Systematic Speedup of Path Integrals of a Generic $N$-fold Discretized Theory

TL;DR

This paper introduces a systematic analytical method to improve the convergence speed of path integrals in discretized theories, enabling more efficient Monte Carlo simulations for quantum models.

Contribution

The authors develop explicit formulas for effective actions that accelerate the convergence of path integrals, with expressions up to level p=9, enhancing computational efficiency.

Findings

Effective actions reduce discretization errors by order 1/N^p.

Explicit formulas for effective actions up to p=9.

Significant speedup in Monte Carlo simulations for tested models.

Abstract

We present and discuss a detailed derivation of a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. We develop an explicit procedure for calculating a set of effective actions $S^{(p)}$, for $p=1,2,3,...$ which have the property that they lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit ever faster. Discretized amplitudes calculated using the $p$ level effective action differ from the continuum limit by a term of order $1/N^p$. We obtain explicit expressions for the effective actions for levels $p\le 9$. We end by analyzing the speedup of Monte Carlo simulations of two different models: an anharmonic oscillator with quartic coupling and a particle in a modified P\"oschl-Teller potential.