High magnetic fields : science and technology. Volume 1, Magnet technology anbd experimental techniques

305 Pages·13.751 MB·English

by  editors Fritz Herlach, Noboru Miura.

Quantum
probability
communications
This page is intentionally left blank
QP-PQ
Volume XI
Quantum
probability
communications
Managing Editors
Stephane Attal, J. Martin Lindsay
Editorial Board
Ph. Biane, F. Fagnola, M. Fannes,
B. Kummerer, H. Maassen, D. Petz
Advisory Board
L. Accardi, L. Gross, T. Hida,
K. R. Parthasarathy, A. Verbeure, D. Voiculescu
V fe World Scientific
wll NNeeww J Jeerrsseeyy •• LLoonnddoonn •• SSii ngapore • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
QUANTUM PROBABILITY COMMUNICATIONS, VOL. XI
Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 981-238-976-8
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Printed in Singapore.
CONTENTS
Contents of QPC XII vi
Preface for Volumes XI & XII vii
Extensions of quantum stochastic calculus 1
Stephane Attal
Quantum Ito algebras: axioms, representations, decompositions 39
Viacheslav Belavkin
Free probability for probabilists 55
Philippe Biane
Conditional expectations on von Neumann algebras 73
Carlo Cecchini
Classical probability theory: an outline of
stochastic integrals and diffusions 87
Michel Emery
Quantum stochastic differential equations 123
Franco Fagnola
Canonical commutation and anticommutation relations 171
Mark Fannes
Quantum and classical stochastic calculus 199
Alexander Holevo
An introduction to quantum stochastic calculus and
some of its applications 221
Robin Hudson
Stationary processes in quantum probability 273
Burkhard Kummerer
v
CONTENTS of QPC XII
Integral-sum kernel operators 1
J. Martin Lindsay
Quantum probability applied to the damped harmonic oscillator 23
Hans Maassen
Quantum probability and strong quantum Markov processes 59
K. R. Parthasarathy
Limit problems for quantum dynamical semigroups — inspired by
scattering theory 139
Rolando Rebolledo
A survey of operator algebras 173
Jean-Luc Sauvageot
Quantum stop times 195
Kalyan B. Sinha
Free calculus 209
Roland Speicher
Continuous kernel processes in quantum probability 237
Wilhelm von Waldenfels
Bibliography for QPC XI & XII 261
vi
PREFACE for QPC Volumes XI & XII
The Grenoble Summer School, organised by one of us, took place in June 1998. Lec­
tures from the School are collected in these two volumes of the Series QP — PQ. All have
been refereed by at least one expert and revised, some extensively, for publication. No
other natural division having emerged, material has been ordered purely alphabetically.
A composite bibliography, containing the references from each of the lectures, is included
in Volume XII. It is our hope that current and future students of quantum probability
will be engaged, informed and inspired by these volumes.
We owe thanks to all the authors for their contribution to the volumes, and for bearing
with us during their prolonged incubation. We are indebted too to the referees, and to
Cathie Shipley whose assistance in our efforts to achieve unity of typographical style has
been invaluable. We would like to acknowledge the continued interest and support of
H.T. Leong, of World Scientific, in bringing this enterprise to fruition.
Stephane Attal
Martin Lindsay
vu
Quantum Probability Communications, Vol. XI (pp. 1-37)
© 2003 World Scientific Publishing Company
EXTENSIONS OF QUANTUM STOCHASTIC CALCULUS
STEPHANE ATTAL
CONTENTS
1. Abstract Ito calculus on Fock space 1
2. Extension of quantum stochastic calculus 13
3. The algebra of regular quantum semimartingales 22
4. Some recent developments 32
Bibliographical Notes 37
References 37
1. ABSTRACT ITO CALCULUS ON FOCK SPACE
1.1. Short notations.
The symmetric spaces. Let V denote the set of finite subsets of R+. That is, V =
