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{{short description|Relativistic wave equation describing massless fermions}}
{{quantum field theory}}
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 possible types of elementary fermions, the other two being the [[Dirac fermion|Dirac]] and the [[Majorana fermion]]s.

None of the [[elementary particle]]s in the [[Standard Model]] are Weyl fermions. Previous to the confirmation of the [[neutrino oscillation]]s, it was considered possible that the [[neutrino]] might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In [[condensed matter physics]], some materials can display [[quasiparticle]]s that behave as Weyl fermions, leading to the notion of [[Weyl semimetal]]s.

Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term.<ref name=Shifman-1999/>

== History ==
The [[Dirac equation]] was published in 1928 by [[Paul Dirac]], and was first used to model [[spin-1/2]] particles in the framework of [[relativistic quantum mechanics]].<ref name=Pal-2011/> Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation.<ref name=Pal-2011/><ref name=Weyl-1929/> [[Wolfgang Pauli]] wrote in 1933 against Weyl's equation because it violated [[Parity (physics)|parity]].<ref name=Bilenky-2005/> However, three years before, Pauli had predicted the existence of a new elementary [[fermion]], the [[neutrino]], to explain the [[beta decay]], which eventually was described using the Weyl equation.

In 1937, [[Conyers Herring]] proposed that Weyl fermions may exist as [[quasiparticles]] in condensed matter.<ref name=Vishwanath-2015/>

Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless).<ref name=Bilenky-2005/> The same year the [[Wu experiment]] showed that [[Parity (physics)|parity]] could be violated by the [[weak interaction]], addressing Pauli's criticism.<ref name="Wu1957" >
{{cite journal
 |last1=Wu |first1=C. S.
 |last2=Ambler |first2=E.
 |last3=Hayward |first3=R. W.
 |last4=Hoppes |first4=D. D.
 |last5=Hudson |first5=R. P.
 |year=1957
 |title=Experimental Test of Parity Conservation in Beta Decay
 |journal=[[Physical Review]]
 |volume=105 |issue=4 |pages=1413–1415
 |bibcode=1957PhRv..105.1413W
 |doi=10.1103/PhysRev.105.1413
 |doi-access=free
}}</ref> This was followed by the measurement of the neutrino's [[Helicity (particle physics)|helicity]] in 1958.<ref name=Bilenky-2005/> As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. Thus, the [[Standard Model]] was built under the assumption that neutrinos were Weyl fermions.<ref name=Bilenky-2005/>

While Italian physicist [[Bruno Pontecorvo]] had proposed in 1957 the possibility of neutrino masses and [[neutrino oscillation]]s,<ref name=Bilenky-2005/> it was not until 1998 that [[Super-Kamiokande]] eventually confirmed the existence of neutrino oscillations, and their non-zero mass.<ref name=Bilenky-2005/> This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos, as the equations can only describe massless particles.<ref name=Pal-2011/>

In 2015, the first [[Weyl semimetal]] was demonstrated experimentally in crystalline [[tantalum arsenide]] (TaAs) by the collaboration of [[M. Zahid Hasan|M.Z. Hasan]]'s ([[Princeton University]]) and H. Ding's ([[Chinese Academy of Sciences]]) teams.<ref name=Vishwanath-2015/> Independently, the same year, [[Marin Soljačić|M. Soljačić]] team ([[Massachusetts Institute of Technology]]) also observed Weyl-like excitations in [[photonic crystal]]s.<ref name=Vishwanath-2015/>

==Equation==
The Weyl equation comes in two forms.  The right-handed form can be written as follows:<ref name=Pearson-2004/><ref name=Woan-2010/><ref name=Peskin-Schroeder-1995/>
:<math> \sigma^\mu\partial_\mu \psi = 0</math>

Expanding this equation, and inserting <math>c</math> for the [[speed of light]], it becomes

