Factoring the Dispersion Relation in the Presence of Lorentz Violation

TL;DR

This paper derives an explicit, factored dispersion relation for the Dirac equation within the Standard Model Extension, employing quaternion algebra to simplify analysis of Lorentz violation effects.

Contribution

It introduces a novel quaternion-based technique to factor the dispersion relation in Lorentz-violating models, simplifying the study of parameter space.

Findings

Dispersion relation can be explicitly factored in special cases.

Splitting parameter space simplifies analysis of Lorentz violation.

Quaternion algebra provides a new tool for theoretical physics.

Abstract

We produce an explicit formula for the dispersion relation for the Dirac Equation in the Standard Model Extension (SME) in the presence of Lorentz violation. Our expression is obtained using a novel techniques which exploit the algebra of quaternions. The dispersion relation is found to conveniently factor in two special cases that each involve a mutually exclusive set of non-vanishing Lorentz-violating parameters. This suggests that a useful approach to studies of Lorentz-violating models is to split the parameter space into two separate pieces, each of which yields a simple, tractable dispersion relation that can be used for analysis.