The Feynman Integral and Feynman's Operational Calculus
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The Feynman Integral and Feynman's Operational Calculus

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Gerald W. Johnson
1116 g
235x159x42 mm
Oxford Mathematical Monographs

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 Schroedinger 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. Generalised 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
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 Feynmanintegral 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 theunitary 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 mathematicsand 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.