Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 4 / i / Solution

Write $P^i=(ϵ/a^2){\hat{p}}^i$ and setwith $A$ first order. The photon on-shell condition isUsing ${\delta }_{ij}{\hat{p}}^i{\hat{p}}^j=1$ and retaining linear terms gives $A=-\delta N+{\hat{p}}^i{\partial }_i\psi$. Hence