Mathematical Feynman Path Integrals and Their Applications

225 Pages·2009·3.404 MB·English

by  Sonia Mazzucchi

Mathematical
Feynman Path Integrals
and their Applications
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Mathematical
Feynman Path Integrals
and their Applications
S o n i a M a z z u c c h i
U n i v e r s i t y o f T r e n t o , I t a l y
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
MATHEMATICAL FEYNMAN PATH INTEGRALS AND THEIR APPLICATIONS
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-283-690-8
ISBN-10 981-283-690-X
Printed in Singapore.
LaiFun - Math'l Feynman Path.pmd 1 3/27/2009, 11:44 AM
March27,2009 10:32 WorldScienti(cid:12)cBook-9inx6in MathematicalFeynman
Preface
Evenifmorethan60yearshavepassedsincetheir(cid:12)rstappearanceinFeyn-
man’s PhD thesis, Feynman path integrals have not lost their fascination
yet.
They givea suggestivedescription of quantum evolution, reintroducing
in quantum mechanics the classical concept of trajectory, which had been
banned from the traditional formulation of the theory. In fact, they can
be recognized as a bridge between the classical description of the physical
world and the quantum one. Not only do they provide a quantization
method, allowing to associate, at least heuristically a quantum evolution
to any classicalLagrangian,but also they make very intuitive the study of
the semiclassicallimit of quantummechanics,i.e. the studyof the detailed
behavior of the wave function when the Planck constant is regarded as a
small parameter convergingto zero.
Nowadays,the physicalapplicationsof Feynman’s ideas gobeyond non
relativistic quantum mechanics and include quantum (cid:12)elds, statistical me-
chanics,quantumgravity,polymerphysics,geometry. Nevertheless,inmost
cases,Feynmanpathintegralsremainamathematicalchallengeastheyare
not well de(cid:12)ned from a mathematical point of view.
Since 1960, a large amount of work has been devoted to the mathe-
matical realization of Feynman path integrals in terms of a well de(cid:12)ned
functional integral. Despite the several interesting results that have been
obtained in the last decades, the feeling that Feynman integrals are only
an heuristic tool is still a widespread belief among mathematicians and
physicists.
The present book provides a detailed and self-contained description of
the rigorous mathematical realization of Feynman path integrals in terms
of in(cid:12)nite dimensional oscillatory integrals, a particular kind of functional
v
March27,2009 10:32 WorldScienti(cid:12)cBook-9inx6in MathematicalFeynman
vi Mathematical Feynman Path Integrals and Applications
integrals that can be recognized as the direct generalization of classical
oscillatory integrals to the case where the integration is performed on an
in(cid:12)nite dimensional space, in particular on a space of paths.
The book describes the mathematical di(cid:14)culties, the (cid:12)rst results
obtained in the 70’s and the 80’s, as well as the more recent develop-
ment andapplications. Special attentionhasbeen paidto enlighteningthe
mathematicaltechniques,includingin(cid:12)nitedimensionalintegrationtheory,
asymptotic expansions and resummation techniques, without losing the
connection with the physical interpretation of the theory.
A large amount of references allows the reader to get a deeper knowl-
edge of the most interesting mathematical results as well as of the modern
physical applications.
I am grateful to many coworkers, friends and colleagues for fruitful
discussions. SpecialthanksareduetoS.Albeverioforhishelpandsupport,
aswellasforreadingthemanuscriptandmakinglotsofusefulcomments. I
am also particularlygratefulto G. Greco,V.Moretti, E.Pagani,M. Toller
and L. Tubaro.
S. Mazzucchi
March27,2009 10:32 WorldScienti(cid:12)cBook-9inx6in MathematicalFeynman
Contents
Preface v
1. Introduction 1
1.1 Wiener’s and Feynman’s integration . . . . . . . . . . . . 4
1.2 The Feynman functional . . . . . . . . . . . . . . . . . . . 10
