Quantum Electrodynamics of Particles on a Plane and the Chern-Simons Theory

TL;DR

This paper develops an effective 2+1 dimensional gauge theory for charged particles on a plane, revealing a natural emergence of Chern-Simons terms from QED and exploring implications for condensed matter physics.

Contribution

It demonstrates how the QED Lagrangian inherently produces Chern-Simons constraints and topological terms in a 2+1D setting, linking gauge theory with condensed matter applications.

Findings

QED on a plane yields an effective Chern-Simons gauge theory.

Topological $ heta$-term induces Chern-Simons contributions.

Potential connections to bosonization and condensed matter systems.

Abstract

We study the electrodynamics of generic charged particles (bosons, fermions, relativistic or not) constrained to move on an infinite plane. An effective gauge theory in 2+1 dimensional spacetime which describes the real electromagnetic interaction of this particles is obtained. The relationship between this effective theory with the Chern-Simons theory is explored. It is shown that the QED lagrangian {\it per se} produces the Chern-Simons constraint relating the current to the effective gauge field in 2+1 D. It is also shown that the geometry of the system unavoidably induces a contribution from the topological $\theta$-term that generates an explicit Chern-Simons term for the effective 2+1 dimensional gauge field as well as a minimal coupling of the matter to it. The possible relation of the effective three dimensional theory with the bosonization of the Dirac fermion field in 2+1 D is briefly discussed as well as the potential applications in Condensed Matter systems.