A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

TL;DR

This paper reveals that different operator modifications with the same GUP commutator can lead to different physical implications, particularly regarding the existence of a minimal length scale in quantum gravity models.

Contribution

It highlights the importance of operator modification choices in GUP models, clarifying which constructions imply a minimal length scale.

Findings

Some GUP modifications do not imply a minimal length.

Operator modifications influence physical outcomes beyond the commutator.

Guidance for constructing GUP models with desired properties.

Abstract

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, $[\hat{X},\hat{P}] = i \hbar (1+\beta p^2)$, may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.