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Use a Minkowski external source term $+\int J\chi$, where $\chi$ is the integration variable, and normalize the vacuum functional:The usual vacuum boundary prescription is implicit. This is the logarithmic convention $Z=e^W$ for the connected generating functional. It makes the given quadratic expression consistent with ${\Delta }_F=⟨T\chi \chi ⟩$. Another common convention is $Z=e^{i{W}_c}$, with $W_c=-iW$; mixing the two conventions would change the factors of $i$ below.

Functional derivative generates the full and connected correlation functions:These are time-ordered products in the vacuum, and $G_1=G_1^c=0$ by the stated assumption. Differentiating $e^W$ four times partitions the four insertions into connected blocks. All partitions with a singleton vanish, leavingThe shorthand arguments stand for spacetime points; this is a functional cumulant identity, not an assumption of a free field.

To construct the quantum effective action, define the source-dependent classical fieldLocally invert this relation to obtain $J[\phi ]$, assuming the response integral kernel is invertible with the chosen regulator and boundary prescription. Take its Legendre transformThe chain-rule terms proportional to $\delta J$ cancel, so $\delta \Gamma /\delta \phi =-J$. A field-independent constant is fixed separately and is immaterial to the vertices. At zero external source the classical field is zero, and there is no linear term.

For the two-point vertex, differentiate the classical field once. If $G=G_2^c$, thenThe vertex coefficients in this problem are derivatives of $i\Gamma$, not of $\Gamma$ alone. ThereforeFor the three-point vertex, the derivative of the connected two-point integral kernel at a general external source isCombine this with $\delta J/\delta \phi =-i{G}^{-1}$ and differentiate the inverse identity, $\delta G^{-1}=-G^{-1}(\delta G)G^{-1}$. This gives the logarithmic-source inverse-kernel vertices relationAll kernels here are finally evaluated at zero external source. The connected three-point function is amputated by its three inverse quantum field theory propagators.

For the specified quadratic $W$, the external source response is $\phi =i{\Delta }_{F}J$, hence $J=-i{\Delta }_F^{-1}\phi$. Substitution in the transform givesFor the standard free quantum field theory propagator ${\tilde{\Delta }}_F(p)=i/(p^2-m^2+i0)$, this is the usual quadratic free action, up to the vacuum prescription. The generating functional is Gaussian, so connected functions beyond second order vanish and there are no interaction vertices or nontrivial scattering amplitudes. Full higher even-point functions can still be nonzero: they are products of two-point functions, as the four-point formula shows.