The Inhomogeneity Expansion for Planar Q.E.D in a Magnetic Background

TL;DR

This paper develops an inhomogeneity expansion method to derive the effective action for planar Q.E.D in magnetic backgrounds, revealing new terms like Chern-Simons and Zeeman contributions in both non-relativistic and relativistic regimes.

Contribution

It introduces a novel inhomogeneity expansion approach to analyze the effective action of planar Q.E.D in magnetic fields, including the derivation of Chern-Simons and Zeeman terms.

Findings

Generated a Chern-Simons like term in non-relativistic case

Derived the effective action for relativistic Q.E.D in magnetic background

Linked four-fermi interactions to Zeeman-like terms

Abstract

The effective action for Q.E.D in external magnetic field is constructed using the method of inhomogeneity expansion. We first treat the non-relativistic case where a Chern-Simons like term is generated. We then consider the full relativistic theory and derive the effective action for the $A_{\mu}$ fields. In the non-relativistic case we also add a 4-fermi type interaction and show that under certain circumstances, it corresponds to a Zeeman type term in the effective action.