Gravità quantistica su reticolo bidimensionale

An unsolved problem in modern physics is that we have quantum field theory and general relativity but we don't know whether we may combine them in a consistent way, however there are several reasons for which a consistent theory must exist. Perturbative quantum gravity is not renormalizable, however even if we are not able to face this problem in this way it is possible to get a better understanding of the problem non-perturbatively by studying the cut-off theory through lattice formulation. The main idea behind this is that if a non-Gaussian fixed point is found the theory is asymptotically safe and an UV completion is possible. The discretized version of the theory can be achieved using two main methods: One is the Regge calculus, where the lattice itself is dynamical, exploiting the geometrical aspect of gravity. The other uses the Vielbein formalism and exploits the analogy with others gauge theories. Classic two dimensional gravity is trivial because the action is a topological invariant, however in the quantum case large quantum fluctuations can change the genus of the surface making the theory highly non-trivial. So reasonably, the bidimensional case may give useful information about the quantum problem in higher dimensions. We provide a numerical analysis of the gauge formulation of bidimensional quantum gravity. Through a Monte Carlo simulation we analyse both pure theory and coupled theory. In this thesis we see if the latter approach permits, in a consistent way, to go beyond the approach used in previous works in which the quantum version of the theory has been achieved summing over random triangulation of the surface. We show that the model admits a phase transition and its properties are studied. Furthermore we explore the topological structures and curvature fluctuations. Future works involve using these results as an aid to formulate a consistent quantum theory in h