Majorana equation

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any (possibly non-relativistic) fermionic particle that is its own anti-particle (and is therefore electrically neutral).

There have been proposals that massive neutrinos are described by Majorana particles; there are various extensions to the Standard Model that enable this. The article on Majorana particles presents status for the experimental searches, including details about neutrinos. This article focuses primarily on the mathematical development of the theory, with attention to its discrete and continuous symmetries. The discrete symmetries are charge conjugation, parity transformation and time reversal; the continuous symmetry is Lorentz invariance.

Charge conjugation plays an outsize role, as it is the key symmetry that allows the Majorana particles to be described as electrically neutral. A particularly remarkable aspect is that electrical neutrality allows several global phases to be freely chosen, one each for the left and right chiral fields. This implies that, without explicit constraints on these phases, the Majorana fields are naturally CP violating. Another aspect of electric neutrality is that the left and right chiral fields can be given distinct masses. That is, electric charge is a Lorentz invariant, and also a constant of motion; whereas chirality is a Lorentz invariant, but is not a constant of motion for massive fields. Electrically neutral fields are thus less constrained than charged fields. Under charge conjugation, the two free global phases appear in the mass terms (as they are Lorentz invariant), and so the Majorana mass is described by a complex matrix, rather than a single number. In short, the discrete symmetries of the Majorana equation are considerably more complicated than those for the Dirac equation, where the electrical charge ${\displaystyle U(1)}$ symmetry constrains and removes these freedoms.

The Majorana equation can be written in several distinct forms:

These three forms are equivalent, and can be derived from one-another. Each offers slightly different insight into the nature of the equation. The first form emphasises that purely real solutions can be found. The second form clarifies the role of charge conjugation. The third form provides the most direct contact with the representation theory of the Lorentz group.

The conventional starting point is to state that "the Dirac equation can be written in Hermitian form", when the gamma matrices are taken in the Majorana representation. The Dirac equation is then written as

with ${\displaystyle {\hat {\alpha }}}$ being purely real 4×4 symmetric matrices, and ${\displaystyle \beta }$ being purely imaginary skew-symmetric; as required to ensure that the operator (that part inside the parentheses) is Hermitian. In this case, purely real 4‑spinor solutions to the equation can be found; these are the Majorana spinors.

The Majorana equation is