Extended Feynman Formula for the Harmonic Oscillator by the Discrete Time Method

TL;DR

This paper presents a more rigorous derivation of the Feynman formula for the harmonic oscillator at caustics using a discrete path integral regularization, improving upon previous methods.

Contribution

It introduces a discrete time method for deriving the Feynman formula at caustics, offering a more reliable and rigorous approach than prior techniques.

Findings

Derived the Feynman formula for the harmonic oscillator at caustics using discrete path integral.

Provided a more rigorous regularization method for path integrals at caustics.

Enhanced the mathematical understanding of quantum harmonic oscillators.

Abstract

We calculate the Feynman formula for the harmonic oscillator beyond and at caustics by the discrete formulation of path integral. The extension has been made by some authors, however, it is not obtained by the method which we consider the most reliable regularization of path integral. It is shown that this method leads to the result with, especially at caustics, more rigorous derivation than previous.