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 formula_1 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 formula_3 being purely real 4×4 symmetric matrices, and formula_4 being purely imaginary skew-symmetric (as required to ensure that the operator between (...) 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 formula_6 written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components. The spinor formula_7 is the charge conjugate of formula_8 By construction, charge conjugates are necessarily given by
where formula_10 denotes the transpose, formula_11 is an arbitrary phase factor formula_12 conventionally taken as formula_13 and formula_14 is a 4×4 matrix, the "charge conjugation matrix". The matrix representation of formula_14 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 formula_17 One states that in any representation of the gamma matrices, including the Dirac, Weyl, and Majorana representations, that formula_18 and so one may write
where formula_20 is the complex conjugate of formula_21 The charge conjugation matrix formula_14 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:
Proof
This form is not entirely obvious, and so deserves a proof. Starting with
Expand formula_26:
Multiply by formula_14 use formula_29:
Charge conjugation transposes the gamma matrices:
Take the complex conjugate:
The matrix formula_33 is Hermitian, formula_34 in all three representations (Dirac, chiral, Majorana):
It is also an involution, taking the Hermitian conjugate: formula_36
Multiply by formula_33, note that formula_39 and make use of formula_40:
The above is just the definition of the conjugate, so conclude that
A detailed discussion of the physical interpretation of matrix formula_43 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 formula_44 is defined as "the charge conjugate" of formula_45 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", formula_46 is defined as
where 
is a vector whose components are the 2×2 identity matrix formula_49 for formula_50 and (minus) the Pauli matrices for formula_51 The formula_52 is an arbitrary phase factor, formula_53 typically taken to be one: formula_54 The formula_55 is a 2×2 matrix that can be interpreted as the symplectic form for the symplectic group formula_56 which is a double covering of the Lorentz group. It is
which happens to be isomorphic to the imaginary unit i.e. formula_58 and formula_59 for formula_60 with the matrix transpose being the analog of complex conjugation.
Finally, the formula_61 is a short-hand reminder to take the complex conjugate. The Majorana equation for a left-handed complex-valued two-component spinor formula_62 is then
or, equivalently,
with formula_65 the complex conjugate of formula_66 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.
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 formula_73 for "μ" = 0 and the Pauli matrices for "μ" = 1,2,3. Under the parity transformation formula_74 one obtains a dual equation
where formula_76. 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 formula_78 is isomorphic to the symplectic group formula_79 Both of these groups are double covers of the Lorentz group formula_80 The Lorentz invariance of the derivative term (from the Weyl equation) is conventionally worded in terms of the action of the group formula_78 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 formula_83 and formula_84 and formula_85 is the Hermitian transpose. This is used to relate the transformation properties of the differentials under a Lorentz transformation formula_86 to the transformation properties of the spinors.
The symplectic group formula_87 is defined as the set of all complex 2x2 matrices formula_88 that satisfy
where 
is a skew-symmetric matrix. It is used to define a symplectic bilinear form on formula_91 Writing a pair of arbitrary two-vectors formula_92 as
the symplectic product is
where formula_95 is the transpose of formula_96 This form is invariant under Lorentz transformations, in that 
The skew matrix takes the Pauli matrices to minus their transpose:
for formula_99 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 formula_86 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
Proof
These transformation properties are not particularly "obvious", and so deserve a careful derivation. Begin with the form
for some unknown formula_107 to be determined. 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. This is easier said than done, as it invokes the identity
where formula_113 is the flat-space Minkowski metric. The above identity is often used to define the elements formula_114 One takes the transpose:
to write
One thus regains the original form if formula_117 that is, formula_118 Performing the same manipulations for the left-handed equation, one concludes that 
with formula_120
Mass term.
The complex conjugate of the right handed spinor field transforms as
The defining relationship for formula_122 can be rewritten as formula_123 From this, one concludes that the skew-complex field transforms as
This is fully compatible with the covariance property of the differential. Taking formula_125 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 formula_127) 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 formula_129 The skew complex conjugate formula_130 can be recognized as the charge conjugate form of formula_131 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 formula_133 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 formula_137 and formula_138 are both Lorentz covariant. The product is explicitly 
Verifying this requires keeping in mind that formula_140 and that formula_141 The RHS reduces to the Klein–Gordon operator provided that formula_142, that is, formula_143 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 formula_144 satisfying formula_145 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 formula_140 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 formula_159 its charge conjugate is
with formula_43 an ordinary 4x4 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 formula_163 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 formula_164 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 formula_1 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 formula_1 gauge theory follows from the "trivial" observation that 
is an automorphism of formula_169 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 formula_169
In the above, formula_43 is a 4x4 matrix, given in the article on the gamma matrices. Its explicit form is representation-dependent. The operator formula_164 cannot be written as a 4x4 matrix, as it is taking the complex conjugate of formula_173, and complex conjugation cannot be achieved with a complex 4x4 matrix. It can be written as a real 8x8 matrix, presuming one also writes formula_173 as a purely real 8-component spinor. Letting formula_133 stand for complex conjugation, so that formula_176 one can then write, for four-component spinors,
It is not hard to show that formula_178 and that formula_179 It follows from the first identity that formula_164 has two eigenvalues, which may be written as
The eigenvectors are readily found in the Weyl basis. From the above, in this basis, formula_164 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.
Proof
That both eigenstates solve the Klein–Gordon equation follows from the earlier identities for the two-component versions. Defining, as before,
As was previously shown
The four-component spinor requires the introduction of
which also obey
Therefore
The chiral representation requires an extra factor of formula_191:
and so one concludes that
That is, both eigenvectors of the charge conjugation operator solve the Klein–Gordon equation. The last identity can also be verified directly, by noting that formula_194 and that formula_195 
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 formula_199 define its conjugate as formula_200 It is not hard to show that formula_201 and that therefore, if formula_202 then also formula_203 and therefore that
or equivalently
This works, because formula_206 and so this reduces to the Dirac equation for 
To conclude, and reiterate, the Majorana equation is
It has four inequivalent, linearly independent solutions, formula_209 Of these, only two are also solutions to the Dirac equation: namely formula_210 and formula_211
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 formula_212 leads to the Majorana equation taking the form 
In the chiral (Weyl) basis, one has that
with formula_215 the Pauli vector. The sign convention here is consistent with the article gamma matrices. Plugging in the positive charge conjugation eigenstate formula_216 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 formula_219 yields the complex conjugate of the Pauli matrices: 
The plane wave solutions can be developed for the energy-momentum formula_221 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 formula_68 and formula_67 in the Majorana equation means that the field formula_68 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, formula_68 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 formula_3 being purely real symmetric matrices, and formula_4 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 formula_233 This stands in contrast to the Dirac spinors, which are only covariant under the action of the complexified spin group formula_234 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 formula_235 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 formula_236 spatial dimensions. In this case, the complexified spin group formula_237 has a double covering by formula_238 with formula_239 the circle. The implication is that formula_240 encodes the generalized Lorentz transformations (of course), while the circle can be identified with the formula_241 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 formula_242 The formula_243 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 formula_243 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 formula_245 (of the underlying manifold that the spinor field sits on) plus the (electromagnetic) field strength formula_246 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 formula_247 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 formula_248 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.