Functional Analysis and the Feynman Operator Calculus

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.

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

Inhalt
Preliminary Background.- Integration on Infinite-dimensional Spaces.- HK-Integral and HK-Spaces.- Analysis on Hilbert Space.- Operators on Banach Space.- Spaces of von Neumann Type.- The Feynman Operator Calculus.- Applications of the Feynman Calculus.