Classical electrodynamics in a space with spin noncommutativity of coordinates

TL;DR

This paper introduces a new Lorentz-invariant spin-noncommutative algebra, develops an electromagnetic theory in this space, and derives exact solutions for specific field configurations, revealing non-abelian features.

Contribution

It constructs a novel spin-noncommutative algebra, formulates electromagnetic field theory within it, and derives exact solutions, advancing the understanding of noncommutative geometries in physics.

Findings

Electromagnetic field becomes non-abelian in spin-noncommutative space.

Exact solutions for plane wave propagation in combined magnetic and electric fields.

Derived nonlinear field equations from the least action principle.

Abstract

We propose a new relativistic Lorentz-invariant spin-noncommutative algebra. Using the Weyl ordering of noncommutative position operators, we build an analogue of the Moyal-Groenewald product for the proposed algebra. The Lagrange function of an electromagnetic field in the space with spin noncommutativity is constructed. In such a space electromagnetic field becomes non-abelian. A gauge transformation law of this field is also obtained. Exact nonlinear field equations of noncommutative electromagnetic field are derived from the least action principle. Within the perturbative approach we consider field of a point charge in a constant magnetic field and interaction of two plane waves. An exact solution of a plane wave propagation in a constant magnetic and electric fields is found.