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

 -i {\partial\!\!\!\big /} \psi + m \psi_c = 0 \qquad \qquad (1)

with the derivative operator {\partial\!\!\!\big /} 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

 \psi_c := i \psi^*.\

Equation (1) can alternatively be expressed as

 i {\partial\!\!\!\big /} \psi_c + m \psi = 0 \qquad \qquad (2) .

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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