Noncommutative Gravitational Quantum Well

TL;DR

This paper investigates the effects of noncommutative geometry on a two-dimensional gravitational quantum well, using experimental neutron data to set bounds on noncommutative parameters and exploring implications for fundamental constants.

Contribution

It introduces a model with noncommutativity in both configuration and momentum spaces and derives experimental bounds on the momentum scale, also analyzing effects on Planck's constant.

Findings

Upper bound on momentum noncommutativity scale: $oxed{ ext{}\sqrt{ ext{ exteta}} ext{ ext{ extless}}1 ext{ ext{ meV/c}}}$

Configuration space noncommutativity has negligible leading-order effects

Correction to Planck's constant at 1 part in $10^{24}$

Abstract

We study noncommutative geometry at the Quantum Mechanics level by means of a model where noncommutativity of both configuration and momentum spaces is considered. We analyze how this model affects the problem of the two-dimensional gravitational quantum well and use the latest experimental results for the two lowest energy states of neutrons in the Earth's gravitational field to establish an upper bound on the fundamental momentum scale introduced by noncommutativity, namely $\sqrt{\eta}\lesssim1\ \mathrm{meV/c}$, a value that can be improved in the future by up to 3 orders of magnitude. We show that the configuration space noncommutativity has, in leading order, no effect on the problem. We also analyze some features introduced by the model, specially a correction to the presently accepted value of Planck's constant to 1 part in $10^{24}$.