Weyl equation

The Weyl equation of particle physics , named after Hermann Weyl , is the Dirac equation for massless particles with spin  1/2. It is used in describing the weak interaction . Correspondingly, fermions that satisfy this equation are called Weyl fermions .

Derivation

The representation of the Lorentz group on Dirac spinors is reducible . In a suitable representation of the Dirac matrices , the Weyl representation , the first two and the last two components of the 4 spinors transform separately, which is why they are also referred to as bispinors :

${\displaystyle \Psi ={\begin{pmatrix}\Psi _{L}\\\Psi _{R}\end{pmatrix}}}$

The 2 spinors and are the left and right handed Weyl spinors . They are the eigenstates of the chirality operator when it is written in the Weyl representation. ${\displaystyle \Psi _{L}}$ ${\displaystyle \Psi _{R}}$ ${\displaystyle \gamma ^{5}}$

${\displaystyle {\begin{aligned}\gamma ^{5}{\begin{pmatrix}\Psi _{L}\\0\end{pmatrix}}&=-{\begin{pmatrix}\Psi _{L}\\0\end{pmatrix}}\\\gamma ^{5}{\begin{pmatrix}0\\\Psi _{R}\end{pmatrix}}&={\begin{pmatrix}0\\\Psi _{R}\end{pmatrix}}\end{aligned}}}$.

In the Dirac equation for a free spin 1/2 particle they are coupled by the mass : ${\displaystyle m}$

${\displaystyle \left(\mathrm {i} \gamma ^{\mu }\partial _{\mu }-m\right)\Psi ={\begin{pmatrix}-m&\mathrm {i} {\bar {\sigma }}^{\mu }\partial _{\mu }\\\mathrm {i} \sigma ^{\mu }\partial _{\mu }&-m\end{pmatrix}}{\begin{pmatrix}\Psi _{L}\\\Psi _{R}\end{pmatrix}}=0}$

Here is and , where the three are Pauli matrices and the two-dimensional identity matrix . ${\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&{\vec {\sigma }}\end{pmatrix}}}$ ${\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}\sigma ^{0}&-{\vec {\sigma }}\end{pmatrix}}}$ ${\displaystyle {\vec {\sigma }}}$ ${\displaystyle \sigma ^{0}}$

If the mass ( ) disappears , the four-dimensional Dirac equation decouples into two two-dimensional equations for the left- and right-handed spinor: ${\displaystyle m=0}$

${\displaystyle {\begin{aligned}\mathrm {i} \sigma ^{\mu }\partial _{\mu }\Psi _{L}&=0\\\mathrm {i} {\bar {\sigma }}^{\mu }\partial _{\mu }\Psi _{R}&=0\end{aligned}}}$

Chiral coupling

To describe the electroweak interaction, it is important that the left- and right-handed spinors can couple differently, but Lorentz covariant , to vector fields ( chiral coupling). The coupling is created by replacing the derivatives with covariant derivatives :

${\displaystyle D_{\mu }=\partial _{\mu }-\mathrm {i} gT^{a}W_{\mu }^{a}}$

Designate

• ${\displaystyle g}$the coupling constant ,
• ${\displaystyle T_{a}}$the generators of the Lie algebra of the gauge group and
• ${\displaystyle W_{\mu }^{a}}$ the components of the calibration fields.

The calibration group can be chosen differently for left- and right-handed particles without the Lorenz covariance being impaired.