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

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.