UCLA

A standard insight of the AdS/CFT correspondence is that some aspects of the geometry of a bulk state are encoded in the entanglement structure of its dual boundary state. As entanglement is not a linear quantum observable, this means that geometry in a quantum theory of gravity should likewise not be a linear observable.

This is seemingly in tension with an understanding of quantum gravity as a gravity path integral summing over geometries, which inherently treats geometry as a linear observable. In a recent work the authors Marolf and Maxfield explore this tension in the context of a simple model of a gravity path integral where it is resolved by the appearance of null states, or in other words geometries that are equal to superpositions of other geometries. In Chapter 1 of this dissertation, we extend their 2d topological gravity model to have as its bulk action any open/closed TQFT obeying Atiyah's axioms. We describe the Hilbert spaces of these more general theories that remain after null states have been accounted for. The holographic duals of these topological gravity models are ensembles of 1d topological theories with random dimension. These holographic interpretations of our gravity models require projecting out negative-norm states from the baby universe Hilbert space, which Marolf and Maxfield achieved by adding a non-local boundary term to the bulk action. We describe the analogous solution in the framework of a TQFT with defects by coupling the boundaries of the gravity models to auxiliary 2d TQFTs in a non-gravitational (i.e. fixed topology) region. The gravity model is then holographically dual to an ensemble of boundary conditions in an open/closed TQFT without gravity. In Chapter 2 we explore linear dependencies between certain states with simple geometric duals: states made up of n copies of a thermofield double state and the states obtained from this one by permuting the n right hand sides. We derive expressions for the maximum fidelity between one such state and a linear combination of the others, and see that this fidelity approaches 1 as the number n of black holes increases. We also consider the possibility of obtaining a single thermofield double state as the partial trace of linear combinations of such states with topologies with no connection between the two untraced sides. We derive lower bounds for the fidelity between the thermofield double state and such partial traces and comment on the conceptual implications of the existence of such states.