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

Perform the infinitesimal local change of integration variablesin the Euclidean path integral for $⟨j^{\mu }(x)\psi (x_1)\bar{\psi }(x_2)⟩$. The vector transformation has no quantum anomaly, so its functional measure is invariant. The action variation supplies $-i\int \alpha {\partial }_{\mu }{j}^{\mu }$, while varying the two charged insertions supplies contact terms at $x_1$ and $x_2$ with opposite signs. Since a change of integration variables cannot change the integral, the coefficient of the arbitrary function $\alpha (x)$ vanishes:This Schwinger-Dyson equation is the position-space Ward-Takahashi identity.