Feynman's Path Integrals as Evolutionary Semigroups

TL;DR

This paper establishes a rigorous mathematical foundation for Feynman's path integrals in certain systems, linking them to evolutionary semigroups and utilizing Henstock's integration technique.

Contribution

It introduces a formal mathematical basis for path integrals using Henstock's integration and connects propagators to evolutionary semigroups for systems with specific Lagrangians.

Findings

Path integrals can be rigorously defined for certain systems.

Propagators are expressible as evolutionary semigroups.

Phase factors relate to Noether's theorem in this context.

Abstract

We show that, for a class of systems described by a Lagrangian L(x,\dot{x},t) = 1/2\dot{x}^{2} - V(x,t) the propagator can be reduced via Noether's Theorem to a standard path integral multiplied by a phase factor. Using Henstock's integration technique, this path integral is given a firm mathematical basis. Finally, we recast the propagator as an evolutionary semigroup.