Majorana equation

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 symmetry constrains and removes these freedoms.
Definition
The Majorana equation can be written in several distinct forms:
As the Dirac equation written so that the Dirac operator is purely Hermitian, thus giving purely real solutions.
As an operator that relates a four-component spinor to its charge conjugate.
As a 2×2 differential equation acting on a complex two-component spinor, resembling the Weyl equation with a properly Lorentz covariant mass term.
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 (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.
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 (Dirac, chiral, Majorana). 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 (minus) 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 (i.e. and for ) 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 (from the Weyl equation) 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.
Differentials
Under the Lorentz transformation the differential term transforms as
provided that the right-handed field transforms as
Similarly, the left-handed differential transforms as
provided that the left-handed spinor transforms as
Mass term
The complex conjugate of the right handed spinor field transforms as
The defining relationship for can be rewritten as From this, one concludes that the skew-complex field transforms as
This is fully compatible with the covariance property of the differential. Taking to be an arbitrary complex phase factor, the linear combination
transforms in a covariant fashion. Setting this to zero gives the complex two-component Majorana equation for the right-handed field. Similarly, the left-chiral Majorana equation (including an arbitrary phase factor ) is
The left and right chiral versions are related by a parity transformation. As shown below, these square to the Klein–Gordon operator only if The skew complex conjugate can be recognized as the charge conjugate form of this is articulated in greater detail below. Thus, the Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form.
Left and right Majorana operators
Define a pair of operators, the Majorana operators,
where is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as
whereas the Weyl spinors transform as
just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take
as a pair of complex 2-spinor Majorana equations.
The products and are both Lorentz covariant. The product is explicitly
Verifying this requires keeping in mind that and that The RHS reduces to the Klein–Gordon operator provided that , that is, These two Majorana operators are thus "square roots" of the Klein–Gordon operator.
Four-component Majorana equation
The real four-component version of the Majorana equation can be constructed from the complex two-component equation as follows. Given the complex field satisfying 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. This is
That is,
Applying this to the 4-spinor
and recalling that one finds that the spinor is an eigenstate of the mass term,
and so, for this particular spinor, the four-component Majorana equation reduces to the Dirac equation
The skew matrix can be identified with the charge conjugation operator (in the Weyl basis). Explicitly, this is
Given an arbitrary four-component spinor its charge conjugate is
with an ordinary 4×4 matrix, having a form explicitly given in the article on gamma matrices. In conclusion, the 4-component Majorana equation can be written as
Charge conjugation and parity
The charge conjugation operator appears directly in the 4-component version of the Majorana equation. When the spinor field is a charge conjugate of itself, that is, when then the Majorana equation reduces to the Dirac equation, and any solution can be interpreted as describing an electrically neutral field. 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.
The charge conjugation operator for a four-component spinor is defined as
A general discussion of the physical interpretation of this operator in terms of electrical charge is given in the article on charge conjugation. Additional discussions are provided by Bjorken & Drell or Itzykson & Zuber. In more abstract terms, it is the spinorial equivalent of complex conjugation of the coupling of the electromagnetic field. This can be seen as follows. If one has a single, real scalar field, it cannot couple to electromagnetism; however, a pair of real scalar fields, arranged as a complex number, can. For scalar fields, charge conjugation is the same as complex conjugation. The discrete symmetries of the gauge theory follows from the "trivial" observation that
is an automorphism of For spinorial fields, the situation is more confusing. Roughly speaking, however, one can say that the Majorana field is electrically neutral, and that taking an appropriate combination of two Majorana fields can be interpreted as a single electrically charged Dirac field. The charge conjugation operator given above corresponds to the automorphism of
In the above, is a 4×4 matrix, given in the article on the gamma matrices. Its explicit form is representation-dependent. The operator cannot be written as a 4×4 matrix, as it is taking the complex conjugate of , and complex conjugation cannot be achieved with a complex 4×4 matrix. It can be written as a real 8×8 matrix, presuming one also writes as a purely real 8-component spinor. Letting stand for complex conjugation, so that one can then write, for four-component spinors,
It is not hard to show that and that It follows from the first identity that has two eigenvalues, which may be written as
The eigenvectors are readily found in the Weyl basis. From the above, in this basis, is explicitly
and thus
Both eigenvectors are clearly solutions to the Majorana equation. However, only the positive eigenvector is a solution to the Dirac equation:
The negative eigenvector "doesn't work", it has the incorrect sign on the Dirac mass term. It still solves the Klein–Gordon equation, however. The negative eigenvector is termed the ELKO spinor.
