The <Dirac spinor> representation is the four-dimensional representation $(\tfrac12,0)\oplus(0,\tfrac12)$ of the connected <Lorentz group>. It is characterized by
$$
S[\Lambda]^{-1}\gamma^\mu S[\Lambda]
=\Lambda^\mu{}_{\nu}\gamma^\nu.
$$
For
$$
\Lambda=\exp\!\left(\frac12\Omega_{\rho\sigma}M^{\rho\sigma}\right),
$$
one may take
$$
S[\Lambda]
=\exp\!\left(\frac18\Omega_{\rho\sigma}[\gamma_\rho,\gamma_\sigma]\right),
$$
with the signs of $\Omega_{\rho\sigma}$ fixed by the displayed vector-representation convention. This formula follows by differentiating the covariance relation and using the <Clifford algebra> $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$; moving both gamma indices upstairs changes the apparent parameter signs according to the metric.

Use the <Weyl representation of the gamma matrices>,
$$
\gamma^0=\begin{pmatrix}0&I_2\\I_2&0\end{pmatrix},
\qquad
\gamma^i=\begin{pmatrix}0&\sigma^i\\-\sigma^i&0\end{pmatrix}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 3 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-301/3/solution