A Rigorous Real Time Feynman Path Integral

TL;DR

This paper develops a rigorous mathematical formulation of the real-time Feynman path integral using improper Riemann integrals and nonstandard analysis, applicable to certain Hamiltonians with discontinuities.

Contribution

It introduces a rigorous, mathematically sound version of the real-time Feynman path integral for specific Hamiltonians, extending previous heuristic approaches.

Findings

Provides a rigorous formulation using improper Riemann integrals.

Includes a nonstandard analysis perspective.

Applicable to Hamiltonians with finite discontinuities and singularities.

Abstract

Using improper Riemann integrals, we will formulate a rigorous version of the real-time, time-sliced Feynman path integral for the $L^2$ transition probability amplitude. We will do this for nonvector potential Hamiltonians with potential which has at most a finite number of discontinuities and singularities. We will also provide a Nonstandard Analysis version of our formulation.