On the numerical evaluation of real-time path integrals: Double exponential integration and the Maslov correction

TL;DR

This paper applies double exponential integration to oscillatory path integrals in real-time quantum mechanics, demonstrating high-precision numerical evaluation of the Maslov correction and discussing prospects for scattering amplitude calculations.

Contribution

It introduces a novel application of Ooura's double exponential integration to real-time path integrals, enabling accurate numerical evaluation of the Maslov correction and phase behavior.

Findings

High-precision numerical evaluation of multi-dimensional Gauss-Fresnel integrals.

Demonstration of the Maslov phase correction in the harmonic oscillator.

Discussion of potential for direct numerical scattering amplitude calculations.

Abstract

Ooura's double exponential integration formula for Fourier transforms is applied to the oscillatory integrals occuring in the path-integral description of real-time Quantum Mechanics. Due to an inherent, implicit regularization multi-dimensional Gauss-Fresnel integrals are obtained numerically with high precision but modest number of function calls. In addition, the Maslov correction for the harmonic oscillator is evaluated numerically with an increasing number of time slices in the path integral thereby clearly demonstrating that the real-time propagator acquires an additional phase $ - \pi/2 $ each time the particle passes through a focal point. However, in the vicinity of these singularities an overall small damping factor is required. Prospects of evaluating scattering amplitudes of finite-range potentials by direct numerical evaluation of a real-time path integral are discussed.