Factoring the Dispersion Relation in the Presence of Lorentz Violation

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.