:<math> I_2 \frac{1}{c}\frac{\partial \psi}{\partial t} + \sigma_x\frac{\partial \psi}{\partial x} + \sigma_y\frac{\partial \psi}{\partial y} + \sigma_z\frac{\partial \psi}{\partial z} = 0</math>

where

:<math> \sigma^\mu = \begin{pmatrix}\sigma^0 & \sigma^1 & \sigma^2 & \sigma^3\end{pmatrix} = \begin{pmatrix}I_2 & \sigma_x & \sigma_y & \sigma_z\end{pmatrix}</math>

is a [[vector (mathematics and physics)|vector]] whose components are the 2×2 [[identity matrix]] <math>I_2</math> for <math>\mu = 0</math> and the [[Pauli matrices]] for <math>\mu = 1, 2, 3,</math> and <math>\psi</math> is the [[wavefunction]] – one of the Weyl [[spinors]]. The left-handed form of the Weyl equation is usually written as:

:<math> \bar{\sigma}^\mu\partial_\mu \psi = 0</math>

where
: <math>\bar{\sigma}^\mu = \begin{pmatrix}I_2 & -\sigma_x & -\sigma_y & -\sigma_z\end{pmatrix} ~.</math>

The solutions of the right- and left-handed Weyl equations are different: they have right- and left-handed [[helicity (particle physics)|helicity]], and thus [[chirality]], respectively. It is convenient to indicate this explicitly, as follows: <math> \sigma^\mu\partial_\mu \psi_{\rm R} = 0</math> and <math>\bar{\sigma}^\mu\partial_\mu \psi_{\rm L} = 0 ~.</math>

==Plane wave solutions==
The [[plane-wave]] solutions to the Weyl equation are referred to as the left and right handed Weyl spinors, each is with two components. Both have the form

:<math>\psi\left(\mathbf{r}, t\right) = \begin{pmatrix}
    \psi_1 \\
    \psi_2 \\ 
  \end{pmatrix} =
  \chi e^{-i(\mathbf{k}\cdot\mathbf{r} - \omega t)} =
  \chi e^{-i(\mathbf{p}\cdot\mathbf{r} - Et)/\hbar}
</math>,

where

:<math>\chi = \begin{pmatrix}
    \chi_1 \\
    \chi_2 \\ 
  \end{pmatrix}
</math>

is a momentum-dependent two-component spinor which satisfies

:<math>
  \sigma^\mu p_\mu \chi =
  \left( I_2 E - \vec{\sigma} \cdot \vec{p} \right) \chi =
  0
</math>

or 
:<math>
  \bar{\sigma}^\mu p_\mu \chi =
  \left( I_2 E + \vec{\sigma} \cdot \vec{p} \right) \chi =
  0
</math>.

By direct manipulation, one obtains that 
:<math>
  \left(\bar{\sigma}^\nu p_\nu\right) \left(\sigma^\mu p_\mu\right) \chi =
  \left(\sigma^\nu p_\nu\right) \left(\bar{\sigma}^\mu p_\mu\right) \chi =
  p_\mu p^\mu \chi =
  \left(E^2 - \vec{p}\cdot\vec{p}\right) \chi = 0
</math>,
and concludes that the equations correspond to a particle that is [[mass]]less. As a result, the magnitude of [[momentum]] <math>\mathbf{p}</math> relates directly to the [[wave-vector]] <math>\mathbf{k}</math> by the [[de Broglie relations]] as:

:<math> |\mathbf{p}| = \hbar |\mathbf{k}| = \frac{\hbar\omega}{c} \, \Rightarrow \, |\mathbf{k}| = \frac{\omega}{c} </math>

The equation can be written in terms of left and right handed spinors as:

:<math>\begin{align}
        \sigma^\mu \partial_\mu \psi_{\rm R} &= 0 \\
  \bar{\sigma}^\mu \partial_\mu \psi_{\rm L} &= 0 
\end{align}</math>

===Helicity===

{{main|Helicity (particle physics)}}

The left and right components correspond to the helicity <math>\lambda</math> of the particles, the projection of [[angular momentum operator]] <math>\mathbf{J}</math> onto the linear momentum <math>\mathbf{p}</math>:

:<math>
  \mathbf{p}\cdot\mathbf{J}\left|\mathbf{p},\lambda\right\rangle =
  \lambda |\mathbf{p}|\left|\mathbf{p},\lambda\right\rangle
</math>

