= Spinor Lorentz generator commutator
{title2=$[\sigma^{\mu\nu},\gamma^\rho]=2i(g^{\nu\rho}\gamma^\mu-g^{\mu\rho}\gamma^\nu)$}
Define $\sigma^{\mu\nu}=\frac i2[\gamma^\mu,\gamma^\nu]$ using the <gamma matrix> <Clifford algebra>. Moving $\gamma^\rho$ through the two factors with $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ proves the displayed <operator commutator>. For real antisymmetric $\omega_{\mu\nu}$ and $S=1-\frac i4\sigma^{\mu\nu}\omega_{\mu\nu}$, it gives $S^{-1}\gamma^\rho S=\gamma^\rho+\omega^\rho{}_{\nu}\gamma^\nu+O(\omega^2)$. This is the infinitesimal <Lorentz covariance> identity required by the <Dirac equation>.
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