Modified Commutators vs Modified Operators in a Quantum Gravity minimal length scale

Abstract

Generic theories of quantum gravity often postulate that at some high energy/momentum scale there will be a fixed, minimal length. Such a minimal length can be phenomenologically investigated by modifying the standard Heisenberg Uncertainty relationship. This is generally done in practice by modifying the commutator between position and momentum operators, which in turn means modifying these operators. However, modifications such that the uncertainty relation changes lead to conflicts with observational data (gamma ray bursts). This arises in the form of a predicted minimal length energy scale that is above the Planck energy rather than below it. As a result there seems to be an implication that there is no minimal length scale in these generic theories. Meanwhile, modifying the operators such that the standard uncertainty relation retains the same form, leads to no such conflict with observational data. We show that it is this modification of the position and momentum operators that is the key determining factor in the existence (or not) of a minimal length scale. By focusing primarily on the role of these operators we also show that one can avoid the constraints from the observations of short gamma ray bursts, which in certain cases seem to push the minimal length scale above the Planck scale.