\begin{equation}
\begin{aligned}
\psi \rightarrow \Lambda_{\frac{1}{2}} \psi &= \left(1-\frac{i}{2} \omega_{\rho \sigma} S^{\rho \sigma}\right) \psi \\
&= \left(1-i \left(\omega_{ij} S^{ij}+\omega_{0i} S^{0i}\right)\right)\psi
\end{aligned}
\end{equation}

where
\( S^{0 i}=\frac{i}{4}\left[\gamma^{0}, \gamma^{i}\right]=-\frac{i}{2}\left(\begin{array}{cc}{\sigma^{i}} & {0} \\ {0} & {-\sigma^{i}}\end{array}\right) \)
\( S^{i j}=\frac{i}{4}\left[\gamma^{i}, \gamma^{j}\right]=\frac{1}{2} \epsilon^{i j k}\left(\begin{array}{cc}{\sigma^{k}} & {0} \\ {0} & {\sigma^{k}}\end{array}\right) \equiv \f…