Dirac Quantization of the Chern-Simons Field Theory in the Coulomb Gauge

TL;DR

This paper analyzes Chern-Simons field theories in the Coulomb gauge using Dirac's formalism, revealing their two-dimensional nature and trivial field commutation relations in the absence of matter couplings.

Contribution

It provides a detailed canonical quantization of Chern-Simons theories in Coulomb gauge, highlighting their reduced dimensionality and the behavior of field commutators with and without matter interactions.

Findings

Chern-Simons theories become two-dimensional models with no time propagation.

Field commutators are trivial at all orders without matter couplings.

With matter couplings, commutators are complex but vanish at tree level.

Abstract

In this letter the Chern-Simons field theories are studied in the Coulomb gauge using the Dirac's canonical formalism for constrained systems. As a strategy, we first work out the constraints and then quantize, replacing the Dirac brackets with quantum commutators. We find that the Chern-Simons field theories become two dimensional models with no propagation along the time direction. Moreover, we prove that, despite of the presence of non-trivial self-interactions in the gauge fixed functional, the commutation relations between the fields are trivial at any order in perturbation theory in the absence of couplings with matter fields. If these couplings are present, instead, the commutation relations become rather involved, but it is still possible to study their main properties and to show that they vanish at the tree level.