Vacuum-normalized effective action of a free scalar field ($\Gamma =S$)

Write $S=\frac{1}{2}\varphi A\varphi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the Gaussian path integral gives $Z[J]/Z[0]=\exp [-iJ{A}^{-1}J/2]$, hence $W=-J{A}^{-1}J/2$, $\phi =-A^{-1}J$ and $\Gamma =\phi A\phi /2=S[\phi ]$. 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