Rigorous Time Slicing Approach to Feynman Path Integrals
This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives o
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Daisuke Fujiwara
Rigorous Time Slicing Approach to Feynman Path Integrals
Mathematical Physics Studies Series editors Giuseppe Dito, Dijon, France Edward Frenkel, Berkeley, CA, USA Sergei Gukov, Pasadena, CA, USA Yasuyuki Kawahigashi, Tokyo, Japan Maxim Kontsevich, Bures-sur-Yvette, France Nicolaas P. Landsman, Nijmegen, The Netherlands
More information about this series at http://www.springer.com/series/6316
Daisuke Fujiwara
Rigorous Time Slicing Approach to Feynman Path Integrals
123
Daisuke Fujiwara Department of Mathematics Gakushuin University Tokyo Japan
ISSN 0921-3767 Mathematical Physics Studies ISBN 978-4-431-56551-2 DOI 10.1007/978-4-431-56553-6
ISSN 2352-3905
(electronic)
ISBN 978-4-431-56553-6
(eBook)
Library of Congress Control Number: 2017939312 © Springer Japan KK 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Japan KK The registered company address is: Chiyoda First Bldg. East, 3-8-1 Nishi-Kanda, Chiyoda-ku, Tokyo 101-0065, Japan
Preface
The Feynman path integral is a method of quantization using the Lagrangian function, while Schrödinger’s quantization uses the Hamiltonian function. Since it provides a different view point from Schrödinger’s, it is a very useful basic tool in quantum physics. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger’s method, there is still much to be done concerning rigorous mathematical treatment of Feynman’s method. The difficulty lies in the fact that the Feynman path integral is not an integral by means of a countably additive measure. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces. To construct this approximating sequence he divided
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