Trace Anomalies from Quantum Mechanics

TL;DR

This paper presents a method to compute trace anomalies for various fields in gravitational and Yang-Mills backgrounds using a path integral approach that maintains gauge covariance and avoids supersymmetric sigma models.

Contribution

It introduces a direct path integral formulation with matrix-valued actions for trace anomalies, avoiding supersymmetric sigma models and clarifying divergence cancellations.

Findings

Computed trace anomalies in 2 and 4 dimensions.

Maintained gauge covariance with a matrix-valued action.

Demonstrated divergence cancellation using vector ghosts.

Abstract

The 1-loop anomalies of a d-dimensional quantum field theory can be computed by evaluating the trace of the regulated path integral jacobian matrix, as shown by Fujikawa. In 1983, Alvarez-Gaum\'e and Witten observed that one can simplify this evaluation by replacing the operators which appear in the regulator and in the jacobian by quantum mechanical operators with the same (anti)commutation relations. By rewriting this quantum mechanical trace as a path integral with periodic boundary conditions for a one-dimensional supersymmetric nonlinear sigma model, they obtained the chiral anomalies for spin 1/2 and 3/2 fields and selfdual antisymmetric tensors in d dimensions. In this article, we treat the case of trace anomalies for spin 0, 1/2 and 1 fields in a gravitational and Yang-Mills background. We do not introduce a supersymmetric sigma model, but keep the original Dirac matrices $\g^\m$ and internal symmetry generators $T^a$ in the path integral. As a result, we get a matrix-valued action. Gauge covariance of the path integral then requires a definition of the exponential of the action by time-ordering. We exponentiate the factors $\sqrt g$ in the path integral measure by using vector ghosts in order to exhibit the cancellation of the sigma model divergences more clearly. We compute the trace anomalies in d=2 and d=4.