Mode Regularization of the Configuration Space Path Integral for a Particle in Curved Space

TL;DR

This paper introduces a mode regularization scheme for the configuration space path integral of a particle in curved space, ensuring a consistent, coordinate-invariant formulation that reproduces known heat kernel results.

Contribution

It proposes a novel regularization method using Fourier sine series, Lee-Yang ghost fields, and an effective potential to handle the path integral measure and maintain invariance.

Findings

Regularization scheme reproduces De Witt's heat kernel expansion

Three-loop computation validates the mode regularization approach

Ghost fields and effective potential ensure finite, invariant path integrals

Abstract

The proper definition and evaluation of the configuration space path integral for the motion of a particle in curved space is a notoriously tricky problem. We discuss a consistent definition which makes use of an expansion in Fourier sine series of the particle paths. Salient features of the regularization are the Lee-Yang ghosts fields and a specific effective potential to be added to the classical action. The Lee-Yang ghost fields are introduced to exponentiate the non-trivial path integral measure and make the perturbative loop expansion finite order by order, whereas the effective potential is necessary to maintain the general coordinate invariance of the model. We also discuss a three loop computation which tests the mode regularization scheme and reproduces consistently De Witt's perturbative solution of the heat kernel.