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

Understand

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.