At the Corner of Quantum and Gravity

Abstract

In the presence of spacetime boundaries, diffeomorphisms in gravitational theories can become physical and acquire non-vanishing Noether charges. These charges obey an algebra which, within the extended phase-space formalism, faithfully realizes diffeomorphism algebra. The corner proposal takes this algebra of physical corner symmetries as a fundamental ingredient of quantum gravity, in close analogy with the role of the Poincaré group in quantum field theory. In this thesis we develop the quantum corner framework in the two-dimensional setting. We give the full representation theory of the two-dimensional extended corner symmetry group, which may be interpreted either as the symmetry group of two-dimensional gravity or as the corner symmetry group relevant for four-dimensional spherically symmetric gravity. Within the corner proposal, the resulting representation spaces are then interpreted as candidate Hilbert spaces for quantum gravity. This representation-theoretic structure naturally enables a description of local subsystems. In particular, we present a gluing procedure that constructs quantum states associated with an entangling corner between two spacetime subregions, and use it to compute the entanglement entropy between the two regions. To connect the quantum observables to the classical corner charges, we construct the coadjoint orbits of the quantum corner symmetry group and relate them to the classical structure through twisted moment maps and to the quantum structure through generalized Perelomov coherent states. This provides a notion of semiclassical limit within the corner framework. Finally, in the context of static, spherically symmetric spacetimes, we show that a distinguished family of coherent states reproduces the horizon area law for entropy in the semiclassical limit, yielding a quantum, symmetry-based explanation of the Bekenstein--Hawking formula.

Figures (7)

• Figure 1: Reproduced from Ciambelli:2023bmn, the symmetries pyramid represents the perspective of the corner proposal as a generalization of Strominger's infrared triangle. In this picture, symmetries are the fundamental structure from which memory effects, soft theorems, and quantum gravity can be derived.
• Figure 2: The three symplectic spaces appearing in the corner proposal: the coadjoint orbits equipped with the Kirillov--Kostant--Souriau form, the field space equipped with the extended symplectic form of the covariant phase-space formalism, and the projective Hilbert space equipped with the Fubini--Study form. The classical limit is obtained by identifying the point on the coadjoint orbit determined by the twisted moment map with the point determined by the Berezin symbols.
• Figure 3: One spacelike segment connected to the boundary $L$. The Hilbert space describing this system is a representation of the corner symmetry group at $L$.
• Figure 4: Two segments are glued together identifying the left and right corners. The Hilbert space associated with the Cauchy slice $\Sigma_G = \Sigma_L \cup \Sigma_R$ is given by a proper subspace of the tensor product of the left and right Hilbert space.
• Figure 5: A three legs Feynman diagram $(A)$ where momentum conservation is enforced by $\delta(p+q-k)$ and a two legs diagram $(B)$ where momentum conservation reads $\delta(p-q)$.