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 fermionic particle that is its own anti-particle.
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, 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 symmetry constrains and removes these freedoms.

Definition

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.

Purely real four-component form

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 being purely real 4×4 symmetric matrices, and being purely imaginary skew-symmetric; as required to ensure that the operator is Hermitian. In this case, purely real 4‑spinor solutions to the equation can be found; these are the Majorana spinors.

Charge-conjugate four-component form

The Majorana equation is
with the derivative operator written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components. The spinor is the charge conjugate of By construction, charge conjugates are necessarily given by
where denotes the transpose, is an arbitrary phase factor conventionally taken as and is a 4×4 matrix, the charge conjugation matrix. The matrix representation of depends on the choice of the representation of the gamma matrices. By convention, the conjugate spinor is written as
A number of algebraic identities follow from the charge conjugation matrix One states that in any representation of the gamma matrices, including the Dirac, Weyl, and Majorana representations, that and so one may write
where is the complex conjugate of The charge conjugation matrix also has the property that
in all representations. From this, and a fair bit of algebra, one may obtain the equivalent equation:
A detailed discussion of the physical interpretation of matrix as charge conjugation can be found in the article on charge conjugation. In short, it is involved in mapping particles to their antiparticles, which includes, among other things, the reversal of the electric charge. Although is defined as "the charge conjugate" of the charge conjugation operator has not one but two eigenvalues. This allows a second spinor, the ELKO spinor to be defined. This is discussed in greater detail below.

Complex two-component form

The Majorana operator, is defined as
where
is a vector whose components are the 2×2 identity matrix for and the Pauli matrices for The is an arbitrary phase factor, typically taken to be one: The is a 2×2 matrix that can be interpreted as the symplectic form for the symplectic group which is a double covering of the Lorentz group. It is
which happens to be isomorphic to the imaginary unit with the matrix transpose being the analog of complex conjugation.
Finally, the is a short-hand reminder to take the complex conjugate. The Majorana equation for a left-handed complex-valued two-component spinor is then
or, equivalently,
with the complex conjugate of The subscript is used throughout this section to denote a left-handed chiral spinor; under a parity transformation, this can be taken to a right-handed spinor, and so one also has a right-handed form of the equation. This applies to the four-component equation as well; further details are presented below.

Key ideas

Some of the properties of the Majorana equation, its solution and its Lagrangian formulation are summarized here.
  • The Majorana equation 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 of a spinor . In contrast, the Weyl equation is for two-component spinor without mass.
  • Solutions to the Majorana equation can be interpreted as electrically neutral particles that are their own anti-particle. By convention, the charge conjugation operator takes particles to their anti-particles, and so the Majorana spinor is conventionally defined as the solution where That is, the Majorana spinor is "its own antiparticle". Insofar as charge conjugation takes an electrically charge particle to its anti-particle with opposite charge, one must conclude that the Majorana spinor is electrically neutral.
  • The Majorana equation is Lorentz covariant, and a variety of Lorentz scalars can be constructed from its spinors. This allows several distinct Lagrangians to be constructed for Majorana fields.
  • When the Lagrangian is expressed in terms of two-component left and right chiral spinors, it may contain three distinct mass terms: left and right Majorana mass terms, and a Dirac mass term. These manifest physically as two distinct masses; this is the key idea of the seesaw mechanism for describing low-mass neutrinos with a left-handed coupling to the Standard model, with the right-handed component corresponding to a sterile neutrino at GUT-scale masses.
  • The discrete symmetries of C, P and T conjugation are intimately controlled by a freely chosen phase factor on the charge conjugation operator. This manifests itself as distinct complex phases on the mass terms. This allows both CP-symmetric and CP-violating Lagrangians to be written.
  • The Majorana fields are CPT invariant, but the invariance is, in a sense "freer" than it is for charged particles. This is because charge is necessarily a Lorentz-invariant property, and is thus constrained for charged fields. The neutral Majorana fields are not constrained in this way, and can mix.

    Two-component Majorana equation

The Majorana equation can be written both in terms of a real four-component spinor, and as a complex two-component spinor. Both can be constructed from the Weyl equation, with the addition of a properly Lorentz-covariant mass term. This section provides an explicit construction and articulation.

Weyl equation

The Weyl equation describes the time evolution of a massless complex-valued two-component spinor. It is conventionally written as
Written out explicitly, it is
The Pauli four-vector is
that is, a vector whose components are the 2 × 2 identity matrix for μ = 0 and the Pauli matrices for μ = 1, 2, 3. Under the parity transformation one obtains a dual equation
where. These are two distinct forms of the Weyl equation; their solutions are distinct as well. It can be shown that the solutions have left-handed and right-handed helicity, and thus chirality. It is conventional to label these two distinct forms explicitly, thus:

Lorentz invariance

The Weyl equation describes a massless particle; the Majorana equation adds a mass term. The mass must be introduced in a Lorentz invariant fashion. This is achieved by observing that the special linear group is isomorphic to the symplectic group Both of these groups are double covers of the Lorentz group The Lorentz invariance of the derivative term is conventionally worded in terms of the action of the group on spinors, whereas the Lorentz invariance of the mass term requires invocation of the defining relation for the symplectic group.
The double-covering of the Lorentz group is given by
where and and is the Hermitian transpose. This is used to relate the transformation properties of the differentials under a Lorentz transformation to the transformation properties of the spinors.
The symplectic group is defined as the set of all complex 2×2 matrices that satisfy
where
is a skew-symmetric matrix. It is used to define a symplectic bilinear form on Writing a pair of arbitrary two-vectors as
the symplectic product is
where is the transpose of This form is invariant under Lorentz transformations, in that
The skew matrix takes the Pauli matrices to minus their transpose:
for The skew matrix can be interpreted as the product of a parity transformation and a transposition acting on two-spinors. However, as will be emphasized in a later section, it can also be interpreted as one of the components of the charge conjugation operator, the other component being complex conjugation. 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.