Hyperbolic Numbers and the Dirac Spinor

Details Hyperbolic Numbers and the Dirac Spinor Hyperbolic Numbers and the Dirac Spinor

1998-12-03

arxiv.org/abs/hep-th/9812036v1

Physics

High Energy Physics

High Energy Physics - Theory

13 pages, LaTex. To appear in "The Photon and Poincare Group", Nova Science Publishers

Scientific paper

A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over the hyperbolic number system. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the transformation properties of the complex Dirac spinor. As an application, we show that there exists an algebra of automorphisms of the complex Dirac spinor that leaves the transformation properties of its eight real components invariant under any given Lorentz transformation. Interestingly, the representation of the Lorentz algebra presented here is naturally embedded in the Lie algebra of a group isomorphic to SO(3,3;R) instead of the conformal group SO(2,4;R).

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