Majorana 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.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.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.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.being purely imaginary skew-symmetric; as required to ensure that the operator (that part inside the parentheses) is Hermitian.written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components.From this, and a fair bit of algebra, one may obtain the equivalent equation: This form is not entirely obvious, and so deserves a proof.In short, it is involved in mapping particles to their antiparticles, which includes, among other things, the reversal of the electric charge.The Weyl equation describes the time evolution of a massless complex-valued two-component spinor.on spinors, whereas the Lorentz invariance of the mass term requires invocation of the defining relation for the symplectic group.This form is invariant under Lorentz transformations, in that The skew matrix takes the Pauli matrices to minus their transpose: forThe skew matrix can be interpreted as the product of a parity transformation and a transposition acting on two-spinors.Applying it to the Lorentz transformation yields These two variants describe the covariance properties of the differentials acting on the left and right spinors, respectively.The Lorentz transform, in coordinates, is or, equivalently, This leads to In order to make use of the Weyl map a few indexes must be raised and lowered.From this, one concludes that the skew-complex field transforms as This is fully compatible with the covariance property of the differential.to be an arbitrary complex phase factor, the linear combination transforms in a covariant fashion.Thus, the matched combinations of these are Lorentz covariant, and one may take as a pair of complex 2-spinor Majorana equations.as above, define Using the algebraic machinery given above, it is not hard to show that Defining a conjugate operator The four-component Majorana equation is then Writing this out in detail, one has Multiplying on the left by brings the above into a matrix form wherein the gamma matrices in the chiral representation can be recognized.However, the charge conjugation operator has not one, but two distinct eigenstates, one of which is the ELKO spinor; it does not solve the Majorana equation, but rather, a sign-flipped version of it.One convenient starting point for writing the solutions is to work in the rest frame way of the spinors.yields the complex conjugate of the Pauli matrices: The plane wave solutions can be developed for the energy-momentumThe Dirac equation can be written in a purely real form, when the gamma matrices are taken in the Majorana representation.This stands in contrast to the Dirac spinors, which are only covariant under the action of the complexified spin groupThe location of this extra degree of freedom is pin-pointed by the charge conjugation operator, and the imposition of the Majorana constraintOnce removed, there cannot be any coupling to the electromagnetic potential, ergo, the Majorana spinor is necessarily electrically neutral.field corresponds to the electromagnetic potential can be seen by noting that (for example) the square of the Dirac operator is the Laplacian plus the scalar curvature