Majorana equation

Definition

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 ψ_c is the charge conjugate of ψ, which can be defined in the Majorana basis as

This relation leads to the alternate expression

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

Similarity to Dirac equation

The Majorana is similar to the Dirac equation, but includes the charge conjugate ψ_c of a spinor ψ.

Charge conservation

The appearance of both ψ and ψ_c in the Majorana equation means 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 = ψ_c 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

Particles corresponding to Majorana spinors are known as Majorana particles, due to the above constraints. 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.