Mathematical Theory of Feynman Path Integrals
Feynman path integrals integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ide
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523 Sergio A. Albeverio Raphael J. Heeqh-Krohn
Mathematical Theory of Feynman Path Integrals
Springer-Verlag Berlin· Heidelberg· New York 1976
Authors
Sergio A Albeverio Raphael J. Hoogh-Krohn Institute of Mathematics University of Oslo Blindern, Oslo 3/Norway
Library of Congress Cataloging in PublicatioD Data
AJ.beverio, Sergio. Mathematical theory of Feynman path integrals. (Lecture notes in mathematics; 523) Bibliography: p. Includes index. 1. Feynman integrals. I. lR:legh-Krohn, Raphael, joirrl author. II. Title. III. Series: Lecture notes in mathematics (Berlin) ; 523. Q,A,3.L28 no. 523 [QC174.17.F45] 510'.8s [515'.43]
76-18919
AMS Subject Classifications (1970): 28 A 40,35 J10,81 A18,81 A 45,81 A81, 82A15 ISBN 3-540-07785-5 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-07785-5 Springer-Verlag New York' Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1976 Printed in Germany / Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Preface In this work we develop a general theory of oscillatory integrals on real Hilbert spaces and apply i t to the mathematical foundation of the so called Feynman path integrals of non relativistic quantum mechanics, quantum statistical mechanics and quantum field theory. The translation invariant integrals we define provide a natural extension of the theory of finite dimensional oscillatory integrals, which has newly undergone an impressive development, and appear to be a suitable tool in infinite dimensional analysis. For one example, on the basis of the present work we have extended the methods of stationary phase, Lagrange immersions and corresponding asymptotic expansions to the infinite dimensional case, covering in particular the expansions around the classical limit of quantum mechanics. A particular case of the oscillatory integrals studied in the present work are the Feynman path integrals used extensively in the physical literature, starting with the basic work on quantum dynamics by Dirac and Feynman, in the forties. In the introduction we give a brief historical sketch and some references concerning previous work on the problem of the mathematical justification of Feynman's heuristic formulation of the integral. However our aim with the present pUblication was in no way to write a review work, but rather to develop from scratch a self contained theory of oscillatory integrals in infinite dimensional spaces, in view of the mathematical and physical applications alluded above. The structure of the work is brieflY as follows. It consists of nine sections. Section 1
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