LAUSR

The main topic of this thesis is a key point in testing the viability of the EPRL/FK spin foam model as a quantum theory of gravity. While it is common knowledge that there are fundamental mathematical inconsistencies between Einstein's General Relativity and Quantum Mechanics, pointing, among other reasons, towards the necessity of such a theory, our current inability to observe the extremely high energies and/or small wavelengths at which quantum effects are expected to appear leaves us with mathematical consistency tests as the only, albeit incomplete, way of separating possibly viable models from incorrect ones. One of the most basic tests available is the study of the model's asymptotics in a semiclassical regime. Indeed, any quantum theory of gravity must be able to reproduce Einstein's model when quantum effects are negligible. With that in mind, we will discuss the asymptotics of spin foam models, in particular the EPRL/FK prescription, and note the non-trivial issues that arise in the course of that study. In order to provide context to the discussion, first we will briefly introduce spin foam models as a state sum formulation of Loop Quantum Gravity, the canonical quantization program of Einstein's theory, giving a short review of the LQG formalism and the issues that led to the construction of spin foams. We will then briefly refer to some historical aspects of this line of study, starting with the original discussion based on BF theory that resulted in the Ponzano-Regge model for 3-dimensional gravity, and proceed to 4-dimensional models and the issues that led to the crafting of the EPRL/FK model. We will then review the calculation of the EPRL vertex amplitude in more detail, before moving on the the topic of asymptotics, the definition of an adequate semiclassical limit to work in, and existing results, with emphasis on the so-called "flatness problem" originally enunciated by Bonzom, as well as a critique of the reasonings that led to it, namely the concept of varying the EPRL action with respect to a discrete variable - the face areas in a given triangulation of spacetime geometry. With the above in mind, and introducing our practical approach to the variation of the face areas, we move on to the main original work presented, a detailed calculation of the zero-order classical equations of motion and their solutions for a concrete triangulation of three 4-simplices, which has been named Delta 3. The goal of said calculation is to assess whether the flatness problem exists or not in a practical example, and ultimately check if the results obtained satisfy what is expected from Einstein gravity. A negative result would, in plain terms, kill the model, or at the very least show it needs modifications, while a positive result, though only a particular case, would be a small step towards the understanding of spin foam asymptotics and possibly hint towards more general properties of the model.