Maxwell-Chern-Simons Theory in Covariant and Coulomb Gauges

TL;DR

This paper quantizes 2+1 dimensional Quantum Electrodynamics with a Chern-Simons term in covariant and Coulomb gauges, analyzing gauge conditions, particle states, and Lorentz symmetry, and showing no fractional statistics arise from Gauss's law.

Contribution

It provides a detailed construction of charged particle states in Maxwell-Chern-Simons theory across different gauges, demonstrating gauge invariance of interactions and the Poincaré algebra.

Findings

Interactions are gauge-independent in the analyzed representations.

Gauss's law does not induce fractional statistics.

The Poincaré algebra is realized explicitly in the constructed states.

Abstract

We quantize Quantum Electrodynamics in $2+1$ dimensions coupled to a Chern-Simons (CS) term and a charged spinor field, in covariant gauges and in the Coulomb gauge. The resulting Maxwell-Chern-Simons (MCS) theory describes charged fermions interacting with each other and with topologically massive propagating photons. We impose Gauss's law and the gauge conditions and investigate their effect on the dynamics and on the statistics of $n$-particle states. We construct charged spinor states that obey Gauss's law and the gauge conditions, and transform the theory to representations in which these states constitute a Fock space. We demonstrate that, in these representations, the nonlocal interactions between charges and between charges and transverse currents, as well as the interactions between currents and massive propagating photons, are identical in the different gauges we analyze in this and in earlier work. We construct the generators of the Poincar\'e group, show that they implement the Poincar\'e algebra, and explicitly demonstrate the effect of rotations and Lorentz boosts on the particle states. We show that the imposition of Gauss's law does not produce any ``exotic'' fractional statistics. In the case of the covariant gauges, this demonstration makes use of unitary transformations that provide charged particles with the gauge fields required by Gauss's law, but that leave the anticommutator algebra of the spinor fields untransformed. In the Coulomb gauge, we show that the anticommutators of the spinor fields apply to the Dirac-Bergmann constraint surfaces, on which Gauss's law and the gauge conditions obtain. We examine MCS theory in the large CS coupling constant limit, and compare that limiting form with CS theory, in which the Maxwell kinetic energy term is not included in the