Quantum effective potential from discarded degrees of freedom

I obtain the quantum correction ΔVeff=(ħ2/8m)[(1−4ξd+1d)(S′)2+2(1−4ξ)S″] that appears in the effective potential whenever a compact d-dimensional subspace (of volume ∝exp⁡[S(x)]) is discarded from the configuration space of a nonrelativistic particle of mass m and curvature coupling parameter ξ. This correction gives rise to a force −〈ΔVeff′〉 that pushes the expectation value 〈x〉 off its classical trajectory. Because ΔVeff 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 S(x).