Dispersion Relations in Noncommutative Theories

TL;DR

This paper investigates how noncommutativity in gauge theories alters wave dispersion relations, energy flow, and wave propagation conditions, revealing differences between Moyal and Seiberg-Witten formulations and their physical implications.

Contribution

It provides a detailed analysis of plane waves in noncommutative gauge theories, highlighting differences in dispersion relations and wave behavior between Moyal and Seiberg-Witten frameworks.

Findings

Dispersion relations are deformed by background fields in noncommutative theories.

In Moyal case, some wave solutions are forbidden when time is noncommutative.

Seiberg-Witten case shows no restrictions on plane wave solutions.

Abstract

We present a detailed study of plane waves in noncommutative abelian gauge theories. The dispersion relation is deformed from its usual form whenever a constant background electromagnetic field is present and is similar to that of an anisotropic medium with no Faraday rotation nor birefringence. When the noncommutativity is induced by the Moyal product we find that for some values of the background magnetic field no plane waves are allowed when time is noncommutative. In the Seiberg-Witten context no restriction is found. We also derive the energy-momentum tensor in the Seiberg-Witten case. We show that the generalized Poynting vector obtained from the energy-momentum tensor, the group velocity and the wave vector all point in different directions. In the absence of a constant electromagnetic background we find that the superposition of plane waves is allowed in the Moyal case if the momenta are parallel or satisfy a sort of quantization condition. We also discuss the relation between the solutions found in the Seiberg-Witten and Moyal cases showing that they are not equivalent.