How `Complex' is the Dirac Equation?

TL;DR

This paper explores a novel algebraic representation of the Lorentz group using 4x4 matrices over a simple algebra, revealing a connection to SO(3,3) and automorphisms of the Dirac spinor.

Contribution

It introduces a new algebraic framework for the Lorentz group and demonstrates its relation to Dirac spinors and automorphisms, differing from traditional complex matrix representations.

Findings

Representation embedded in SO(3,3) instead of SO(2,4)

Existence of automorphisms preserving Lorentz transformation properties

Equivalence to standard complex Dirac spinor transformations

Abstract

A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over a simple non-division algebra. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the transformation properties of the usual complex Dirac spinor. As an application, we show that there exists an algebra of automorphisms of the complex Dirac spinor that leave the transformation properties of its eight real components invariant under any given Lorentz transformation. Interestingly, the representation of the Lorentz group presented here has a natural embedding in SO(3,3) instead of the conformal symmetry SO(2,4).