LIn'Pn where Vo = {0} and V is the set of n-element subsets of R+, n > 1. By
n
ordering elements of a a = {t\,t<i,...,t} G P we identify V with E = {0 <
n n n n
*i < h < • • • < t} C (M+). This way V inherits the measure space structure of
n n
(R+)n. By putting the Dirac measure 5$ on V, we have denned a a-finite measure
n
space structure on V (which, I insist, is the n-dimensional Lebesgue measure on
each V) whose only atom is {0}. The elements of V are denoted by lower case
n
Greek letters a, w,r,..., the associated measure is denoted da, dio, dr,..., (keeping,
in mind, that a = {t\ < t% < ■ ■ ■ < £„} and da = dt\dt2 ■ ■ ■ dt„). It is now clear
that L2(V) is isomorphic to $, the symmetric Fock space over L2(R ). Indeed,
+
L2(V) = © L2{V) is isomorphic to © L2(S) (with S = {0}) that is $. In order
n n n n 0
to be really clear, the isomorphism between $ and L2(V) can be explicitly written
as:
where / = £ /„ and
n
[F/1(fT)={ /!(ti,... A) if {*!<■..< in}-
For example, a coherent vector e(u), seen in L2(V), satisfies
(where an empty product equals 1). Let us fix some notations on V.
If a ^ 0 we put Vc = max a and a— = a \ {VCT}.
1
2 Stephane Attal
If t 6 a then a\t denotes a \ {t}.
If {t £ a} then a U < denotes cr U {£}.
If 0 < s < t then <7 ) = a n [0, s[, (T(,t) = cfljs, i[, and tfy = aH]t, +oo[.
S s
l < means 1 if cr C [0, t], and 0 otherwise.
CT t
If 0 < s < t then Ps> = {a € P; <r C [0,s[}, P<-s^ = {a e P;a c]s,t[} and
P"={ireP;(7C]t,+oo[}.
#c is the cardinality of a.
With the notations of R.L. Hudson's course, it is clear that
<3> ~ L2(Ps)), $ * L2(P^) and $„ ~ L2(7>«).
s] M
In the following we make several identifications:
• $ is not distinguished from L2(P) (and the same holds for <£,,] and L2(PS^),
etc ....)
• L2{Vs)), L2(P<-3^), and L2(V{t)) are seen as subspaces of L2(P): the subspace
of / € L2(P) such that f (a) = 0 for all a such that a <t [0, s] (resp. a gt [s, t],
resp. a gt [t, +oo]).
Integral-sum lemma. The following lemma is a very important and useful combinato­
rial result that we will use quite often in the sequel. What this lemma says is mainly
the following: consider the Wick product on $:
[/:ff]Mde^f '£ /(«M*\a)
then this product behaves like a convolution.
Lemma 1.1 (¥))• Let f be a measurable positive {resp. integrable) function on
V x V. Define a function g onP by
aCcr
Then g is measurable positive (resp. integrable) and
f g{a)da = f f{a,f3)dad[3.
JV JVxV
Proof. By density arguments one can restrict ourselves to the case where f(a,/3) =
h (a) k (/?) and where h = s{u) and k = s(v) are coherent vectors, with u, v £
L1 n L2(R+). In this case one has
f f(a, (3) dad/3 = [ e{u){a) da f e(v) {(3) d(3
JVxV JV JV
Extensions of quantum stochastic calculus 3
and
/^/( ,a\a)da = /^]}«(*) II w(a) ^
a
= f V(u(s) + u(a)) d<r = Jo~ »(»>+»(»> ds.
e
a
As seen in Hudson's course we have, for all t, an isomorphism between $ and
$j ® $[. In terms of this short notation, and with the help of the Y^-Lemma, the
t t
isomorphism is nicely described.
Theorem 1.2. The mapping
with h{a) = f{a))g{ar(t) defines an isomorphism between <&| <g> $[ and 4>.
t t t
Proof.
f\h(a)\2da = f\f(a )\2\g(a )\2 da
t) (t
Jv Jv
= / / lac[0,t]lg\aC[t,+oo[ |/(a)|'| (a\a)|'d<7
-u J fl
|/(a)|2 |(/?)|2 dad/? (by the ^-Lemma)
5
= / \f(a)\2da[ \g((3)\2dP
Jv'i Jvi*
= ll/®ffl|2-
a
1.2. Ito calculus on Fock space. We are now ready to define the main ingredi­
ents for developing our quantum stochastic calculus, namely several differential and
integral operators on the Fock space.
Conditional expectations. For all t > 0 define the operator P from $ to $ by
t
[Ptf(a) = /Wk .
M
It is very easy to check that P is actually the orthogonal projector from $ onto <E>].
t (
For t = 0we define PQ by
[PofM = /(0)l =a
CT
which is the orthogonal projection onto L2{VQ) = Cl where 1 is the vacuum (1(a) =