Canonical Quantization of $(2+1)$ Dimensional Qed with Topological Mass Term

TL;DR

This paper presents a canonical quantization approach for 2+1 dimensional QED with a topological Chern-Simons mass term, analyzing gauge fixing, photon propagator, and angular momentum properties of charged particles.

Contribution

It introduces a gauge-fixing scheme that avoids primary constraints, derives the photon propagator, and clarifies the angular momentum and rotation behavior of charged particles in topologically massive QED.

Findings

Photon propagator derived from canonical theory

Charged particles behave as fermions under rotations

Separation of electric and magnetic fields reflects massive photon presence

Abstract

We discuss the canonical quantization of Quantum Electrodynamics in $2+1$ dimensions, with a Chern-Simons topological mass term and gauge-covariant coupling to a Dirac spinor field. A gauge-fixing term is used which generates a canonical momentum for $A_0$, so that there are no primary constraints on operator-valued fields. Gauss's Law and the gauge condition, $A_0=0$, are implemented by embedding the formulation in an appropriate physical subspace, in which state vectors remain naturally, in the course of time evolution. The photon propagator is derived from the canonical theory. The electric and magnetic fields are separated into parts that reflect the presence of massive photons, and other parts that are rigidly attached to charged fermions and do not consist of any observable, propagating particle excitations. The effect of rotations on charged particle states is analyzed, and the relation between the canonical and the Belinfante ``symmetric'' angular momentum is discussed. It is shown that the rotation operator can be consistently formulated so that charged particles behave like fermions, and do not acquire any arbitrary phases during rotations, even when they are dressed in the electromagnetic fields required for them to obey Gauss's law.