Born-Infeld electrodynamics: Clifford number and spinor representations

TL;DR

This paper develops a Clifford number formalism for Maxwell and nonlinear Born-Infeld electrodynamics, introducing a Clifford imaginary unit and spinor representations, and explores invariance properties under Lorentz transformations.

Contribution

It presents a novel Clifford number and spinor formalism for Maxwell and Born-Infeld nonlinear electrodynamics, including new invariant representations and a nonlinear Dirac-like equation.

Findings

Clifford imaginary unit for space-time introduced

Representation of Maxwell equations as nonlinear Dirac equations

Invariant formulations under Lorentz transformations

Abstract

Clifford number formalism for Maxwell equations is considered. The Clifford imaginary unit for space-time is introduced as coordinate independent form of fully antisymmetric fourth-rank tensor. The representation of Maxwell equations in massless Dirac equation form is considered; we also consider two approaches to the invariance of Dirac equation in respect of the Lorentz transformations. According to the first approach, the unknown column is invariant and according to the second approach it has the transformation properties known as spinorial ones. Clifford number representation for nonlinear electrodynamics equations is obtained. From this representation, we obtain the nonlinear like Dirac equation which is the form of nonlinear electrodynamics equations. As a special case we have the appropriate representations for Born-Infeld nonlinear electrodynamics.