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Path Integrals in Physics Volume I Stochastic Processes and Quantum Mechanics Path Integrals in Physics Volume I Stochastic Processes and Quantum Mechanics M Chaichian Department of Physics, University of Helsinki and Helsinki Institute of Physics, Finland and A Demichev Institute of Nuclear Physics, Moscow State University, Russia Institute of Physics Publishing Bristol and Philadelphia c IOP Publishing Ltd 2001 All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 7503 0801 X (Vol I) 7503 0802 (Vol II) 7503 0713 (2 Vol set) Library of Congress Cataloging-in-Publication Data are available Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing Executive: Colin Fenton Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in TEX using the IOP Bookmaker Macros Printed in the UK by Bookcraft, Midsomer Norton, Bath Fate has imposed upon our writing this tome the yoke of a foreign tongue in which we were not sung lullabies Freely adapted from Hermann Weyl Contents Preface Introduction Notational conventions ix Path integrals in classical theory 1.1 Brownian motion: introduction to the concept of path integration 1.1.1 Brownian motion of a free particle, diffusion equation and Markov chain 1.1.2 Wiener’s treatment of Brownian motion: Wiener path integrals 1.1.3 Wiener’s theorem and the integration of functionals 1.1.4 Methods and examples for the calculation of path integrals 1.1.5 Change of variables in path integrals 1.1.6 Problems 1.2 Wiener path integrals and stochastic processes 1.2.1 A short excursion into the theory of stochastic processes 1.2.2 Brownian particles in the field of an external force: treatment by functional change of variables in the path integral 1.2.3 Brownian particles with interactions 1.2.4 Brownian particles with inertia: a Wiener path integral with constraint and in the space of velocities 1.2.5 Brownian motion with absorption and in the field of an external deterministic force: the Bloch equation and Feynman–Kac formula 1.2.6 Variational methods of path-integral calculations: semiclassical and quadratic approximations and the method of hopping paths 1.2.7 More technicalities for path-integral calculations: finite-difference calculus and Fourier decomposition 1.2.8 Generating (or characteristic) functionals for Wiener integrals 1.2.9 Physics of macromolecules: an application of path integration 1.2.10 Problems 12 12 12 22 29 36 45 49 56 56 94 101 108 111 Path integrals in quantum mechanics 2.1 Feynman path integrals 2.1.1 Some basic facts about quantum mechanics and the Schră dinger equation o 2.1.2 FeynmanKac formula in quantum mechanics 2.1.3 Properties of Hamiltonian operators from the Feynman–Kac formula 2.1.4 Bohr–Sommerfeld (semiclassical) quantization condition from path integrals 2.1.5 Problems 2.2 Path integrals in the Hamiltonian formalism 122 123 123 137 141 144 149 153 63 66 69 72 78 Contents viii 2.2.1 2.2.2 2.3 2.4 2.5 2.6 Derivation of path integrals from operator formalism in quantum mechanics Calculation of path integrals for the simplest quantum-mechanical systems: a free particle and a harmonic oscillator 2.2.3 Semiclassical (WKB) approximation in quantum mechanics and the stationaryphase method 2.2.4 Derivation of the Bohr–Sommerfeld condition via the phase-space path integral, periodic orbit theory and quantization of systems with chaotic classical dynamics 2.2.5 Particles in a magnetic field: the Ito integral, midpoint prescription and gauge invariance 2.2.6 Applications of path integrals to optical problems based on a formal analogy with quantum mechanics 2.2.7 Problems Quantization, the operator ordering problem and path integrals 2.3.1 Symbols of operators and quantization 2.3.2 General concept of path integrals over trajectories in phase space 2.3.3 Normal symbol for the evolution operator, coherent-state path integrals, perturbation expansion and scattering operator 2.3.4 Problems Path integrals and quantization in spaces with topological constraints 2.4.1 Point particles in a box and on a half-line 2.4.2 Point particles on a circle and with a torus-shaped phase space 2.4.3 Problems Path integrals in curved spaces, spacetime transformations and the Coulomb problem 2.5.1 Path integrals in curved spaces and the ordering problem 2.5.2 Spacetime transformations