Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 1 / ii / a

Insert the mode expansion of a free field into the classical expression for $\mathbf{P}$. The $aa$ and $a^{†}{a}^{†}$ terms cancel after $\mathbf{p}\mapsto -\mathbf{p}$, while the mixed terms give, after the stated normal ordering,Thus the momentum operator counts each occupied mode with weight $\mathbf{p}$.