Dirac spinors, covariant currents and the Lorentz group over a finite field

Abstract

After a brief review of the proper rotation and Lorentz groups over a finite geometry, the paper is devoted to analysing, in this framework, Dirac spinors and covariant currents. First, we show that the representations become unitary with respect to the rotation group only if thought of as ray representations. We then extend the proper Lorentz group through the space inversion: as a typical feature of the finite version of the Lorentz group, it is shown that there exist two kinds of Dirac spinors, say ψ(e),e=0, 1, which transform under inequivalent ray representations. Looking at the bispinor sesquilinear forms (or currents) it is then found that the covariance properties are different according to whether the current is built with spinors of the kinde=0 ore=1; in particular, for the choicee=1 only vector or axial-vector currents may be formed.