Introduction

4.1 Introduction

This quantization approach has received a lot of attention since the early nineties [2

, 1

, 11

], inspired by analogous studies in two-dimensional gravity, where dynamical triangulation methods have been a valuable tool in complementing analytical results (see, for example, [85 , 5 ]). I will here exclusively concentrate on the 4d results. Other overview material is contained in [6 , 136 , 195 , 4 , 3 , 17 , 59 , 73 , 144 , 135 , 8

, 9 , 54 ].

Dynamical triangulations are a variant of quantum Regge calculus, where the dynamical variables are not the edge lengths of a given simplicial complex, but its connectivity. A precursor is Weingarten’s [193 ] prescription for computing transition amplitudes between three-geometries, by summing over all interpolating four-geometries, built from equilateral 4d hypercubes living on an imbedding $p$ -dimensional hypercubic lattice with lattice spacing $a$ . Evaluating the Einstein action on such a configuration amounts to a simple counting of hypercubes of dimension 2 and 4, cf. (21 ).

To avoid a potential overcounting in the usual Regge calculus, Römer and Zähringer [184 ] proposed a gauge-fixing procedure for Regge geometries. They argued for an essentially unique association of Riemannian manifolds and equilateral triangulations that in a certain sense are best approximations to the continuum manifolds. The resulting “rigid Regge calculus” is essentially the same structure that nowadays goes by the name of “dynamical triangulations”. In this ansatz one studies the statistical mechanical ensemble of triangulated four-manifolds with fixed edge lengths, weighted by the Euclideanized Regge action, with a cosmological constant term, and optionally higher-derivative contributions. Each configuration represents a discrete geometry, i.e., the discrete version of a Riemannian four-metric modulo diffeomorphisms. At least for fixed total volume, the state sum converges for appropriate values of the bare coupling constants, if one restricts the topology (usually to that of a sphere $S4$ ).