOC.Providence, RI, 1997. 33. M. Takesaki: Structure of factors and automorphism groups, Published for the Conference Board of the Mathematical Sciences, Washington, D.C., 1983. 34. Y. Ueda: A minimal action of the compact quantum group SU,(n) o n a fulZ factor, J. Math. SOC.Japan 51 (1999), No. 2, 449-461. 35. Y. Ueda: Amalgamated free product over Cartan subalgebra, Pacific J.
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Math. 191 (1999), No. 2, 359-392. 36. Y. Ueda: O n the jixed-point algebra under a minimal free product-type action of the quantum group SU,(2), Internat. Math. Res. Notices 2000, NO. 1, 35-56. 37. Y. Ueda: Remarks on free products with respect to non-tracial states, Math. Scand. 88 (2001), No. 1, 111-125. 38. Y. Ueda: Amalgamated free product over Cartan subalgebra, 11: Supplementary results l3 examples, preprint 2000, to appear. 39. Y. Ueda: Fullness, Connes ’ x-groups, and amalgamated free products over Cartan subalgebras, Trans. Amer. Math. SOC. 355 (2003), 349371. 40. D. Voiculescu: Entropy of dynamical systems and perturbations of operators, Ergodic Theory Dynam. Systems 11 (1991), No. 4,779-786; Vol. 11, Huston J. Math. 17 (1991), No. 4, 651-661. 41. D. Voiculescu: The analogues of entropy and of Fisher’s information measure in free probability theory. 11, Invent. Math. 118 (1994), No. 3, 411440. 42. D. Voiculescu: Free probability theory: random matrices and von Neum a n n algebras, in “Proceedings of the International Congress of Mathematicians,” Vol. 1,2 (Zurich, 1994), pp. 227-241, Birlhauser, Basel, 1995. 43. D. Voiculescu: Lectures o n free probability theory, in “Lectures on probability theory and statistics (Saint-Flour, 1998),” pp. 279-349, Springer, Berlin, 2000. 44. A. Wassermann: Coactions and Yang-Baxter equations f o r ergodic actions and subfactors, in “Operator algebras and applications,” Vol. 2,pp. 203-236, Cambridge Univ. Press, Cambridge, 1988. 45. S. L. Woronowicz: Twisted SU(2) group. An example of a noncommutative diferential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), No. 1, 117-181. 46. S. L. Woronowicz: Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), No. 4, 613-665. 47. S. L. Woronowicz: Tannaka-Krein duality f o r compact matrix pseudogroups. T w i s t e d S U ( N ) groups, Invent. Math. 93 (1988), No. 1,35-76. 48. T. Yamanouchi: O n dominancy of minimal actions of compact Kac algebras and certain automorphisms in Aut(A/A”), Math. Scand. 84 (1999), NO. 2, 297-319.
REMARKS ON THE S-FREE CONVOLUTION HIROAKI Y OSHIDA Department of Information Sciences Ochanomku University 2-1-1, Otsuka, Bunkyo, Tokyo, 112-8610 Japan E-mail: yoshidaOis.ocha.ac.jp This paper will be devoted to the study of the s-free convolution introduced by Boiejko. The s-free Fock space is constructed, which provides a concrete example of general constructions of the interacting Fock spaces. The s-free Gaussian random variables will be given by the position operators, that is, the sum of the s-creation and the s-annihilation operators, on the s-free Fock space together with the vacuum expectation. Family of non-commutative random variables will be also constructed, which have the distributions given by the s-free Poisson measures.
1
Introduction
In [7], Bozejko has introduced the r-free product of states on the free product of C*-algebras for 0 5 T 5 1, which will take the reduced free product of states of Voiculescu in [23] (see also [25]), if T = 1. In the case T = 0 then it will be reduced to the regular free (Boolean) product of states in [5] and [S]. Using the construction of the r-free product of states, he has also introduced an associative convolution of probability measures on R, called the r-free convolution, of which idea is coming from the papers [8] and [lo]. Based on the conditionally free products of states in [8], he has also introduced a large class of deformed free convolutions, called the A-convolution, in which the r-free convolution can be realized as the special case. The Adeformed moments-cumulants formula has been given explicitly in [26] by introducing the weight function on non-crossing partitions associated with the A-sequence, which can be regarded as the generalized set partition statistic on non-crossing partitions. Furthermore, the A-free Gaussian and Poisson distributions have been also investigated in [26]. The s-free convolution has been investigated as another interesting example of the A-convolution in [7]. This paper is also devoted to the study of the s-free convolution. We first recall the weight function on non-crossing partitions for the s-free deformed moments-cumulants formula and make it clear that this weight function is based on the set partition statistic rs on non-crossing partitions studied in [MI. In the subsequent section, we will construct the s-free Fock space (a certain deformed full Fock space), as well as the s-creation and the s-annihilation
412
413
