Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2018 / 3 / a / i

Because $({\gamma }_5{)}^2=1$, its even and odd powers are ${\gamma }_5^{2r}=1$ and ${\gamma }_5^{2r+1}={\gamma }_5$. Separate the matrix exponential into these two power series:No representation of the gamma matrices is needed. Hermiticity of ${\gamma }_5$ also makes this exponential unitary for real $\alpha$, while the projectors $(1\pm {\gamma }_5)/2$ acquire opposite phases.