of Hamiltonians 2.5.3 Path integrals in polar coordinates 2.5.4 Path integral for the hydrogen atom: the Coulomb problem 2.5.5 Path integrals on group manifolds 2.5.6 Problems Path integrals over anticommuting variables for fermions and generalizations 2.6.1 Path integrals over anticommuting (Grassmann) variables for fermionic systems 2.6.2 Path integrals with generalized Grassmann variables 2.6.3 Localization techniques for the calculation of a certain class of path integrals 2.6.4 Problems 154 161 169 176 183 187 190 200 200 209 216 226 230 231 238 243 245 245 251 258 266 272 282 286 286 298 304 315 Appendices A General pattern of different ways of construction and applications of path integrals B Proof of the inequality used for the study of the spectra of Hamiltonians C Proof of lemma 2.1 used to derive the Bohr–Sommerfeld quantization condition D Tauberian theorem 318 318 318 322 326 Bibliography 328 Index 333 Preface The importance of path-integral methods in theoretical physics can hardly be disputed Their applications in most branches of modern physics have proved to be extremely fruitful not only for solving already existing problems but also as a guide for the formulation and development of essentially new ideas and approaches in the description of physical phenomena This book expounds the fundamentals of path integrals, of both the Wiener and Feynman type, and their numerous applications in different fields of physics The book has emerged as a result of many courses given by the authors for students in physics and mathematics, as well as for researchers, over more than 25 years and is based on the experience obtained from their lectures The mathematical foundations of path integrals are summarized in a number of books But many results, especially those concerning physical applications, are scattered in a variety of original papers and reviews, often rather difficult for a first reading In writing this book, the authors’ aim was twofold: first, to outline the basic ideas underlying the concept, construction and methods for calculating the Wiener, Feynman and phase-space quantum-mechanical path integrals; and second, to acquaint the reader with different aspects concerning the technique and applications of path integrals It is necessary to note that, despite having almost an 80-year history, the theory and applications of path integrals are still a vigorously developing area In this book we have selected for presentation the more or less traditional and commonly accepted material At the same time, we have tried to include some major achievements in this area of recent years However, we are well aware of the fact that many important topics have been either left out or are only briefly mentioned We hope that this is partially compensated by references in our book to the original papers and appropriate reviews The book is intended for those who are familiar with basic facts from classical and quantum mechanics as well as from statistical physics We would like to stress that the book is not just a linearly ordered set of facts about path integrals and their applications, but the reader may find more effective ways to learn a desired topic Each chapter is self-contained and can be considered as an independent textbook: it contains general physical background, the concepts of the path-integral approach used, followed by most of the typical and important applications presented in detail In writing this book, we have endeavored to make it as comprehensive as possible and to avoid statements such as ‘it can be shown’ or ‘it is left as an exercise for the reader’, as much as it could be done A beginner can start with any of the first two chapters in volume I (which contain the basic concepts of path integrals in the theory of stochastic processes and quantum mechanics together with essential examples considered in full detail) and then switch to his/her field of interest A more educated user, however, can start directly with his/her preferred field in more advanced areas of quantum field theory and statistical physics (volume II), and eventually return to the early chapters if necessary For the reader’s convenience, each chapter of the book is preceded by a short introductory section containing some background knowledge of the field Some sections of the book