# Weyl equation

In physics, particularly quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. It is named after the German physicist Hermann Weyl.

## Equation

The general equation can be written: [1][2][3]

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

explicitly in SI units:

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

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 for μ = 0 and the Pauli matrices for μ = 1,2,3, and ψ is the wavefunction - one of the Weyl spinors.

### Weyl spinors

There are left and right handed Weyl spinors, each with two components. Both have the form

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

where

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

is a constant two-component spinor which satisfies

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

Since the particles are massless, i.e. m = 0, 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 _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}}

where ${\displaystyle {\bar {\sigma }}^{\mu }=(I_{2},-\sigma _{x},-\sigma _{y},-\sigma _{z})}$.

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

## Derivation

The equations are obtained from the Lagrangian densities

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

By treating the spinor and its conjugate (denoted by ${\displaystyle \dagger }$) as independent variables, the relevant Weyl equation is obtained.