The Semi-Classical Limit of Quantum Gravity on Corners

TL;DR

The paper develops a geometric framework that connects quantum states governed by the quantum corner symmetry group to a classical system realizing the same symmetry, using coadjoint orbits, moment maps, and Berezin quantization. It shows how generalized Perelomov coherent states provide a bridge between representation-theoretic data and classical observables, with a precise semiclassical expansion and a Casimir invariant that matches between quantum and classical pictures. The construction yields an exact link for linear observables and a deformation-quantization description for higher orders, and it demonstrates invariance of the Casimir under moment-map changes. This provides a mathematically well-defined route to a semiclassical gravity within the corner symmetry framework and connects to entanglement entropy area laws, with planned exploration of Verlinde–Zurek relations in future work.

Abstract

We study quantum and classical systems defined by the quantum corner symmetry group $QCS = \widetilde{SL}(2,\mathbb{R}) \ltimes H_3$, which arises in the context of quantum gravity. In particular, we relate the quantum observables, defined by representation-theoretic data, to their classical counterparts through generalized Perelomov coherent states and the framework of Berezin quantization. The resulting procedure provides a mathematically well-defined notion of the semi-classical limit of quantum gravity, viewed as the representation theory of the corner symmetry group.