A rigorous real time Feynman Path Integral and Propagator

TL;DR

This paper rigorously derives a real-time Feynman path integral and propagator for non-relativistic quantum systems, including explicit formulas, for systems with certain singularities, and applies nonstandard analysis to compute the harmonic oscillator propagator.

Contribution

It introduces a rigorous real-time Feynman path integral formulation applicable to a broad class of Hamiltonians, including explicit computation for the harmonic oscillator using nonstandard analysis.

Findings

Explicit real-time propagator for non-relativistic quantum systems.

Path integral representation valid for Hamiltonians with finite singularities.

Computed harmonic oscillator propagator without classical assumptions.

Abstract

We will derive a rigorous real time propagator for the Non-relativistic Quantum Mechanic $L^2$ transition probability amplitude and for the Non-relativistic wave function. The propagator will be explicitly given in terms of the time evolution operator. The derivation will be for all self-adjoint nonvector potential Hamiltonians. For systems with potential that carries at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real time, time sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in Nonstandard Analysis Formulation. Finally, we will compute the propagator for the harmonic oscillator using the Nonstandard Analysis Feynman path integral formuluation; we will compute the propagator without using any knowledge of classical properties of the harmonic oscillator.