Perform the infinitesimal local change of integration variables
$$
\delta\psi(x)=i\alpha(x)\psi(x),
\qquad
\delta\bar\psi(x)=-i\alpha(x)\bar\psi(x)
$$
in the Euclidean <path integral> for $\langle j^\mu(x)\psi(x_1)\bar\psi(x_2)\rangle$. 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:
$$
\boxed{
\partial_\mu\langle j^\mu(x)\psi(x_1)\bar\psi(x_2)\rangle
=-\bigl[\delta^{(4)}(x-x_1)-\delta^{(4)}(x-x_2)\bigr]
\langle\psi(x_1)\bar\psi(x_2)\rangle }.
$$
This <Schwinger-Dyson equation> is the position-space <Ward-Takahashi identity>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-304/3/b/solution