# Majorana equation

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

## Definition

The Majorana equation is

$-i{\partial \!\!\!{\big /}}\psi +m\psi _{c}=0\qquad \qquad (1)$ with the derivative operator ${\partial \!\!\!{\big /}}$ written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components.

In this equation, ${\textstyle \psi _{c}}$ is the charge conjugate of ${\textstyle \psi }$ , which can be defined in the Majorana basis as

$\psi _{c}:=i\psi ^{*}.\$ This relation leads to the alternate expression

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

In both cases, the quantity ${\textstyle m}$ is called the Majorana mass.

## Properties

### Similarity to Dirac equation

The Majorana is similar to the Dirac equation in the sense that it involves four-component spinors, gamma matrices, and mass terms, but includes the charge conjugate ${\textstyle \psi _{c}}$ of a spinor ${\textstyle \psi }$ . In contrast, the Weyl equation is for two-component spinor without mass.

### Charge conservation

The appearance of both ${\textstyle \psi }$ and ${\textstyle \psi _{c}}$ in the Majorana equation means that the field ${\textstyle \psi }$ cannot be coupled to a charged electromagnetic field without violating charge conservation, since particles have the opposite charge to their own antiparticles. To satisfy this restriction, ${\textstyle \psi }$ must be 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 ${\textstyle \Psi =\Psi _{c}}$ results in a single neutral particle, in which case ${\textstyle \psi }$ is known as a Majorana spinor. For a Majorana spinor, the Majorana equation is equivalent to the Dirac equation.

### Majorana particle

Particles corresponding to Majorana spinors are known as Majorana particles, due to the above self-conjugacy constraint. All the fermions included in the Standard Model have been excluded as Majorana fermions (since they have non-zero electric charge they cannot be antiparticles of themselves) with the exception of the neutrino (which is neutral).

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