= Lorentz-spinor generators from a Clifford algebra
{c}
{title2=$M^{\rho\sigma}=\frac14[\gamma^\rho,\gamma^\sigma]$}
In a real-generator convention, $M^{\rho\sigma}=\frac14[\gamma^\rho,\gamma^\sigma]$ obeys $[M^{\rho\sigma},\gamma^\mu]=g^{\sigma\mu}\gamma^\rho-g^{\rho\mu}\gamma^\sigma$. Applying this identity to the commutator of two gamma matrices proves the <Lorentz algebra> relations. With the matching vector generators and real antisymmetric parameters, $S=\exp(\frac12\Omega_{\rho\sigma}M^{\rho\sigma})$ obeys $S^{-1}\gamma^\mu S=\Lambda^\mu{}_{\nu}\gamma^\nu$. A convention using Hermitian Lorentz generators instead must consistently insert factors of $i$.
Back to article page
