Horizon entropy with loop quantum gravity methods
Daniele Pranzetti, Hanno Sahlmann
TL;DR
The paper recasts the isolated horizon boundary in loop quantum gravity terms by expressing its presymplectic structure with a 2+1 gravity-like first-order formalism. It then quantizes the horizon degrees of freedom using LQG techniques, treating the boundary as a non-commutative system with a flat-connection formulation and regularized punctures that carry SU(2) spins, leading to a Chern-Simons/3d-gravity–inspired boundary Hilbert space. The resulting state counting, implemented via a physical scalar product and quantum-group recouplings, yields a horizon entropy that reproduces the Bekenstein-Hawking area law in the limit of imaginary Barbero-Immirzi parameter ($\beta=i$), and clarifies ambiguities related to bulk-boundary coupling. The work also connects to alternative quantization schemes, including a 3d gravity perspective with $\Lambda=0$, and emphasizes the interpretation of horizon punctures as 2+1 gravity particles. Overall, it provides a coherent LQG-based framework for black hole entropy that avoids previous Chern-Simons coupling ambiguities and deepens the link between horizon thermodynamics and lower-dimensional gravity.
Abstract
We show that the spherically symmetric isolated horizon can be described in terms of an SU(2) connection and a su(2) valued one form, obeying certain constraints. The horizon symplectic structure is precisely the one of 3d gravity in a first order formulation. We quantize the horizon degrees of freedom in the framework of loop quantum gravity, with methods recently developed for 3d gravity with non-vanishing cosmological constant. Bulk excitations ending on the horizon act very similar to particles in 3d gravity. The Bekenstein-Hawking law is recovered in the limit of imaginary Barbero-Immirzi parameter. Alternative methods of quantization are also discussed.