Minimal and maximal lengths of quantum gravity from non-Hermitian position-dependent noncommutativity

TL;DR

This paper explores how non-Hermitian, position-dependent noncommutativity can simultaneously produce minimal and maximal length scales in quantum gravity, revealing a connection between non-Hermitian operators and fundamental length uncertainties.

Contribution

It provides an alternative derivation of quantum gravity length scales from non-Hermitian noncommutativity and demonstrates their transformation into Hermitian systems via a Dyson map.

Findings

Existence of discrete space families with both minimal and maximal lengths.

Minimal uncertainties linked to non-Hermitian operators.

Transformation to Hermitian variables preserves quantum gravity properties.

Abstract

A minimum length scale of the order of Planck length is a feature of many models of quantum gravity that seek to unify quantum mechanics and gravitation. Recently, Perivolaropoulos in his seminal work [Phys. Rev.D 95, 103523 (2017)] predicted the simultaneous existence of minimal and maximal length measurements of quantum gravity. More recently, we have shown that both measurable lengths can be obtained from position-dependent noncommutativity [J. Phys. A: Math.Theor. 53, 115303 (2020)]. In this paper, we present an alternative derivation of these lengths from non-Hermitian position-dependent noncommutativity. We show that a simultaneous measurement of both lengths form a family of discrete spaces. In one hand, we show the similarities between the maximal uncertainty measurement and the classical properties of gravity. On the other hand, the connection between the minimal uncertainties and the non-Hermicity quantum mechanic scenarios. The existence of minimal uncertainties are the consequences of non-Hermicities of some operators that are generators of this noncommutativity. With an appropriate Dyson map, we demonstrate by a similarity transformation that the physically meaningfulness of dynamical quantum systems is generated by a hidden Hermitian position-dependent noncommutativity. This transformation preserves the properties of quantum gravity but removes the fuzziness induced by minimal uncertainty measurements at this scale. Finally, we study the eigenvalue problem of a free particle in a square-well potential in these new Hermitian variables.