Generalization of the Majorana equation for real spinors

TL;DR

This paper presents a new way to decompose the Dirac equation for real spinors into two simpler equations, clarifying the structure of Majorana spinors and extending the formalism to complex spinors.

Contribution

It introduces a novel decomposition of the Dirac equation for real spinors, unifying the Majorana equation within this framework and allowing for extensions to complex spinors.

Findings

Decomposition of Dirac equation into two first-order equations

Majorana equation as a special case of the derived system

Extension of formalism to complex (charged) spinors

Abstract

We show that the Dirac equation for real spinors can be naturally decomposed into a system of two first-order relativistic wave equations. The decomposition separates in a transparent way the real and imaginary parts of the Dirac equation by means of two algebraic differential operators, allowing to describe real spinors in any representation of the Dirac matrices maintaining the reality condition $\tilde{\Psi}=\tilde{\Psi}^{*}$ unaltered. In addition, it is shown that the Majorana wave equation is a particular case of the relativistic system of equations deduced in this paper. We also briefly discuss how the formalism can be extended to deal with complex (charged) spinors.