On Clifford Subalgebras, Spacetime Splittings and Applications

TL;DR

This paper explores Clifford algebra gradings and spacetime splittings, providing a detailed mathematical framework for Dirac operators and spinor fields, with potential applications in physics.

Contribution

It introduces an alpha-grading based on automorphisms related to spacetime splitting and expresses the Dirac equation in terms of this splitting, revealing new structural insights.

Findings

Dirac operator split into parallel and orthogonal components

Dirac spinor decomposed into two quaternions

Potential applications in physical theories

Abstract

Z2-gradings of Clifford algebras are reviewed and we shall be concerned with an alpha-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced alpha-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.