Quantum Groups, Gravity, and the Generalized Uncertainty Principle

TL;DR

This paper explores how quantum deformation of the Poincaré algebra leads to a generalized uncertainty principle, revealing a natural emergence of a minimal observable length in quantum gravity models.

Contribution

It establishes a direct link between the generalized uncertainty principle and the quantum deformation of the Poincaré algebra, highlighting the emergence of a minimal length scale.

Findings

Deformed Newton-Wigner position operator obeys a deformed Heisenberg algebra.

The deformed Poincaré algebra implies a minimal observable length.

Generalized uncertainty principle arises naturally from quantum deformation.

Abstract

We investigate the relationship between the generalized uncertainty principle in quantum gravity and the quantum deformation of the Poincar\'e algebra. We find that a deformed Newton-Wigner position operator and the generators of spatial translations and rotations of the deformed Poincar\'e algebra obey a deformed Heisenberg algebra from which the generalized uncertainty principle follows. The result indicates that in the $\kappa$-deformed Poincar\'e algebra a minimal observable length emerges naturally.