On wave equations for the Majorana particle in (3+1) and (1+1) dimensions

TL;DR

This paper analyzes various wave equations used to describe Majorana particles in (3+1) and (1+1) dimensions, comparing their forms, representations, and algebraic structures, and introduces new derivations for lower dimensions.

Contribution

It provides a comprehensive algebraic comparison of Majorana wave equations across dimensions and representations, and introduces a novel derivation method in (1+1) dimensions.

Findings

Different forms of Majorana equations depend on representation and dimension.

The wave function for Majorana particles involves four or two real quantities.

New derivation method for Majorana equations in (1+1) dimensions is proposed.

Abstract

In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain.