Note on the geometrical interpretation of quantum groups and noncommutative spaces in gravity

Quantum groups and noncommutative spaces have been repeatedly utilized in approaches to quantum gravity. They provide a mathematically elegant cutoff, often interpreted as related to the Planck-scale quantum uncertainty in position. We consider here a different geometrical interpretation of this cutoff, where the relevant noncommutative space is the space of directions around any spacetime point. The limitations in angular resolution express 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 $q=e^{i\Lambda l_{\mathrm{Planck}}^2}$ and the cosmological constant, and resolves the difficulty of more conventional interpretations of the physical geometry described by quantum groups or fuzzy spaces.

Figure 1

The fuzzy sphere is the local celestial sphere.