Charged Particle in Lie-Poisson Electrodynamics

TL;DR

This paper explores the dynamics of a charged particle in Lie-Poisson electrodynamics, deriving gauge-invariant variables, formulating equations of motion, and analyzing specific solvable problems within non-commutative gauge theories.

Contribution

It provides explicit gauge-invariant variables and classical equations of motion for a charged particle in Lie-Poisson electrodynamics, including solvable examples like the Kepler problem.

Findings

Derived gauge-invariant position variables.

Formulated classical action and equations of motion.

Analyzed solvable examples such as the Kepler problem.

Abstract

Lie-Poisson electrodynamics describes the semi-classical limit of non-commutative $U(1)$ gauge theory, characterized by Lie-algebra-type non-commutativity. We focus on the mechanics of a charged point-like particle moving in a given gauge background. First, we derive explicit expressions for gauge-invariant variables representing the particle's position. Second, we provide a detailed formulation of the classical action and the corresponding equations of motion, which recover standard relativistic dynamics in the commutative limit. We illustrate our findings by exploring the exactly solvable Kepler problem in the context of the $\lambda$-Minkowski (or the angular) non-commutativity, along with other examples.