Problems with modified commutators

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 and, crucially, that the action scale characterising the quantum effects of gravity, β, must be many orders of magnitude smaller than Planck’s constant, β ∼ 10–61 × ℏ, 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.

Generalised uncertainty relations from finite-accuracy measurements

“…Heuristically, the non-canonical term β(k′ − k) can be interpreted as an additional momentum "kick, " transferred to the canonical wave function by a quantum fluctuation of the background. The interested reader is 10.3389/fspas.2023.1087724 referred to (Lake, 2020;Lake, 2019;Lake et al, 2019;Lake et al, 2020;Lake, 2021a;Lake, 2021b;Lake et al, 2023) for further details.…”

“…By contrast, the canonical Hamiltonian is determined by the canonical (projective) position and momentum operators, x and p , via Ĥ = p 2 /(2m) + V( x ), where the former obey the canonical Heisenberg algebra: [ x i , p j ] = iℏδ i j Î , [ x i , x j ] = 0, [ p i , p j ] = 0. This leaves the canonical Heisenberg and Schrödinger equations unchanged and neatly evades the pathologies that afflict modified commutator models (Lake, 2020;Hossenfelder, 2013;Tawfik and Diab, 2014;Tawfik and Diab, 2015;Lake et al, 2023).…”

“…In fact, the situation is even worse since even a redefinition of the position-coordinate origin alters the value of the bound on the right-hand side. This gives rise to a coordinate-dependent "minimum" length, which is clearly unphysical, and which strongly suggests that GUR models based on modified commutation relations are not mathematically self-consistent (Lake, 2020;Lake et al, 2023).…”