We study Quantum Field Theory (QFT) on a background de Sitter spacetime dS d+1 . Our main tool is the Hilbert space decomposition in irreducible unitarity representations of its isometry group SO(d + 1, 1). Throughout this work, we focus on the late-time physics of dS d+1 , in particular on the boundary operators that appear in the late-time expansion of bulk local operators. As a first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of bulk two-point functions. In the process, we exhibit a relation between poles in the corresponding spectral densities and boundary CFT data. Next, we study the conformal partial wave decomposition of four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity properties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS2.

We study quantum field theory on a de Sitter spacetime dSd+1 background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group SO(d + 1, 1). As the first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity proper- ties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS2.

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, ϕ4 theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.

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