Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 1

For a scalar field with classical action $S[\varphi ]$, the Euclidean source convention used throughout this question givesThe generating functional produces correlation functions through functional derivatives. For example,More generally, each derivative brings down $-\varphi /\hslash$, so an $n$-point function carries $(-\hslash {)}^n/Z[0]$.

The classical action $S[\varphi ]$ supplies the vertices and quadratic quantum field theory propagator in the path integral. The connected generating functional isand its first derivative is the source-dependent mean fieldThe quantum effective action is the Legendre transformwhere $J$ is eliminated in favor of $\Phi$. At vanishing source, stationary points of $\Gamma$ are the quantum equations of motion. This $W[J]$ is the connected, or Schwinger, functional; a Wilsonian effective action instead integrates out modes above a momentum scale.

The perturbative expansion of $Z[J]$ contains arbitrary Feynman diagrams, including disconnected products. The linked-cluster theorem givesbecause the factorials from repeated connected components reproduce the exponential series. Therefore $W[J]=-\hslash \log Z[J]$ is the sum of connected Feynman diagrams.

The Legendre transform removes diagrams that disconnect upon cutting one internal line. Equivalently, every connected diagram is a tree whose vertices are exact one-particle-irreducible vertices and whose edges are exact propagators. Thusand its functional derivatives are the one-particle-irreducible correlation functions. Algebraically, differentiating the Legendre relations givesso an exact quantum field theory propagator joining two proper vertices is precisely the inverse Hessian matrix needed to reconstruct connected diagrams.

Change variables in the defining integral from the fluctuation to the total field, $\phi ={\varphi }_0+\eta$. Thenand henceIt follows nonperturbatively thatThus the same source $J_{\chi }$ corresponds at zero background to the mean field $\Phi =\chi +{\varphi }_0$. Using the source-sign-compatible Legendre transform $\Gamma (\chi ;{\varphi }_0)=W(J_{\chi };{\varphi }_0)-J_{\chi }\chi$,Renaming $\chi$ as $\eta$ gives the requested identity