Abstract
We first derive without recourse to the Dirac equation the two-component Majorana equation with a mass term by a direct linearization of the relativistic dispersion relation of a massive particle. Thereby, we make only use of the complex conjugation operator and the Pauli spin matrices, corresponding to the irreducible representation of the Lorentz group. Then we derive the complex two-component eigenfunctions of the Majorana equation and the related quantum fields in a concise way, by exploiting the so-called chirality conjugation operator that involves the spin-flip operator. Subsequently, the four-component spinor solutions of the real Majorana equation are derived, and their intrinsic relations with the spinors of the complex two-component version of the Majorana equation are revealed and discussed extensively.
Highlights
In this paper we first present a new derivation 1 of the two-component Majorana equation including a mass term by a direct linearization of the relativistic dispersion relation of a massive particle, in a way similar to that used originally by Dirac 2
The principal goal of this paper was to rederive and discuss the two-component and four-component Majorana equations completely on their own rather than as a spinoff of the chiral Dirac equation. We obtained these equations including a mass term in a new way by a direct linearization of the relativistic dispersion relation of a massive particle
We calculated the eigenfunctions of the complex two-component Majorana equation and its related quantum field in a transparent way, exploiting the spin-flip or related chirality conjugation operator
Summary
In this paper we first present a new derivation 1 of the two-component Majorana equation including a mass term by a direct linearization of the relativistic dispersion relation of a massive particle, in a way similar to that used originally by Dirac 2. One intention of this work is to obtain the Majorana equation “completely on its own rather than as an afterthought when treating the Dirac equation,” as it was phrased by Case 6 , who first reformulated half a century ago the Majorana theory of the neutrino Another main objective is to address the key question of whether Majorana particles are their own antiparticles, and they are according to common. Our main motivation is to contribute to the ongoing discussion of whether massive neutrinos are Dirac or Majorana fermions and to better understand the latter theoretically in terms of the two-component theory and its connection to the four-component real version. In this respect we complement the old pedagogical review on Majorana masses by Mannheim 7
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