A unified field theory from a complexified quaternion-octonion Dirac equation

TL;DR

This paper develops a novel algebraic framework using complexified quaternions and octonions to extend the Dirac equation, aiming to construct a unified field theory with properties similar to the Standard Model.

Contribution

It introduces a new algebraic scheme based on Clifford algebra projectors and ladder operators to formulate a toy unified matter field model extending the Dirac equation.

Findings

Describes fermion fields with properties akin to one Standard Model generation

Exhibits U(1), SU(2), and SU(3) symmetries in the model

Shows SU(2) induces frame fields

Abstract

It is set manifest an underlying algebraic structure of Dirac equation and solutions, in terms of Cl2 Clifford algebra projectors and ladder operators. From it, a scheme is proposed for constructing unified field theories by enlarging the pointed algebra. A toy unified matter field model is formulated, modifying Dirac equation with complex quaternions and octonions. The result describes a set of fermion fields with reminiscent properties of one standard model particle generation, exhibiting U(1) electromagnetism, SU(2) flavor and SU(3) color like symmetries, remarkably SU(2) induces frame fields, though further explorations are needed.