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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.

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# Reps of lorentz group and pauli and gamma matrices

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