Rigorous Real-Time Feynman Path Integral for Vector Potentials

TL;DR

This paper establishes a rigorous mathematical framework for real-time Feynman path integrals involving vector potentials, ensuring existence and uniqueness under certain conditions, and deriving the formulation via improper Riemann integrals.

Contribution

It introduces a novel rigorous formulation of the real-time Feynman path integral for systems with vector potentials, handling singularities and discontinuities.

Findings

Proves existence and uniqueness of the path integral formulation.

Derives the formulation on the $L^2$ transition probability amplitude.

Handles vector potentials with finite singularities and discontinuities.

Abstract

we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the $L^2$ transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities.