Probing deformed quantum commutators

TL;DR

This paper investigates the quantum limits of measuring a minimal length predicted by quantum gravity theories, using harmonic oscillator experiments to analyze the effects of deformed commutators on measurement precision.

Contribution

It introduces an analytical framework for assessing the quantum bounds on estimating minimal length via deformed quantum commutators in harmonic oscillators.

Findings

Derived analytical expressions for measurement bounds

Identified the impact of deformed algebra on uncertainty limits

Provided insights into experimental feasibility of minimal length detection

Abstract

Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.