Analytical continuation of black hole entropy in Loop Quantum Gravity: Lessons from black hole thermodynamics

Abstract

This contribution is devoted to summarize the recent results obtained in the construction of an “analytic continuation” of Loop Quantum Gravity (LQG). By this we mean that we construct analytic continuation of physical quantities in LQG from real values of the Barbero-Immirzi pa- rameter g to the purely imaginary g = i . This should allow us to define a quantization of gravity with self dual Ashtekar variables. We first realized in [ 1 ] that this procedure, when applied to compute the entropy of a spherical black hole in LQG for g = i , allows to reproduce exactly the Bekenstein Hawking area law at the semi classical limit. The rigorous construction of the analytic continuation of spherical black hole entropy has been done in [ 2 ]. Here we start with a review of the main steps of this construction: we recall that our prescription turns out to be unique (under natural assumptions) and leads to the right semi-classical limit with its logarithmic quantum corrections. Futhermore, the discrete and g -dependent area spectrum of the black hole horizon becomes continuous and obviously g -independent. Then, we review how this analytic continuation could be interpreted in terms of the analytic continuation from the compact gauge group SU ( 2 ) to the non compact gauge group SU ( 1 ; 1 ) relying on an analysis of three dimensional quantum gravity.