Octonionic Dirac Equation

TL;DR

This paper develops an octonionic quantum mechanics framework by introducing new operators, translating octonionic numbers into matrix forms, and formulating an octonionic Dirac-like wave equation with solutions analogous to the standard Dirac equation.

Contribution

It introduces left-right barred operators for octonionic quantum mechanics and formulates an octonionic relativistic wave equation with Dirac-like solutions.

Findings

Established translation rules between octonions and matrices

Developed an octonionic relativistic wave equation

Found four independent solutions similar to Dirac equation

Abstract

In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a translation is also given for $4\times 4$ complex matrices). We develop an octonionic relativistic free wave equation, linear in the derivatives. Even if the wave functions are only one-component we show that four independent solutions, corresponding to those of the Dirac equation, exist.