Generalized Uncertainty Principle in the Presence of Extra Dimensions

We argue that in the generalized uncertainty principle (GUP) model, the parameter β₀ whose square root, multiplied by Planck length ℓ_p, approximates the minimum measurable distance, varies with energy scales. Since the minimal measurable length and extra dimensions are both suggested by quantum gravity theories, we investigate the models based on the GUP and one extra dimension, compactified with radius ρ. We obtain an inspiring relation √β₀ ℓ_p/ρ∼ O(1). This relation is also consistent with the predictions at Planck scale and the usual quantum mechanics scale. We also estimate the application range of the GUP model. It turns out that the minimum measurable length is exactly the compactification radius of the extra dimension.