Fractional Dirac Equations from Polynomial Linearization: Solutions and Difficulties

TL;DR

This paper explores fractional Dirac equations derived from polynomial linearization and generalized Clifford algebras, analyzing their solutions and highlighting conceptual difficulties in interpreting these fractional models physically.

Contribution

It introduces a novel approach to fractional Dirac equations using polynomial linearization and generalized Clifford algebras, and discusses the challenges in their physical interpretation.

Findings

Derived fractional Dirac equations from polynomial linearization.

Identified mathematical solutions to these fractional equations.

Highlighted conceptual difficulties in physical interpretation.

Abstract

The linearization of a quadratic form gives rise to a Clifford algebra structure, as seen in Dirac's factorization of the d'Alembert operator. A similar structure known as a generalized Clifford algebra arises from the continuation of this procedure to higher order forms. This technique combined with the existence of a fractional derivative satisfying the semi-group property can be used to factor the d'Alembert operator further, producing a fractional partial differential matrix equation that has a similar form to Dirac's equation. We examine these equations, their solutions, and point out difficulties when attempting to make physical sense of them.