Towards an octonionic world

TL;DR

This paper develops a consistent framework for octonionic quantum mechanics by introducing new operators, translating octonionic numbers into matrix forms, and formulating an octonionic relativistic wave equation with solutions analogous to Dirac's.

Contribution

It introduces left-right barred operators and a complex geometric approach to resolve hermiticity issues in octonionic quantum mechanics, leading to a new relativistic wave equation.

Findings

Established translation rules between octonions and matrices

Defined a momentum operator within OQM using complex geometry

Derived an octonionic relativistic wave equation with Dirac-like solutions

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). The use of a complex geometry allows us to overcome the hermiticity problem and define an appropriate momentum operator within OQM. As an application of our results, 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.