ETD system

Electronic theses and dissertations repository


Tesi etd-08312017-101600

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