A New Route to the Majorana Equation

Eckart Marsch

doi:10.3390/sym5040271

Abstract

In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.

Highlights

We describe a novel route to the complex two-component Majorana equation, including a mass term
The expectation value of any Hermitian operator, O, for the Majorana quantum field Φ is defined through the scalar product of the related wave functions
We calculated the eigenfunctions of the two-component complex Majorana equation and, thereby, exploited the properties of the spin-flip or chiral conjugation operator

Summary

Introduction

We describe a novel route to the complex two-component Majorana equation, including a mass term. This equation is obtained here by a direct linearization of the Klein-Gordon equation (see, e.g., [1]) for a relativistic massive particle and requires the introduction of what we call rho matrices to be defined below. With the results of this paper, we want to contribute to the ongoing discussion of whether massive neutrinos are Dirac or Majorana fermions and help in better understanding the latter theoretically in terms of the two-component spinor theory, which has extensively been reviewed in the recent report by Dreiner et al [13]. The physical content of this new equation is essentially the same as that of the standard Majorana equation, insofar as it describes uncharged massive fermions of opposite mean kinetic helicity

A New Route to the Majorana Equation

Symmetries and Lorentz Transformation of the Majorana Equation

Derivation of the Real Four-Component Majorana Equation

Summary and Conclusions