The Path Integral for a Particle in Curved Spaces and Weyl Anomalies

TL;DR

This paper develops a covariant method using ghost fields to compute Weyl anomalies via path integrals for particles in curved spaces, providing a new approach for anomaly calculations in quantum field theory.

Contribution

It introduces a ghost field technique for covariant path integral evaluation of Weyl anomalies, enhancing the computational framework for anomalies in curved backgrounds.

Findings

Derived the Hamiltonian for the evolution kernel in curved space

Successfully computed the trace for Weyl anomalies in two dimensions

Provided a perturbative expansion method for path integrals with ghost fields

Abstract

The computation of anomalies in quantum field theory may be carried out by evaluating path integral Jacobians, as first shown by Fujikawa. The evaluation of these Jacobians can be cast in the form of a quantum mechanical problem, whose solution has a path integral representation. For the case of Weyl anomalies, also called trace anomalies, one is immediately led to study the path integral for a particle moving in curved spaces. We analyze the latter in a manifestly covariant way and by making use of ghost fields. The introduction of the ghost fields allows us to represent the path integral measure in a form suitable for performing the perturbative expansion. We employ our method to compute the Hamiltonian associated with the evolution kernel given by the path integral with fixed boundary conditions, and use this result to evaluate the trace needed in field theoretic computation of Weyl anomalies in two dimensions.