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

The Dirac spinor representation is the four-dimensional representation $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ of the connected Lorentz group. It is characterized byForone may takewith 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,