Majorana equation

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

Definition

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

\psi _{c}:=i\psi ^{*}.\

This relation leads to the alternate expression

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

.

In both cases, the quantity $m$ is called the Majorana mass.^[1]

Properties

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 ψ_c of a spinor ψ. In contrast, the Weyl equation is for two-component spinor without mass.

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, since particles have the opposite charge to their own antiparticles. To satisfy this restriction, ψ must be 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 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]

References

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

External links

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[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