Holography and Entanglement in Quantum Field Theory

Quantum field theory (QFT) and general relativity are individually successful but hard to reconcile. The holographic principle, realized concretely in the AdS/CFT correspondence, offers a route to a reconciliation. However, this correspondence is rigorously established only for special, highly supersymmetric string theory setups, and the holographic duals of most QFTs, including QCD, are unknown. Meanwhile, entanglement is central both to the emergence of spacetime in holography and to non-perturbative QFT. This thesis studies both problems in two largely independent parts. Part I asks whether holography can be derived entirely within QFT itself. The central idea is that the Feynman-diagram expansion can be reorganized into a string-theory amplitude. In diagrams with many interaction vertices, particle worldlines effectively cover a two-dimensional surface, which becomes the worldsheet of an emergent string description. Two constructions are presented. In the first, for diagrams with many vertices, the graph data are lifted to a Riemann surface by treating them as a discretization of continuous functions, using discrete exterior calculus. A worldsheet action is identified whose discretization reproduces each diagram's amplitude. This action describes strings propagating in an AdS space with one more dimension than the QFT, and the emergent holographic direction is identified with the local energy scale around a vertex. In the second, the large-vertex requirement is dropped yielding a much more general and formal approach. Each diagram's amplitude is written as a correlation function in a worldsheet conformal field theory, using the correspondence between graphs and the moduli space of Riemann surfaces. The moduli-space integral is promoted to a path integral over metrics, which introduces the usual ghost system and a dilaton. For massless scalar QFTs with non-derivative interactions, the worldsheet CFT must contain one worldsheet scalar per spacetime direction, a large-N matrix model, ghosts, a dilaton, and auxiliary fields. A fully re-summed string theory could not be obtained as gravitational backreaction was not taken into account, however, the construction indicates that a string theoretic description can be constructed for such theories, though probably without a semiclassical gravity description. This gives a clear microscopic mechanism for the emergence of a holographic description and opens the possibility of constructing holographic duals to much more general QFTs. Part II develops a method for computing mutual information in general conformal field theories, without assuming a holographic dual. Replica twist operators are expanded in terms of integrals of local and multi-local primary operators, with coefficients fixed by requiring that the twist operator act as modular flow. This re-sums the contributions of all replica operators built from a given set of primaries into expressions labelled only by primaries of the original theory. This procedure is much more efficient than existing approaches, re-summing infinitely many terms. Moreover, it reproduces the long-distance expansion of the mutual information and, for a generalized free field in a large number of dimensions, matches the short-distance divergence, thereby interpolating between the two regimes. Together, the two parts show how holography and the entanglement structure of QFT can be studied from within QFT itself.