The generalized uncertainty principle in the presence of extra dimensions

TL;DR

This paper explores how the Generalized Uncertainty Principle (GUP) interacts with extra dimensions, revealing that the minimal measurable length corresponds to the compactification radius of the extra dimension, linking quantum gravity concepts.

Contribution

It introduces a model connecting GUP parameter variations with extra dimensions, showing the minimal measurable length equals the compactification radius.

Findings

Minimum measurable length equals the compactification radius of the extra dimension.

The GUP parameter varies with energy scales and relates to extra dimension size.

The relation $rac{ oot{eta_0} \, ext{length}_p}{ ho} \, ext{is of order 1}$.

Abstract

We argue that in the Generalized Uncertainty Principle (GUP) model, the parameter $\beta_0$ whose square root, multiplied by Planck length $\ell_p$, approximates the minimum measurable distance, varies with energy scales. Since minimal measurable length and extra dimensions are both suggested by quantum gravity theories, we investigate models based on GUP and one extra dimension, compactified with radius $\rho$. We obtain an inspiring relation $\sqrt{\beta_0} \ell_p/\rho \sim {\cal O}(1)$. This relation is also consistent with predictions at Planck scale and usual quantum mechanics scale. We also make estimations on the application range of the GUP model. It turns out that the minimum measurable length is exactly the compactification radius of the extra dimension.