The Dirac \(\gamma\) matrices
Show that the commutator \([\sigma_{\mu\nu},\sigma_{\alpha\beta}]\) can be written in terms of the matrices \(\sigma_{\mu\nu}\). Find the coefficients of this expansion.
Compute the trace \[\operatorname{Tr}\left(\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_\alpha\gamma_\beta\gamma_5\right).\]
Compute \[\sigma_{\mu\nu}\sigma^{\mu\nu}, \quad \sigma_{\mu\nu}\gamma^5\sigma^{\mu\nu}, \quad \sigma_{\mu\nu}\gamma^\alpha\sigma^{\mu\nu}.\]