Linear and angular momentum spaces for Majorana spinors

TL;DR

This paper demonstrates that Majorana spinors, represented as real solutions to the Dirac equation, can describe the energy, linear, and angular momenta of free spin-1/2 particles, and explores their mathematical properties.

Contribution

It establishes Majorana spinors as irreducible representations of Lorentz groups and defines Fourier and Hankel transforms related to their momentum properties.

Findings

Majorana spinors can describe particle momenta.

Majorana spinors form irreducible Lorentz group representations.

Defined Fourier-Majorana and Hankel-Majorana transforms.

Abstract

In a Majorana basis, the Dirac equation for a free spin one-half particle is a 4x4 real matrix differential equation. The solution can be a Majorana spinor, a 4x1 real column matrix, whose entries are real functions of the space-time. Can a Majorana spinor, whose entries are real functions of the space-time, describe the energy, linear and angular momentums of a free spin one-half particle? We show that it can. We show that the Majorana spinor is an irreducible representation of the double cover of the proper orthochronous Lorentz group and of the full Lorentz group. The Fourier-Majorana and Hankel-Majorana transforms are defined and related to the linear and angular momentums of a free spin one-half particle.