Here <math display="inline">\lambda = \pm\frac{1}{2} ~.</math>

==Lorentz invariance==
Both equations are [[Lorentz invariant]] under the [[Lorentz transformation]] <math>x \mapsto x^\prime = \Lambda x</math> where <math>\Lambda \in \mathrm{SO}(1,3)~.</math> More precisely, the equations transform as
:<math>\sigma^\mu\frac{\partial}{\partial x^\mu} \psi_{\rm R}(x) 
 \mapsto \sigma^\mu\frac{\partial}{\partial x^{\prime\mu}} \psi^\prime_{\rm R}\left(x^\prime\right)
 = \left(S^{-1}\right)^\dagger \sigma^\mu\frac{\partial}{\partial x^\mu} \psi_{\rm R}(x)
</math>

where <math>S^\dagger</math> is the [[Hermitian transpose]], provided that the right-handed field transforms as

:<math>\psi_{\rm R}(x) \mapsto \psi^\prime_{\rm R}\left(x^\prime\right) = S\psi_{\rm R}(x)</math>

The matrix <math>S \in SL(2,\mathbb{C})</math> is related to the Lorentz transform by means of the [[cover (mathematics)|double covering]] of the [[Lorentz group]] by the [[special linear group]] <math>\mathrm{SL}(2, \mathbb{C})</math> given by
:<math>\sigma_\mu {\Lambda^\mu}_\nu = \left(S^{-1}\right)^\dagger \sigma_\nu S^{-1}</math>

Thus, if the untransformed differential vanishes in one Lorentz frame, then it also vanishes in another. Similarly

:<math>\overline{\sigma}^\mu\frac{\partial}{\partial x^\mu} \psi_{\rm L}(x) 
 \mapsto \overline{\sigma}^\mu\frac{\partial}{\partial x^{\prime\mu}} \psi^\prime_{\rm L}\left(x^\prime\right)
 = S \overline{\sigma}^\mu\frac{\partial}{\partial x^\mu} \psi_{\rm L}(x)
</math>

provided that the left-handed field transforms as

:<math>\psi_{\rm L}(x) \mapsto \psi^\prime_{\rm L}\left(x^\prime\right) = \left(S^\dagger\right)^{-1}\psi_{\rm L}(x)~.</math>

Proof: Neither of these transformation properties are in any way "obvious", and so deserve a careful derivation.  Begin with the form

:<math>\psi_{\rm R}(x) \mapsto \psi^\prime_{\rm R}\left(x^\prime\right) = R\psi_{\rm R}(x)</math>

for some unknown <math>R \in \mathrm{SL}(2, \mathbb{C})</math> to be determined. The Lorentz transform, in coordinates, is
:<math>x^{\prime\mu} = {\Lambda^\mu}_\nu x^\nu</math>
or, equivalently,
:<math>x^\nu = {\left(\Lambda^{-1}\right)^\nu}_\mu x^{\prime\mu}</math>

This leads to
:<math>\begin{align}
     \sigma^\mu \partial^\prime_\mu \psi^\prime_{\rm R}\left(x^\prime\right)
  &= \sigma^\mu\frac{\partial}{\partial x^{\prime\mu}} \psi^\prime_{\rm R}\left(x^\prime\right) \\
  &= \sigma^\mu \frac{\partial x^\nu}{\partial x^{\prime\mu}} \frac{\partial}{\partial x^\nu} R \psi_{\rm R}(x) \\
  &= \sigma^\mu {\left(\Lambda^{-1}\right)^\nu}_\mu \frac{\partial}{\partial x^\nu} R\psi_{\rm R}(x) \\
  &= \sigma^\mu {\left(\Lambda^{-1}\right)^\nu}_\mu \partial_\nu R\psi_{\rm R}(x)
\end{align}</math>

