Quaternionic Formulation of the Dirac Equation

TL;DR

This paper introduces a quaternionic approach to the Dirac equation with Lorentz violation, simplifying the dispersion relation analysis and identifying parameter subsets with easily solvable relations.

Contribution

It presents a novel quaternionic formulation of the Dirac equation that reduces complexity and clarifies the structure of Lorentz-violating dispersion relations.

Findings

Quaternionic formulation simplifies the Dirac equation analysis.

Identifies two parameter subsets with solvable dispersion relations.

All Lorentz-violating parameters are included in the analysis.

Abstract

The Dirac equation with Lorentz violation involves additional coefficients and yields a fourth-order polynomial that must be solved to yield the dispersion relation. The conventional method of taking the determinant of $4\times 4$ matrices of complex numbers often yields unwieldy dispersion relations. By using quaternions, the Dirac equation may be reduced to $2 \times 2$ form in which the structure of the dispersion relations become more transparent. In particular, it is found that there are two subsets of Lorentz-violating parameter sets for which the dispersion relation is easily solvable. Each subset contains half of the parameter space so that all parameters are included.