Modified commutators are not sufficient to determine a quantum gravity minimal length scale

TL;DR

This paper challenges the common assumption that a modified commutator alone determines a minimal length scale in quantum gravity, showing that the specific form of operator modifications is crucial.

Contribution

It demonstrates that different operator modifications can produce the same commutator but different minimal length scales, emphasizing the importance of operator form.

Findings

Modified commutator does not uniquely determine minimal length.

Operator form critically influences the existence of a minimal length.

Different operator pairs can yield identical commutators with varying minimal lengths.

Abstract

In quantum gravity it is generally thought that a modified commutator of the form $[{\hat x}, {\hat p}] = i \hbar (1 + \beta p^2)$ is sufficient to give rise to a minimum length scale. We test this assumption and find that different pairs of modified operators can lead to the same modified commutator and yet give different or even no minimal length. The conclusion is that the modification of the operators is the main factor in determining whether there is a minimal length. This fact - that it is the specific form of the modified operators which determine the existence or not of a minimal length scale - can be used to keep or reject specific modifications of the position and momentum operators in theory of quantum gravity.