A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity

TL;DR

This paper offers a new geometric interpretation of quantum groups and non-commutative spaces in quantum gravity, linking the quantum deformation parameter to the cosmological constant through angular resolution limitations.

Contribution

It introduces a novel geometrical perspective where non-commutative spaces represent directions around spacetime points, connecting quantum deformation to cosmological parameters.

Findings

Relates quantum deformation parameter to the cosmological constant.

Provides a geometric interpretation of non-commutative spaces in gravity.

Resolves issues in conventional interpretations of quantum groups in physical geometry.

Abstract

Quantum groups and non-commutative spaces have been repeatedly utilized in approaches to quantum gravity. They provide a mathematically elegant cut-off, often interpreted as related to the Planck-scale quantum uncertainty in position. We consider here a different geometrical interpretation of this cut-off, where the relevant non-commutative space is the space of directions around any spacetime point. The limitations in angular resolution expresses the finiteness of the angular size of a Planck-scale minimal surface at a maximum distance 1/\sqrt{Lambda} related the cosmological constant Lambda. This yields a simple geometrical interpretation for the relation between the quantum deformation parameter q=exp[i Lambda l_{Planck}^2 ] and the cosmological constant, and resolves a difficulty of more conventional interpretations of the physical geometry described by quantum groups or fuzzy spaces.