A Monte Carlo Approach to the Worldline Formalism in Curved Space

TL;DR

This paper introduces a Monte Carlo numerical method for evaluating worldline path integrals in curved space, incorporating a modified YLOOPS algorithm and proper time discretization, validated against known heat kernel results.

Contribution

The paper develops a novel Monte Carlo approach with a modified YLOOPS algorithm for worldline integrals in curved space, including proper time discretization and counter-terms.

Findings

Successfully tested against analytic heat kernel calculations in symmetric spaces

Demonstrated improved convergence and stability of the numerical method

Validated the approach as a viable tool for quantum field theory in curved backgrounds

Abstract

We present a numerical method to evaluate worldline (WL) path integrals defined on a curved Euclidean space, sampled with Monte Carlo (MC) techniques. In particular, we adopt an algorithm known as YLOOPS with a slight modification due to the introduction of a quadratic term which has the function of stabilizing and speeding up the convergence. Our method, as the perturbative counterparts, treats the non-trivial measure and deviation of the kinetic term from flat, as interaction terms. Moreover, the numerical discretization adopted in the present WLMC is realized with respect to the proper time of the associated bosonic point-particle, hence such procedure may be seen as an analogue of the time-slicing (TS) discretization already introduced to construct quantum path integrals in curved space. As a result, a TS counter-term is taken into account during the computation. The method is tested against existing analytic calculations of the heat kernel for a free bosonic point-particle in a D-dimensional maximally symmetric space.