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

TL;DR

The paper investigates quantum mechanical path integrals in curved spaces on an infinite time interval using dimensional regularization (DR). It compares DR with mode and time-slicing regularizations, showing that DR does not require noncovariant counterterms but does require a covariant two-loop counterterm $V_{DR} = {\hbar^2\over 8} R$ to reproduce known results, while an infrared mass regulator breaks covariance in the final expression. The authors perform 3-loop calculations in Riemann normal coordinates and 2-loop calculations in general coordinates to demonstrate the necessity and form of the covariant counterterm, establishing the covariance properties of DR at these loop orders. The findings clarify the role of regularization in preserving general coordinate invariance in worldline quantum mechanics and point to extensions to finite propagation time relevant for anomalies in quantum field theory.

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.