# Majorana equation

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

## DefinitionEdit

The Majorana equation is

with the derivative operator written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components.

In this equation, *is the charge conjugate of* , which can be defined in the Majorana basis as

This relation leads to the alternate expression

- .

In both cases, the quantity is called the **Majorana mass**.^{[1]}

## PropertiesEdit

### Similarity to Dirac equationEdit

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 of a spinor . In contrast, the Weyl equation is for two-component spinor without mass.

### Charge conservationEdit

The appearance of both and in the Majorana equation means that the field 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, must be taken to be neutral.

## Field quantaEdit

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

### Majorana particleEdit

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.^{[2]}

## ReferencesEdit

**^**Cheng, T.-P.; Li, L.-F. (1983).*Gauge Theory of Elementary Particle Physics*. Oxford University Press. ISBN 0-19-851961-3.**^**A. Franklin,*Are There Really Neutrinos?: An Evidential History*(Westview Press, 2004), p. 186