LITMIR.BIZ

This edition first published copyright

      © 2021 John Wiley & Sons

      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, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

      The right of Patrick Joseph Muldowney to be identified as the author of this work has been asserted in accordance with law.

       Registered Office

      John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA

       Editorial Office

      111 River Street, Hoboken, NJ 07030, USA

      For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.

      Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats.

       Limit of Liability/Disclaimer of Warranty

      While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

       Library of Congress Cataloging‐in‐Publication Data

      Names: Muldowney, P. (Patrick), 1946- author.

      Title: Gauge integral structures for stochastic calculus and quantum electrodynamics / Patrick Muldowney.

      Description: Hoboken, NJ :Wiley, [2020] | Includes bibliographical references and index.

      Identifiers: LCCN 2020016333 (print) | LCCN 2020016334 (ebook) | ISBN 9781119595496 (cloth) | ISBN 9781119595502 (adobe pdf) | ISBN 9781119595526 (epub)

      Subjects: LCSH: Stochastic analysis. | Henstock-Kurzweil integral. | Feynman integrals. | Quantum electrodynamics‐Mathematics.

      Classification: LCC QA274.2 .M85 2020 (print) | LCC QA274.2 (ebook) | DDC 519.2/2‐dc23

      LC record available at https://lccn.loc.gov/2020016333

      LC ebook record available at https://lccn.loc.gov/2020016334

      Cover Design: Wiley

      Cover Image: © bannerwega/Getty Images

      Set in 9.5/12.5pt STIXTwoText by SPi Global, Chennai, India

Preface

      This book is about infinite‐dimensional integration in stochastic calculus and in quantum electrodynamics, using the gauge integral technique pioneered by R. Henstock and J. Kurzweil.

      A link between stochastic calculus and quantum mechanics is provided in a previous book by the author ([121], A Modern Theory of Random Variation, or [MTRV] for short), which establishes a mathematical connection between large scale Brownian motion on the one hand and, on the other, small scale quantum level phenomena of particle motion subject to a conservative external mechanical force. In [MTRV] each of the two subjects is a special case of

‐Brownian motion.

      The present book is a continuation of [MTRV], in the sense that it develops and extends some of the themes of that book. On the other hand this book is a stand‐alone introduction to particular problems of integration in the probabilistic theory of stochastic calculus, and in the probability‐like theory of quantum mechanics.

      Between [MTRV] and this book there is a significant difference in style of exposition. Practically all the underlying mathematical theory is already set out in [MTRV]. The present book includes motivational explanation of the key points of the underlying mathematical theory, along with ample illustrations of the calculus—the routine procedures—of the gauge theory of integration.

      But because the “mathematical heavy lifting” (or rigorous mathematical underpinning) is already accomplished in [MTRV], the present book can take a more gradual, relaxed, and discursive approach which seeks to engage the reader with the subject by exploring a much smaller range of chosen themes.

      Thus there is hardly anything of the formal Theorem‐Proof structure in this book. Instead the text is organised around Examples with accompanying introductions and explanation, illustrating themes from probability and physics which can be difficult and taxing. Particular areas of interest in the book can be selected and read without engaging with other topics. Its relatively self‐contained component parts can easily be “dipped into”.

      In addition to [MTRV], two principal physics sources for this book are [39], Space‐time approach to non‐relativistic quantum mechanics (cited as [F1] for short) by R. Feynman; and [46], Quantum Mechanics and Path Integrals (cited as [FH] for short) by R. Feynman and A. Hibbs.

Certain modes of expression used by physicist R. Feynman are highly illuminating—but from a physics perspective. For instance: Integrate [some expression] over all degrees of freedom [all variables] of the [physical] system. This statement does not specify the mathematical domain

for this integration process, nor how Скачать книгу