Dimensional regularization of the path integral in curved space on an infinite time interval

TL;DR

This paper applies dimensional regularization to quantum mechanical path integrals in curved spaces, demonstrating covariant renormalization at two and three loops and analyzing the impact of mass terms on covariance.

Contribution

It introduces a covariant two-loop counterterm for path integrals in curved space using dimensional regularization, aligning results with previous schemes.

Findings

Covariant two-loop counterterm V_{DR} = R/8 identified.

Dimensional regularization yields consistent results with other methods.

Mass term breaks covariance explicitly in the final result.

Abstract

We use dimensional regularization to evaluate quantum mechanical path integrals in arbitrary curved spaces on an infinite time interval. We perform 3-loop calculations in Riemann normal coordinates, and 2-loop calculations in general coordinates. It is shown that one only needs a covariant two-loop counterterm (V_{DR} = R/8) to obtain the same results as obtained earlier in other regularization schemes. It is also shown that the mass term needed in order to avoid infrared divergences explicitly breaks general covariance in the final result.