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

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

1999-08-30

arxiv.org/abs/hep-th/9908200v1

Int.J.Theor.Phys.40:399-411,2001

Physics

High Energy Physics

High Energy Physics - Theory

10 pages, nothing special, Submitted to IJTP special issue about the Ixtapa conference on `Clifford algebras and their applica

Scientific paper

10.1023/A:1004128131072

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.

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