Overview
Provides a mathematically rigorous formulation of both the path integral and the operator calculus in the manner intended by Feynman
Unifies the theory of partial differential equations with time-dependent coefficients using Feynman's theory
Provides the first introduction to Lebesgue integration on Banach spaces
Includes a direct approach to the study of time-dependent evolution equations in both finite and infinite-dimensional settings
Solves the second of the two open Dyson conjectures
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Table of contents (8 chapters)
Keywords
About this book
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. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers.
Reviews
“The book is a self-contained treatise on the mathematical foundation of Feynman operational calculus and Feynman path integrals. … It contains a large amount of original material which cannot be found elsewhere in book form; in fact, most of the original results are due to the authors themselves. … The book will be of interest to both graduate students and researchers in pure or applied mathematics.” (Sonia Mazzucchi, Mathematical Reviews, November, 2016)
“This book carries Fujiwara's insight to provide the mathematical frameworks for Feynman's operator calculus and for the Feynman path integral. … The book is intended for advanced graduate students and researchers and can be used as a text for advanced courses in functional analysis, operator theory, mathematical physics, or related subjects.” (Miyeon Kwon, zbMATH 1345.81001, 2016)>
Authors and Affiliations
Bibliographic Information
Book Title: Functional Analysis and the Feynman Operator Calculus
Authors: Tepper L. Gill, Woodford W. Zachary
DOI: https://doi.org/10.1007/978-3-319-27595-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2016
Hardcover ISBN: 978-3-319-27593-2Published: 08 April 2016
Softcover ISBN: 978-3-319-80180-3Published: 25 April 2018
eBook ISBN: 978-3-319-27595-6Published: 30 March 2016
Edition Number: 1
Number of Pages: XIX, 354
Number of Illustrations: 3 illustrations in colour
Topics: Functional Analysis, Mathematical Methods in Physics, Operator Theory