1.3 In(cid:12)nite dimensional oscillatory integrals . . . . . . . . . . 12
2. In(cid:12)nite Dimensional Oscillatory Integrals 15
2.1 Finite dimensional oscillatory integrals . . . . . . . . . . . 15
2.2 The Parsevaltype equality . . . . . . . . . . . . . . . . . 19
2.3 Generalized Fresnel integrals . . . . . . . . . . . . . . . . 23
2.4 In(cid:12)nite dimensional oscillatory integrals . . . . . . . . . . 32
2.5 Polynomial phase functions . . . . . . . . . . . . . . . . . 42
3. Feynman Path Integrals and the Schro(cid:127)dinger Equation 57
3.1 The anharmonic oscillator with a bounded anharmonic
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Time dependent potentials. . . . . . . . . . . . . . . . . . 68
3.3 Phase space Feynman path integrals . . . . . . . . . . . . 76
3.4 Magnetic (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Quartic potential . . . . . . . . . . . . . . . . . . . . . . . 88
4. The Stationary Phase Method and the Semiclassical
Limit of Quantum Mechanics 101
4.1 Asymptotic expansions . . . . . . . . . . . . . . . . . . . . 101
4.2 The stationary phase method. Finite dimensional case . . 106
vii
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viii Mathematical Feynman Path Integrals and Applications
4.3 The stationary phase method. In(cid:12)nite dimensional case . 115
4.4 The semiclassical limit of quantum mechanics . . . . . . . 128
4.5 The trace formula . . . . . . . . . . . . . . . . . . . . . . 138
5. Open Quantum Systems 147
5.1 Feynman path integrals and open quantum systems . . . 147
5.2 The Feynman-Vernon in(cid:13)uence functional . . . . . . . . . 155
5.3 The stochastic Schro(cid:127)dinger equation . . . . . . . . . . . . 165
6. Alternative Approaches to Feynman Path Integration 177
6.1 Analytic continuation of Wiener integrals . . . . . . . . . 177
6.2 The sequential approach . . . . . . . . . . . . . . . . . . . 181
6.3 White noise calculus . . . . . . . . . . . . . . . . . . . . . 184
6.4 Poissonprocesses . . . . . . . . . . . . . . . . . . . . . . . 188
6.5 Further approaches and results . . . . . . . . . . . . . . . 189
Appendix A Abstract Wiener Spaces 191
A.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 The classical Wiener space. . . . . . . . . . . . . . . . . . 195
Bibliography 197
Index 215
March27,2009 10:32 WorldScienti(cid:12)cBook-9inx6in MathematicalFeynman
Chapter 1
Introduction
One of the most challenging problems of modern physics is the connection
between the macroscopic and the microscopic world, that is between clas-
sicalandquantummechanics. Inprincipleamacroscopicsystemshould be
described as a collection of microscopic ones, so that classical mechanics
should be deduced from quantum theory by means of suitable approxima-
tions. At a (cid:12)rst glance the solution of the problem is not straightforward:
indeed there are deep di(cid:11)erences between the classical and the quantum
description of the physical world.
In classical mechanics the state of an elementary physical system, for
instance, a point particle is given by specifying its position q (a point in
its con(cid:12)guration space) and its velocity q_. The time evolution in the time
interval [0;t] is described by a path q(s) in the con(cid:12)guration space.
s [0;t]
2
The dynamics of the particle under the action of a force (cid:12)eld described by
the real-valued potential V is determined by the classical Lagrangian:
m
(q(s);q_(s)):= q_2 V(q); (1.1)
L 2 (cid:0)
where m is the mass of the particle. By the Hamilton’s least action princi-
ple, the Euler-Lagrangeequation of motion
d @ @
L L =0
dt @q_ (cid:0) @q
followbyavariationalargument. Thetrajectoryof the particleconnecting
a point x at time t to a point y at time t is the path making stationary
0
the action functional S:
t
(cid:14)S (q)=0; S (q)= (q(s);q_(s))ds: (1.2)
t t
L
Z0
The quantum description of a point particle appears at a (cid:12)rst glance
completely di(cid:11)erent. First of all the concept of trajectory is meaningless.
1