Canonical Quantization of the Maxwell-Chern-Simons Theory in the Coulomb Gauge

TL;DR

This paper presents a canonical quantization of the Maxwell-Chern-Simons theory in Coulomb gauge, establishing a consistent framework for polarization vectors, Poincaré densities, and scattering amplitudes, ensuring gauge invariance and anomaly freedom.

Contribution

It introduces a detailed quantization procedure in Coulomb gauge, constructs anomaly-free Poincaré densities, and proves the equivalence of Coulomb and covariant Feynman rules for scattering amplitudes.

Findings

Polarization vector structure explains gauge boson spin.

Coulomb gauge Feynman rules match covariant results.

Infrared safe photon propagator simplifies interaction Hamiltonian.

Abstract

The Maxwell-Chern-Simons theory is canonically quantized in the Coulomb gauge by using the Dirac bracket quantization procedure. The determination of the Coulomb gauge polarization vector turns out to be intrincate. A set of quantum Poincar\'e densities obeying the Dirac-Schwinger algebra, and, therefore, free of anomalies, is constructed. The peculiar analytical structure of the polarization vector is shown to be at the root for the existence of spin of the massive gauge quanta.The Coulomb gauge Feynman rules are used to compute the M\"oller scattering amplitude in the lowest order of perturbation theory. The result coincides with that obtained by using covariant Feynman rules. This proof of equivalence is, afterwards, extended to all orders of perturbation theory. The so called infrared safe photon propagator emerges as an effective propagator which allows for replacing all the terms in the interaction Hamiltonian of the Coulomb gauge by the standard field-current minimal interaction Hamiltonian.