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

TL;DR

This paper analyzes the microcanonical entropy of quantum isolated horizons considering the puncture count as a macroscopic parameter, revealing bounds on the Barbero-Immirzi parameter based on entropy positivity.

Contribution

It introduces a strict microcanonical analysis including puncture number as a macroscopic parameter, deriving bounds on the Barbero-Immirzi parameter for quantum isolated horizons.

Findings

Entropy depends on puncture number and BI parameter

Bounds on BI parameter are derived from entropy positivity

Entropy form confirms the significance of puncture count in quantum horizon thermodynamics

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.