Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / f

Use the sign convention $E(\varphi )=\delta S_0/\delta \varphi$ from part b, and letIn the path integral, make the local field redefinition ${\varphi }_{\mathrm{old}}=\chi -\eta f(\chi )$. The functional Taylor expansion of the action givesThe two first-order terms cancel. Neglecting the functional Jacobian as permitted,Thus $fE$ is a redundant operator at first order, by the equivalence theorem for field redefinitions. The same argument works for a local derivative-dependent $f$, which is needed to remove $(\square \varphi {)}^2$. Insertions and sources must be transformed too: arbitrary untransformed off shell correlation functions need not be equal.

Sign qualification. If instead $E_{+}=\square \varphi +m^2\varphi +\lambda {\varphi }^3/6$ is used, then $E_{+}=-\delta S_0/\delta \varphi$. An added $+\eta f{E}_{+}$ is removed by ${\varphi }_{\mathrm{old}}=\chi +\eta f(\chi )$. The minus shift requested in the source therefore fixes which sign is meant by its otherwise unspecified $E$.