Operator Algebras and Third Quantization

Abstract

In quantum gravity, the gravitational path integral involves a sum over topologies, representing the joining and splitting of multiple universes. To account for topology change, one is led to allow the creation and annihilation of closed and open universes in a framework often called third quantization or universe field theory. We argue that since topology change in gravity is a rare event, its contribution to exponentially late-time physics is universally described by a Poisson process. This universal Poisson process is responsible for the plateau in the multi-boundary generalization of the spectral form factor at late times. In the Fock space of closed baby universes, this universality implies that the statistics of the total number of baby universes is captured by a coherent state. Allowing for the creation of asymptotic open universes calls for a noncommutative generalization of a Poisson process. We propose such an operator algebraic framework, called Poissonization, which takes as input the observable algebra and a (unnormalized) state of a quantum system and outputs a von Neumann algebra of a many-body theory represented on its symmetric Fock space. Physically, Poissonization is a generalization of the coherent state vacua of bipartite quantum systems or matrix quantum mechanics. The multi-boundary correlators of the Marolf-Maxfield toy model of baby universes, and the sum over bordisms of closed-open 2D topological quantum field theory, are entirely captured by Poissonization. In the Jackiew-Teitelboim gravity, the universal Poisson process reproduces the late-time correlators in the $τ$-scaling limit.