Two dimensional representation of the Dirac equation in Non associative algebra

TL;DR

This paper introduces a higher-dimensional algebra extending complex numbers to formulate a 2D Dirac equation, revealing a sub-algebra where associativity is restored, thus offering new mathematical insights into Dirac equations.

Contribution

It proposes a novel algebraic extension for the Dirac equation in two dimensions and identifies a sub-algebra with associative properties.

Findings

Formulation of a 2D Dirac equation using the new algebra

Solution of the 2D Dirac equation within this framework

Identification of a sub-algebra with associative properties

Abstract

In this note a simple extension of the complex algebra to higher dimension is proposed. Using the postulated algebra a two dimensional Dirac equation is formulated and its solution is calculated. It is found that there is a sub-algebra where the associative nature can be recovered.