operators, for the realization of non-commutative random variable with distribution given by the s-free Gaussian measure. Our construction of the s-free Fock space will provide another concrete example of general constructions of the interacting Fock spaces introduced in [2] and studied in [l],like as the t-free Fock space introduced in [lo]. Our interest is also focused on constructing non-commutative random variable with the s-free Poisson distribution in the last section. Our s-free Poisson random variable has the similar form as of the free case in [20], but the gauge operator and the identity operator should be replaced by some deformed corresponding operators. The recurrence formula for the orthogonal polynomial with respect to the s-free Poisson measure will be also given. 2
The s-Free Convolution and the Set Partition Statistic
The notion of conditionally free products of states was introduced in [8] and the corresponding convolution was also investigated deeply both in combinatorial and in analytic. By the formula for the conditionally free convolution of a pair of probability measures, the s-free convolution can be formulated as follows: Definition 2.1 Let Pc(R) be the set of compactly supported probability measures on R. For 0 5 s 5 l and v E Pc(R), we consider the s-dilation map D , ( Y ) ,that is,
mn(Ds(v)) = snmn(v) ( n 2 01, where mn(D,(v))and rn,(v) are the nth moments of the probability measures D , ( Y ) and Y, respectively. Associated with the dilation map D,,we define the s-free convolution p of the probability measures p1 and p2 by ( P , D S ( P l ) Ds(cl2)) = ( P l , D S ( P l ) ) ( c l 2 , D s ( p 2 ) ) , with the helps of the formula of the conditionally free convolution in [8]. We denote simply the above situation by p = p1 E18 p2. The positivity of such an s-free convolution has been shown already in [8] (see also (91). Applying the results of analytical investigations on the conditionally free convolution in [8], we can obtain the s-deformed R-transfom, R t )( z ) , by the following formula:
where G , ( z ) and Go,(,)(z) are the Cauchy transforms of the probability measures p and D s ( p ) ,respectively. As in the free case of Voiculescu [24], the
414
s-deformed R-transform makes the s-convolution linearlize that Pl&P2
+ RI",)(z).
( z ) = Rt,)(z)
The s-deformed R-transform R t ) ( z )can be reformulated as the recurrence formula (see [8] and [7]) that n
n
k=l
((1). L(1)+
... , t ( - k ) > O ...+t(k)=n-k
In [26], we solved such a recurrence relation for more general case of A-convolution, and gave the A-deformed moments-cumulants formula using the certain weight function on non-crossing partitions. Here we shall recall the weight function for the s-free convolution and write the s-free momentscumulants formula, which will be, of course, realized as the special case of the A-sequence { 6 n } ~ = oby putting 6, = s". Let K = { B l ,B2, . . . ,Bk} be a partition of the ordered set {1,2,. . . ,n } , that is, Bi's are non-empty and disjoint sets, of which union is { 1 , 2 , . . . ,n}. We shall call Bi E K a bZock and the number of elements in a block Bi is denoted by IBil. The block will be called singleton, if IBI = 1. The set of all partitions of the ordered set { 1 , 2 , . .. ,n} will be denoted by P({1,2,. . . ,n}) or, simply, P ( n ) . We call the partition K crossing, if there exist two blocks Bi # Bj in K and elements b 1 , b E Bi, c1,cz E Bj such that bl < c1 < b2 < c2. A partition is called non-crossing, if it is not crossing. We denote the set of all non-crossing partitions of the ordered set {1,2,. . . , n } by NC({l, 2,. . . ,n } ) or, simply, NC(n). This notion of non-crossing partition was first introduced in [13]. For more about non-crossing partitions, see, for instance [15,18,19,21].
415
For 0 5 s 5 1, we shall introduce the weight wt,(n) of a non-crossing partition K E NC({1,2,. . . ,n})as follows: Definition 2.2 Let K be a non-crossing partition in NC({l,2,. . . ,n})and let B be a block in K. If the block B is not singleton (i.e. IBI 2 2) then we put B = { b l , b 2 , . . . , b l ~ ~where }, bl < 4 < . .. < b p i , and make (IBI - 1) connections like bridges (bl,b),(b2, b 3 ) , .. . , ( b l B l - l , b l B l ) , successively. They are called arcs of the block B. It is clear that there is no pair of arcs which will cross, in a non-crossing partition. For an arc p = (c,d ) of a block B , we shall call the number ( d - c - 1) the number of inner points of the arc p. We shall give the weight to the arc p = ( c , d ) by s to the number of inner points,
= Sd-c-l. Then we define the weight of the block B , w t s ( B ) ,by the product of the weights of the arcs of the block B , that is, Wt&)
wts(B) =
wts(p). p is an arc
of B
If the block B is a singleton then we give the weight by 1. That is, for the block B = {bl, b 2 , . . . ,b l ~ l } the , weight of the block B , is defined as
if IBI = 1. Finally, we define the weight of a non-crossing partition product of the weights of the blocks in n, that is, wt,(n) =
n
K,
wts(n), by the
WtJB).
BET
Remark 2.3 For a block B = {bl, weight wts(B) can be rewritten as
b2,.
