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Géométrie quantique dans les mousses de spins : de la théorie topologique BF vers la relativité générale

Loop quantum gravity has provided us with a canonical framework especially devised for background independent and diffeomorphism invariant gauge field theories. In this quantization the fundamental excitations are called spin network states, and in the context of general relativity, they give a meaning to quantum geometry. Spin foams are a sort of path integral for spin network states, supposed to enable the computations of transition amplitudes between these states. The spin foam quantization has proved very efficient for topological field theories, like 2d Yang-Mills, 3d gravity or BF theories. Different models have also been proposed for 4-dimensional quantum gravity. In this PhD manuscript, I discuss several methods to study spin foam models. In particular, I present some recurrence relations on spin foam amplitudes, which generically encode classical symmetries at the quantum level, and are likely to help fill the gap with the Hamiltonian constraints. These relations can be naturally interpreted in terms of elementary deformations of discrete geometric structures, like simplicial geometries. Another interesting method consists in exploring the way spin foam models can be written as path integrals for systems of geometries on a lattice, taking inspiration from topological models and Regge calculus. This leads to a very geometric view on spin foams, and gives classical action principles which are studied in details.