The Microcanonical Entropy of Quantum Isolated Horizon, `quantum hair' $N$ and the Barbero-Immirzi parameter fixation

Abstract

{\it If} the total number of punctures($N$) of a quantum isolated horizon is considered to be a macroscopic parameter alongside the Chern-Simons level($k$) or equivalently classical area$(A_{cl})$ a strict analysis of the {\it microcanonical} ensemble reveals that the {\it microcanonical} entropy has the form $S_{MC}=A_{cl}/4\ell_p^2+N\sigma(\gamma)$, only for values of the Barbero-Immirzi(BI) parameter$(\gamma)$ greater than a certain number. It is argued that the term $N\sigma(\gamma)$ must be negative definite, which leads to the bound on the BI parameter.