Tepper L. Gill and Woodford W. Zachary
Feynman Operator Calculus: The Constructive Theory
(196K, laTeX)
ABSTRACT. In this paper, we survey progress on the constructive foundations for the Feynman operator calculus in which operators acting at different times commute. We begin with the development of an operator version of the Henstock-Kurzweil integral. After developing our time-ordered operator theory we extend a few of the important theorems of semigroup theory, including the Hille-Yosida theorem. This means that our approach is a natural extension of basic operator theory to the time-ordered setting. As an application, we unify and extend the theory of time-dependent parabolic and hyperbolic evolution equations. We then develop a general perturbation theory and use it to prove that all theories generated by semigroups are asympotic in the operator-valued sense of Poincar\'{e}. This allows us to provide a general theory for the interaction representation of relativistic quantum theory. We then show that our theory can be reformulated as a physically motivated sum over paths, and use this version to define the Feynman path integral and to include general interactions. Our approach is independent of the space measures and the space of continuous functions, and thus makes it clear that the need for a measure is more of a natural expectation based on past experience than a death blow to the foundations for the Feynman path integral.
In addition, we provide a simple and direct solution to the problem of disentanglement, a method used by Feynman to relate his theory to conventional analysis. Using our disentanglement approach, we extend the Trotter-Kato theory to include the case where the intersection of the domains of two operators may be zero.