Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2018 / 2 / a / i

Use the source action $S_{\mathrm{src}}=\int d^{4}x[J\varphi +{\bar{\eta }}_{\alpha }{\psi }_{\alpha }+{\bar{\psi }}_{\alpha }{\eta }_{\alpha }]$, with independent Grassmann variables used as sources $\eta ,\bar{\eta }$ and left functional derivatives. A scalar insertion is $(1/i)\delta /\delta J$. For the odd fields the ordered rules areThe middle sign is not arbitrary: differentiating the rightmost odd source in $\bar{\psi }\eta$ from the left gives ${\delta }_L(\bar{\psi }\eta )/\delta \eta =-\bar{\psi }$. Derivative ordering must consequently match the displayed order of the fields; the rightmost derivative acts first.

Expand $\exp [ig\int \varphi \bar{\psi }\psi ]$ in its power series. Replace each field in every term by its source insertion, preserving the fermion order. These derivatives can be pulled outside the free path integral. Re-exponentiation gives the unnormalized generating functionalSummation over the spinor index is understood. Normalized correlation functions require division by the same interacting $Z[0,0,0]$, which cancels vacuum factors; that division is separate from this operator identity.