Split-Quaternions and the Dirac Equation

Abstract

We show that Dirac 4-spinors admit an entirely equivalent formulation in terms of 2-spinors defined over the split-quaternions. In this formalism, a Lorentz transformation is represented as a $2\times 2$ unitary matrix over the split-quaternions. The corresponding Dirac equation is then derived in terms of these 2-spinors. In this framework the $SO(3,2;\mathbf{R})$ symmetry of the Lorentz invariant scalar $\overline{\psi }\psi$ is manifest.

17 pages; corrected misprint in eqn (70); corrected sign for Weyl basis in Section 5; minor improvements in layout

References in corpus (1)

• Split Quaternionic Analysis and Separation of the Series for SL(2,R) and SL(2,C)/SL(2,R)