On relativistic quantum mechanics of the Majorana particle: quaternions, paired plane waves, and orthogonal representations of the Poincaré group

H Arodź,Z Świerczyński

doi:10.1088/1361-6471/abc969

Abstract

The standard momentum operator −i∇ has the trivial domain (the null vector) if the L 2 Hilbert space consists of only real-valued functions. In consequence, it is useless in quantum mechanics of the relativistic Majorana particle which is formulated in such a Hilbert space. Instead, one can consider the axial momentum operator introduced in (2019) Phys. Lett. A 383 1242. In the present paper we report several new results which elucidate usability of the axial momentum observable. First, a new motivation for the axial momentum is given, and the Heisenberg uncertainty relation checked. Next, we show that the general solution of time evolution equation written in the axial momentum basis has a connection with quaternions. Furthermore, it turns out that in the case of massive Majorana particles, single traveling monochromatic plane waves are not possible, but there exist solutions which have the form of two plane waves traveling in opposite directions. Another issue discussed here in detail is relativistic invariance. A single real, orthogonal and irreducible representation of the Poincaré group—consistent with the lack of antiparticle—is unveiled.

Highlights

The discovery of non vanishing mass of neutrinos has led to many conjectures about the nature of these particles
We have shown that the axial momentum operator for the Majorana particle is related to the ordinary momentum for the Weyl particle by the one-to-one mapping between the two models, and that it obeys the Heisenberg uncertainty relation
Using the eigenfunctions and eigenvalues of the axial momentum operator, we have written the general solution of the Dirac equation for the real bispinor in the form of superposition of traveling plane waves, with the eigenvalues p of the axial momentum playing the role of wave vectors, i.e., giving the wave length and the direction of propagation

Summary

Introduction

The discovery of non vanishing mass of neutrinos has led to many conjectures about the nature of these particles. Using the mapping inverse to M one can transform Eq (1) for the Majorana bispinor ψ to the space of right-handed Weyl bispinors – we obtain iγμ∂μφ − mφ∗ = 0, which is known as the Dirac equation for φ with the Majorana mass term (recall that φ∗ is the charge conjugation of φ). This last equation can not be accepted as a quantum mechanical evolution equation for the Weyl bispinor φ because it is not linear over C – it is linear only over R. This aspect is discussed in detail in [10]

Time evolution of the axial momentum amplitudes and quaternions

Relation with irreducible representations of the Poincaré group

Summary and remarks