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

2011/05/13 by Eugenio Bianchi, Carlo Rovelli · 1 citation
Physics and Astronomy · #gr-qc #hep-th

paper · pdf · doi:10.1103/physrevd.84.027502

published as Phys.Rev.D84:027502,2011 · 2 pages, 1 figure

arxiv created 2011/05/13 · arxiv updated 2011/08/09

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/√(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_Planck² ] and the cosmological constant, and resolves a difficulty of more conventional interpretations of the physical geometry described by quantum groups or fuzzy spaces.

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