Quantum Field Theory in Anti-de Sitter and de Sitter Space.

This PhD thesis is about Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) and de Sitter (dS) space. In recent years there has been a lot of progress in understanding Conformal Field Theories (CFTs), through the conformal bootstrap program. However many quantum field theory of interest do not have conformal symmetry, and we refer to them as massive. One motivation to study quantum field in AdS space is that it builds a bridge between conformal field theories and massive quantum field theories. This is because any quantum field theory on the AdS geometry admits a set of asymptotic observables, akin to the S-matrix in flat space, that are defined as correlation functions of local operators at the boundary of space, and are symmetric under the conformal group, meaning that they can be studied using CFT techniques. Other motivations to study QFT in AdS are the nice geometric properties of this background such as the curvature that acts as infrared (IR) regulator, and at the same time the presence of an infinite volume which allows for phase transitions and symmetry breaking. Moreover there is an analytical continuation of the observables in AdS to the dS geometry that describes a maximally symmetric universe with accelerated expansion, a property in common with our Universe and also with the inflationary phase in early cosmology. As a result, studying QFT in AdS also leads to new techniques for QFT in dS, potentially relevant for applications to cosmology. In this thesis we study gauge theories, and in particular Quantum Electrodynamics (QED) with bosonic or fermionic matter in spacetime dimensions d less or equal to 4, both in AdS and in dS. In AdS we are using combination of Large N and bootstrap methods to go beyond standard perturbation theory. In dS we set up the basics of how to define and compute gauge-invariant late-time correlation functions in scalar QED. In chapter 1, we provide the results present in the literatue in context of bootstrap and large N and discuss the motivation to study these techniques in AdS and dS. In Chapter 2 we review some background material that is used in the rest of the thesis. We introduce the embedding formalism for both the AdS and dS space. We review the spectral representation which allows to map two-point correlators from coordinate space to functions of a spectral parameter , much like Fourier transformation in flat space. We also show how to analytically rotate the Lagrangian from dS to AdS for a scalar field theory.In Chapter 3 we review the O(N) model, first in flat space. We then review the computation in the AdS case, where one uses both the bootstrap and large N techniques. We review both the phases found in this theory. We also present results in the literature for the O(N)model in dS space.In Chapter 4 we discuss scalar QED at large N in flat space, studying both phases: the Coulomb phase and the Higgs phase. We also consider the CFT which separates the two said phase. In Chapter 5 we present the results obtained by applying both the large N and analytical bootstrap methods to scalar QED in AdS. We studied the realization of the different phases in AdS, and we also identify AdS analogue of resonance in flat space in the Higgs phase. We also discuss the case with bulk conformal symmetry, and the issue of IR divergences.In Chapter 6 we discuss the relation between vector propagators in AdS and dS space and then we present the rotation of the Lagrangian of scalar QED from dS to AdS. We discuss the strategy to compute the bubble in dS from this rotated lagrangian.In Chapter 7 we discuss fermionic QED in flat space, pointing out the existence of a bound state in the scattering amplitude of the fermions in the massive phase. We then outline a strategy to study this theory in AdS. In Chapter 8 we summarize the thesis and presented the possible future directions.In appendices we have discussed the flat space limit and other important results in spectral representation.