Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3 / c / Solution

Fourier transformation sends ${\partial }_{\mu }$ to $i{q}_{\mu }$. Write the connected current three-point function as the two full Dirac propagators joined to the amputated vertex:The two contact terms remove one propagator at a time. Multiplying the transformed identity by $G^{-1}(p_1)$ on the left and $G^{-1}(p_2)$ on the right yieldsThis is the momentum-space Ward-Takahashi identity for the full vertex and full propagator.