Past exam of the quantum field theory course of the University of Oxford / 2023 / 3 / a / Solution

In terms of the Pauli matrices, ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$. The Pauli matrix multiplication law implies ${\sigma }_j^2=I_2$ and ${\sigma }_j{\sigma }_k=-{\sigma }_k{\sigma }_j$ for $j\ne k$. ConsequentlyThese diagonal and off-diagonal cases are exactly the Clifford algebra relationThe identity matrix is implicit on the right-hand side in the source. Useful conventions for the remaining parts are $({\gamma }^0{)}^{†}={\gamma }^0$, $({\gamma }^j{)}^{†}=-{\gamma }^j$ for $j=1,2$, and ${\gamma }^{0\ast }={\gamma }^0$, ${\gamma }^{1\ast }=-{\gamma }^1$, ${\gamma }^{2\ast }={\gamma }^2$.