.. ,b p ~ } , it is easy to see that the
wt,(B) = ~ ( b l B l - b l - ( l B l - l ) ) , which is valid even for the case of a singleton by regarding b l ~ l= b l . Thus, for a non-crossing partition K = {Bl, Bz, .. . ,B k } of k blocks, we have k
Wt,(r)=
I'I
s(&-fi-(lBil-1))
i=l
- ,((Xi[.)-(Xi fi)-n+k)
416
where fi and .ti are the first and the last elements of the block Bi, respectively. Example 2.4 (a) r = {{1,2,6}, {3,5}, (4));
A 1
2
3
4
5
wt,(r) = s4.
wt,({1,2,6}) = s 3 , wt,({3,5}) = s1, wt,({4}) = 1.
wts(r) = s7.
wt,({1,7}) = s 5 , wt,({2,5,6)) = s2, wt,({3,4}) = s o . (4
7r
6
= {{1,2), (31, (4,576,711;
f 1
i
2
3
4
l
wt,((1,2}) = so, wt,({3}) = 1, wt,({4,5,6,7}) = SO.
5
6
m
7
wt,(r) = 1.
Then the s-free deformed moments-cumulants formula can be obtained by using the above weight (see [26]) that .Ir€NC(n)
BET
The A-free Gaussian and Poisson distributions, and the recurrence relations for their moments were investigated in [26]. Here we shall recall them in the s-free case. The standard s-free Gaussian distribution (the central limit measure with respect to the s-free convolution) p (i ’ can be characterized by the s-deformed R-transform as
from which, on the moments of p!), we have
417
where AfCz(2m) denotes the set of all non-crossing pair-partitions of 2m elements. This formula for the moments, of course, can be also derived by putting a:) = 1 and a!) = 0 (k # 2) in the s-free deformed momentscumulants formula. Furthermore, the orthogonal polynomial {T?)(z)}with respect to the measure p k ) has been obtained in [7] that {T?)(z)}satisfy the recurrence relation,
T,(")(z) = 1, T,(")(Z)= z,
~ , f $ ~(z)= z ~ ? (z) ) - s2n-z~:j1
(z)
(n 2 1).
It is natural to consider that the s-free Poisson distribution p t ) of parameter A should be characterized as the distribution all of which s-free cumulants equal to A, just as for the usual and the free cases. Hence the s-deformed R-series for the s-free Poisson distribution R(;?, (z) can be given by PP
which implies, on the nth moments of the s-free Poisson distribution p t ) , rnn(p?)) =
c
Wt"(7r)A'y
rENC(n)
where I7rI stands the number of the blocks in the non-crossing partition 7r. Of course, the above formula can be obtained by putting a?) = A (k 2 1) in the s-free deformed moments-cumulants formula. The recurrence relation for the orthogonal polynomials with respect to the s-free Poisson distribution p g ) will be presented later in this paper (see Theorem 4.7). Remark 2.5 Before ending this section, we would like to give an important remark that the s-free convolution should be based on the set partition statistic T S on non-crossing partitions investigated in [18]. Now we shall recall the definition of the set partition statistic T S which is arisen from certain inversions. Let T = {B1, B2, . .. ,Bk} be a partition of the ordered set { 1,2,. . . ,n}, where its blocks Bi's are indexed in increasing order of their first (minimum) elements. A partition may be represented by its restricted growth function introduced in [14], w : {1,2 ,... , n } -+ {1,2 ,... ,n},
418
where w ( i ) = the index of the block of ?r which contains i. We write the restricted growth function for a partition a by the word w(a) = wlw2 --.wn, where wi = w(i). Thus, for example, a = {{1,7},{2,5,6},{3,4}} has the restricted growth function W ( B ) = 1 2 3 3 2 2 1. Given a partition a E P(n), let w(a) = w 1 w 2 . . . w n be its restricted growth function. Then the set partition statistic rs(a) is defined by n
w(a) = p { w j
I wj < wi, j > i}.
i= I
It has been proven in [18, Lemma 2.11 that, for a non-crossing partition K = {Bi, B2,. . . ,B k } E n/C(n) of k blocks, the set partition statistics rs(a) can be written as k
k
i=l
i=l
where fi and Ci are the first and the last elements in the block Bi as we have put before. Combining with Remark 2.3, it follows that wts(a) = s7.474, namely, the s-free deformed moments-cumulants formula, thus, the s-free convolution, is based on the set partition statistic rs on non-crossing partitions.
3 The s-Free Fock Space and the s-Free Gaussian Random Variables In this section, we are going to construct the family of operators, of which distributions will be given by the s-free Gaussian distributions. Our construction depends on a certain deformation of a full Fock space, and of creation and annihilation operators there. Definition 3.1 For 0 5 s 5 1 and given a Hilbert space 'U with the scalar product ( . 1 . ), the s-free Fock space is defined as the full Fock space, W
FS(3c) = (CR eT3 @ 3CBn, n=l
completed with respect to the following scalar product: n
(51
8 52 8 . . . B ~n
I 71 8 72 8 . . ' 871,). = bn,m sn(n-l)n(<j 1~ j j=1
(0I
w,= 1,
) ,
419
where $2 is the distinguished unit vector called the vacuum vector. For a vector J E X,we define the s-creation operator at(<) on .F,(X)by a!(<)
a!("
8 * * * 8 Jn = J 8 J1
52
8 ... 8 Jn
= J,
and the s-annihilation operator a(<) on
F,(X)by
as(<)
= (Jl as(<)fl = 0,
as(J)<1
( n L 11,
(n L 21,
I J)Q
where < I , & , .. . ,Jn are arbitrary vectors in X. By definition, it can be easily obtained that the operators a:(<) and a,(<) are bounded operators with norms
llad(J~11= Ilas(E)II = IIJIIProposition 3.2 The operators a!(<) and a,(J) are adjoint each other with respect to the scalar product ( . 1 . ),. That is, for any vectors z,y E 3,(31), and for any J E X,we have (as(+
I Y),
= I.(
at,(J)y),
.
