Generalization of Euler's Summation Formula to Path Integrals

TL;DR

This paper extends Euler's summation formula to path integrals, enabling faster convergence from 1/N to 1/N^p, with Monte Carlo simulations confirming the theoretical improvements.

Contribution

It introduces a generalized Euler's summation formula for path integrals that systematically improves convergence rates.

Findings

Convergence improves from 1/N to 1/N^p with the new formula.

Monte Carlo simulations confirm the theoretical speedup.

The method applies to various models, demonstrating broad effectiveness.

Abstract

A recently developed analytical method for systematic improvement of the convergence of path integrals is used to derive a generalization of Euler's summation formula for path integrals. The first $p$ terms in this formula improve convergence of path integrals to the continuum limit from 1/N to $1/N^p$, where $N$ is the coarseness of the discretization. Monte Carlo simulations performed on several different models show that the analytically derived speedup holds.