Octonionic Version of Dirac Equations

TL;DR

This paper introduces an octonionic formulation of Dirac equations using split octonions, where wave functions are real split octonions and probability amplitudes require four components, offering a novel algebraic perspective.

Contribution

It presents a new octonionic framework for Dirac equations with real wave functions and a four-component probability amplitude structure, differing from traditional complex spinor approaches.

Findings

Equivalence to standard Dirac equations via split octonion algebra

Wave functions are real split octonions, not bi-spinors

Four split octonions are needed for positive probability amplitudes

Abstract

It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that are equivalent to the standard Dirac equations. In our approach the particle mass and electro-magnetic potentials are part of an octonionic gradient function together with the space-time derivatives. As distinct from previous attempts to translate the Dirac equations into different number systems here the wave functions are real split octonions and not bi-spinors. To formulate positively defined probability amplitudes four different split octonions (transforming into each other by discrete transformations) are necessary, rather then two complex wave functions which correspond to particles and antiparticles in usual Dirac theory.