Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 305 / 1 / b / Solution

Apply the parity symmetry in quantum field theory to the given mode expansion of a Dirac field:The stated Dirac spinor identities are equivalently $u^s(p)={\gamma }^0{u}^s(p_P)$ and $v^s(p)=-{\gamma }^0{v}^s(p_P)$. Relabel the momentum sum by $p\mapsto p_P$ and use $p_P\cdot x=p\cdot x_P$ to obtainThe phase $|{\eta }_P|=1$ is the intrinsic parity convention.