The original contributions of this thesis are contained in chapter three and four. In chapter three we discuss the elongated phase of 4D simplicial quantum gravity by exploiting recent analytical results. In particular using Walkup's theorem we prove that the dominating configurations in the elongated phase are tree-like struc tures called "Staked Spheres" . Such configurations can be mapped into branched polymers and baby universes argument are used in order to analyse the critical behaviour of the theory in the weak coupling regime. We prove that the string susceptibility exponent yst• is smaller than 1 for the statistical ensemble of stacked spheres. An argument from the theory of random surfaces shows a strong evidence that stacked spheres correspond to a mean field phase ystr = so that any attempt of performing a continuum limit in this phase, even if we assume the existence of the limiting Schwinger functions, will give a trivial theory. The numerical evidence for a first order phase transition and the triviality of the elongated phase suggest that a new approach to simplicial quantum gravity might be useful. Along this line following the work of various authors we study, in chapter four, a first order version of Regge calculus formulated as a local theory of the Poincare group. We recall and improve the definition of this Wilson-like action for Regge calculus. We prove that it is invariant under the plaquette orientation. Following the approach of a previous work a first order principle, in the sense of Palatini, is defined on lattice in the same spirit of the continuum theory of General Relativity in the Cartan formalism. We derive the first order field equations in the approximation of small deficit angles and we prove that (second order) Regge calculus is a solution. This is the main new result of this chapter. Successively we derive the general first order field equations by taking carefully into account the constraints of the theory. An invariant measure for the path integral of this theory is defined. The coupling with matter, in particular fermions, is also discussed in analogy to the continuum theory.
|Titolo:||Discrete Approaches Towards the Definition of a Quantum Theory of Gravity|
|Data di pubblicazione:||10-dic-1998|
|Appare nelle tipologie:||8.1 PhD thesis|