In order to make use of the Weyl map
:<math>\sigma_\mu{\Lambda^\mu}_\nu = \left(S^{-1}\right)^\dagger\sigma_\nu S^{-1}</math>
a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity
:<math>\eta\Lambda^\mathsf{T}\eta = \Lambda^{-1}</math>
where <math>\eta = \mbox{diag}(+1, -1, -1, -1)</math> is the flat-space [[Minkowski metric]]. The above identity is often used to define the elements <math>\Lambda\in \mathrm{SO}(1,3).</math> One takes the transpose:
:<math>{\left(\Lambda^{-1}\right)^\nu}_\mu = {\left(\Lambda^{-1\mathsf{T}}\right)_\mu}^\nu</math>
to write
:<math>\begin{align}
    \sigma^\mu {\left(\Lambda^{-1}\right)^\nu}_\mu \partial_\nu R\psi_{\rm R}(x) 
 &= \sigma^\mu {\left(\Lambda^{-1\mathsf{T}}\right)_\mu}^\nu \partial_\nu R\psi_{\rm R}(x) \\
 &= \sigma_\mu {\Lambda^\mu}_\nu \partial^\nu R\psi_{\rm R}(x) \\
 &= \left(S^{-1}\right)^\dagger \sigma_\mu \partial^\mu S^{-1} R\psi_{\rm R}(x)
\end{align}</math>
One thus regains the original form if <math>S^{-1} R = 1,</math> that is, <math>R = S.</math> Performing the same manipulations for the left-handed equation, one concludes that
:<math>\psi_{\rm L}(x)\mapsto \psi^\prime_{\rm L}\left(x^\prime\right) = L\psi_{\rm L}(x)</math>
with <math>L = \left(S^\dagger\right)^{-1}.</math>{{efn|The results presented here are identical to those of Aste (2010)<ref name=Aste-2010/> equations&nbsp;52 and 57, although the derivation performed here is completely different. The double-covering used here is also identical to Aste's equation&nbsp;48, and to the current version (December 2020) of the Wikipedia article on the [[Lorentz group]].}}

===Relationship to Majorana===
The Weyl equation is conventionally interpreted as describing a massless particle. However, with a slight alteration, one may obtain a two-component version of the [[Majorana equation]].<ref name=Aste-2010/> This arises because the [[special linear group]] <math>\mathrm{SL}(2,\mathbb{C})</math> is [[isomorphic]] to the [[symplectic group]] <math>\mathrm{Sp}(2,\mathbb{C})~.</math> The symplectic group is defined as the set of all complex 2×2 matrices that satisfy

:<math> S^\mathsf{T} \omega S = \omega</math>

where
:<math>\omega = i\sigma_2 = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}</math>

The defining relationship can be rewritten as <math>\omega S^* = \left( S^\dagger \right)^{-1}  \omega</math> where <math>S^*</math> is the [[complex conjugate]]. The right handed field, as noted earlier, transforms as

:<math>\psi_{\rm R}(x) \mapsto \psi^\prime_{\rm R}\left(x^\prime\right) = S\psi_{\rm R}(x)</math>

and so the complex conjugate field transforms as

:<math>\psi^*_{\rm R}(x) \mapsto \psi^{\prime *}_{\rm R}\left(x^\prime\right) = S^*\psi^*_{\rm R}(x)</math>

Applying the defining relationship, one concludes that

:<math>m \omega \psi^*_{\rm R}(x) \mapsto m \omega \psi^{\prime *}_{\rm R} \left(x^\prime\right) = \left(S^\dagger\right)^{-1} m \omega \psi^*_{\rm R}(x)</math>

which is exactly the same Lorentz covariance property noted earlier. Thus, the linear combination, using an arbitrary complex phase factor <math>\eta = e^{i\phi}</math>

:<math>i\sigma^\mu \partial_\mu \psi_{\rm R}(x) + \eta m \omega \psi^*_{\rm R}(x)</math>

transforms in a covariant fashion; setting this to zero gives the complex two-component [[Majorana equation]]. The Majorana equation is conventionally written as a four-component real equation, rather than a two-component complex equation; the above can be brought into four-component form (see that article for details). Similarly, the left-chiral Majorana equation (including an arbitrary phase factor <math>\zeta</math>) is

:<math>i\overline{\sigma}^\mu \partial_\mu \psi_{\rm L}(x) + \zeta m \omega \psi^*_{\rm L}(x) = 0</math>

As noted earlier, the left and right chiral versions are related by a parity transformation. The skew complex conjugate <math>\omega\psi^* = i\sigma^2\psi</math> can be recognized as the [[charge conjugation|charge conjugate]] form of <math>\psi ~.</math> Thus, the Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form. The two distinct phases on the mass term are related to the two distinct eigenvalues of the charge conjugation operator; see [[charge conjugation]] and Majorana equation for details.

