The Feynman Integral and Feynman's Operational Calculus

by Johnson, Gerald W.; Lapidus, Michel L.

• ISBN13: 9780198535744
• ISBN10: 0198535740
• Format: Hardcover
• Copyright: 2000-05-11
• Publisher: Oxford University Press

Summary

The aim of this book is to make accessible to mathematicians, physicistsand other scientists interested in qunatum theory, the beautiful butmathematically difficult subjects of the Feynman integral and Feynman'soperational calculus. Some advantages of the approaches to the Feynmanintegral which are treated in detail in this book are the following: theexistence of the Feynman integral is established for very general potentials inall four cases; under more restrictive but still broad conditions, three ofthese Feynman integrals agree with one another and with the unitary group fromthe usual approach to quantum dynamics; these same three Feynman integralspossess pleasant stability properties. Much of the material covered here waspreviously available only in the research literature, and the book also containssome new results. The background material in mathematics and physics thatmotivates the study of the Feynman integral and Feynman's operational calculusis discussed, and detailed proofs are provided for the central results.

Table of Contents

Introduction

The physical phenomena of Brownian motion

Wiener measure

Scaling in Wiener space and the analytic Feynman integral

Stochastic processes and the Wiener process

Quantum dynamics and the Schrodinger equation

The Feynman integral: heuristic ideas and mathematical difficulties

Semigroups of operators: an informal introduction

Linear semigroups of operators

Unbounded self-adjoint operators and quadratic forms

Product formulas with applications to the Feynman integral

The Feynman-Kac formula

Analytic-in-time or -mass operator-valued Feynman integrals

Feynman's operational calculus for noncommuting operaors: an introduction

Generalized Dyson series, the Feynman integral and Feynman's operational calculus

Stability results

The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus

Noncommutative operations on Wiener functionals, disentangling algebras and Feynman's operational calculus

Feynman's operational calculus and evolution equations

Further work on or related to the Feynman integral

References

Index of Symbols

Author index

Subject index

Table of Contents provided by Publisher. All Rights Reserved.