= Vacuum-normalized effective action of a free scalar field
{title2=$\Gamma=S$}
Write $S=\tfrac12\phi A\phi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the <Gaussian path integral> gives $Z[J]/Z[0]=\exp[-iJA^{-1}J/2]$, hence $W=-JA^{-1}J/2$, $\varphi=-A^{-1}J$ and $\Gamma=\varphi A\varphi/2=S[\varphi]$. The source-independent <determinant> cancels in vacuum normalization. Without that normalization, the equality holds up to a field-independent vacuum constant.
// Target: perturbative-quantum-field-theory.bigb
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