The algebraic structure of the generalized uncertainty principle

TL;DR

This paper demonstrates that a specific deformation of the Heisenberg algebra underpins the generalized uncertainty principle in quantum gravity, uniquely determined by simple assumptions, and connects to string theory results.

Contribution

It identifies a unique algebraic structure underlying the generalized uncertainty principle and relates it to string theory and potential Lorentz invariance implications.

Findings

Deformed algebra fixed by simple assumptions

Reproduces string theory results at first order

Suggests possible Lorentz invariance modifications

Abstract

We show that a deformation of the Heisenberg algebra which depends on a dimensionful parameter $\kappa$ is the algebraic structure which underlies the generalized uncertainty principle in quantum gravity. The deformed algebra and therefore the form of the generalized uncertainty principle are fixed uniquely by rather simple assumptions. The string theory result is reproduced expanding our result at first order in $\Delta p/M_{\rm PL}$. We also briefly comment on possible implications for Lorentz invariance at the Planck scale.