[QUE/QFT-06001]

Let \(\Gamma\) denote a product of Dirac matrices. Define \(\widetilde{\Gamma}\) by \begin{equation*} \bar{u}(s,q) \widetilde{\Gamma}u(r,p) = (\bar{u}(r,p)^\dagger\Gamma u(s,q))^* . \end{equation*} Use the above definition of \(\widetilde{\Gamma}\) and show that

1. \(\widetilde{\Gamma} = \gamma_0 \Gamma^\dagger \gamma_0. \)
2.      \(\widetilde{\gamma^\mu} = \gamma^\mu \)
3.     \(\widetilde{\gamma_5} = -\gamma_5\)
4.     \(\widetilde{\gamma^\mu \gamma_5}= - \gamma^\mu \gamma_5\)

Compute \(\widetilde{\sigma}_{0k}\) and \(\widetilde{\sigma}_{ij}\) and hence show      that \(\widetilde{\sigma}_{\mu\nu}=\sigma_{\mu\nu}\)