A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum

TL;DR

This paper introduces a higher order GUP that incorporates both a minimal length and a maximal momentum, aligning with quantum gravity theories and affecting the energy spectrum of quantum systems.

Contribution

It proposes a new form of GUP consistent with multiple quantum gravity frameworks, predicting a maximal momentum and its implications on quantum spectra.

Findings

Maximal momentum leads to an upper energy bound.

The GUP form aligns with string theory, loop quantum gravity, and doubly special relativity.

Impacts on the energy spectrum of quantum systems like the harmonic oscillator.

Abstract

We present a higher order generalized (gravitational) uncertainty principle (GUP) in the form $[X,P]=i\hbar/(1-\beta P^2)$. This form of GUP is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. We show that the presence of the maximal momentum results in an upper bound on the energy spectrum of the momentum eigenstates and the harmonic oscillator.