UC Santa Barbara

Holographic dualities like AdS/CFT provide a non-perturbative definition of a bulk theory of quantum gravity in terms of a lower-dimensional boundary quantum field theory. Elucidating quantum gravity thus becomes the challenge of deciphering the dictionary between bulk and boundary physics, and using it to translate basic properties of the latter into lessons about the former. A crucial intermediary between these two realms is the gravitational path integral, which defines the bulk theory in terms of an integral over spacetimes subject to conditions from the boundary theory. Though shallowly understood and rather formal, this piece of technology has thus far been able to provide deep insights into quantum gravity. This thesis is organized in three parts, each focusing on a different basic property in quantum theory and its consequences for quantum gravity through the gravitational path integral: entanglement, causality, and factorization.

Part I addresses the emergence of spacetime from entanglement, with a focus on understanding constraints on the entanglement structure of quantum states to posses classical geometries as holographic bulk duals. These constraints can be expressed as linear inequalities and used to define the holographic entropy cone (HEC). A systematic study of the HEC is accomplished by reformulating the holographic computation of von Neumann entropies as a graph-theoretic one, thereby recasting a complicated problem in differential geometry as a purely combinatorial one. This allows to prove important properties of the HEC, devise proof methods and algorithms for constructing it, derive precise relations to other polytope structures, and ultimately pursue a top-down understanding of the HEC from the universal quantum inequality of subadditivity. This part concludes with an exploration of how the machinery involved in the study of the HEC may also be generalized to settings where the von Neumann entropy receives contributions from bulk quantum fields, a regime where the graph-theoretic apparatus has to be upgraded.

Part II presents an alternative perspective on spacetime emergence, both in classical and quantum regimes, based on causality. Starting at a classical level, we explain how the conformal bulk geometry can be reconstructed by encoding its causal structure in data accessible from the boundary field theory. Through the use of field theory correlators, we propose a method for obtaining the full-dimensional bulk geometry up to a conformal factor. This generalizes the approach to bulk metric reconstruction based on light-cone cuts to a prescription which allows for recovering even those dimensions which become compact asymptotically. Moving away from the classical limit, we then resolve a known puzzle that arises from a tension between the bulk and boundary causal structures when the bulk theory is understood as a genuine gravitational path integral over spacetimes.

Finally, part III delves into the consequences of the lack of factorization that occurs in holography when wormholes are included in the gravitational path integral. In particular, we study generating functionals in quantum gravity and propose a recipe for their computation which accounts for the contribution of such connected topologies. This allows to differentiate between quenched and annealed quantities in quantum gravity, a distinction which may be used as a consistency test-ground for foundational aspects of the gravitational path integral regarding summing over topologies.