New Directions for Causal Dynamical Triangulations

The purpose of this study is to investigate and propose new algorithms and methods of analysis in the context of the Causal Dynamical Triangulations (CDT) approach to Quantum Gravity. Grounded upon Markov Chain Monte-Carlo methods and physical insight from the Wilsonian Renormalization Group framework, the CDT research program is acquir- ing growing interest because of recent observations strongly supporting the presence of continuous order critical points in the phase diagram of 4D simulations. This could validate the asymptotic safety conjecture advanced by Weinberg in 1976, stat- ing the existence of a non-Gaussian UV fixed point around which one could renor- malize non-perturbatively the Einstein-Hilbert gravity with cosmological constant, then opening the possibility to explore quantum-gravitational effects by lattice reg- ularization. Research in this direction is currently being undertaken by Ambjorn et al.. In the first part of this thesis, an overview of the CDT program is presented, and numerical methods are discussed. The algorithm currently adopted by the CDT community is analyzed in detail and implemented in C++; moreover, a new class of algorithms, generalizing the stan- dard one and named TBlocked, is proposed in order to cure the presence of slow modes in a region of the phase diagram, but gives also the opportunity to parallelize the standard algorithm. Simulation results and a validation of some standard results are presented, and the standard and TBlocked implementations are compared. In the last part of this thesis the current lack of definitions for observables in pure- gauge gravity is discussed; in particular, no observable encoding geometric features in a satisfactory way has been found in the foregoing literature on CDT. Driven by this need, a new and almost complete class of observables characterizing geometric properties of the spatial slices is proposed, based upon the analysis of eigenvalues and eigenvectors of the Laplace-Beltrami matrices associated with the graph dual to the slices. This method, that actually pertains to the realm of spectral graph theory, acts like a Fourier transform generalized on graphs, and gives sense to previously inaccessible concepts in CDT, like a coarse-grained definition of the scalar curvature or of any microscopically defined observable, straightforward to de- fine for regular lattices but not so for random lattices. Main results are obtained and discussed, but this work does not exhaust all the types of analysis that are enabled by this method. It is in the hopes of the author to investigate further applications and generalizations of this method in the future