  The spinfoam framework is a proposal for a regularized path integral for
quantum gravity. Spinfoams define quantum space-time structures describing the
evolution in time of the spin network states for quantum geometry derived from
Loop Quantum Gravity (LQG). The construction of this covariant approach is
based on the formulation of General Relativity as a topological theory plus the
so-called simplicity constraints which introduce local degrees of freedom. The
simplicity constraints are essential in turning the non-physical topological
theory into 4d gravity. In this PhD manuscript, an original way to impose the
simplicity constraints in 4d Euclidean gravity using harmonic oscillators is
proposed and new coherent states, solutions of the constraints, are given.
Moreover, a consistent spinfoam model for quantum gravity has to be connected
to LQG and must have the right semi-classical limit. An explicit map between
the spin network states of LQG and the boundary states of spinfoam models is
given connecting the canonical and the covariant approaches. Finally, new
techniques to compute semiclassical asymptotic expressions for the transition
amplitudes of 3d quantum gravity and to extract semi-classical information from
a spinfoam model are introduced. Explicit computations based on approximation
methods and on the use of recurrence relations on spinfoam amplitudes have been
performed. The results are relevant to derive quantum corrections to the
dynamics of the gravitational field.