Define a pair of operators, the Majorana operators,
:<math>
  D_{\rm L} = i\overline{\sigma}^\mu \partial_\mu +\zeta m\omega K\qquad
  D_{\rm R} = i\sigma^\mu \partial_\mu + \eta m\omega K
</math>
where <math>K</math> is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as
:<math>
  D_{\rm L} \mapsto D^\prime_{\rm L} = S D_{\rm L} S^\dagger \qquad
  D_{\rm R} \mapsto D^\prime_{\rm R} = \left(S^\dagger\right)^{-1} D_{\rm R} S^{-1}
</math>
whereas the Weyl spinors transform as
:<math>
  \psi_{\rm L} \mapsto \psi^\prime_{\rm L} = \left(S^\dagger\right)^{-1} \psi_{\rm L} \qquad
  \psi_{\rm R} \mapsto \psi^\prime_{\rm R} = S \psi_{\rm R}
</math>
just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take
:<math> D_{\rm L} \psi_{\rm L} = 0 \qquad D_{\rm R} \psi_{\rm R} = 0 </math>

as a pair of complex 2-spinor Majorana equations.

The products <math>D_{\rm L} D_{\rm R}</math> and <math>D_{\rm R} D_{\rm L}</math> are both Lorentz covariant. The product is explicitly 
:<math>
     D_{\rm R}D_{\rm L}
   = \left(i\sigma^\mu \partial_\mu + \eta m\omega K\right) \left(i\overline{\sigma}^\mu \partial_\mu + \zeta m\omega K\right)
   = -\left(\partial_t^2 - \vec\nabla \cdot \vec\nabla + \eta\zeta^* m^2\right)
   = -\left(\square + \eta\zeta^* m^2\right)
</math>
Verifying this requires keeping in mind that <math>\omega^2 = -1</math> and that <math>K i = -i K ~.</math> The RHS reduces to the [[Klein–Gordon equation|Klein–Gordon operator]] provided that <math>\eta\zeta^* = 1</math>, that is <math>\eta = \zeta~.</math> These two Majorana operators are thus "square roots" of the Klein–Gordon operator.

==Lagrangian densities==

The equations are obtained from the [[lagrangian density|Lagrangian densities]]

:<math> \mathcal L = i \psi_{\rm R}^\dagger \sigma^\mu \partial_\mu \psi_{\rm R} ~,</math>

:<math> \mathcal L = i \psi_{\rm L}^\dagger \bar\sigma^\mu \partial_\mu \psi_{\rm L} ~.</math>

By treating the spinor and its [[Hermitian adjoint|conjugate]] (denoted by <math> \dagger </math>) as independent variables, the relevant Weyl equation is obtained.

==Weyl spinors==
The term '''Weyl spinor''' is also frequently used in a more general setting, as an element of a [[Clifford module]].  This is closely related to the solutions given above, and gives a natural geometric interpretation to [[spinor]]s as geometric objects living on a [[manifold]]. This general setting has multiple strengths: it clarifies their interpretation as [[fermion]]s in physics, and it shows precisely how to define spin in [[General Relativity]], or, indeed, for any [[Riemannian manifold]] or [[pseudo-Riemannian manifold]]. This is informally sketched as follows.

The Weyl equation is [[Covariance and contravariance of vectors|invariant]] under the action of the Lorentz group. This means that, as [[Lorentz boost|boosts]] and [[rotation]]s are applied, the form of the equation itself does not change. However, the form of the [[spinor]] <math>\psi</math> itself does change. Ignoring [[spacetime]] entirely, the algebra of the spinors is described by a (complexified) [[Clifford algebra]]. The spinors transform under the action of the [[spin group]]. This is entirely analogous to how one might talk about a vector, and how it transforms under the [[rotation group]], except that now, it has been adapted to the case of spinors.

