Generalized uncertainty principles and localization in discrete space

TL;DR

This paper derives generalized uncertainty principles for quantum systems with discrete space and periodic momentum, revealing corrections to standard relations applicable to models like loop quantum cosmology.

Contribution

It introduces a moment expansion method to derive uncertainty relations in discrete space, highlighting differences from traditional assumptions.

Findings

Derived correction terms to standard uncertainty relations.

Applicable to discrete models of matter and space-time.

Potential relevance to loop quantum cosmology.

Abstract

Generalized uncertainty principles are able to serve as useful descriptions of some of the phenomenology of quantum gravity effects, providing an intuitive grasp on non-trivial space-time structures such as a fundamental discreteness of space, a universal bandlimit or an irreducible extendedness of elementary particles. In this article, uncertainty relations are derived by a moment expansion of states for quantum systems with a discrete coordinate, and correspondingly a periodic momentum. Corrections to standard uncertainty relations are found, with some similarities but also key differences to what is often assumed in this context. The relations provided can be applied to discrete models of matter or space-time, including loop quantum cosmology.