Quantum Theory of Half-integer Spin Free Particles from the Perspective of the Majorana Equation

TL;DR

This paper explores the solutions of the Majorana equation for half-integer spin particles, comparing them with Dirac solutions, and investigates the physicality and potential transitions to tachyonic states at high energies.

Contribution

It provides a detailed analysis of Majorana equation solutions for half-integer spins and examines their physical relevance and implications for high-energy particle states.

Findings

Majorana solutions differ from Dirac solutions in the relativistic limit.

Unphysical solutions increase in probability with particle velocity.

High-energy, small-mass particles may transition to tachyonic states.

Abstract

In this study, the Majorana equation for particles with arbitrary spin is solved for a half-integer spin free particle. The solution for the fundamental state, corresponding to the reference frame in which the particle is at rest, is compared with that obtained using the Dirac equation, especially as regards the approximation in the relativistic limit, in which the speed of the particle is close to that of light. Furthermore, the solutions that Majorana defines unphysical, proving that their occupation probability increases with the particle velocity, are taken into consideration. The anomalous behavior exhibited by these states also shows that for high-energy particles with small mass, transitions from a bradyonic state to a tachyonic state become possible.