Functional Analysis and the Feynman Operator Calculus
This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting.
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unctional Analysis and the Feynman Operator Calculus
Functional Analysis and the Feynman Operator Calculus
Tepper L. Gill • Woodford W. Zachary
Functional Analysis and the Feynman Operator Calculus
123
Woodford W. Zachary Departments of Electrical and Computer Engineering Howard University Washington, DC, USA
Tepper L. Gill Departments of Electrical and Computer Engineering Howard University Washington, DC, USA
ISBN 978-3-319-27593-2 DOI 10.1007/978-3-319-27595-6
ISBN 978-3-319-27595-6 (eBook)
Library of Congress Control Number: 2015957963 Mathematics Subject Classification (2010): 28A35, 35K15, 35L15, 47D06, 46B99, 46F25, 43A25 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2016 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. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
Preface
Two approaches to the mathematical foundations of relativistic quantum theory began in the USA. Both evolved from the application of quantum field methods to electron theory in the late 1940s by Feynman, Schwinger, and Tomonaga (see [SC1]). The first program is well known and was begun in the early 1950s by Professor A.S. Wightman of Princeton University (1922–2013). Following a tradition inspired by Hilbert, the program was called axiomatic field theory. It sought to provide rigorous justification for the complicated and difficult method of renormalization successfully employed by the physics community (see [SW] and [GJ]). Professor Wightman is considered the founding father of modern mathematical physics, but he also strongly influenced a number of other areas in mathematics. In 1982, Sokal notice some difficulties with the constructive approach to field theory (the concrete version of axiomatic field theory) and conjectured that this approach may not work as expected in four space-time dimensions (see [SO]). His conjecture was later verified by Aizenman
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