Analytical map between EPRL spin foam models in loop quantum gravity

Loop Quantum Gravity (LQG) is one of the theoretical frameworks attempting to build a quantum theory of gravitation. Spin Foam theory provides a regularized, background-independent, and Lorentz covariant path integral for the transition amplitudes between LQG kinematical states. The state of the art of the theory is the EPRL model, formulated with the Euclidean and the Lorentzian signatures. They differ by their gauge group structures, which are respectively SO(4,R) and SO(1,3). The first is a compact group: it has finite-dimensional unitary irreducible representations, and the integral on the group manifold is simple. The second is non-compact. Therefore, the computations in the Lorentzian EPRL model are more complicated than in the Euclidean one. The Euclidean model is the preferred choice for physical calculations. Given their similarities it has been so far assumed, as a strong hypothesis, that the results obtained in the Euclidean model also hold for the Lorentzian one. This work's primary goal is to present the main characteristics of the models and a set of prescriptions to map the structure and, at least in a qualitative way, the results obtained in the Euclidean model into the Lorentzian one. Chapter 1 provides an overview of the basic ingredients of the discussion: General Relativity, BF theories and LQG transition amplitudes between quantum states of spacetime. Chapters 2 and 3 are respectively a description of the Euclidean and Lorentzian EPRL models, from the representation theory of their gauge groups to the construction of the EPRL transition amplitudes. Chapter 4 portrays the current state of research in EPRL Spin Foam theory, with a qualitative description of the main results achieved in both models. The main topic of the thesis and my original work is contained in Chapter 5, in which, from a set of prescriptions, the group structure of the Euclidean model is mapped into the Lorentzian one, allowing a comparison between the transition amplitudes.