The intermediate fermionic species created by $SO(3)$ rotation in the representation of the Dirac equation

TL;DR

This paper investigates how different choices of gamma matrices in the Dirac equation lead to a family of intermediate fermion species, revealing their transformation properties under SO(3) and connections to Majorana fermions.

Contribution

It introduces the concept of intermediate fermion species arising from gamma matrix variations and analyzes their transformation and properties, including links to Majorana fermions.

Findings

All intermediate fermion species transform via SO(3) similarity transformations.

Eigenvalue problems and boosts are consistent across species.

Sub-representations generate Majorana fermions, linked to U(1) symmetry.

Abstract

The question of how does the Dirac equation depend on the choice of the $\gamma$ matrices has partially been addressed and explored in the literature. In this paper we focus on this question by considering a general form of $\gamma$ matrices, and call the resulting spin $\frac{1}{2}$ fermions as \textit{intermediate fermion species} (IFS). By inspecting the properties of IFS, we find that all species transform to each other by a $SO(3)$ similarity transformation in the space of parameters, that are the entities of the $\gamma$ matrices. Many properties, like eigenvalue problem and boost are tested for IFS. We find also sub-representations that generate Majorana fermions, which is isomorphism to $U(1)$ group.