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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 and the Majorana fermions.

None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, 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 quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals.

Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term.^[1]

History

The Dirac equation was published in 1928 by Paul Dirac, and was first used to model spin-½ particles in the framework of relativistic quantum mechanics.^[2] Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation.^[2][3] Wolfgang Pauli wrote in 1933 against Weyl's equation because it violated parity.^[4] 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.^[5]

Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless).^[4] The same year the Wu experiment showed that parity could be violated by the weak interaction, addressing Pauli's criticism.^[6] This was followed by the measurement of the neutrino's helicity in 1958.^[4] 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.^[4]

While Italian physicist Bruno Pontecorvo had proposed in 1957 the possibility of neutrino masses and neutrino oscillations,^[4] it was not until 1998 that Super-Kamiokande eventually confirmed the existence of neutrino oscillations, and their non-zero mass.^[4] This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos, as the equations can only describe massless particles.^[2]

In 2015, the first Weyl semimetal was demonstrated experimentally in crystalline tantalum arsenide (TaAs) by the collaboration of M.Z. Hasan's (Princeton University) and H. Ding's (Chinese Academy of Sciences) teams.^[5] Independently, the same year, M. Soljačić team (Massachusetts Institute of Technology) also observed Weyl-like excitations in photonic crystals.^[5]

Equation

The Weyl equation comes in two forms. The right-handed form can be written as follows:^[7][8][9]

${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}$

Expanding this equation, and inserting ${\displaystyle c}$ for the speed of light, it becomes

${\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}$

where

${\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}$

is a vector whose components are the 2×2 identity matrix ${\displaystyle I_{2}}$ for ${\displaystyle \mu =0}$ and the Pauli matrices for ${\displaystyle \mu =1,2,3,}$ and ${\displaystyle \psi }$ is the wavefunction – one of the Weyl spinors. The left-handed form of the Weyl equation is usually written as:

${\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}$

where

${\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}~.}$

The solutions of the right- and left-handed Weyl equations are different: they have right- and left-handed helicity, and thus chirality, respectively. It is convenient to indicate this explicitly, as follows: ${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0}$ and ${\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}$

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

${\displaystyle \psi \left(\mathbf {r} ,t\right)={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }}$,

where

${\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}$

is a momentum-dependent two-component spinor which satisfies

${\displaystyle \sigma ^{\mu }p_{\mu }\chi =\left(I_{2}E-{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}$

or

${\displaystyle {\bar {\sigma }}^{\mu }p_{\mu }\chi =\left(I_{2}E+{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}$.

By direct manipulation, one obtains that

${\displaystyle \left({\bar {\sigma }}^{\nu }p_{\nu }\right)\left(\sigma ^{\mu }p_{\mu }\right)\chi =\left(\sigma ^{\nu }p_{\nu }\right)\left({\bar {\sigma }}^{\mu }p_{\mu }\right)\chi =p_{\mu }p^{\mu }\chi =\left(E^{2}-{\vec {p}}\cdot {\vec {p}}\right)\chi =0}$,

and concludes that the equations correspond to a particle that is massless. As a result, the magnitude of momentum ${\displaystyle \mathbf {p} }$ relates directly to the wave-vector ${\displaystyle \mathbf {k} }$ by the de Broglie relations as:

${\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |={\frac {\hbar \omega }{c}}\,\Rightarrow \,|\mathbf {k} |={\frac {\omega }{c}}}$

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

${\displaystyle {\begin{aligned}\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}&=0\\{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}&=0\end{aligned}}}$

Lorentz invariance

Both equations are Lorentz invariant under the Lorentz transformation ${\displaystyle x\mapsto x^{\prime }=\Lambda x}$ where ${\displaystyle \Lambda \in \mathrm {SO} (1,3)~.}$ More precisely, the equations transform as

${\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}$

where ${\displaystyle S^{\dagger }}$ is the Hermitian transpose, provided that the right-handed field transforms as

${\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x)}$

The matrix ${\displaystyle S\in SL(2,\mathbb {C} )}$ is related to the Lorentz transform by means of the double covering of the Lorentz group by the special linear group ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ given by

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}$

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

${\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}$

provided that the left-handed field transforms as

${\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)~.}$

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

${\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=R\psi _{\rm {R}}(x)}$

for some unknown ${\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )}$ to be determined. The Lorentz transform, in coordinates, is

${\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}$

or, equivalently,

${\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}$

This leads to

${\displaystyle {\begin{aligned}\sigma ^{\mu }\partial _{\mu }^{\prime }\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)&=\sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)\\&=\sigma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\&=\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\&=\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\end{aligned}}}$

