= Solution
The classical <action> $S[\phi]$ supplies the vertices and quadratic <quantum field theory propagator> in the <path integral>. The <connected generating functional> is
$$
W[J]=-\hbar\log Z[J],
$$
and its first derivative is the source-dependent mean field
$$
\Phi(x)=\frac{\delta W}{\delta J(x)}=\langle\phi(x)\rangle_J.
$$
The <quantum effective action> is the <Legendre transform>
$$
\boxed{\Gamma[\Phi]=W[J]-\int d^dx\,J(x)\Phi(x),
\qquad \frac{\delta\Gamma}{\delta\Phi(x)}=-J(x),}
$$
where $J$ 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.
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