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Annales mathématiques Blaise Pascal

The Feynman integral and Feynman's operational calculus : a heuristic and mathematical introduction

Michel L. Lapidus

Published Jan 1, 1996• 0 citations
Publisher
DOI
10.5802/ambp.55

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It is known that Wiener and Feynman path integrals provide one way of making Feynman’s heuristic operational calculus for noncommuting operators mathematically rigorous. The disentangling process and associated operator orderings are central to Feynman’s ideas. We begin here to study the effects of time maps in clarifying the disentangling process and in altering the operator orderings in certain prescribed ways. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract The first part of the introduction provides a (short) historical overview of Feynman’s development of his operational calculus, i.e. the formation of functions of several not necessarily commuting operators. This is done using quotes from Feynman’s Nobel lecture, as well as from biographical information and parts of Feynman’s early papers. This portion of the introduction will aid the reader not familiar with Feynman’s ideas to gain some insight into their intellectual and physical origins. After this historical sketch of the origins of the operational calculus, we take the time to introduce, via simple examples, Feynman’s time-ordering convention and Feynman’s heuristic “rules.” The remainder of the introduction is devoted to a rather detailed chapter-by-chapter description of the contents of this volume.
Abstract The aim of this book is to make accessible to mathematicians, physicists and other scientists interested in qunatum theory, the beautiful but mathematically difficult subjects of the Feynman integral and Feynman's operational calculus. Some advantages of the approaches to the Feynman integral which are treated in detail in this book are the following: the existence of the Feynman integral is established for very general potentials in all four cases; under more restrictive but still broad conditions, three of these Feynman integrals agree with one another and with the unitary group from the usual approach to quantum dynamics; these same three Feynman integrals possess pleasant stability properties. Much of the material covered here was previously available only in the research literature, and the book also contains some new results. The background material in mathematics and physics that motivates the study of the Feynman integral and Feynman's operational calculus is discussed, and detailed proofs are provided for the central results.