Revisiting the algebraic structure of the generalized uncertainty principle

TL;DR

This paper compares algebraic formulations of the generalized uncertainty principle, highlighting that one formulation is more universally valid across spins and more constrained, with implications for minimum length and quantum-classical transition.

Contribution

It demonstrates that the earlier algebraic formulation by one of the authors is more fundamental and valid for all spins, unlike the KMM formulation which is limited to spin zero.

Findings

KMM formulation satisfies Jacobi identities only for spin zero.

The earlier formulation has only two solutions: minimum length and quantum-to-classical transition.

The algebraic structure suggests a discretized time at the Planck scale.

Abstract

We compare different formulations of the generalized uncertainty principle that have an underlying algebraic structure. We show that the formulation by Kempf, Mangano and Mann (KMM) [Phys. Rev. D 52 (1995)], quite popular for phenomenological studies, satisfies the Jacobi identities only for spin zero particles. In contrast, the formulation proposed earlier by one of us (MM) [Phys. Lett. B 319 (1993)] has an underlying algebraic structure valid for particles of all spins, and is in this sense more fundamental. The latter is also much more constrained, resulting into only two possible solutions, one expressing the existence of a minimum length, and the other expressing a form of quantum-to-classical transition. We also discuss how this more stringent algebraic formulation has an intriguing physical interpretation in terms of a discretized time at the Planck scale.