420
and 8 <2 8 ... 8 t n I a!(<)~z8 773 8 ..*8 ~ = (
(
n
)
~
<
n
- Sn(n-l)(
1%).
j=2
I In the rest of this paper, we shall write a!(<)as a,(<)*. Remark 3.3 The above construction of the s-free Fock space is still a concrete example of the interacting Fock spaces studied in [2] and [l]like as the t-free Fock space in [lo] but, of course, the (An)-sequence is not the same for t-free case. In the general construction of the interacting Fock spaces, take A, = d n - l ) ( n 2 1) then we obtain scalar product on our s-free Fock space, as well as the s-creation and the s-annihilation operators (cf. [lo]). We shall define the vacuum state cp on all bounded operators b on the Fock space FS(31) as cp(b) = ( b Q I and, for
< E 31, the position operator as
ws,
9s(<) = as(<)+ as(<)*
which is, of course, a bounded self-adjoint operator. Now we shall see that the position operator gives our desired s-free Gaussian random variable. Theorem 3.4 For a unit vector E 31, the distribution of the position operator gS(<) with respect to the vacuum state cp is given b y the standard s-free Gaussian distribution p p ) . That is, the operator gs(<) (11<11 = 1) is the standard Gaussian random variable.
<
PROOF.As we have mentioned in the previous section that the orthogonal polynomial for the standard s-free Gaussian distribution p p ) satisfy the recurrence relations:
{
T,(")(2) = 1, T,(")(z) = 2,
T Z 1(z)= ZTp (z) - s 2 n - W n-I(Z) s)
We shall see by induction on n that TP(9s(<)W=
Pn .(
2 01,
( n 2 1).
42 1
where T?) is the monic polynomial of degree n defined by the above recurrence relation, and means the vacuum vector R. It is clear that <@'
T,(")(g,(<))R = 1R = R, Assume T,(")(g,(<))R =
T,(")(g,(<))R = g R = <.
cBk for k 5 n. Then it follows that
T2,(gs(<))n = g s ( E ) T P ( g s ( O ) - s2"-2T?-'1(%(r)) = g,(<)<@'" - S2n--2 I Wn-1)
= (a,(<)<@" +a,(<)*<@") - s2*-2<@(n--1)
- s2n-2
s@(n-1) + <@(n+l)
<
# n then
The above facts ensure that if m (T~)(gs(<))T~)(gs(<))R
<
- S2n-2 @(n-1) = @ ( n + l )
I
Q
S
= (T%s(<))R I T P ( g s ( 0 ) Q ) s = (<@" I pn), = 0,
because the element gs(<) is self-adjoint with respect to the inner product ( . I . ),. This means that for all polynomial f
Hence the distribution of the operator gs([)
is given by the s-free Gaussian
I
distribution p!). 4
Model of the s-Free Poisson Random Variables
In this section, we are going to construct the s-free Poisson random variable on the s-free Fock space F,(X).For this purpose, we shall define two more special operators on F,(X). Definition 4.1 We define the operator n, by n,R = 0, n,
(TI_>
I),
and the operator k, by
k,R = R, k,& @ E2 @
. - @ E,,
= sn
where
(n _> l),
422
We shall list some properties on the operators n, and k, without proofs because they are direct consequences of the definitions. Proposition 4.2 (a) For 0 5 s 5 1, the operators k , and ns are bounded with norms llnsll = Ilks11 = 1.
< and q in U , we have the relation,
(b) For vectors
as(v)as(O*= (E 117) k,2.
(c) In the case s = 1, kl
= l F , ( X ) , n1 = PFs(X)0@R,
where 1F,(3t)is the identity operator on F,(R), and PF,(X)e@n is the projection onto the closed subspace F'(R) 8 (CR Now we shall introduce the special operators on Fs(3t). For a vector [ in U and X > 0, we put the bounded self-adjoint operator
PA<; = ns + &(
+~ k ,
+ d%z,(<)* + Xk,,
which will give our desired s-free Poisson random variable in the case ll
and consider the vacuum expectation in a term wise. (&,(S)*), and (Xk,) We call a product of operators (n,), (Act,(<)), admissible, if it has non-zero vacuum expectation. Although (Xk,) commutes with ( f i a , ( < ) ) and ( f i a , ( < ) * ) , we shall treat them as non-commutative operators. That is, for instance, the products (fia,(<))(fia,(<)*)(Xk,)and (Xk,)(fia,(f))(fia,(<)*) should be distinguished. For a given product of length n y = z,z,-1
* *.
qz]
where zk E
{(ns),
(fias(c)),
(A%(<)*), (%)} (k=
we put the sets as
Cy = { k I zk = ( h ~ s ( < ) * ) } , A y = { k I zk = (&(<))}, Ky= { k I zk = (Xk,)}. Ny= { k 1 zk = (ns)},
,n),
423
We should note that the factors are labeled from the right. We shall define the level of the kth factor Z k , !(k) (1 5 k 5 n),as k-1
W )= 0, W )= CX(i)(k 2 21, j=1
where x(j) is the step function given by
{
x(j) =
if j E C,, if j E A,, if j E Ny U K,.
