Minkowski inverse-Hessian relation for an effective action (${\Gamma }^{(2)}{W}^{(2)}=-I$)

With $W=-i\log Z$, $\phi =\delta W/\delta J$ and $\Gamma =W-\int J\phi$, variation gives $\delta \Gamma /\delta \phi =-J$. Differentiating through the source-to-field map givesThe inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=i{C}_2$, the full connected propagator is determined by the inverse effective-action quadratic kernel. Factors of $i$ and the minus sign depend on the explicitly chosen Minkowski Legendre transform convention.

// Target: perturbative-quantum-field-theory.bigb