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

Similarly $[\mathbf{P},a_{\mathbf{p}}]=-\mathbf{p},a_{\mathbf{p}}$. Transforming both terms of the field expansion givesTherefore the spatial translation operator generated by $\mathbf{P}$ translates the field argument. Equivalently, its action on states translates a wavefunction in the opposite argument convention.