Majorana equation

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The Majorana equation is a relativistic wave equation similar to the Dirac equation but includes the charge conjugate ψc of a spinor ψ. It is named after the Italian physicist Ettore Majorana, and it 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 ψc is the charge conjugate of ψ, which can be defined in the Majorana basis as

Equation (1) can alternatively be expressed as


In either case, the quantity m in the equation is called the Majorana mass.

The appearance of both ψ and ψc in the Majorana equation means that the field ψ cannot be coupled to an electromagnetic field without violating charge conservation, so ψ is taken to be neutrally charged. Nonetheless, the quanta of the Majorana equation given here are two particle species, a neutral particle and its neutral antiparticle. The Majorana equation is frequently supplemented by the condition that ψ = ψc (in which case one says that ψ is a Majorana spinor); this results in a single neutral particle. For a Majorana spinor, the Majorana equation is equivalent to the Dirac equation.

Particles corresponding to Majorana spinors are aptly called Majorana particles. Such a particle is its own antiparticle. Thus far, of all the fermions included in the Standard Model, none is a Majorana fermion. However, there is the possibility that the neutrino is of a Majorana nature. 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.[1]


  1. ^ A. Franklin, Are There Really Neutrinos?: An Evidential History (Westview Press, 2004), p. 186

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