Evolution Processes and the Feynman-Kac Formula

244 Pages·1996·6.38 MB·English

by  Brian Jefferies (auth.)

Evolution Processes and the Feynman-Kac Formula
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre/or Mathematics arui Computer Science, Amsterdam, The Netherlaruis
Volume 353
Evolution Processes and the
Feynman-Kac Formula
by
Brian Jefferies
School ofM athematics,
The University ofN ew South Wales,
Sydney, New South Wales, Australia
Springer- Science+B usiness Media, B. V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4650-5 ISBN 978-94-015-8660-3 (eBook)
DOI 10.1007/978-94-015-8660-3
Printed on acid-free paper
All Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996.
Softcover reprint of the hardcover 1s t edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
For Igor
Contents
Preface ix
Introduction
Chapter 1. Vector Measures and Function Spaces 7
1.1 Terminology and Notation 7
1.2 Vector Measures 11
1.3 Vector Integration 14
1.4 Function Spaces 16
1.5 Convergence 21
1.6 Notes 24
Chapter 2. Evolution Processes 25
2.1 Markov Evolution Processes 25
2.2 Construction of a -Additive Processes 31
2.3 Sufficient Conditions for Boundedness 35
2.4 Bounded Finite Dimensional Distributions for Convolution Groups 41
2.5 Bounded Finite Dimensional Distributions and Regular Operators 49
2.6 Notes 57
Chapter 3. Feynman-Kac Formulae 59
3.1 Multiplicative Functionals 59
3.2 Continuity of the Perturbed Seruigroup 62
3.3 A General Feynman-Kac Formula for Bounded Processes 65
3.4 The Feynman-Kac Formula for Probabilistic Markov Processes 71
3.5 Notes 79
Chapter 4. Bilinear Integration 81
4.1 Bilinear Integration in Tensor Products 81
4.2 Semivariation 87
4.3 Integration of Operator Valued Functions with respect to Operator
Valued Measures 91
4.4 Bilinear Integration for LP -Spaces 99
4.5 Semivariation in LP-Spaces 101
4.6 Examples of Bilinear Integrals 105
4.7 Notes 110
Chapter 5. Random Evolutions 112
5.1 Multiplicative Operator Functiona1s 113
5.2 A Noncommutative Feynman-Kac Formula 120
5.3 Representation of Evolutions 129
viii CONTENTS
5.4 Notes 132
Chapter 6. Some Bounded Evolution Processes 134
6.1 More Bilinear Integrals 135
6.2 Operator Valued Transition Functions 138
6.3 Semigroups on L with a Bounded Generator 142
00
6.4 The Direct Sum of Dynamical Systems 147
6.5 Support Properties 159
6.6 Notes 169
Chapter 7. Integration with respect to Unbounded Set Functions 171
7.1 Integration with respect to Closable Systems of Set Functions 171
7.2 Integration Structures 178
7.3 A Non-closable Bimeasure 187
7.4 Notes 190
Chapter 8. The SchrOdinger Process 191
8.1 The Schr6dinger Process 191
8.2 The Feynman Representation for the Schr6dinger Process 197
8.3 Notes 202
Chapter 9. The Radial Dirac Process 204
9.1 The Radially Symmetric Dirac Process 204
9.2 Construction of the Cut-Off Measures R}k,E), E > 0 207
9.3 The Support of the Cut-Off Measures R}k,E) , E > 0 208
9.4 Integration with Respect toMt and M?), E > 0 218
9.5 The Feynman Representation for the Radial Dirac Process 221
9.6 Notes 227
Bibliography 229
Index 233
Preface
This book is an outgrowth of ideas originating from 1. Kluvanek. Unfortunately,
Professor Kluvanek did not live to contribute to the project of writing up in a systematic
form, the circle of ideas to which the present work is devoted. It is more than likely
that with his input, the approach and areas of emphasis of the resulting exposition would
have been quite different from what we have here. Nevertheless, the stamp of Kluvanek's
thought and philosophy (but not necessarily his approval) abounds throughout this book.
