Quantum effective potential from discarded degrees of freedom

TL;DR

This paper derives a universal quantum correction to the effective potential for a particle when certain degrees of freedom are ignored, showing how this correction influences the particle's expected trajectory.

Contribution

It provides a general formula for the quantum correction to the effective potential due to discarded degrees of freedom with maximum entropy, independent of specific subspace details.

Findings

Quantum correction derived for discarded degrees of freedom

Correction influences the expected particle trajectory

Results are model-independent and based on entropy considerations

Abstract

I obtain the quantum correction $\Delta V_\mathrm{eff}= (\hbar^2/8m) [(1- 4\xi \frac{d+1}{d})(\mathcal{S}')^2 + 2(1-4\xi)\mathcal{S}"]$ 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 $-\langle\Delta V_\mathrm{eff}'\rangle$ that pushes the expectation value $\langle x\rangle$ 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 $\mathcal{S}(x)$.