Constructive Representation Theory for the Feynman Operator Calculus

TL;DR

This paper develops a constructive operator theory for the Feynman path integral, extending its mathematical foundation and applying it to quantum and evolution equations with a focus on operator commutativity and integrals.

Contribution

It introduces a new constructive approach to Feynman operator calculus, including an operator Henstock-Kurzweil integral and a Hilbert space framework for path integrals.

Findings

Unified theory of time-dependent evolution equations.

Extended semigroup theorems like Hille-Yosida.

Generalized Feynman path integral with broader interactions.

Abstract

In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. (The theory is constructive in that, operators acting at different times, actually commute.) We first develop an operator version of the Henstock-Kurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner originally envisioned by Feynman. After developing our time-ordered operator theory we extend a few of the important theorems of semigroup theory, including the Hille-Yosida theorem. 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 Poincare. 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 extend the Feynman path integral to include more general interactions. Our approach is independent of the space of continuous functions and thus makes the question of the existence of a measure more of a natural expectation than a death blow to the foundations for the Feynman integral.