Beecroft Building, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU
Tom Philip Yildirim, University of Oxford, Theoretical Physics
Andrei Constantin (andrei.constantin@physics.ox.ac.uk)
Abstract
Recent developments have stressed the importance of including non-perturbative wormhole saddles in the gravitational path integral. For example, such saddles can account for the unitarity of black hole evaporation and reveal that infinite sets of seemingly orthogonal black hole microstates in reality only consist of a Beckenstein-Hawking entropy’s worth of independent states. However, these wormhole contributions suggest that the gravitational path integral should be understood as computing coarse-grained or ensemble averaged observables of an underlying fine-grained theory. Additionally, the gravity path integral itself is hard to evaluate, and in realistic models of gravity results are often limited to the saddle-point approximation. There are thus two distinct difficulties: 1) the gravity path integral only has access coarse-grained distributions of fine-grained observables and 2) the saddle-point approximation results in an approximate distribution for these fine-grained observables. In this talk we demonstrate that both these difficulties can be overcome in demonstrating exact equivalences between fine-grained quantities in quantum gravity. To do so we introduce a toolkit for the Euclidean path integral that allows fine-grained equalities to be extracted arbitrarily far beyond the saddle point approximation and without any assumptions about holographic duality. This toolkit involves three ingredients: (1) a way of resolving the identity with an over-complete basis of microstates (2) a drastic simplification of the sum over topologies in the limit where the basis is infinitely over-complete, and (3) a way of cutting and splicing geometries to demonstrate equality between two different gravitational path integrals even if neither can be explicitly computed. We arrive at this toolkit by answering the question of how a given set of gravity states can be shown to provide a complete basis. One implication of our results is that universes containing a horizon can sometimes be understood as superpositions of horizonless geometries entangled with a closed universe. We will then go on to discuss how this toolkit can be used to address the Hilbert space factorisation problem and the question of whether the Gibbons-Hawking prescription for the gravity thermal partition function is in-fact tracing over the states of the theory.