Majorana equation

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


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]


Similarity to Dirac equation[edit]

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 conservation[edit]

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 quanta[edit]

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 particle[edit]

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]


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

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