Quantum effective potential from discarded degrees of freedom

I obtain the quantum correction $Delta V_mathrm{eff}= (hbar^2/8m) [(1- 4xi frac{d+1}{d})(mathcal{S})^2 + 2(1-4xi)mathcal{S}]$ 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 $-langleDelta V_mathrm{eff}rangle$ that pushes the expectation value $langle xrangle$ off its classical trajectory. Because $Delta V_mathrm{eff}$ 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 $mathcal{S}(x)$.