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 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_- 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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92. T. Okamoto and K. Yajima: Complex scaling technique in nonrelativistic massive QED, Ann. Inst. Henri Poincar6 42 (1985), 311-327. 93. W. Pauli and M. Fierz: Zur Theorie der Emission langwelliger Lichtquanten, Nuovo Cimento 15 (1938), 167-188. 94. M. Reed and B. Simon: “Methods of Modern Mathematical Physics I Functional Analysis,” Academic Press, New York, 1972. 95. M. Reed and B. Simon: “Methods of Modern Mathematical Physics 11: Fourier Analysis, Self-Adjointness,” Academic Press, New York, 1975. 96. M. Reed and B. Simon: “Methods of Modern Mathematical Physics 111: Scattering Theory,” Academic Press, New York, 1979. 97. M. Reed and B. Simon: “Methods of Modern Mathematical Physics IV: Analysis of Operators,” Academic Press, New York, 1978. 98. B. Simon: “The P(4)z Euclidean (Quantum) Field Theory,” Princeton University Press, Princeton, NJ, 1974. 99. B. Simon: Universal diamagnetism of spinless boson systems, Phys. Rev. Lett. 36 (1976), 804-806. 100. B. Simon: “Functional Integration and Quantum Physics,” Academic Press, New York, 1979. 101. B. Simon and R. Hqkgh-Krohn: Hypercontractive semigroups and twodimensional self-coupled Bose fields, J. h n c t . Anal. 9 (1972), 121-180. 102. E. Skibsted: Spectral analysis of N-body s y s t e m coupled t o a bosonic field, Rev. Math. Phys. 7 (1998), 989-1026. 103. H. Spohn: Ground state(s) of the spin-boson Hamiltonian, Commun. Math. Phys. 123 (1989), 277-304. 104. H. Spohn: Asymptotic completeness for Rayleigh scattering, J. Math. Phys. 38 (1997), 2281-2296. 105. H. Spohn: Ground state of a quantum particle coupled to a scalar Bose field, Lett. Math. Phys. 44 (1998), 9-16. 106. H. Spohn, R. Stiickl and W. Wreszinski: Localization for the spin Jboson Hamiltonian, Ann. Inst. Henri Poincar6 53 (1990), 225-244. 107. B. Thaller: “The Dirac Equation,” Springer, Berlin, Heidelberg, New York 1992, 108. T. A. Welton: Some observable effects of the quantum mechanical fluetuations of the electromagnetic field, Phys. Rev. 74 (1948), 1157-1167.
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 Col121fp(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 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 < ~ 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,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) 11z110on 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 ~ ) ~_
(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 = +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.
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References 1. L. Accardi: O n the quantum Feynman-Kac formula, Rend. Sem. Mat. Fis. Milano XLVIII (1978), 135-180. 2. L. Accardi, A. F‘rigerio and J. T. Lewis: Quantum stochastic processes, Publ. R.I.M.S. Kyoto Univ. 18 (1982), 97-133. 3. L. Accardi, A. Frigerio and Y. G. Lu: The weak coupling limit as a quantum functional central limit, Comm. Math. Phys. 131 (1990), 537570. 4. L. Accardi, S. V. Kozyrev: O n the structure of Markov flows, in “ h e versibility, Probability and Complexity (Les Treilles/Clausthal, 1999),” Chaos Solitons Fractals 12 (2001), 2639-2655. 5. L. Accardi, Y. G. Lu and I. Volovich: “Quantum Theory and its Stochastic Limit,” Springer-Verlag, Berlin, 2002. 6. R. Alicki and K. Lendi: “Quantum Dynamical Semigroups and Applications,” Lect. Notes in Phys. Vol. 286, Springer-Verlag, Berlin, 1987. 7. A. Barchielli: Quantum stochastic differential equations: a n application to the electron shelving eflect, J. Phys. A 20 (1987), 634145355. 8. A. Barchielli: Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics, Quantum Opt. 2 (1990), 42341. 9. A. Barchielli and G. Lupieri: Quantum stochastic models of two-level atoms and electromagnetic cross sections, J. Math. Phys. 41 (ZOOO), 7181-7205. 10. C. Barnett, R.F. Streater and I.F. Wilde: Quaszfiee quantum stochastic integrals f o r the CAR and CCR, J. F’unct. Anal. 52 (1983), 1 9 4 7 . 11. V.P. Belavkin: Quantum stochastic positive evolutions: characterization, construction, dilation, Comm. Math. Phys. 184 (1997), 533-566. 12. V.P. Belavkin: Measurement, filtering and control in quantum open dynamical systems, Rep. Math. Phys. 43 (1999), 405-425. 13. A. Ben Ghorbal, H. Dogan and M. Schiirmann: Non-commutative stochastic integration for Boolean-adapted processes, preprint. 14. B. V. Rajarama Bhat and M. Skeide: Tensor product systems of Hilbert modules and dilations of completely positave semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519-575. 15. A. M. Chebotarev and F. Fagnola: Suficient conditions for conservativity of minimal quantum dynamical semigroups, J. F’unct. Anal. 153 (1998), 382-404. 16. E. B. Davies: “Quantum Theory of Open Systems,” Academic Press, London-New York, 1976.
