Tests and applications of Migdal's particle path-integral representation for the Dirac Propagator

TL;DR

This paper explores Migdal's particle path-integral approach to the Dirac propagator, deriving non-perturbative results in lower dimensions, including the chiral anomaly and Chern-Simons current, and analyzing particle propagation in electromagnetic backgrounds.

Contribution

It provides new non-perturbative derivations of the chiral anomaly and Chern-Simons current using Migdal's path-integral formulation in 1+1 and 2+1 dimensions.

Findings

Derivation of the chiral anomaly in 1+1 dimensions.

Derivation of the Chern-Simons current in 2+1 dimensions.

Analysis of particle propagation in constant electromagnetic fields.

Abstract

We derive some non-perturbative results in 1+1 and 2+1 dimensions within the context of the particle path-integral representation for a Dirac field propagator in the presence of an external field, in a formulation introduced by Migdal in 1986. We consider the specific properties of the path-integral expressions corresponding to the 1+1 and 2+1 dimensional cases, presenting a derivation of the chiral anomaly in the former and of the Chern-Simons current in the latter. We also discuss particle propagation in constant electromagnetic field backgrounds.