Parity
Under parity, the left-handed spinors transform to right-handed spinors. The two eigenvectors of the charge conjugation operator, again in the Weyl basis, are
As before, both solve the four-component Majorana equation, but only one also solves the Dirac equation. This can be shown by constructing the parity-dual four-component equation. This takes the form
where
Given the two-component spinor define its conjugate as It is not hard to show that and that therefore, if then also and therefore that
or equivalently
This works, because and so this reduces to the Dirac equation for
To conclude, and reiterate, the Majorana equation is
It has four inequivalent, linearly independent solutions, Of these, only two are also solutions to the Dirac equation: namely and
Solutions
Spin eigenstates
One convenient starting point for writing the solutions is to work in the rest frame way of the spinors. Writing the quantum Hamiltonian with the conventional sign convention leads to the Majorana equation taking the form
In the chiral (Weyl) basis, one has that
with the Pauli vector. The sign convention here is consistent with the article gamma matrices. Plugging in the positive charge conjugation eigenstate given above, one obtains an equation for the two-component spinor
and likewise
These two are in fact the same equation, which can be verified by noting that yields the complex conjugate of the Pauli matrices:
The plane wave solutions can be developed for the energy-momentum and are most easily stated in the rest frame. The spin-up rest-frame solution is
while the spin-down solution is
That these are being correctly interpreted can be seen by re-expressing them in the Dirac basis, as Dirac spinors. In this case, they take the form
and
These are the rest-frame spinors. They can be seen as a linear combination of both the positive and the negative-energy solutions to the Dirac equation. These are the only two solutions; the Majorana equation has only two linearly independent solutions, unlike the Dirac equation, which has four. The doubling of the degrees of freedom of the Dirac equation can be ascribed to the Dirac spinors carrying charge.
Momentum eigenstates
In a general momentum frame, the Majorana spinor can be written as
Electric charge
The appearance of both and in the Majorana equation means that the field 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, must be taken to be electrically neutral. This can be articulated in greater detail.
The Dirac equation can be written in a purely real form, when the gamma matrices are taken in the Majorana representation. The Dirac equation can then be written as
with being purely real symmetric matrices, and being purely imaginary skew-symmetric. In this case, purely real solutions to the equation can be found; these are the Majorana spinors. Under the action of Lorentz transformations, these transform under the (purely real) spin group This stands in contrast to the Dirac spinors, which are only covariant under the action of the complexified spin group The interpretation is that complexified spin group encodes the electromagnetic potential, the real spin group does not.
This can also be stated in a different way: the Dirac equation, and the Dirac spinors contain a sufficient amount of gauge freedom to naturally encode electromagnetic interactions. This can be seen by noting that the electromagnetic potential can very simply be added to the Dirac equation without requiring any additional modifications or extensions to either the equation or the spinor. The location of this extra degree of freedom is pin-pointed by the charge conjugation operator, and the imposition of the Majorana constraint removes this extra degree of freedom. Once removed, there cannot be any coupling to the electromagnetic potential, ergo, the Majorana spinor is necessarily electrically neutral. An electromagnetic coupling can only be obtained by adding back in a complex-number-valued phase factor, and coupling this phase factor to the electromagnetic potential.
The above can be further sharpened by examining the situation in spatial dimensions. In this case, the complexified spin group has a double covering by with the circle. The implication is that encodes the generalized Lorentz transformations (of course), while the circle can be identified with the action of the gauge group on electric charges. That is, the gauge-group action of the complexified spin group on a Dirac spinor can be split into a purely-real Lorentzian part, and an electromagnetic part. This can be further elaborated on non-flat (non-Minkowski-flat) spin manifolds. In this case, the Dirac operator acts on the spinor bundle. Decomposed into distinct terms, it includes the usual covariant derivative The field can be seen to arise directly from the curvature of the complexified part of the spin bundle, in that the gauge transformations couple to the complexified part, and not the real-spinor part. That the 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 (of the underlying manifold that the spinor field sits on) plus the (electromagnetic) field strength For the Majorana case, one has only the Lorentz transformations acting on the Majorana spinor; the complexification plays no role. A detailed treatment of these topics can be found in Jost while the case is articulated in Bleeker. Unfortunately, neither text explicitly articulates the Majorana spinor in direct form.
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 corresponds to the Majorana spinor.
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.
Notes
References
Additional reading
"Majorana Legacy in Contemporary Physics", Electronic Journal of Theoretical Physics (EJTP) Volume 3, Issue 10 (April 2006) Special issue for the Centenary of Ettore Majorana (1906-1938?). ISSN 1729-5254
Frank Wilczek, (2009) "Majorana returns", Nature Physics Vol. 5 pages 614–618.
Eponymous equations of physics
Quantum field theory
Spinors