Problems with Modified Commutators

TL;DR

This paper critiques the reliance on modified commutation relations in generalized uncertainty relations, proposing an alternative approach that avoids these modifications but has profound implications for quantum gravity and space-time quantization.

Contribution

It introduces a simple assumption that generates GURs without modifying the canonical algebra, impacting quantum gravity and space-time models.

Findings

Modified commutator models face theoretical issues.

An alternative assumption can generate GURs without algebra modification.

Implications include a quantum space-time as a quantum reference frame and a tiny action scale.

Abstract

The purpose of this paper is to challenge the existing paradigm on which contemporary models of generalised uncertainty relations (GURs) are based, that is, the assumption of modified commutation relations. We review an array of theoretical problems that arise in modified commutator models, including those that have been discussed in depth and others that have received comparatively little attention, or have not been considered at all in the existing literature, with the aim of stimulating discussion on these topics. We then show how an apparently simple assumption can solve, or, more precisely, evade these issues, by generating GURs without modifying the basic form of the canonical Heisenberg algebra. This simplicity is deceptive, however, as the necessary assumption is found to have huge implications for the quantisation of space-time and, therefore, gravity. These include the view that quantum space-time should be considered as a quantum reference frame (QRF) and, crucially, that the action scale characterising the quantum effects of gravity, $\beta$, must be many orders of magnitude smaller than Planck's constant, $\beta \sim 10^{-61} \times \hbar$, in order to recover the present day dark energy density. We argue that these proposals should be taken seriously, as a potential solution to the pathologies that plague minimum length models based on modified commutators, and that their implications should be explored as thoroughly as those of the existing paradigm, which has dominated research in this area for almost three decades.