Quantum deformation of quantum gravity

Abstract

We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, depends on the cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2)q spin networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally, eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform.