On the equivalence of Daviau's space Clifford algebraic, Hestenes' and Parra's formulations of (real) Dirac theory

TL;DR

This paper demonstrates the equivalence of various formulations of the Dirac theory—Clifford algebraic, Hestenes', and Parra's—showing they are interconnected through isomorphisms and automorphisms, unifying different mathematical perspectives.

Contribution

It proves that Daviau's map is an isomorphism linking matrix and Clifford algebra formulations and shows the equivalence of Hestenes' and Parra's formulations to Daviau's approach.

Findings

Daviau's map theta is an isomorphism between C^4 and M_2(C)

Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation

The connection involves outer automorphisms and relates left/right actions to bi-module structures

Abstract

Recently Daviau showed the equivalence of ordinary matrix based Dirac theory -formulated within a spinor bundle S_x \simeq C^4_x-, to a Clifford algebraic formulation within space Clifford algebra CL(R^3,delta) \simeq M_2(C) \simeq P \simeq Pauli algebra (matrices) \simeq H \oplu H \simeq biquaternions. We will show, that Daviau's map theta : C^4 \mapsto M_2(C) is an isomorphism. Furthermore it is shown that Hestenes' and Parra's formulations are equivalent to Daviau's space Clifford algebra formulation, which however uses outer automorphisms. The connection between such different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra P^e \simeq P\otimes P^T on P. The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau and others to interprete the right action as the iso-spin freedom.