Quantum effective potential from discarded degrees of freedom

I obtain the quantum correction $\Delta V_$\mathrm{e}\mathrm{f}\mathrm{f}$= (\hbar^2/8m) [(1- 4\xi \frac{d+1}{d})($\mathcal{S}$')^2 + 2(1-4\xi)$\mathcal{S}$"]$ that appears in the effective potential whenever a compact $d$-dimensional subspace (of volume $\propto \exp [\mathcal{S}(x)]$) is discarded from the configuration space of a nonrelativistic particle of mass $m$ and curvature coupling parameter $\xi$. This correction gives rise to a force $-⟨\mathrm{\Delta }{V}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{\prime }}⟩$ that pushes the expectation value $⟨x⟩$ off its classical trajectory. Because $\mathrm{\Delta }{V}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ does not depend on the details of the discarded subspace, these results constitute a generic model of the quantum effect of discarded variables with maximum entropy/information capacity $\mathcal{S}(x)$.