A higher order GUP with minimal length uncertainty and maximal momentum

Abstract

We present a higher order generalized (gravitational) uncertainty principle (GUP) in the form [X,P]=iℏ/(1−βP2). 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.