User's guide to Monte Carlo methods for evaluating path integrals

TL;DR

This paper introduces Monte Carlo techniques for evaluating path integrals on a lattice, focusing on the quantum harmonic oscillator, addressing autocorrelation issues, and providing computational guidance for broader applications.

Contribution

It offers a detailed computational framework and pseudocode for lattice path integrals, including autocorrelation mitigation strategies like over-relaxation.

Findings

Explicit Monte Carlo setup for path integrals

Effective over-relaxation technique to reduce autocorrelations

Guidance on error estimation and extension to other systems

Abstract

We give an introduction to the calculation of path integrals on a lattice, with the quantum harmonic oscillator as an example. In addition to providing an explicit computational setup and corresponding pseudocode, we pay particular attention to the existence of autocorrelations and the calculation of reliable errors. The over-relaxation technique is presented as a way to counter strong autocorrelations. The simulation methods can be extended to compute observables for path integrals in other settings.