Poisson Gauge Theories in Three Dimensions: Exact Solutions and Conservation Laws

TL;DR

This paper studies Maxwell-Chern-Simons theory on noncommutative 3D spacetime, finding exact solutions with finite energy, novel flux properties, and a generalized Gauss's law, highlighting noncommutativity as a natural regulator.

Contribution

It constructs exact classical solutions in noncommutative 3D gauge theory, demonstrating noncommutativity's role in regularizing energy and enabling flux generation.

Findings

Noncommutativity ensures finite electromagnetic energy.

Solutions exhibit non-perturbative dependence on noncommutativity.

A noncommutative Gauss's law is formulated.

Abstract

We investigate Maxwell-Chern-Simons theory on a three-dimensional noncommutative spacetime endowed with a constant spacelike Poisson structure. By exploiting the residual rotational symmetry, we construct exact classical solutions corresponding to pointlike electric and magnetic charges. We demonstrate that noncommutativity acts as a natural regulator, ensuring a finite total electromagnetic energy and thereby resolving the classical self-energy divergence. Furthermore, some of these solutions exhibit a non-perturbative dependence on the noncommutativity parameter and allow for the generation of an arbitrary magnetic flux. We also present a noncommutative generalization of Gauss's law, providing a robust framework for the physical interpretation of these exact solutions.