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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@,112< I. -
with the inner product ( 6 , ~ ) := CgEP, 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 )
%(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
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 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 , 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
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 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,(
-.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.
0 there exists a probability measure plrz on E* uniquely specified by
(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 %(.
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( 0, V," 5 Crt with
For nl
X;'
.. . X i d . 1 ,
ek 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>= $ (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)
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
251
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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4. G. J . Milburn and D. E. Walls: Phys. Rev. A 30 (1984), 56-60. 5. H. Carmichael: “Open Systems in Quantum Optics,” Lect. Notes in Phys. Vol. 18, Springer-Verlag, 1986. 6. P. Zoller, M. A. Marte and D. F. Walls: Phys. Rev. A 35 (1987), 198207. 7. A. Barchielli: J. Phys. A: Math. Gen., 20 (1987), 6341-6355. 8. C. A. Holmes, G. J . Milburn and D. F. Walls: Phys. Rev. A 39(1989), 2493-2501. 9. M. Ueda: Phys. Rev. A 41 (1990), 3875-3890. 10. G. Milburn and Gagen: Phys. Rev. A 46 (1992), 1578. 11. Ph. Blanchard and A. Jadczyk: Event-enhanced-quantum theory and piecewise deterministic dynamics, Annalen der Physik, 4 (1995), 583599. 12. Ph. Blanchard and A. Jadchyk: Relativistic quantum events, Found. Phys. 26 (1996), 1669-1681. 13. Ph. Blanchard and A. Jadchyk: Time and events, Int. J. Theor. Phys. 37 (1998), 227-233. 14. V. P. Belavkin: in “Modelling and Control of Systems (A. BlaquiBre, Ed.),” pp. 245-265, Lect. Notes in Control and Information Sciences Vol. 121, Springer-Verlag, 1988. 15. V. P. Belavkin: A continuous counting observation and posterior quantum dynamics, J. Phys. A: Math. Gen. 22 (1989), L1109-L1114.
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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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quantum measurement projection postulate, Int. J. Theor. Phys. 35 (1996), 2215-2228. C. W. Gardiner: “Quantum Noise,” Springer-Verlag, 1991. V. P. Belavkin: Quantum stochastic calculus and quantum nonlinear filtering, J. Multivariate Anal. 42 (1992), 171-201. V. P. Belavkin and P. Staszewski: Rep. Math. Phys., 29 (1991), 213. V. P. Belavkin and 0. Melsheimer: A stochastic Humiltonian approach f o r quantum jumps, spontaneous localizations, and continuous trajectories, Quantum and Semiclassical Optics 8 (1996), 167-187. A. Barchielli and V. P. Belavkin: Measurements continuous in time and a posteriori states in quantum mechanics, J. Phys. A: Math. Gen. 24 (1991), 1495-1514. I. Persival: “Quantum State Diffusion,” Cambridge UP, 1999. V. P. Belavkin: A dynamical theory of quantum continuous measurements and spontaneous localizations, Russ. J. Math. Phys. 3 (1995), 3-24. V. P. Belavkin and P. Staszewski: Quantum stochastic diflerential equation for unstable systems, J. Math. Phys. 41 (2000), 7220-7233. V. P. Belavkin: In “Evolution Equations and Their Applications in Physical and Life Sciences (G. Lumer and L. Weis Eds.),” pp. 311-327, Lect. Notes in Pure and Appl. Math. Vol. 215, Marcel Dekker, Inc., 2001. V. P. Belavkin and V. N. Kolokoltsov: Stochastic evolutions as boundary value problems, RIMS Kokyuroku 1227 (2001), 83-95. E. Schrodinger: Sitzberg Press Akad. Wiss. Phys.-Math. K1. (1931), 144-153. J. G. Cramer: Rev. Mod. Phys., 58 (1986), 647-687.
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 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 = 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.
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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 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
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).
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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< =
( 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.
41 1
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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( 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 = Ao) 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