Given an arbitrary [[pseudo-Riemannian manifold]] <math>M</math> of dimension <math>(p,q)</math>, one may consider its [[tangent bundle]] <math>TM</math>. At any given point <math>x \in M,</math> the [[tangent space]] <math>T_x M</math> is a <math>(p,q)</math> dimensional [[vector space]]. Given this vector space, one can construct the Clifford algebra <math>\mathrm{Cl}(p,q)</math> on it. If <math>\{e_i\}</math> are a [[vector space basis]] on <math>T_x M</math>, one may construct a pair of Weyl spinors as<ref>{{cite book |first=Jurgen |last=Jost |year=2002 |title=Riemannian Geometry and Geometric Analysis |edition=3rd |publisher=Springer Universitext}}</ref>

:<math>w_j = \frac{1}{\sqrt{2}} \left(e_{2j} + ie_{2j+1}\right)</math>
and 
:<math>w_j^* = \frac{1}{\sqrt{2}} \left(e_{2j} - ie_{2j+1}\right)</math>

When properly examined in light of the Clifford algebra, these are naturally [[Anticommutative property|anti-commuting]], that is, one has that <math>w_j w_m = -w_m w_j~.</math> This can be happily interpreted as the mathematical realization of the [[Pauli exclusion principle]], thus allowing these abstractly defined formal structures to be interpreted as fermions. For <math>(p,q)=(1,3)</math> dimensional [[Minkowski space-time]], there are only two such spinors possible, by convention labelled "left" and "right", as described above. A more formal, general presentation of Weyl spinors can be found in the article on the [[spin group]].

The abstract, general-relativistic form of the Weyl equation can be understood as follows: given a pseudo-Riemannian manifold <math>M,</math> one constructs a [[fiber bundle]] above it, with the spin group as the fiber. The spin group <math>\mathrm{Spin}(p,q)</math> is a [[Cover (mathematics)|double cover]] of the [[special orthogonal group]] <math>\mathrm{SO}(p,q)</math>, and so one can identify the spin group fiber-wise with the [[frame bundle]] over <math>M~.</math> When this is done, the resulting structure is called a [[spin structure]].

Selecting a single point on the fiber corresponds to selecting a [[Local reference frame|local coordinate frame]] for spacetime; two different points on the fiber are related by a (Lorentz) boost/rotation, that is, by a local change of coordinates. The natural inhabitants of the spin structure are the Weyl spinors, in that the spin structure completely describes how the spinors behave under (Lorentz) boosts/rotations.

Given a [[spin manifold]], the analog of the [[metric connection]] is the [[spin connection]]; this is effectively "the same thing" as the normal connection, just with spin indexes attached to it in a consistent fashion. The [[covariant derivative]] can be defined in terms of the connection in an entirely conventional way. It acts naturally on the [[Clifford bundle]]; the Clifford bundle is the space in which the spinors live. The general exploration of such structures and their relationships is termed [[spin geometry]].

=== Mathematical definition ===
{{See also |classification of Clifford algebras}}
For even <math>n</math>, the even subalgebra <math>\mathbb{C}l^0(n)</math> of the complex Clifford algebra <math>\mathbb{C}l(n)</math> is isomorphic to <math>\mathrm{End}(\mathbb{C}^{N/2}) \oplus \mathrm{End}(\mathbb{C}^{N/2}) =: \Delta^+_n \oplus \Delta^-_n</math>, where <math>N = 2^{n/2}</math>. A left-handed (respectively, right-handed) '''complex Weyl spinor''' in <math>n</math>-dimensional space is an element of <math>\Delta^+_n</math> (respectively, <math>\Delta^-_n</math>).

