= Inverse Lorentz action on gamma matrices
{title2=$S\gamma^\rho S^{-1}=(\Lambda^{-1})^\rho{}_{\mu}\gamma^\mu$}
With $\{\gamma_\mu,\gamma_\nu\}=2\eta_{\mu\nu}$, $\Omega=\omega^{\mu\nu}\gamma_{\mu\nu}/4$, and $\Lambda=e^\omega$, the <Clifford algebra> gives $[\Omega,\gamma^\rho]=-\omega^\rho{}_{\mu}\gamma^\mu$. Exponentiation proves the inverse Lorentz action displayed above, equivalently $S^{-1}\gamma^\rho S=\Lambda^\rho{}_{\mu}\gamma^\mu$. This second order is what combines with $\psi\mapsto S\psi$ and the <Dirac adjoint> action $\bar\psi\mapsto\bar\psi S^{-1}$ to make the gamma bilinears transform as ordinary contravariant <tensors>.
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