1, -1, 0,
Then it can be seen by rather routine argument that the monomial y is admissible if and only if the levels l ( k ) (1 k _< n) satisfy the following Motzkin path conditions:
<
n
t(k) 2 0 for 1 5 k _< n, t ( k ) 2 1 if k E IVY, and
c x ( j )= 0. j=1
If the monomial y is an admissible product then the level l ( k ) reflects the fact that (Zk-IZb-2
’. .Z1) a E (Cc@e(k),
where tS0means the vacuum vector 0. It should be aware of that the operators (fiu,(C)*) and ( A u s ( [ ) )make a complete parenthesization in an admissible product. Thus we can have the non-crossing partition ~ ( y in ) NC({1 , 2 , . . . ,n } ) associated with an admissible product y of length n as follows: Consider the sets C,, A,, N,, and K, as above. Each element in the set K, makes a singleton. The elements in the sets C, and A, will be used for the first and the last elements of blocks, respectively. The elements in the set IVY will be used for intermediate elements of blocks. Of course, it will be automatically determined by non-crossingness that, in which block each of elements in Ny should be contained, because the elements in the sets C, and A, are completely parenthesized. Example 4.3 (a) For the admissible product of length 6 , Yo
------
= (fias(t>> (fias(t)) 28
25
( ~ k s(JT;as(<>*) ) (ns)(JT;us(t)*), 24
23
22
21
we have C,, = {1,3}, Aya = { 5 , 6 ) , Nya= (21, and Kga = (4). Thus we obtain the non-crossing partition, 4 Y O )
= {{1,2,61, (3, 51, (4)).
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(b) For the admissible product of length 7,
= 4. Thus we we have Cyb= {1,2,3}, AYb= {4,6,7}, Nyb = { 5 } , and Ky,, obtain the non-crossing partition,
In order to evaluate the vacuum expectation of an admissible product, we shall use the cards arrangement, which is similar technique as in [12] for juggling patterns but we have to prepare different kinds of cards. [The creation cards] We prepare the cards Ci (i = 0,1,2,. . .) for the s-creation operator a s ( t ) * .The card Ci has i inflow lines from the left and (i 1) outflow lines to the right, where one new line starts from the middle point on the ground level. For each j 2 1, the inflow line of the j t h level goes through out to the ( j 1)st level without any crossing. We call the card Ci the creation card of level i. Each of the creation cards has the weight 1.
+
+
Level 0
Level 1
Level 2
...
Level i
...
[The annihilation cards] We shall make the cards Ai (i = 1,2,3,. . .) for the s-annihilation operator a s ( [ ) . The card Ai has i inflow lines from the left and ( i - 1) outflow lines to the right. On the card Ai, only the line of the lowest level goes down to the middle point on the ground level and will be annihilated. For each j 2 2, the inflow line of the j t h level goes through out to the ( j - 1)st level without any crossing. We call the card Ai the annihilation cards of level i . We shall give the weight to the card Ai by s to twice the
425
number of the through out lines, that is, s2(i-1). Level 1
Level 2
Level 3
e .
M 1. A3
A2
..
Level i
Ai
i-1
...
Remark 4.4 The creation card Ci represents the relation,
<
(i 2 01,
a,(<) * @i -
where we regard = R and the number of lines corresponds to the power of tensor product. Similarly, the annihilation card Ai represents the relation, <@'
as(<)<
@i
- 2(i-1) -s
<
@(i-1)
(i 2 1).
Since a,(<)R = 0, thus the annihilation card of level 0 is not available. Here we shall prepare some more cards which will correspond to the operators n, and k, introduced in this section. [The intermediate cards] We consider the cards Ni (i = 1 , 2 , 3 , .. . ) for the operator n,. The card Ni has i inflow lines and i, the same number of, outflow lines. Only the line of the lowest level goes down to the middle point on the ground and continues its flow as the lowest line again. The rest of inflow lines will keep their levels. We call the card Ni the intermediate card of level i. We shall give the weight to the card Ni by s to the number of the directly through out lines, di-l). Level 1
Level 2
Level 3
.-.
Level i
...
S
... The intermediate card N; represents the relation, n,<@i
= &l)<@i
(i 2 11,
...
426
and the intermediate card of level 0 is not available because n,0 = 0. [The singleton cards] Furthermore, we shall make the singleton cards Ki (i = 0 , 1 , 2 , . . ,) for the operator k,. The card Ki has i horizontally parallel lines and the short pole at the middle point on the ground. We call the card Ki the singleton card of level i and give the weight by s to the number of through out lines, s'. Level 0 Level 1 Level 2 -.. Level i ...
