= Minkowski inverse-Hessian relation for an effective action
{c}
{title2=$\Gamma^{(2)}W^{(2)}=-I$}
With $W=-i\log Z$, $\varphi=\delta W/\delta J$ and $\Gamma=W-\int J\varphi$, variation gives $\delta\Gamma/\delta\varphi=-J$. Differentiating through the source-to-field map gives
$$
\int dz\,\Gamma^{(2)}(x,z)W^{(2)}(z,y)=-\delta(x-y).
$$
The inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=iC_2$, 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
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