The Dirac Equation and the Majorana Dirac Equation

TL;DR

This paper explores the structure of the Dirac and Majorana Dirac equations, linking discrete physics, nilpotent structures, and Majorana fermions, and provides solutions using nilpotent techniques.

Contribution

It introduces a split quaternion expression for the Majorana Dirac equation and analyzes its structure and solutions, connecting it with prior theoretical frameworks.

Findings

Derived a split quaternion form of the Majorana Dirac equation

Identified basic solutions using nilpotent techniques

Linked Majorana fermions with nilpotent structures

Abstract

We discuss the structure of the Dirac equation and how the nilpotent and the Majorana operators arise naturally in this context. This provides a link between Kauffman's work on discrete physics, iterants and Majorana Fermions and the work on nilpotent structures and the Dirac equation of Peter Rowlands. We give an expression in split quaternions for the Majorana Dirac equation in one dimension of time and three dimensions of space. Majorana discovered a version of the Dirac equation that can be expressed entirely over the real numbers. This led him to speculate that the solutions to his version of the Dirac equation would correspond to particles that are their own anti-particles. It is the purpose of this paper to examine the structure of this Majorana-Dirac Equation, and to find basic solutions to it by using the nilpotent technique. We succeed in this aim and describe our results.