Although the title gives no indication, integration theory in vector spaces is a cen
tral topic of this work. However, the various notions of integration developed here are
intimately connected with a specific application-the representation of evolutions by func
tional integrals. The representation of a perturbation to the heat semigroup in terms of
Wiener measure is known as the Feynman-Kac formula, but the term has a wider meaning
in the present work. Traditionally, such representations have been used to obtain analytic
information about perturbations to free evolutions as an alternative to arguments with a
more operator-theoretic flavour. No applications of this type are given here. It is an un
derlying assumption of the presentation of this material that representations of the nature
of the Feynman-Kac formula are worth obtaining, and in the process of obtaining them,
we may be led to new, possibly fertile mathematical structures-a view largely motivated
by the pervasive use of path integrals in quantum physics.
There is an uncomfortable gap between notions of path integration employed computa
tionally as a matter of course by theoretical physicists and what mathematicians understand
by integration. The gap remains after this book. The purpose here is to distil some common
features of a number of different representations of evolving systems in terms offunctional
integrals. In the tradition of functional analysis, the development of mathematical ideas
here is inspired by potential applications to physical problems, but it is the mathematics
itself which is of primary concern. It should come as no surprise that concepts related to
integration in vector spaces seem especially important for the treatment of path integrals.
I hope that this book persu~t he reader that the study of vector integration may prove
fruitful for exposing some of the Underlying mathematical structure of quantum physics.
I thank W. Ricker and B. Goldys for reading Chapters 1 and 2 and S. Okada for his
collaboration in the writing of Chapter 4. A grant from the Australian Research Council for
this work is gratefully acknowledged. It is widely recognised that the economic orthodoxy
of the late twentieth century has put increasing pressure on academic researchers. Without
the support of my wife Bronwyn Eather, I would not have found the time to write in
between other urgent professional duties.
September, 1995 Brian Jefferies
Sydney
ix
Introduction
The mathematical description of an evolving physical system must formulate two
fundamental aspects of physical reasoning-a description of the dynamics of the system
and a method for making observations.
A natural way to describe the dynamics of many systems is by a semigroup of con
tinuous linear operators acting on a vector space of states of the system. The semi group
property embodies the principle of determinism and the continuity of the linear operators
is a stability condition for initial data. The linearity of the operators corresponds to the
principle of superposition of states. From the mathematical viewpoint, the idea of making
observations of a physical system is best formulated in terms of a spectral measure. This
viewpoint of the mathematical foundations of physical systems is set out systematically
in the monograph of G. Mackey [M].
Associated with any semigroup S of continuous linear operators acting on a vector
space X of states and a spectral measure Q acting on X, there is an operator valued set
function Mt acting on X, corresponding to each time t ~ o. The family (Mt)t,,=o of operator
valued set functions and its properties is a mathematical codification of the properties of the
evolving physical system. In the domain of quantum physics, the dynamics of interacting
systems may be described in terms ofthe integrals offunctionals with respect to the operator
valued set functions Mt, t ~ 0, associated with the freely evolving system. Allowing a
great deal of mathematical licence, this viewpoint is essentially adopted by R. Feynman in
[Fe] and subsequent works [Fe-H].1t is worthwhile to spell out these points more precisely
in the terminology of [M].
