Entropy in the Classical and Quantum Polymer Black Hole Models

TL;DR

This paper compares classical and quantum polymer models in loop quantum gravity to compute black hole entropy, showing that classical polyhedra counting captures key features of quantum intertwiner counting, simplifying entropy analysis.

Contribution

It introduces a classical polyhedra model as an effective approximation for quantum intertwiners in black hole entropy calculations in LQG.

Findings

Classical polyhedra counting reproduces leading order entropy.

Logarithmic corrections are captured by the classical model.

Exact formulas for intertwiner space dimensions are derived.

Abstract

We investigate the entropy counting for black hole horizons in loop quantum gravity (LQG). We argue that the space of 3d closed polyhedra is the classical counterpart of the space of SU(2) intertwiners at the quantum level. Then computing the entropy for the boundary horizon amounts to calculating the density of polyhedra or the number of intertwiners at fixed total area. Following the previous work arXiv:1011.5628, we dub these the classical and quantum polymer models for isolated horizons in LQG. We provide exact micro-canonical calculations for both models and we show that the classical counting of polyhedra accounts for most of the features of the intertwiner counting (leading order entropy and log-correction), thus providing us with a simpler model to further investigate correlations and dynamics. To illustrate this, we also produce an exact formula for the dimension of the intertwiner space as a density of "almost-closed polyhedra".