Split-Quaternions and the Dirac Equation

TL;DR

This paper presents an alternative formulation of the Dirac equation using split-quaternions, revealing new symmetry properties and providing a different mathematical perspective on relativistic quantum mechanics.

Contribution

It introduces a split-quaternionic formalism for Dirac 4-spinors, making Lorentz transformations and symmetries more explicit in this framework.

Findings

Dirac 4-spinors can be equivalently formulated with split-quaternions.

Lorentz transformations are represented as 2x2 split-quaternionic unitary matrices.

The $SO(3,2; f R)$ symmetry of the scalar $ar{\psi}\psi$ is explicitly shown.

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 $SO(3,2; {\bf R})$ symmetry of the Lorentz invariant scalar $\overline{\psi}\psi$ is manifest.