From the Corner Proposal to the Area Law

TL;DR

Addresses explicit realization of the Corner Proposal in four-dimensional, spherically symmetric spacetimes via the Extended Corner Symmetry (ECS) algebra, $\mathfrak{ecs} = \mathfrak{diff}(S) (\mathfrak{sl}(2,\mathbb{R})\mathbb{R}^2)^S$, and its quantum realization (QCS). Constructs Perelomov coherent states on the QCS group, defines a class of classical states with $c_\zeta = \bar{c}_\zeta = -s \frac{\pi}{2}$, and relates quantum charges to classical corner data via Berezin symbols. For classical states, the entanglement entropy scales as $S^{\mathrm{cl}} \sim 2\pi s + \frac{1}{2}\ln(2\pi s)$ in the large-$s$ limit, and, by identifying $s$ with geometric data, derives the area law $S \sim \frac{A}{4 l_p^2 \epsilon} + \frac{1}{2}\ln\left(\frac{A}{4 l_p^2 \epsilon}\right) + \mathcal{O}(\epsilon)$ for Schwarzschild, de Sitter, and Minkowski causal diamonds. This demonstrates that the Corner Proposal can reproduce the semiclassical area scaling directly from the quantum algebra of boundary observables and outlines future work on multiple corners and quantum corrections.

Abstract

We provide an explicit realization of the Corner Proposal for Quantum Gravity in the case of spherically symmetric spacetimes in four dimensions, or equivalently, two-dimensional dilaton gravity. We construct coherent states of the Quantum Corner Symmetry group and compute the entanglement entropy relative to these states. We derive the classical corner charges and relate them to operator expectation values in coherent states. For a subset of coherent states that we call classical states, we find that the entanglement entropy exhibits a leading term proportional to the area, recovering the Bekenstein-Hawking area law in the semiclassical limit.