Maxwell's Theory on Non-Commutative Spaces and Quaternions

TL;DR

This paper explores Maxwell's electromagnetic theory within non-commutative spaces, deriving non-linear field equations in quaternion form, analyzing wave solutions, energy-momentum tensors, and symmetry breaking effects.

Contribution

It introduces a quaternion-based formulation of non-commutative Maxwell equations, deriving new non-linear wave equations and analyzing energy-momentum tensor properties.

Findings

Plane electromagnetic waves are solutions to the non-linear equations.

Energy-momentum tensors have non-zero traces, indicating trace anomalies.

Dual symmetry transformations are broken in non-commutative spaces.

Abstract

The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quaternion) form. It was shown that the plane electromagnetic wave is the solution of the system of non-linear wave equations of the second order for the electric and magnetic induction fields. We have found the canonical and symmetrical energy-momentum tensors and their non-zero traces. So, the trace anomaly of the energy-momentum tensor was obtained in electrodynamics on non-commutative spaces. It was noted that the dual transformations of electromagnetic fields on non-commutative spaces are broken.