SPIN FOAM MODELS OF N-DIMENSIONAL QUANTUM GRAVITY, AND NON- ARCHIMEDAN AND NON-COMMUTATIVE FORMULATIONS

Abstract. This paper is twofold. First of all a complete unified picture of n-dimensional quantum gravity is proposed in the following sense: In spin foam models of quantum gravity the evaluation of spin networks play a very important role. These evaluations correspond to amplitudes which contribute in a state sum model of quantum gravity. In [6], the evaluation of spin networks as integrals over internal spaces was described. This evaluation was restricted to evaluations of spin networks in n-dimensional Euclidean quantum gravity. Here we propose that a similar method can be considered to include Lorentzian quantum gravity. We therefore describe the the evaluation of spin networks in the Lorentzian framework of spin foam models. We also include a limit of the Euclidean and Lorentzian spin foam models which we call Newtonian. This Newtonian limit was also considered in [10]. Secondly, we propose an alternative formulation of spin foam models of quantum gravity with its corresponding evaluation of spin networks. This alternative formulation is a non-archimedean or p-adic spin foam model. The interest on this description is that it is based on a discrete space-time, which is the expected situation we might have at the Planck length; this description might lead us to an alternative regularisation of quantum gravity. Moreover a non-commutative formulation follows from the non-archimedean one. 1 1