Weyl equation

In physics, particularly a 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 that the neutrino might be a Weyl fermion (it is now considered to be either a Dirac or a Majorana fermion). In condensed matter physics, some materials that can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals.

History

The Dirac equation, was published in 1928 by Paul Dirac, first describing spin-½ particles in the framework of relativistic quantum mechanics.[1] German mathematician and mathematical physicist, Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation.[1][2] Wolfgang Pauli wrote in 1933 against Weyl's equation because it violated parity.[3] However, three years before, Pauli had predicted the existence of a new elementary fermion, the neutrino, to explain the beta decay, which will eventually be described by the same equation.

In 1937, Conyers Herring proposed the idea of Weyl fermions quasiparticles in condensed matter.[4]

Neutrinos were finally confirmed in 1956 as particles with vanishing masses.[3] The same year the Wu experiment, showed that parity was violated by the weak interaction. Followed by the experimental discovery of the neutrino fixed helicity in 1958.[3] Additionally, as experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. The Standard Model, was thus build under the assumption that neutrinos were Weyl fermions.[3]

While Italian physicist Bruno Pontecorvo had proposed in 1957 the possibility of neutrino masses and neutrino oscillations,[3] it was not until 1998 that Super-Kamiokande eventually confirmed its existence.[3] This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos.[1]

In 2015, the first Weyl semimetal was demonstrated experimentally in crystalline tantalum arsenide (${\displaystyle {\ce {TaAs}}}$) by the collaboration of M.Z. Hasan's (Princeton University) and H. Ding's (Chinese Academy of Sciences) teams.[4] Independently, the same year, M. Soljačić team (Massachusetts Institute of Technology) also observed Weyl like excitation is photonic crystals.[4]

Equation

The Weyl equation can be written as[5][6][7]

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

Expanding out the above, and inserting ${\displaystyle c}$ for the speed of light:

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

where

${\displaystyle \sigma ^{\mu }=(\sigma ^{0},\sigma ^{1},\sigma ^{2},\sigma ^{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})}$

is a vector whose components are the 2 × 2 identity matrix ${\displaystyle I_{2}}$ for μ = 0 and the Pauli matrices for μ = 1,2,3, and ψ is the wavefunction – one of the Weyl spinors. A dual form of the equation is usually written as:

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

where ${\displaystyle {\bar {\sigma }}^{\mu }=(I_{2},-\sigma _{x},-\sigma _{y},-\sigma _{z})}$. These two are 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 convenient to label these two explicitly; the labeling is ${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0}$ 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

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

where

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

is a momentum-dependent two-component spinor which satisfies

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

or

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

By direct manipulation, one obtains that

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

and concludes that the equations correspond to a particle that is massless. As a result, the magnitude of momentum p relates directly to the wave-vector k by the De Broglie relations as:

${\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |=\hbar \omega /c\,\rightarrow \,|\mathbf {k} |=\omega /c}$

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

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

Helicity

The left and right components correspond to the helicity λ of the particles, the projection of angular momentum operator J onto the linear momentum p:

${\displaystyle \mathbf {p} \cdot \mathbf {J} \left|\mathbf {p} ,\lambda \right\rangle =\lambda |\mathbf {p} |\left|\mathbf {p} ,\lambda \right\rangle }$

Here ${\displaystyle \lambda =\pm 1/2}$.

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

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

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

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

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

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

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

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

provided that the left-handed field transforms as

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

Proof

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

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

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,

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

This leads to

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

In order to make use of the Weyl map

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

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

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

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:

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

to write

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

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

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

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

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.[8] This arises because the special linear group ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ is isomorphic to the symplectic group ${\displaystyle \mathrm {Sp} (2,\mathbb {C} ).}$ The symplectic groups is defined as the set of all complex 2x2 matrices that satisfy

${\displaystyle \omega ^{-1}S^{T}\omega =S^{-1}}$

where

${\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$

The defining relationship can be rewritten as ${\displaystyle \omega S^{*}=(S^{\dagger })^{-1}\omega }$ where ${\displaystyle S^{*}}$ is the complex conjugate. The right handed field, as noted earlier, transforms as

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

and so the complex conjugate field transforms as

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

Applying the defining relationship, one concludes that

${\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}(x^{\prime })=(S^{\dagger })^{-1}m\omega \psi _{\rm {R}}^{*}(x)}$

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

${\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}$

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 ${\displaystyle \zeta }$) is