1
n
j
KO
Ki
i
..
Of course, the singleton card Ki represents the relation,
k,tai = sitai (i 2 l), and the card KO means k,0 = 0. Depending on the factors in an admissible product y, we will arrange the cards along with the following rule: If k E C,, that is, if Z k = ( f i u , ( < ) * ) then we will put the creation card of level C(k) with the fi-multiplicated weight at the kth position. If k E A,, that is, if z k = (&,(<)) then we will put the annihilation card of level C(k) at the kth position. The weights should be also multiplicated by 6, If k E N,, that is, if z k = (n,)then we will use the intermediate card of level C(k) with the original weight. If k E K,, that is, if Zk = (Xk,) then the singleton card of level C(k) with the A-multiplicated weight will be put at the kth position. Then the non-crossing partition ~ ( y can ) be obtained by connected lines because the arcs of ~ ( y are ) naturally drawn on the cards in the arrangement. Now we shall see that the vacuum expectation of an admissible product y can be given by 4 9 ) = wts('.(y))X'*(Y)',
where wt, is the weight function on non-crossing partitions introduced in the section 2 and In(y)I denotes the number of the blocks in the non-crossing partition ~ ( y ) . First we recall again that the operators (&,(()*) and ( f i u , ( < ) ) make a complete parenthesization. Corresponding this parenthesization, we can find
427
pairs of the creation card Ci and the annihilation card A;+1, on which the same number of through out lines are drawn. We have given the weight to the annihilation card by s to the twice number of through out lines on the card. Secondary, we also note that the weights of the intermediate card and the singleton card have been given by s to the number of directly through out lines on cards. Furthermore, in the cards arrangement, we have given the multiplicated the weight for a singleton card by A, and for the creation and the annihilation cards by 6.
@Ii
tt
i
Ai+i
Ci
From the above observation, it can be said that the product of the weights of the cards appeared in the cards arrangements should be given by p(A)#CY
( A ) " " Y
f K Y
=,ypY/2+#AY/2+#KYI
= ,NA(#CV+#KYI
where N is the sum of all the number of through out lines on the cards used in the arrangement. Of course, #Cy, #A,, and #Kystand for the cardinalities of the sets C,, A,, and K,, respectively. By the nature of the figure obtained from the cards arrangement, it can be ensured that the number N is the same as the sum of the numbers of inner points for all the arcs in the non-crossing partition n(y), and that #Ky+#Cy= In(y)1, which imply that cp(y) = wt,(n(y))A'"(Y)'.
Example 4.5 (a) For the admissible product yo in the previous example, we have the following card arrangement: A
1
1
2
(co)
(Nil
JT;
3 (Ci)
s2A
4 (Kz)
S 2 d i
5 ("2)
428
The product of the cards is given by s4X3. The corresponding non-crossing partition
the weight wt,(n(ya)) = s4 as we have shown in Example 2.4. (b) For the admissible product Yb in the previous example, we have the following card arrangement:
(CO)
(Cl)
(CZ)
(-43)
(N2)
(-42)
(-41)
The product of the cards is given by s7X3. The corresponding non-crossing part it ion n(yb) = {{1,7}, {2,5,6), (37 4))
has the weight Wt,(r(yb)) = s7 as we have shown in Example 2.4. Conversely, given a non-crossing partition n E N C ( { I , 2 , . . . ,n } ) ,we can make the admissible product y(a) of length n by the following manner: If {k} is a singleton in the partition n then we put the operator (Xlc,) as the kth factor. If k is the first (resp. last) element of blocks then we use the operator (&,(<)*) (resp. (&,(<)) ) as the kth factor. For the rest case, that is, k is an intermediate element of a block, we adopt the operator (n,) as the kth factor in our product. Using the cards arrangement again, it is easy to see that such a product y(n) has a non-zero vacuum expectation, which can be evaluated as the product of the weights of the cards appeared in the arrangement. There is, hence, one-to-one correspondence between admissible products of length n and the non-crossing partitions of n elements NC(n). Now we have reached that admissible product y of length n
*"(
n)
The right hand side in the above equation is nothing else but the nth moment of the s-free Poisson distribution of parameter A, thus we have
429
Theorem 4.6 For a vector of the operator
< E 3.1 with 11<11
p S ( t ;A) = n ,
= 1 and X
> 0, the distribution
+ ILL,(<) + dL(<)* +~ k ,
with respect to the uacuum state is given by the s-free Poisson distribution of parameter A. On the orthogonal polynomials for the s-freePoisson distribution, we can obtain the following theorem by the similar proof as in [17] (see also [4]): Theorem 4.7 The orthogonal polynomials {C?'(x)} for the s-free Poisson distribution of parameter X are determined by the following recurrence relation:
PROOF.As we have shown above, the s-free Poisson random variable of parameter X is given by p , ( < ; X ) , with a unit vector [ E 3.1 and X > 0. We simply write the operator p s ( < ;A) by p , then it suffices to show that ( n 2 01,
c?)(~)R =
where [@'O = R. We shall show this by induction on n. It is clear that
CiS'(p)R = 1 R = R, Assume Cf'(p)R=
c:;,(P)R
c y ( p ) R = pR - X1R = A<.
