Quaternionic Formulation of the Dirac Equation

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.

Presented at the Fifth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, June 28-July 2, 2010