Discrete Approaches Towards the Definition of a Quantum Theory of Gravity
Abstract
We study the elongated phase of 4D Dynamical Triangulations. In the case of the sphere topology by using the Walkup's theorem we show that the dominating configurations are stacked spheres. These stacked spheres can be mapped into treelike graphs (branched polymers). By using BabyUniverses arguments and an antsatz on the universality class between the stacked spheres and a model coming from the theory of random surfaces we argument that this elongated phase is a trivial phase. 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 a first order version of Regge calculus formulated as a local theory of the Poincare` group. This first order formalism has the effects of smoothing out some pathological configurations, like "spikes", which prevent the theory from having a smooth continuum limit. These confingurations are in fact in the region of large deficit angles where the first order formalism and the secon order formalism are not equivalent on lattice. We derive the first order field equations in the approximation of "small deficit angles" and prove that (second order) Regge calculus is a solution. Successively we derive the general first order field equations by taking into account the constraints of the theory. An invariant measure for the pathintegral of this theory is defined. The coupling with matter, in particular fermions, is also discussed in analogy to the continuum theory.
 Publication:

arXiv eprints
 Pub Date:
 December 1998
 DOI:
 10.48550/arXiv.grqc/9812080
 arXiv:
 arXiv:grqc/9812080
 Bibcode:
 1998gr.qc....12080G
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 81 pages, 8 figures