Synopses & Reviews
The aim of this book is to make accessible to mathematicians, physicists and other scientists interested in quantum theory, the mathematically beautiful but difficult subjects of the Feynman integral and Feynman's operational calculus. Some advantages of the four approaches to the Feynman integral which are given detailed treatment 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 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.
Review
"A most scholarly text which is comprehensive, detailed and very clearly written. It embraces the whole of the topic not just one part of it, and the historical references give an insight into the development of the ideas behind this fascinating approach to quantum theory. Written by experts who are also good teachers."--Aslib Book Guide
"The idea behind the Feynman path integral goes back to a paper by P. A. M. Dirac published in 1933 in Physikalische Zeitschrift der Sowjetunion. It formed the core of Richard Feynman's space-time approach to quantum mechanics and quantum electrodynamics. Although the path integral was not mathematically well defined, it was widely used in quantum field theory, statistical mechanics, and string theory. Recently, path integrals have been the heuristic guide to spectacular developments in pure mathematics. It was clear to Feynman that his 'path integral' was no integral in the ordinary sense of the word, and that what he called its 'summation over histories' did not involve a measure in the usual sense. ... The book by Johnson and Lapidus deals with various approaches to making the Feynman path integral into a mathematically meaningful object. ... I would recommend this book to serious students of the subject ..."--Physics Today
Table of Contents
1. Introduction
2. The physical phenomenon of Brownian motion
3. Wiener measure
4. Scaling in Wiener space and the analytic Feynman integral
5. Stochastic processes and the Wiener process
6. Quantum dynamics and the Schrödinger equation
7. The Feynman integral: heuristic ideas and mathematical difficulties
8. Semigroups of operators: an informal introduction
9. Linear semigroups of operators
10. Unbounded self-adjoint operators and quadratic forms
11. Product formulas with applications to the Feynman integral
12. The Feynman-Kac formula
13. Analytic-in-time or -mass operator-valued Feynman integrals
14. Feynman's operational calculus for noncommuting operators: an introduction
15. Generalized Dyson series, the Feynman integral and Feynman's operational calculus
16. Stability results
17. The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus
18. Noncommutative operations on Wiener functionals, disentangling algebras and Feynman's operational calculus
19. Feynman's operational calculus and evolution equations
20. Further work on or related to the Feynman integral
References
Index of symbols
Author index
Subject index