${\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}$

As noted earlier, the left and right chiral versions are related by a parity transformation. The skew complex conjugate ${\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }$ can be recognized as the charge conjugate form of ${\displaystyle \psi ~.}$ 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,

${\displaystyle D_{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\qquad D_{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K}$

where ${\displaystyle K}$ is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as

${\displaystyle D_{\rm {L}}\mapsto D_{\rm {L}}^{\prime }=SD_{\rm {L}}S^{\dagger }\qquad D_{\rm {R}}\mapsto D_{\rm {R}}^{\prime }=(S^{\dagger })^{-1}D_{\rm {R}}S^{-1}}$

whereas the Weyl spinors transform as

${\displaystyle \psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }=(S^{\dagger })^{-1}\psi _{\rm {L}}\qquad \psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }=S\psi _{\rm {R}}}$

just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take

${\displaystyle D_{\rm {L}}\psi _{\rm {L}}=0\qquad D_{\rm {R}}\psi _{\rm {R}}=0}$

as a pair of complex 2-spinor Majorana equations.

The products ${\displaystyle D_{\rm {L}}D_{\rm {R}}}$ and ${\displaystyle D_{\rm {R}}D_{\rm {L}}}$ are both Lorentz covariant. The product is explicitly

${\displaystyle D_{\rm {R}}D_{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2})=-(\square +\eta \zeta ^{*}m^{2})}$

Verifying this requires keeping in mind that ${\displaystyle \omega ^{2}=-1}$ and that ${\displaystyle Ki=-iK~.}$ The RHS reduces to the Klein–Gordon operator provided that ${\displaystyle \eta \zeta ^{*}=1}$, that is, ${\displaystyle \eta =\zeta ~.}$ These two Majorana operators are thus "square roots" of the Klein–Gordon operator.

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 a certain element of a Clifford algebra. 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 it's 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[9]

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

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 absractly 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.

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 ${\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}$ This group has the structure

${\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q)\cong \mathrm {Spin} (p,q)\times _{\mathbb {Z} _{2}}S^{1}}$

where ${\displaystyle S^{1}\cong \mathrm {U} (1)}$ is the circle, and can be identified with the U(1) of electromagnetism. The product ${\displaystyle \times _{\mathbb {Z} _{2}}}$ is just fancy notation denoting the product ${\displaystyle \mathrm {Spin} (p,q)\times S^{1}}$ with opposite points ${\displaystyle (s,u)=(-s,-u)}$ 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 ${\displaystyle \mathrm {spin} ^{\mathbb {C} }}$ group. That is, it transforms as a spinor, but transversally, such that it is invariant under the ${\displaystyle \mathrm {U} (1)}$ action of the spinc 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

1. The results presented here are identical to those of Aste, op. cit., equations 52 and 57, although the derivation performed here is completely different. The double-covering used here is also identical to Aste equations 48, and to the current version (December 2020) of the article on Lorentz group.

References

1. Pal, Palash B. (2011). "Dirac, Majorana, and Weyl fermions". American Journal of Physics. 79 (5): 485–498. arXiv:1006.1718. doi:10.1119/1.3549729. ISSN 0002-9505.
2. Weyl, Hermann (1929-04-15). "Gravitation and the electron". Proceedings of the National Academy of Sciences. 15 (4): 323–334. doi:10.1073/pnas.15.4.323. ISSN 0027-8424. PMC 522457. PMID 16587474.
3. Bilenky, S M (2005-01-01). "The History of Neutrino Oscillations". Physica Scripta. T121: 17–22. arXiv:hep-ph/0410090. doi:10.1088/0031-8949/2005/T121/001. ISSN 0031-8949.
4. Vishwanath, Ashvin (2015-09-08). "Where the Weyl Things Are". APS Physics. 8.
5. Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
6. The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
7. An Introduction to Quantum Field Theory, M.E. Peskin, D.V. Schroeder, Addison-Wesley, 1995, ISBN 0-201-50397-2
8. Andreas Aste, (2010) "A Direct Road to Majorana Fields", Symmetry 2010(2) 1776-1809; doi:10.3390/sym2041776 ISSN 2073-8994.
9. Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis (3rd edition)" Springer Universitext.

Further reading

• Quantum Field Theory Demystified, D. McMahon, McGraw-Hill (USA), 2008, ISBN 978-0-07-154382-8
• Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
  • 3rd edition, 2013
• Supersymmetry Demystified, P. LaBelle, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
• The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1
• 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. 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: 1140.

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