Quantum Groups, Gravity, and the Generalized Uncertainty Principle

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

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