The Feynman Integral and Feynman's Operational Calculus [Paperback]

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.

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 Schr?dinger 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