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\begin{equation} \lambda^{\mu}_{\;\nu}\gamma^{\nu} = \gamma^{\mu}T-T\gamma^{\mu} \end{equation} \begin{equation} T=\frac{1}{8}\lambda^{\mu\nu}\left(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}\right)=-\frac{i}{2}\lambda^{\mu\nu}\Sigma_{\mu\nu} \end{equation} Sigma is just for convenience. The origin of some variables: \begin{align*} a^\mu_{\;\nu}&=(\delta_\nu^{\mu}+\epsilon\lambda^{\mu}_\nu)\quad \text{$a$ is a Lorentz transform}\\ S&=I+\epsilon T\quad \bar{\gamma}^{\mu} = S^{-1}\gamma^{\mu}S \end{align*} The variables are: \begin{align*} \lambda^\mu_{\;\nu}&\rightarrow\text{characteristic of the infinitesimal L.T.}\\ \gamma^{\mu}&\rightarrow\text{Dirac matrices }\mu=0,1,2,3\\ T&\rightarrow \text{Inf. gen. of S, the $\Delta$ of basis matrix} \end{align*}
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Thu, 22 Apr 2021 21:30 GMT