The Majorana equation is
In this equation, ψc 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 in the sense that it involves four-component spinors, gamma matrices and mass terms, but includes the charge conjugate ψc of a spinor ψ. In contrast, the Weyl equation is for two-component spinor without mass.
The appearance of both ψ and ψc in the Majorana equation means that the field ψ 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, ψ must be taken to be neutral.
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.
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.