Summary

4.15 Summary

In the dynamical triangulations approach, one studies the properties of a statistical ensemble of simplicial four-geometries à la Regge of fixed edge lengths. By summing over such discrete configurations according to (21

), one has implicitly assumed that this leads to a uniform sampling of the space of smooth Riemannian manifolds. There is no obvious weak-field limit, but this is no obstacle in principle to the path-integral construction. Numerical simulations indicate the existence of a well-defined phase for sufficiently small $κ2$ (inverse Newton’s constant) and a sufficiently large cosmological constant. For small $κ2$ , one finds a “crumpled” phase, with small average curvature and a large Hausdorff dimension, and for large $κ2$ an elongated, effectively two-dimensional polymer phase. At present, the consensus seems to be that the corresponding phase transition is of first order, with a finite average curvature at the transition point.

Almost all simulations have been done on simplicial manifolds with $S4$ -topology. Neither the inclusion of factors of $(det g)ρ$ in the measure nor the addition of higher-order curvature terms to the action seem to have a substantial influence on the phase structure. Also matter coupling to spinorial and scalar fields does not seem to lead to a change of universality class, although the inclusion of several gauge fields may have a more drastic effect. The study of singular structures (vertices of high coordination number) has led to a qualitative understanding of the phase structure of the model.