Chaos in Observable Algebras of Quantum Gravity

The Hawking-Page phase transition in holography is associated with the emergence of new symmetries and dynamical properties of correlators. Above the transition point: (1) Correlators cluster in time (Maldacena's information loss) (2) There exists a coarse-grained entropy that grows monotonically in time (Second law) (3) A large class of correlators decay exponentially (Quasi-normal modes) (4) There is an emergent approximate Lie group (near-horizon symmetries).

I discuss all the properties above from the point of view of the modular flow of the observable algebra of quantum gravity on the boundary. I prove that property (1) implies that the observable algebra is a type III_1 von Neumann factor. I point out that there is a class of quantum ergodic systems (K-systems) characterized by the existence of future and past subalgebras. I demonstrate that modular K-systems satisfy all the four properties above. In other words, modular K-systems are maximally chaotic. I comment on the implications of the results above for the emergence of spacetime in holography, and the generalizations of these results beyond modular flow, and the quantum ergodic hierarchy.