# Majorana equation

The Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana.

## Definition

The Majorana equation is

${\displaystyle -i{\partial \!\!\!{\big /}}\psi +m\psi _{c}=0\qquad \qquad (1)}$

with the derivative operator ${\displaystyle {\partial \!\!\!{\big /}}}$ written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components. In this equation ψc is the charge conjugate of ψ, which can be defined in the Majorana basis as

${\displaystyle \psi _{c}:=i\psi ^{*}.\ }$

This relation leads to the alternate expression

${\displaystyle i{\partial \!\!\!{\big /}}\psi _{c}+m\psi =0\qquad \qquad (2)}$.

In both cases, the quantity m is called the Majorana mass.

## Properties

### Similarity to Dirac equation

The Majorana is similar to the Dirac equation[how?], but includes the charge conjugate ψc of a spinor ψ.

### Charge conservation

The appearance of both ψ and ψc in the Majorana equation means[why?] that the field ψ cannot be coupled to a charged electromagnetic field without violating charge conservation. To satisfy this restriction, ψ is taken to be neutral.

## Field quanta

The quanta of the Majorana equation allow for two classes of particles, a neutral particle and its neutral antiparticle. The frequently applied supplemental condition ψ = ψc results in a single neutral particle, in which case ψ is known as a Majorana spinor. For a Majorana spinor, the Majorana equation is equivalent to the Dirac equation.

### Majorana particle

Main article: Majorana particle

Particles corresponding to Majorana spinors are known as Majorana particles, due to the above constraints[which?]. All the fermions included in the Standard Model have been excluded as Majorana fermions[how?] with the exception[clarification needed] of the neutrino.

Theoretically, the neutrino is a possible exception to this pattern.[why?]. If so, neutrinoless double-beta decay, as well as a range of lepton-number violating meson and charged lepton decays, are possible[according to whom?]. A number of experiments[which?] probing whether the neutrino is a Majorana particle are currently underway.[1]

## References

1. ^ A. Franklin, Are There Really Neutrinos?: An Evidential History (Westview Press, 2004), p. 186