In
physics, the Majorana equation is a
relativistic wave equation
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the conte ...
. It is named after the Italian physicist
Ettore Majorana, who proposed it in 1937 as a means of describing
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s that are their own
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
. Particles corresponding to this equation are termed
Majorana particle
A Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Et ...
s, 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 particle
A Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Et ...
s 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
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
,
parity transformation
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
and
time reversal; the continuous symmetry is
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation
In physics, the Lorentz transformations are a six-parameter famil ...
.
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
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
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 In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
; 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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
, 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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
written so that the
Dirac operator is purely Hermitian, thus giving purely real solutions.
* As an operator that relates a four-component
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
to its
charge conjugate.
* As a 2×2 differential equation acting on a complex two-component spinor, resembling the
Weyl equation
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
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
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
. 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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
can be written in Hermitian form", when the
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
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
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form),
: \ \stackrel\ \gamma^1 A_1 + ...
to include the
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
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
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. 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
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
, including the Dirac, Weyl, and Majorana representations, that
and so one may write
:
where
is the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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 physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
. In short, it is involved in mapping
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
s to their
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
s, 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
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
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
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
. 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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
.
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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of
The subscript is used throughout this section to denote a ''left''-handed chiral spinor; under a
parity transformation
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
, 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 particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
, in the sense that it involves four-component spinors, gamma matrices, and mass terms, but includes the
charge conjugate of a
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
. In contrast, the
Weyl equation
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
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
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
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
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
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
Sterile neutrinos (or inert neutrinos) are hypothetical particles (neutral leptons – neutrinos) that are believed to interact only via gravity and not via any of the other fundamental interactions of the Standard Model. The term ''sterile neutrin ...
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
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
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
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
, with the addition of a properly Lorentz-covariant mass term.
[
Andreas Aste, (2010) "A Direct Road to Majorana Fields", ''Symmetry'' 2010(2) 1776-1809; doi:10.3390/sym2041776 ISSN 2073-8994.
] 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
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
. 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
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
for ''μ'' = 1, 2, 3. Under the
parity transformation
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
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
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
. 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
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
The
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation
In physics, the Lorentz transformations are a six-parameter famil ...
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
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
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
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
operator, the other component being
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
. 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
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
. 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 mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
:
The skew matrix can be identified with the
charge conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
operator (in the
Weyl basis
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
). 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 mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. 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
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
is given in the article on
charge conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
. 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
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...
, 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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
. The
discrete symmetries
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square ...
of the
gauge theory follows from the "trivial" observation that
:
is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
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
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. 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
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. 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
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
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 spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
s. 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
In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is al ...
, 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
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
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 transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s, these transform under the (purely real)
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
This stands in contrast to the
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
s, 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 manifold In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
s. 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
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
(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
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
. The frequently applied supplemental condition
corresponds to the Majorana spinor.
Majorana particle
Particles corresponding to Majorana spinors are known as
Majorana particle
A Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Et ...
s, due to the above self-conjugacy constraint. All the fermions included in the
Standard Model have been excluded as
Majorana fermion
A Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Et ...
s (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
In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neut ...
decays, are possible. A number of experiments probing whether the neutrino is a Majorana particle are currently underway.
[A. Franklin, ''Are There Really Neutrinos?: An Evidential History'' (Westview Press, 2004), p. 186]
Notes
{{notelist
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)
, ''Nature Physics'' Vol. 5 pages 614–618.
Quantum field theory
Spinors
Equations