- #1
bartadam
- 41
- 0
I'm very confused
By performing a lorentz transformation on a spinor [tex]\psi\rightarrow S(\Lambda)\psi(\Lambda x)[/tex] and imposing covariance on the Dirac equation [tex]i\gamma^{\mu}\partial_{\mu}\psi=0[/tex] we deduce that the gamma matrices transform as
[tex]S(\Lambda)\gamma^{\mu} S^{-1}(\Lambda)=\Lambda^{\mu}_{\nu}\gamma^{\nu}[/tex]
I understand that.
Now the Gamma matrices can be given by
[tex]\gamma^{\mu}=\left[ \begin{array}{cccc} 0&\sigma^{\mu}\\ \bar{\sigma}^{\mu} & 0\end{array} \right][/tex]
with [tex]\sigma^{\mu}=(1,\sigma^1,\sigma^2,\sigma^3)[/tex] and [tex]\bar{\sigma}^{\mu}=(-1,\sigma^1,\sigma^2,\sigma^3)[/tex]
and the dirac equation is reducible into the weyl equations.
[tex]i\sigma^{\mu}\partial_{\mu}\psi_L=0[/tex] and [tex]i\bar{\sigma}^{\mu}\partial_{\mu}\psi_R=0[/tex]
What is the way to write the lorentz transformations in this case, and how to the pauli matrices transform.
By performing a lorentz transformation on a spinor [tex]\psi\rightarrow S(\Lambda)\psi(\Lambda x)[/tex] and imposing covariance on the Dirac equation [tex]i\gamma^{\mu}\partial_{\mu}\psi=0[/tex] we deduce that the gamma matrices transform as
[tex]S(\Lambda)\gamma^{\mu} S^{-1}(\Lambda)=\Lambda^{\mu}_{\nu}\gamma^{\nu}[/tex]
I understand that.
Now the Gamma matrices can be given by
[tex]\gamma^{\mu}=\left[ \begin{array}{cccc} 0&\sigma^{\mu}\\ \bar{\sigma}^{\mu} & 0\end{array} \right][/tex]
with [tex]\sigma^{\mu}=(1,\sigma^1,\sigma^2,\sigma^3)[/tex] and [tex]\bar{\sigma}^{\mu}=(-1,\sigma^1,\sigma^2,\sigma^3)[/tex]
and the dirac equation is reducible into the weyl equations.
[tex]i\sigma^{\mu}\partial_{\mu}\psi_L=0[/tex] and [tex]i\bar{\sigma}^{\mu}\partial_{\mu}\psi_R=0[/tex]
What is the way to write the lorentz transformations in this case, and how to the pauli matrices transform.