Minimal length, maximal momentum and Hilbert space representation of quantum mechanics

TL;DR

This paper extends Kempf et al.'s Hilbert space formulation of quantum mechanics to include both a minimal length and a maximal momentum, revealing new features in the quantum framework.

Contribution

It generalizes the existing model to incorporate an upper bound on momentum, combining minimal length and maximal momentum in quantum mechanics.

Findings

Introduction of an upper momentum bound leads to novel quantum features.

Generalization of Kempf's framework to include maximal momentum.

Potential implications for quantum gravity and high-energy physics.

Abstract

Kempf et al. in Ref. [1] have formulated a Hilbert space representation of quantum mechanics with a minimal measurable length. Recently it has been revealed, in the context of doubly special relativity, that a test particles' momentum cannot be arbitrarily imprecise and therefore there is an upper bound for momentum fluctuations. Taking this achievement into account, we generalize the seminal work of Kempf {\it et al.} to the case that there is also a maximal particles' momentum. Existence of an upper bound for the test particles' momentum provides several novel and interesting features, some of which are studied in this paper.