On the Majorana equation - Relations between its complex two-component and real four-component eigenfunctions

TL;DR

This paper derives the Majorana equation directly from relativistic principles, explores its eigenfunctions, and elucidates the relationship between complex two-component and real four-component solutions.

Contribution

It presents a novel derivation of the Majorana equation without using the Dirac equation and clarifies the connection between its complex and real spinor solutions.

Findings

Derived the two-component Majorana equation from first principles.

Explicitly constructed the eigenfunctions of the Majorana equation.

Revealed the intrinsic relations between complex and real Majorana spinors.

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.