Geometric temperature and entropy of quantum isolated horizons
Daniele Pranzetti
TL;DR
The paper addresses the microscopic origin of black hole entropy by restoring Lorentz invariance in canonical LQG and defining a geometric horizon temperature through a KMS state for quantum isolated horizons. This requires an analytic continuation to self-dual Ashtekar variables, establishing a direct link between horizon thermality and local Lorentz symmetry, and interpreting the temperature via the Connes–Rovelli thermal time framework. The authors derive a horizon inverse temperature $\beta=2\pi\left(1-\frac{1}{k}\right)$, show that entanglement and Boltzmann entropies coincide, and recover the Bekenstein–Hawking result $S_{BH}=a_H/(4\ell_P^2)$ in the semiclassical limit, with a small quantum correction controlled by the Chern-Simons level $k$. This work bridges entanglement-based and state-counting perspectives on black hole entropy and highlights a geometric, boundary-driven origin of thermality within a hybrid LQG–CS framework, with implications for the information paradox and the thermal time program.
Abstract
By reintroducing Lorentz invariance in canonical loop quantum gravity, we define a geometrical notion of temperature for quantum isolated horizons. This is done by demanding that the horizon state satisfying the boundary conditions be a Kubo-Martin-Schwinger state. The exact formula for the temperature can be derived by imposing the reality conditions in the form of the linear simplicity constraints for an imaginary Barbero-Immirzi parameter. Thus, our analysis reveals the connection between the analytic continuation to the Ashtekar self-dual variables and the thermality of the horizon. The horizon thermal equilibrium state can then be used to compute both the entanglement and the Boltzmann entropies. We show that the two provide the same finite answer, which allows us to recover the Bekenstein-Hawking formula in the semi-classical limit. In this way, we shed new light on the microscopic origin of black hole entropy by revealing the equivalence between the near-horizon degrees of freedom entanglement proposal and the state-counting interpretation. The connection with the Connes-Rovelli thermal time proposal for a general relativistic statistical mechanics is worked out.