The possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator

TL;DR

This paper explores a non-perturbative correction to Mehler's formula for the harmonic oscillator propagator by evaluating a Wiener measure integral with a quartic term, offering an alternative to traditional perturbation methods.

Contribution

It introduces a novel non-perturbative approach to modify Mehler's formula, expanding the linear term in the exponent, and provides analytical results for different frequencies.

Findings

Derived analytical expressions for the corrected propagator.

Demonstrated the method for both positive and negative frequencies.

Offered an alternative to perturbative techniques for anharmonic corrections.

Abstract

We find the possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator. We evaluate the conditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series the term linear in the integration variable in the exponent. We discuss the case, when the starting point of the propagator is zero. We present the results in analytical form for positive and negative frequency.