On Mode Regularization of the Configuration Space Path Integral in Curved Space

TL;DR

This paper reviews mode regularization for the path integral in curved space, demonstrating its consistency through a three-loop computation and comparing it with time discretization methods.

Contribution

It introduces and validates a three-loop computation method for the configuration space path integral in curved space using mode regularization with Lee-Yang ghost fields.

Findings

Mode regularization is consistent at three loops in curved space.

The method reproduces results obtained via time discretization.

Effective potentials are identified at two loops for both regularization schemes.

Abstract

The path integral representation of the transition amplitude for a particle moving in curved space has presented unexpected challenges since the introduction of path integrals by Feynman fifty years ago. In this paper we discuss and review mode regularization of the configuration space path integral, and present a three loop computation of the transition amplitude to test with success the consistency of such a regularization. Key features of the method are the use of the Lee-Yang ghost fields, which guarantee a consistent treatment of the non-trivial path integral measure at higher loops, and an effective potential specific to mode regularization which arises at two loops. We also perform the computation of the transition amplitude using the regularization of the path integral by time discretization, which also makes use of Lee-Yang ghost fields and needs its own specific effective potential. This computation is shown to reproduce the same final result as the one performed in mode regularization.