In order to make use of the Weyl map

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}$

a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity

${\displaystyle \eta \Lambda ^{\mathsf {T}}\eta =\Lambda ^{-1}}$

where ${\displaystyle \eta ={\mbox{diag}}(+1,-1,-1,-1)}$ is the flat-space Minkowski metric. The above identity is often used to define the elements ${\displaystyle \Lambda \in \mathrm {SO} (1,3).}$ One takes the transpose:

${\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }}$

to write

${\displaystyle {\begin{aligned}\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)&=\sigma ^{\mu }{\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }\partial _{\nu }R\psi _{\rm {R}}(x)\\&=\sigma _{\mu }{\Lambda ^{\mu }}_{\nu }\partial ^{\nu }R\psi _{\rm {R}}(x)\\&=\left(S^{-1}\right)^{\dagger }\sigma _{\mu }\partial ^{\mu }S^{-1}R\psi _{\rm {R}}(x)\end{aligned}}}$

One thus regains the original form if ${\displaystyle S^{-1}R=1,}$ that is, ${\displaystyle R=S.}$ Performing the same manipulations for the left-handed equation, one concludes that

${\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=L\psi _{\rm {L}}(x)}$

with ${\displaystyle L=\left(S^{\dagger }\right)^{-1}.}$^{[lower-alpha 1]}

Lagrangian densities

The equations are obtained from the Lagrangian densities

${\displaystyle {\mathcal {L}}=i\psi _{\rm {R}}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}~,}$

${\displaystyle {\mathcal {L}}=i\psi _{\rm {L}}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}~.}$

By treating the spinor and its conjugate (denoted by ${\displaystyle \dagger }$) 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 spinors as geometric objects living on a manifold. This general setting has multiple strengths: it clarifies their interpretation as fermions 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 invariant under the action of the Lorentz group. This means that, as boosts and rotations are applied, the form of the equation itself does not change. However, the form of the spinor ${\displaystyle \psi }$ 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 ${\displaystyle M}$ of dimension ${\displaystyle (p,q)}$, one may consider its tangent bundle ${\displaystyle TM}$. At any given point ${\displaystyle x\in M,}$ the tangent space ${\displaystyle T_{x}M}$ is a ${\displaystyle (p,q)}$ dimensional vector space. Given this vector space, one can construct the Clifford algebra ${\displaystyle \mathrm {Cl} (p,q)}$ on it. If ${\displaystyle \{e_{i}\}}$ are a vector space basis on ${\displaystyle T_{x}M}$, one may construct a pair of Weyl spinors as^[11]

${\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}+ie_{2j+1}\right)}$

and

${\displaystyle w_{j}^{*}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}$

When properly examined in light of the Clifford algebra, these are naturally anti-commuting, that is, one has that ${\displaystyle w_{j}w_{m}=-w_{m}w_{j}~.}$ 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 ${\displaystyle (p,q)=(1,3)}$ 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 ${\displaystyle M,}$ one constructs a fiber bundle above it, with the spin group as the fiber. The spin group ${\displaystyle \mathrm {Spin} (p,q)}$ is a double cover of the special orthogonal group ${\displaystyle \mathrm {SO} (p,q)}$, and so one can identify the spin group fiber-wise with the frame bundle over ${\displaystyle M~.}$ When this is done, the resulting structure is called a spin structure.

Selecting a single point on the fiber corresponds to selecting a 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.

Notes

References

Further reading

• McMahon, D. (2008). Quantum Field Theory Demystified. USA: McGraw-Hill. ISBN 978-0-07-154382-8.
• Martin, B.R.; Shaw, G. (2008). Particle Physics. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-470-03294-7.
  • Martin, Brian R.; Shaw, Graham (2013). Particle Physics (3rd ed.). ISBN 9781118681664 – via Google Books.
• LaBelle, P. (2010). Supersymmetry Demystified. USA: McGraw-Hill. ISBN 978-0-07-163641-4 – via Google Books.
• Penrose, Roger (2007). The Road to Reality. Vintage Books. ISBN 978-0-679-77631-4.
• Johnston, Hamish (23 July 2015). "Weyl fermions are spotted at long last". Physics World. Retrieved 22 November 2018.
• Ciudad, David (20 August 2015). "Massless yet real". Nature Materials. 14 (9): 863. doi:10.1038/nmat4411. ISSN 1476-1122. PMID 26288972.
• Vishwanath, Ashvin (8 September 2015). "Where the Weyl things are". APS Physics. Vol. 8. Retrieved 22 November 2018.
• Jia, Shuang; Xu, Su-Yang; Hasan, M. Zahid (25 October 2016). "Weyl semimetals, Fermi arcs and chiral anomaly". Nature Materials. 15 (11): 1140–1144. arXiv:1612.00416. Bibcode:2016NatMa..15.1140J. doi:10.1038/nmat4787. PMID 27777402. S2CID 1115349.

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