G t s kfor k 5 n.
+ 1 ) l ) cp(p)- s2("-I)Xc:21(p))
= ( ( P - sn-'(sX
+q
= p ( f i C @ n ) - s"-l(sX = (n,
Then we obtain that
fipn
R
- ,2(n-1)X-<@(n-1)
+ L i u , (t)+ f i u , (<)* + ~k,) -S
- S n - l f i < ~ n - s2(n-1),&Tit@(n-1)
n m < @ n
= p-lfi[@n
+s2(n-1),/5ipn-1
+,&Tipn+1
-s = -pn+1.
n
p
+S
n m < @ n
p
-p-1fipn
n
-S2(n--1)mpn--l)
430
# n then (cy@)ck)@)n 1 a), = (cF(p)nI C?)@)Q)S
Hence we have that if m
=(
f i p m
I fip,)s= 0. I
Remark 4.8 In the case s = 1, the recurrence relations of the Orthogonal polynomials {Cil)(x)} becomes
( co: :
(z) = 1,
c,(1) (2)= z-A,
Cn+,(z) = (z- (A + 1)) C?)(z) - Ac:2,(z)
(n 2 l ) ,
which is, of course, ones for the free Poisson distribution (see [3,16,25]), and of constant coefficients type in [ll]. In the case s = 0, the recurrence relation for the polynomials {C,?'(z)} is finite that
Cf)(z) = 1,
c,( 0 ) (z)= z 2 - (A + l)x.
c,( 0 )(2)= $ - A ,
The corresponding orthogonalizing probability measure is given by 1 A Pjp'(dt) = --s(O) -h(l +A), 1+A 1+A where 6 ( t ) denotes the Dirac unit mass at t , which is the Boolean Poisson distribution (see [22]). According to the recurrence relation (more precisely, the Jacobi parameters) for the orthogonal polynomials with respect to the s-free Poisson distribution, we can construct the model of the s-free Poisson random variable by the weighted shift operators on L2(Z>o) as follows: For 0 5 s 5 1 and A > 0, we s h d define the weighted shift S, by
+
(n 2 01, where {en},=, is the completely orthonormal system in L 2 ( Z 2 0 ) . The adjoint operator Sf of S, is determined by Ssen = Sn+'dien+i
00
SZeo = 0,
SZe, = sndXen--l (n 2 1).
Here we consider the Hilbert space L 2 ( Z-> 0 ) is endowed with the canonical inner product. Furthermore we shall define the diagonal operator D, by
+
Dsen = sn-l(sA l)e, (n 2 1). DseO = Aeo, Then it is straightforward to obtain the following theorem by routine argument:
43 1
Theorem 4.9 The distribution of the self-adjoint operator
+ S,* + D, on .t2(ZLo) with respect to the vector state q5( - ) = S,
eoleo) is the s-free Poisson distribution. We can actually work with the matrix of the truncation of the operator (S, + S,* D,)with respect t o the canonical basis, in calculations of the moments for finite orders. This matrix is the tridiagonal of the form ( a
+
x ifi
sfi
0
(SX+l)
s 2 f i
2 6
s(sX+l)
s 3 6
Remark 4.10 We shall here give the final remark on our s-free Poisson random variable introduced in this paper. The s-free Poisson random variable, pg(t;A> = n,
+ JJ;a,(t>+ Jxas(t)*+ ~ k , ,
on the s-free Fock space F,(lfl) has the similar form as of the free Poisson random variable on the full Fock space in [20], and of the q-Poisson random variable on the q-Fock space in [17]. In the s-free Poisson random variable, we have used the operator n,,a deformed projection onto the closed subspace F,(lfl) 8 CR, instead of the gauge operator in the free case, and of the qnumber operator in the q-case. Simultaneously,the identity operator has been replaced by the deformed identity operator k, which, however, still commutes with the Gaussian part.