===Special cases===
There are three important special cases that can be constructed from Weyl spinors. One is the [[Dirac spinor]], which can be taken to be a pair of Weyl spinors, one left-handed, and one right-handed. These are coupled together in such a way as to represent an electrically charged fermion field. The electric charge arises because the Dirac field transforms under the action of the complexified spin group <math>\mathrm{Spin}^\mathbb{C}(p,q).</math> This group has the structure 
:<math>\mathrm{Spin}^\mathbb{C}(p,q)\cong\mathrm{Spin}(p,q)\times_{\mathbb{Z}_2} S^1</math>

where <math>S^1\cong \mathrm{U}(1)</math> is the circle, and can be identified with the <math>\mathrm{U}(1)</math> of [[electromagnetism]]. The product <math>\times_{\mathbb{Z}_2}</math> is just fancy notation denoting the product <math>\mathrm{Spin}(p,q)\times S^1</math> with opposite points <math>(s,u) = (-s,-u)</math> identified (a double covering).

The [[Majorana spinor]] is again a pair of Weyl spinors, but this time arranged so that the left-handed spinor is the charge conjugate of the right-handed spinor. The result is a field with two less degrees of freedom than the Dirac spinor. It is unable to interact with the electromagnetic field, since it transforms as a scalar under the action of the <math>\mathrm{spin}^\mathbb{C}</math> group. That is, it transforms as a spinor, but transversally, such that it is invariant under the <math>\mathrm{U}(1)</math> action of the spin group.

The third special case is the [[ELKO spinor]], constructed much as the Majorana spinor, except with an additional minus sign between the charge-conjugate pair. This again renders it electrically neutral, but introduces a number of other quite surprising properties.

==Notes==
{{notelist}}

==References==
{{reflist|25em|refs=

<ref name=Aste-2010>
{{cite magazine
 |first=Andreas |last=Aste
 |year=2010
 |title=A direct road to Majorana fields
 |magazine=Symmetry
 |volume=2010 |issue=2 |pages=1776–1809
 |doi=10.3390/sym2041776 |issn=2073-8994
|doi-access=free
 }}
</ref>

<ref name=Bilenky-2005>
{{cite journal
 |last=Bilenky |first=S.M.
 |year=2005
 |title=The history of neutrino oscillations
 |journal=Physica Scripta
 |volume=T121 |pages=17–22
 |doi=10.1088/0031-8949/2005/T121/001
 |issn=0031-8949 |arxiv=hep-ph/0410090
 |bibcode=2005PhST..121...17B
 |s2cid=119341278
 |url=https://iopscience.iop.org/article/10.1088/0031-8949/2005/T121/001
}}
</ref>

<ref name=Pal-2011>
{{cite journal
 |last=Pal |first=Palash B.
 |year=2011
 |title=Dirac, Majorana, and Weyl fermions
 |journal=American Journal of Physics
 |volume=79 |issue=5 |pages=485–498
 |doi=10.1119/1.3549729
 |arxiv=1006.1718 |bibcode=2011AmJPh..79..485P
 |s2cid=118685467
 |issn=0002-9505
 |url=http://aapt.scitation.org/doi/10.1119/1.3549729
 |language=en
}}
</ref>

<ref name=Pearson-2004>
{{cite book
 |editor-first=E. Abers |editor-last=Pearson
 |year=2004
 |title=Quantum Mechanics
 |publisher=Addison Wesley, Prentice Hall Inc
 |isbn=978-0-13-146100-0
}}
</ref>

<ref name=Peskin-Schroeder-1995>
{{cite book
 |url=https://books.google.com/books?id=_H-oPv1raioC
 |via=Google Books
 |title=An Introduction to Quantum Field Theory |first1=M.E. |last1=Peskin |first2=D.V. |last2=Schroeder |publisher=Addison-Wesley |year=1995 |isbn=0-201-50397-2
}}
</ref>

<ref name=Shifman-1999>
{{cite book
 |last=Shifman |first=Mikhail
 |year=1999
 |title=ITEP Lectures on Particle Physics and Field Theory
 |volume=1 |page=292
 |isbn=9789810239480
}}
</ref>

<ref name=Vishwanath-2015>
{{cite magazine
 |last=Vishwanath |first=Ashvin
 |date=2015-09-08
 |title=Where the Weyl things are
 |magazine=APS Physics
 |volume=8
 |url=https://physics.aps.org/articles/v8/84
 |lang=en
}}
</ref>

