Quantum gravity from simplices: analytical investigations of causal dynamical triangulations

Abstract

A potentially powerful approach to quantum gravity has been developed over the last few years under the name of Causal Dynamical Triangulations.

Although these models can be solved exactly in a variety of ways in the case of pure gravity in (1+1) dimensions,it is difficult to extend any of the analytical treatments to the case of higher dimensions and/or to that of matter coupling.

The aim of this thesis is to try to take a small step beyond this impasse.

After reviewing in Chapter 1 the methods and the results of causal dynamical triangulations, we proceed in the next two chapters to illustrate in detail two original pieces of work which both represent a small step away from the special case of pure gravity in two dimensions.

In Chapter 2 we tackle the problem of coupling matter to causal dynamical triangulations in two dimensions. To fix a starting point we choose one of the simplest, and certainly the most notorious, matter fields on a lattice, the Ising model.We show how an old method of investigation for spin systems on a lattice, the high- and low-temperature expansion, can be applied on the dynamical lattice represented by the triangulation. The application is not trivial as it requires an ensemble average over the lattices for the graphs appearing in the expansion, but we will develop a scheme that allows the computation of such averages.

The method is quite general and can in principle be used for other kinds of matter models.

The work presented in Chapter 3 goes into the realm of higher dimensions.

We introduce a particular model of three-dimensional causal dynamical triangulations and show that it can be solved in an asymptotic limit and a continuum quantum Hamiltonian is found.

This is an important step in the understanding of these kinds of models, because this is the first case in which, for a dimension larger than two, a continuum limit has been obtained analytically, although many things remain to be understood and some approximations still removed.

Together, the works presented in Chapter 2 and 3 constitute an advance in the analytical understanding of causal dynamical triangulations and have the appealing feature of indicating directions for further development.