= Inverse Hessian relation for a connected two-point function
{title2=$\int dz\,\Gamma^{(2)}(x,z)G(z,y)=\delta(x-y)$}
In the <scalar-field source Legendre transform>, differentiating $h=\delta\Gamma/\delta m$ with respect to the source and using $G=\delta m/\delta h$ yields this inverse-kernel relation. For a scalar quartic <Landau-Ginzburg theory>, $\Gamma_{\rm L}^{(2)}=-\nabla^2+r_0+u_0m^2/2$, so the connected response is the Green kernel of that operator. A one-momentum representation requires a translationally invariant background.
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