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Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics
Patrick Muldowney
This edition first published copyright
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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
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
