Axial momentum for the relativistic Majorana particle

TL;DR

This paper introduces the axial momentum operator for the relativistic Majorana particle, enabling a new basis for solutions and revealing oscillatory behavior that distinguishes massive from massless particles.

Contribution

It proposes the axial momentum operator as a new observable for Majorana particles, providing a basis for solutions and highlighting unique oscillatory features.

Findings

Axial momentum operator is well-defined for Majorana particles.

Eigenvectors of axial momentum form a useful basis for solutions.

Massive Majorana particles exhibit mode coupling and oscillations.

Abstract

The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, and the usual momentum operator $- i \nabla$ can not be defined in this space. For this reason, we introduce the axial momentum operator, $ - i \gamma_5 \nabla$ as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to $m/E$, where $E$ is the energy and $m$ the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. We show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.