Suppose that ~ is the phase space of a system in classical mechanics. An observable
is a Borel function on ~. A state is identified with a probability measure on (the Borel
a -algebra of) ~. It is convenient to consider the vector space E of all signed measures
on ~ with the topology of weak convergence as the state space. An element a of phase
space is identified with the unit point mass Ou E E at a, so embedding ~ in E. The
dynamical group Set), t E JR, associates the state S(t)/-L at time t E JR with the state /-L at
time O. In classical mechanics, the dynamical group has the special form S(t)/-L = /-L 0 Ut- I
where Ut : ~ --+ ~,t E JR, is the dynamical group on the phase space of the system. For
every Borel subset A of ~, there is associated a projection Q(A) : E --+ E defined by
Q(A)/-L = XA/-L. Here XA/-L is the measure defined by [XA/-L](B) = /-L(A n B) for every
Borel subset B of ~. The characteristic function XA is an observable for which XA (a)
corresponds to the 'question' whether or not an arbitrary element a of phase space ~ lies
in A. The projection Q (A) omits from a state those elements of phase space not in A.
Let Q be the space of continuous functions w : [0, (0) --+ ~. Now fix t > 0, let
o
< tl < t2 < ... < tn < t be times before t, and let B1, .•• , Bn be Borel subsets of ~.
2 INTRODUCTION
For every cylinder set
(0.1)
the operator Mt(A) : E ~ E is defined by
The operator Mt(A) has the following physical interpretation. If the initial state f.l
evolves under the action of the dynamical group S up to time tl and then we omit from the
resulting distribution those elements of ~ not in BI, let the resulting measure evolve for
time t2 - tl and omit from the resulting distribution those elements of ~ not in B2, and
so on, then we obtain the measure Mt(A)f.l- the distribution obtained after making the
'observations' XB" ... , XBn at times tl, ... , tn respectively.
n
So defined, the set function Mt is the restriction to cylindrical subsets of of an
operator valued measure Mt. It is easy to write down the extension Mt of Mt explicitly;
y : n
if ~ ~ is the map which associates a point a with its orbit Usa, s ~ 0, then
Mt(A) = S(t)Q (y-I(A») for every cylinder set A. The operator valued measure Mt =
S(t)Q 0 y-I, is therefore concentrated on the flow of the dynamical system. The operator
M
valued measures t, t ~ 0, offer a perverse description of a system in classical mechanics
equivalent to the usual one.
A more interesting example is provided by quantum mechanics. A (pure) state in a
quantum mechanical system is, up to a complex factor of norm one, a normalised vector
in a Hilbert space E. An observable corresponds to a selfadjoint operator acting in E. Let
S be the dynamical group of the system in the SchrMinger picture, that is, S is a group of
unitary operators acting on E. Let Q be the joint spectral measure acting on E associated
=
with the position operators. In the case of an n-particle system in IR3, E L 2(IR3n) and
Q(A) is multiplication by XA for every Borel subset A of IR3n.
n
Let ~ be the domain of Q - some finite dimensional Euclidean° sp ace. Let be the
space of continuous functions (J) : [0, 00) ~ ~. Now fix t > 0, let < tl < t2 < ... <
tn < t be times before t, and let B1, ••• , Bn be Borel subsets of~. For every cylinder set
A of the form (0.1) the operator Mt(A) : E ~ E is defined by formula (0.2)
The operator Mt (A) has the following physical interpretation, analogous to the case of
classical mechanics. If the initial state 1/1 evolves under the action of the dynamical group
S up to time tl and then we ask whether the position coordinates of the system belongs B I,
let the resulting state evolve for time t2 - tl and ask whether the position coordinates of
the system belongs B2, and so on, then we obtain the vector Mt(A)1/I - the state obtained
after making the 'observations' Q(B1), ••• , Q(Bn) at times tl, ... ,tn respectively.
From this viewpoint, the passage from classical to quantum mechanics is, perhaps,
less startling than it seems. However, the set function M is no longer the restriction to
t
n
cylindrical subsets of of an operator valued measure, and it makes no sense to say that
M is concentrated on a flow.
t
A more familiar process is provided by Brownian motion. Let E be the vector space
of all scalar measures on IRd equipped with the topology of weak convergence. Let
Set), t ~ 0, be the semi group of operators defined by
=
[S(t)f.lHA) (27rt)-d/21 [ e-lx-YI2/2tdf.l(y)dx,
J'R
A d