The Two-Component Majorana Equation-Novel Derivations and Known Symmetries

Eckart Marsch

doi:10.4236/jmp.2011.210137

Abstract

We revisit the two-component Majorana equation and derive it in a new form by linearizing the relativistic dispersion relation of a massive particle, in a way similar to that used to derive the Dirac equation. We are using thereby the Pauli spin matrices, corresponding to an irreducible representation of the Lorentz group, and a lucid and transparent algebraic approach exploiting the newly introduced spin-flip operator. Thus we can readily build up the Majorana version of the Dirac equation in its chiral representation. The Lorentz-invariant complex conjugation operation involves the spin-flip operator, and its connection to chiral symmetry is discussed. The eigenfunctions of the Majorana equation are calculated in a concise way.

Highlights

The so called two-component Majorana equation [1,2] permitting a finite mass term has been used in modern quantum field theory for the description of massive neutrinos, ever since convincing empirical evidence [3] for their finite masses and associated oscillations has been found in the past decade
In this paper we present among other topics a novel derivation of the two-component Majorana equation with a mass term, whereby we linearize the relativistic dispersion relation of a massive particle in a way similar to that used in deriving the Dirac equation
By making use of the spin-flip operator, we derived this equation with a mass term in a novel and manifestly covariant way, which revealed its intimate relations to the Dirac equation in its chiral as well as standard form

Summary

Introduction

The so called two-component Majorana equation [1,2] permitting a finite mass term has been used in modern quantum field theory for the description of massive neutrinos, ever since convincing empirical evidence [3] for their finite masses and associated oscillations has been found in the past decade. The corresponding neutrinos are conventionally described by the massless Weyl [5] equation involving two-component Pauli [6] spinors. A consistent interpretation for the Dirac equation is given concerning its property of covariant complex conjugation [7]. It emerges as an intrinsic basic symmetry of the reducible Dirac equation with its four-component spinors, and includes the charge exchange operation for the Dirac equation [8] when being coupled to a gauge field like the electromagnetic field. We use a concise algebraic nomenclature for the Dirac equation, and in a lucid way present its classical symmetries such as parity, time reversal and charge exchange

Dirac Equation in a Concise Algebraic Form

New Derivation of the Majorana Equation

Symmetries of the Dirac Equation Revisited

Chiral Symmetry and Complex Conjugation

Summary and Conclusions