Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta

TL;DR

This paper extends the analysis of maximal localised states in quantum mechanics to include minimal uncertainties in both positions and momenta, considering noncommutative geometric corrections relevant for quantum gravity and string theory.

Contribution

It generalizes previous work by deriving maximal localisation states when both position and momentum uncertainties are minimal, incorporating noncommutative geometry effects.

Findings

Derived states with minimal uncertainties in both positions and momenta.

Extended the framework to include noncommutative geometric corrections.

Provided insights into the concept of locality under quantum gravity modifications.

Abstract

Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small `noncommutative geometric' corrections to the canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta.