Generalized uncertainty principles and localization of a particle in discrete space

Generalized uncertainty principles are able to serve as useful descriptions of some of the phenomenology of quantum gravity effects, providing an intuitive grasp on nontrivial space-time structures such as a fundamental discreteness of space, a universal band limit or an irreducible extendedness of elementary particles. In this article, uncertainty relations for single-particle quantum mechanics 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.