<ref name=Weyl-1929>
{{cite journal
 |last=Weyl |first=Hermann |author-link=Hermann Weyl
 |date=1929-04-15
 |title=Gravitation and the electron
 |journal=Proceedings of the National Academy of Sciences |language=en
 |volume=15 |issue=4 |pages=323–334
 |doi=10.1073/pnas.15.4.323
 |issn=0027-8424 |pmc=522457 |pmid=16587474
 |bibcode=1929PNAS...15..323W |doi-access=free }}
</ref>

<ref name=Woan-2010>
{{cite book
 |title=The Cambridge Handbook of Physics Formulas |editor-first=G. |editor-last=Woan |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-57507-2}}
</ref>
}} <!-- end "refs=" -->

==Further reading==

* {{cite book
 |first=D. |last=McMahon
 |year=2008
 |title=Quantum Field Theory Demystified
 |publisher=McGraw-Hill
 |place=USA
 |isbn=978-0-07-154382-8
 |url=https://books.google.com/books?id=SoAZVbdyY8AC
}} 
* {{cite book
 |first1=B.R. |last1=Martin
 |first2=G.   |last2=Shaw
 |year=2008
 |title=Particle Physics
 |edition=2nd 
 |series=Manchester Physics
 |publisher=John Wiley & Sons
 |isbn=978-0-470-03294-7
}}
** {{cite book
 |title=Particle Physics
 |edition=3rd
 |year=2013
 |isbn=9781118681664
 |url=https://books.google.com/books?id=7tsVcsm84rEC
 |via=Google Books
|last1=Martin
 |first1=Brian R.
 |last2=Shaw
 |first2=Graham
 }}
* {{cite book
 |first=P. |last=LaBelle
 |year=2010
 |title=Supersymmetry Demystified
 |publisher=McGraw-Hill
 |place=USA
 |isbn=978-0-07-163641-4
 |url=https://books.google.com/books?id=SWHhdGbxukkC
 |via=Google Books
}}
* {{cite book
 |first=Roger |last=Penrose
 |year=2007
 |title=[[The Road to Reality]]
 |publisher=Vintage Books
 |isbn=978-0-679-77631-4
}}
*{{cite web
 |last=Johnston |first=Hamish
 |date=23 July 2015
 |title=Weyl fermions are spotted at long last
 |website=Physics World
 |url=http://physicsworld.com/cws/article/news/2015/jul/23/weyl-fermions-are-spotted-at-long-last
 |access-date=22 November 2018
}}
*{{cite journal
 |last=Ciudad |first=David
 |date=20 August 2015
 |title=Massless yet real
 |journal=Nature Materials
 |volume=14 |issue=9 |page=863
 |doi=10.1038/nmat4411 |doi-access=free
 |pmid=26288972 |issn=1476-1122
}}
*{{cite magazine
 |last=Vishwanath |first=Ashvin
 |date=8 September 2015
 |title=Where the Weyl things are
 |magazine=APS Physics
 |volume=8
 |url=http://physics.aps.org/articles/v8/84
 |access-date=22 November 2018
}}
*{{cite journal
 |last1=Jia   |first1=Shuang
 |last2=Xu    |first2=Su-Yang
 |last3=Hasan |first3=M. Zahid
 |date=25 October 2016
 |title=Weyl semimetals, Fermi arcs and chiral anomaly
 |journal=Nature Materials
 |volume=15 |issue=11
 |pages=1140–1144
 |doi=10.1038/nmat4787
 |pmid=27777402
 |arxiv=1612.00416
 |bibcode=2016NatMa..15.1140J
 |s2cid=1115349
 |url=https://www.nature.com/articles/nmat4787
}}

==External links==
* http://aesop.phys.utk.edu/qft/2004-5/2-2.pdf
* http://www.nbi.dk/~kleppe/random/ll/l2.html
* http://www.tfkp.physik.uni-erlangen.de/download/research/DW-derivation.pdf
* http://www.weylmann.com/weyldirac.pdf

[[Category:Quantum mechanics]]