Minimal Lengths in 3D via the Generalized Uncertainty Principle

TL;DR

This paper extends the Generalized Uncertainty Principle to three dimensions, exploring minimal length scales and directional dependencies in quantum systems, with implications for understanding quantum gravity effects.

Contribution

It introduces a coordinate-independent 3D GUP framework with bounded momentum, revealing novel directional minimal length phenomena not seen in 1D.

Findings

Minimal length bounds in all coordinate directions.

Directional differences in minimal length when momentum is aligned or orthogonal.

Distinct phenomena in 3D GUP compared to 1D cases.

Abstract

We investigate an extension of the Generalized Uncertainty Principle (GUP) in three dimensions by modifying the three dimensional position and momentum operators in a manner that remains coordinate-independent and retains as much of the standard position-momentum commutators as possible. Moreover, we bound the physical momentum which leads to an effective minimal length in every coordinate direction. The physical consequences of these modified operators are explored in two scenarios: (i) when a spherically-symmetric wave function is `compressed' into the smallest possible volume; (ii) when the momentum is directed in a single direction. In case (ii), we find that the three dimensional GUP exhibits interesting phenomena that do not occur in one dimension: the minimal distance in the direction parallel to a particle's momentum is different from the minimal distance in the orthogonal directions.