The Feynman integral as a limit of complex measures

TL;DR

This paper presents a novel approach to defining the Feynman integral as a limit of complex measures derived from smooth approximations of the Schrödinger equation's fundamental solution, providing a rigorous mathematical framework.

Contribution

It introduces a method to approximate the Feynman integral using complex measures associated with smooth functions, establishing convergence for certain test functions.

Findings

The Feynman integral can be rigorously defined as a limit of complex measures.

Convergence of integrals holds for specific classes of test functions.

Provides a new mathematical foundation for path integral formulations.

Abstract

The fundamental solution of the Schr\"odinger equation for a free particle is a distribution. This distribution can be approximated by a sequence of smooth functions. It is defined for each one of these functions, a complex measure on the space of paths. For certain test functions, the limit of the integrals of a test function with respect to the complex measures, exists. We define the Feynman integral of one such function by this limit.