In physics, particularly in quantum field theory, the Weyl equation ( VILE) is a relativistic wave equation for describing massless spin-1/2 particles which have an inherent handedness, or chirality, 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.
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. Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation. Wolfgang Pauli wrote in 1933 against Weyl's equation because it violated parity. 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. Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless). The same year the Wu experiment showed that parity could be violated by the weak interaction, addressing Pauli's criticism. This was followed by the measurement of the neutrino's helicity in 1958. 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. While Italian physicist Bruno Pontecorvo had proposed in 1957 the possibility of neutrino masses and neutrino oscillations, it was not until 1998 that Super-Kamiokande eventually confirmed the existence of neutrino oscillations, and their non-zero mass. This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos, as the equations can only describe massless particles. 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. Independently, the same year, M. Soljačić team (Massachusetts Institute of Technology) also observed Weyl-like excitations in photonic crystals.
Equation The Weyl equation comes in two forms. The right-handed form can be written as follows:
σ μ ∂ μ ψ = 0. {\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0.}
Expanding this equation, and inserting c {\displaystyle c} for the speed of light, it becomes
I 2 1 c ∂ ψ ∂ t + σ x ∂ ψ ∂ x + σ y ∂ ψ ∂ y + σ z ∂ ψ ∂ z = 0 , {\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
σ μ = ( σ 0 σ 1 σ 2 σ 3 ) = ( I 2 σ x σ y σ z ) {\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 I 2 {\displaystyle I_{2}} for μ = 0 {\displaystyle \mu =0} and the Pauli matrices for μ = 1 , 2 , 3 {\displaystyle \mu =1,2,3} , and ψ {\displaystyle \psi } is the wavefunction – one of the Weyl spinors. The left-handed form of the Weyl equation is usually written as
σ ¯ μ ∂ μ ψ = 0 , {\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0,}
where
σ ¯ μ = ( I 2 − σ x − σ y − σ z ) . {\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: σ μ ∂ μ ψ R = 0 {\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0} and σ ¯ μ ∂ μ ψ L = 0 {\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
ψ ( r , t ) = ( ψ 1 ψ 2 ) = χ e − i ( k ⋅ r − ω t ) = χ e − i ( p ⋅ r − E t ) / ℏ , {\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
χ = ( χ 1 χ 2 ) {\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}
is a momentum-dependent two-component spinor which satisfies
σ μ p μ χ = ( I 2 E − σ → ⋅ p → ) χ = 0 {\displaystyle \sigma ^{\mu }p_{\mu }\chi =\left(I_{2}E-{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}
or
σ ¯ μ p μ χ = ( I 2 E + σ → ⋅ p → ) χ = 0. {\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
( σ ¯ ν p ν ) ( σ μ p μ ) χ = ( σ ν p ν ) ( σ ¯ μ p μ ) χ = p μ p μ χ = ( E 2 − p → ⋅ p → ) χ = 0 , {\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 p {\displaystyle \mathbf {p} } relates directly to the wave-vector k {\displaystyle \mathbf {k} } by the de Broglie relations as
| p | = ℏ | k | = ℏ ω c ⇒ | k | = ω c . {\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
σ μ ∂ μ ψ R = 0 , σ ¯ μ ∂ μ ψ L = 0. {\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 {\displaystyle \lambda } of the particles, the projection of angular momentum operator J {\displaystyle \mathbf {J} } onto the linear momentum p {\displaystyle \mathbf {p} } :
p ⋅ J | p , λ ⟩ = λ | p | | p , λ ⟩ . {\displaystyle \mathbf {p} \cdot \mathbf {J} \left|\mathbf {p} ,\lambda \right\rangle =\lambda |\mathbf {p} |\left|\mathbf {p} ,\lambda \right\rangle .}
Here λ = ± 1 2 {\displaystyle \textstyle \lambda =\pm {\frac {1}{2}}} .
Lorentz invariance Both equations are Lorentz invariant under the Lorentz transformation x ↦ x ′ = Λ x {\displaystyle x\mapsto x^{\prime }=\Lambda x} where Λ ∈ S O ( 1 , 3 ) {\displaystyle \Lambda \in \mathrm {SO} (1,3)} . More precisely, the equations transform as
σ μ ∂ ∂ x μ ψ R ( x ) ↦ σ μ ∂ ∂ x ′ μ ψ R ′ ( x ′ ) = ( S − 1 ) † σ μ ∂ ∂ x μ ψ R ( x ) , {\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 S † {\displaystyle S^{\dagger }} is the Hermitian transpose, provided that the right-handed field transforms as
ψ R ( x ) ↦ ψ R ′ ( x ′ ) = S ψ R ( x ) . {\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x).}
The matrix S ∈ S L ( 2 , C ) {\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 S L ( 2 , C ) {\displaystyle \mathrm {SL} (2,\mathbb {C} )} given by
σ μ Λ μ ν = ( S − 1 ) † σ ν S − 1 . {\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
σ ¯ μ ∂ ∂ x μ ψ L ( x ) ↦ σ ¯ μ ∂ ∂ x ′ μ ψ L ′ ( x ′ ) = S σ ¯ μ ∂ ∂ x μ ψ L ( x ) , {\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
ψ L ( x ) ↦ ψ L ′ ( x ′ ) = ( S † ) − 1 ψ L ( x ) . {\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
ψ R ( x ) ↦ ψ R ′ ( x ′ ) = R ψ R ( x ) {\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=R\psi _{\rm {R}}(x)}
for some unknown R ∈ S L ( 2 , C ) {\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )} to be determined. The Lorentz transform, in coordinates, is
x ′ μ = Λ μ ν x ν , {\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu },}
or, equivalently,
x ν = ( Λ − 1 ) ν μ x ′ μ . {\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }.}
This leads to
σ μ ∂ μ ′ ψ R ′ ( x ′ ) = σ μ ∂ ∂ x ′ μ ψ R ′ ( x ′ ) = σ μ ∂ x ν ∂ x ′ μ ∂ ∂ x ν R ψ R ( x ) = σ μ ( Λ − 1 ) ν μ ∂ ∂ x ν R ψ R ( x ) = σ μ ( Λ − 1 ) ν μ ∂ ν R ψ R ( x ) . {\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
σ μ Λ μ ν = ( S − 1 ) † σ ν S − 1 , {\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
η Λ T η = Λ − 1 , {\displaystyle \eta \Lambda ^{\mathsf {T}}\eta =\Lambda ^{-1},}
where η = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta ={\mbox{diag}}(+1,-1,-1,-1)} is the flat-space Minkowski metric. The above identity is often used to define the elements Λ ∈ S O ( 1 , 3 ) {\displaystyle \Lambda \in \mathrm {SO} (1,3)} . One takes the transpose
( Λ − 1 ) ν μ = ( Λ − 1 T ) μ ν {\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }}
to write
σ μ ( Λ − 1 ) ν μ ∂ ν R ψ R ( x ) = σ μ ( Λ − 1