Asymptotic analysis and numerical evaluations of vertex amplitudes in spinfoam models

The program of Loop Quantum Gravity is to find a background independent quantum theory of general relativity. The strategy is to follow the program of Dirac and Bergmann for the quantization of classical theories with constraints. In this perspective, in Loop Quantum Gravity the fundamental quanta are not gravitons, but quanta of space itself, whose geometry turns out to be non-commuting. They show discrete spectra, with minimal excitations proportional to the Planck scale. Our attention has been focused on the spinfoam approach, which provides a background independent path integral formulation for quantum gravity. The path integral provides the matrix elements of the projector on the kernel of the quantized Hamiltonian constraint and it is based on an action principle for general relativity due to Plebanski, which relates Einstein's theory to a topological theory with internal gauge group SL(2,C) plus constraints. Amplitudes are expressed as integrals of group holonomies, and the constraints are implemented as a selection of the representation labels over which to sum. The most interesting proposal up to now is the EPRL model. Works by Barrett and collaborators have shown that, performing a stationary phase approximation, the EPRL model has a clear semiclassical connection with a discretization of general relativity, Regge calculus, at least for the vertex amplitude. In this thesis we have revisited these results with a two-fold goal: on the one hand, to study the feasibility of a numerical evaluation of the vertex amplitude; on the other hand, to provide the first confirmation of the saddle point approximation, and test its validity. This has led us to develop a Mathematica/C code for the numerical part and to obtain original results concerning the evaluation of SU(2) and SL(2,C) invariants, expanding the existing analytical techniques. First we studied the SU(2) BF vertex, which is a toy model for the EPRL vertex and shows a similar asymptotic behavior. We have extended known results giving explicitly the stationary points of the action and computing its Hessian for general boundary configurations. To perform the numerical evaluation, we solved the integrals in terms of Clebsch-Gordan coefficients. We can compute numerically the complete amplitude and we find that the asymptotic is reached quite fast for two different configurations that we have considered. We have also computed the Hessian for the EPRL model, and considered the case of Euclidean and Lorentzian boundary data. The expression of the integrals as a sum of Clebsch-Gordan coefficients is a convergent series, due to the infinite dimension of Lorentz unitary irreducible representations. The factorization of the vertex amplitude shows that it can be written with a core given by the SU(2) amplitude and SL(2,C) invariant tensors at the boundary. We have done a stationary phase approximation for these tensors, finding conditions on the existence of critical points that select the terms in the series that are not exponentially suppressed. We have computed the truncated amplitudes and analyzed the convergence of the series at low spins. The asymptotic behavior is much more evident in the Euclidean case than the Lorentzian one, mainly because in the Lorentzian case an exponentially suppressed term still dominates at low spins