‘Quantum hairs’ and entropy of the quantum isolated horizon from Chern–Simons theory
We articulate the fact that the loop quantum gravity (LQG) description of the quantum macrostates of black hole horizons, modeled as quantum isolated horizons (QIHs), is completely characterized in terms of two independent integer-valued ‘quantum hairs’, viz, the coupling constant ( k ) ?> of the quantum SU ( 2 ) ?> Chern–Simons (CS) theory describing QIH dynamics, and the number of punctures ( N ) ?> produced by the bulk spin network edges piercing the isolated horizon (which act as pointlike sources for the CS fields). We demonstrate that the microcanonical entropy of macroscopic (both parameters assuming very large values) QIHs can be obtained directly from the microstates of this CS theory using standard statistical mechanical methods, without having to additionally postulate the horizon as an ideal gas of punctures, or incorporate any additional classical or semiclassical input from general relativity vis-a-vis the functional dependence of the isolated horizon mass on its area, or indeed, without having to restrict to any special class of spins. Requiring the validity of the Bekenstein–Hawking area law relates these two parameters (as an equilibrium ‘equation of state’), and consequently allows the Barbero–Immirzi parameter to take any real and positive value depending on the value of k / N ?> . The logarithmic correction to the area law obtained a decade ago by R Kaul and one of us (PM), ensues straightforwardly, with precisely the coefficient − 3 / 2 ?> , making it a signature of the LQG approach to black hole entropy.