require also a knowledge of the elements of group theory and differential (mainly Riemann) geometry To make the reading easier, we have added to the text a few supplements containing some basic concepts and facts from these ix Appendices 322 which, together with (B.10), gives x2 exp − t0 dW x(s) > d x0 √ πt −L {0;x t =0,t } L exp − (2L−|x0|) t d x0 −2 √ πt −L exp − ξt dξ √ =1− πt |ξ |>L or L −L −2 −2L exp − ξt dξ √ πt exp − ξt dW x(s) > − dξ √ πt |ξ |>L {0;x t =0,t } exp − ξt dξ √ πt 2L + L Taking into account (B.9) this means, in turn, that exp − ξt dW x(s) < dξ √ πt |ξ |>L {0;x t =0,t } (B.13) and, using (B.4), (B.7) and (B.8), we derive {0;x t ,t } t dW x(s) exp −2 ds V (x(s)) < ε + ε (L) − − → −− |x t |→∞ (B.14) (because at |x t | → ∞, we may take L arbitrarily large) Finally, with the help of the inequalities (B.1) and (B.14), we can conclude that ∞ ψ (x t ) < −∞ d x ψ0 (x ) exp − (xt −x0 ) t √ πt −− (ε + ε ) − − → |x t |→∞ (B.15) i.e ψ(x t ) uniformly tends to zero at large values of x t and the second condition (2.1.110) has been proved C Proof of lemma 2.1 used to derive the Bohr–Sommerfeld quantization condition The proof of the lemma starts from the inequality n exp − V (x i ) i=1 t n ≤ t n exp{−t V (x i )} i=1 t n (C.1) which, in turn, is the particular case of the following proposition Proposition 2.2 (Jensen inequality) Let φ(x) be a convex function (so that φ (x) > 0) defined on the interval [a, b] ∈ Ê and x i , i = 1, , n be some points in the interval: x i ∈ [a, b] Then n n αi φ(x i ) ≥ φ i=1 where n αi = i=1 αi x i (C.2) i=1 αi ∈ Ê+ (positive numbers) (C.3) Proof of lemma 2.1 used to derive the Bohr–Sommerfeld quantization condition def n i=1 Proof of the Jensen inequality It is clear that x ≡ Taylor series with the remainder term 323 αi x i belongs to the interval [a, b] Consider the φ(x i ) = φ(x) + (x i − x)φ (x) + (x i − x)2 φ (ξi ) i = 1, , n where ξi ∈ [a, b] Since the function φ is convex, we have φ (x) > Multiplying each of these series by αi and summing them up over i , we obtain the required inequality n αi φ(x i ) ≥ φ(x) i=1 Corollary 2.2 In the particular case αi = 1/n, φ(x) = e−x , we introduce = e−xi and the Jensen inequality becomes equivalent to the relation: (a1 a2 · · · an )1/n ≤ (a1 + a2 + · · · + an ) n ∈ Ê+ (C.4) Corollary 2.3 The set of the positive numbers α = {αi } can be considered as a probability distribution due to condition (C.3), so that the Jensen inequality can be written in the more general form φ( x ) ≤ φ(x) (C.5) where · denotes the mean value with respect to the distribution α = {αi } Relation (C.1) follows from (C.4) if we put = e−t V (xi ) Since the inequality (C.1) is correct for arbitrary n, we can take the limit n → ∞ with the result exp − t t ds V (x(s)) ≤ t t ds exp{−t V (x(s))} (C.6) and since the latter is correct for arbitrary x(s), we can integrate it with the Wiener measure to get K B (x, t|x, 0) = ≤ {x,0;x,t } t t dW x(s) exp − ds V (x(s)) t {x,0;x,t } dW x(s) ds exp{−t V (x(s))} (C.7) Changing the order of integration and using the ESKC relation, we can derive from (C.7) the inequality ∞ −∞ d x K B (x, t|x, 0) ≤ t ∞ −∞ = √ πt t dξ ∞ −∞ ds exp{−t V (ξ )} √ πt dξ exp{−t V (ξ )} (C.8) (problem 2.1.8) This relation is correct for arbitrary t Now we shall prove that for small values of t, there exists the inequality with the opposite sign Thus, actually, (C.8) is an equality and this proves the lemma Appendices 324 Let us make the substitution: x(s) −→ x + x(s) x = x(t) = x(0) in the Wiener integral {x,0;x,t } t dW x(s) exp − ds V (x(s)) = {0,0;0,t } t dW x(s) exp − ds V (x + x(s)) (C.9) The integrand of the latter path integral can be rewritten with the help of the step-function θ via the Stieltjes integral t exp − ∞ ds V (x + x(s)) = t e−u du θ u − due to the well-known relation ds V (x + x(s)) (C.10) d θ (u − a) = δ(u − a) du (C.11) Using (C.10), we can write (changing the order of the integration) {0,0;0,t } = ∞ ∞ ≥ e−u du {0,0;0,t } e−u du t dW x(s) θ u − δ {0,0;0,t } δ {0, 0; 0, t} ds V (x(s)) 0 where t dW x(s) exp − ds V (x + x(s)) t dW x(s) θ u − ds V (x + x(s)) (C.12) is a subset of {0, 0; 0, t} such that |x(s)| < δ The inequality in (C.12) follows from the positivity of the Wiener measure and the positive semidefiniteness of the step-function θ With the help of the inequality t sup |x(s)