References
1. L. Accardi and M. Bozejko: Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 663-670.
432
2. L. Accardi, Y.G. Lu, and I. Volovich: Interacting Fock spaces and Halbert module extensions of the Heisenberg commutation relations, Publications of IIAS, kyoto, 1997. 3. M. Akiyama and H. Yoshida: The distributions for linear combinations of a free family of projections and their orthogonal polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 627-643. 4. M. Anshelevich: Partition-dependent stochastic measures and q-deformed cumuhnts, M S R I preprint, Berkeley, 2001. 5. M. Boiejko: Positive definite functions on the free group and noncommutative Rieszproduct, Boll. Un. Mat. Ital. A(6) 5 (1986), 13-21. 6. M. Bozejko: Uniformly bounded representation of free groups, J . Reine. Angew. Math. 377 (1997), 170-186. 7. M. Boiejko: Deformed free probability of Voiculescu, preprint, 2001. 8. M. Boiejko, M. Leinert, and R. Speicher: Convolution and limit theorems for conditionally free random variables, Pacific J . Math. 175 (1996), 357388. 9. M. Boiejko and R. Speicher: $ independent and symmetrized white noise, in “Quantum Probability and Related Topics, VI (L. Accardi Ed.),” pp. 219-236 World Scientific, Singapole, 1991. 10. M. Bozejko and J. Wysoczanski: Remarks on t-transformations of measures and convolutions, Ann. Inst. H . Poincax6 Probab. Statist. 37 (2001), 737-761. 11. J. M. Cohen and A. R. Trenholme: Orthogonal polynomials with constant recursion formula and an application to harmonic analysis, J . Funct. Anal. 59 (1984), 175-184. 12. R. Ehrenborg and M. Readdy: Juggling and application to q-analogues, Discrete Math. 157 (1996), 107-125. 13. G . Kreweras: Sur les partitions non-croise‘es d’un cycle, Discr. Math. 1 (1972), 333-350. 14. S. C. Milne: Restricted growth functions, rank row matchings of partition lattices and q-Stirling numbers, Adv. Math. 43 (1982), 173-196. 15. A. Nica: R-transforms of free joint distributions and non-crossing partitions, J . Funct. Anal. 135 (1996), 271-296. 16. N. Saitoh and H. Yoshida: A q-deformed Poisson distribution based on orthogonal polynomials, J . Phys. A: Math. Gen. 33 (2000), 1435-1444. 17. N . Saitoh and H. Yoshida: q-deformed Poisson random variables o n qFock space, J . Math. Phys. 41 (2000), 5767-5772. 18. R. Simion: Combinatorial Statistics on non-crossing partitions, J . Combin. Theory Ser. A 66 (1994), 270-301. 19. R. Simion and D. Ullman: O n the structure of the lattice of noncrossing
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partitions, Discr. Math. 98 (1991), 193-206. 20. R. Speicher: A new example of ’Independence’ and ’White Noise’, Probab. Theory Relat. Fields 84 (1990), 141-159. 21. R. Speicher: Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611-628. 22. R. Speicher and R. Woroudi: Boolean convolution, in “Free Probability Theory (D. V. Voiculescu Ed.),” pp. 267-280, Fields Inst. Commun. 12. Providence RI: Amer. Math. SOC.,1997. 23. D. Voiculescu: Symmetries of some reduced free product C*-algebras, in “Operator algebras and Their Connections with Topology and Ergodic Theory,” pp. 556-588, Lect. Notes in Math. Vol. 1132, BerlinHeidelberg-New York, Springer, 1985, . 24. D. Voiculescu: Addition of certain non-commutative random variables, J. Funct. Anal. 66 (1986), 323-346. 25. D. Voiculescu, K. Dykema, and A. Nica: “Ree random variables,” CMR Monograph Series 1,Providence RI: Amer. Math. SOC.,1992. 26. H. Yoshida: The weight function on non-crossing partitions for the Aconvolution, preprint, 2002.
Memorandum: The titles of the past RIMS workshops
1. White noise analysis and applications Organizer: Nobuaki Obata (Nagoya University) 1993.3.22-24. 2. White noise analysis and quantum probability Organizer: Nobuaki Obata (Nagoya University) 1993.12.6-9. RIMS Kbkyiiroku Vol. 874. 3. Analysis of operators on Gaussian space and quantum probability theory Organizer: Nobuaki Obata (Nagoya University) 1995. 3.29-31. RIMS Kbkyiiroku Vol. 923. 4. Quantum stochastic analysis and related fields Organizer: Nobuaki Obata (Nagoya University) 1995.11.27-29. RIMS Kbkyiiroku Vol. 957. 5. Quantum stochastic analysis and related fields Organizer: Taku Matsui (Tokyo Metropolitan University) 1996.10.23-25. 6. Recent trends in infinite dimensional non-commutative analysis Organizer: Taku Matsui (Kyushu University) 1997.10.14-17. RIMS KGkyiiroku Vol. 1035. 7. Development of infinite-dimensional noncommutative analysis Organizer: Akihito Hora (Okayama University) 1998.10.14-16. RIMS KbkyGroku Vol. 1099. 8. New development of infinite-dimensional analysis and quantum probability Organizer: Akihito Hora (Okayama University) 1999.9.16-17. RIMS Kbkyiiroku Vol. 1139. 9. Infinite dimensional analysis and quantum probability theory Organizer: Nobuaki Obata (Nagoya University) 2000.11.20-22. RIMS Kbkyiiroku Vol. 1227. 10. Trends in infinite dimensional analysis and quantum probability Organizer: Nobuaki Obata (Tohoku University) 2001.11.20-22. RIMS Kbkyiiroku Vol. 1278.
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Author Index
Accardi, L. 192 Arai, A. 1 Arimitsu, T. 206
Lhandre, R. 329
Belavkin, V.P. 225
Nahni, M. 192
Fagnola, F. 51 Franz, U. 254
Obata, N. 143 Obata, N. 360
Hashimoto, Y. 275 Hiai, F. 97 Hida, T. 288 Hirai, E. 296 Hirai, T. 296 Hora, A. 318
Saito, K. 360 Shimada, Y. 346 Suzuki, M. 374
Matsui, T. 346
Ueda, Y. 388 Yoshida, H. 412
Ji, U.C. 143
437
