Past exam of the quantum field theory course of the University of Oxford / 2020 / 2 / a / Solution

Write ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$ and $d_{\mathbf{p}}=a_{\mathbf{p}}-b_{\mathbf{p}}$. The nonzero canonical commutation relations arewith cross-species brackets zero. The commutator product rule givesLinearity of the commutator now yields, for the real coefficient $s$,Strictly, the printed $Q$ contains an infinite constant in continuum Fock space. Work first with finitely many modes, or replace it by its normal ordering $:Q:={\int }_{\mathbf{p}}{d}_{\mathbf{p}}^{†}{d}_{\mathbf{p}}$. Their difference is a constant, so every displayed commutator and every conjugation below is unchanged. The latter convention fixes the vacuum state phase without an infinite phase ambiguity.