4.2 Path integral for dynamical 4 Dynamical triangulations4 Dynamical triangulations

4.1 Introduction

This quantization approach has received a lot of attention since the early nineties [2Jump To The Next Citation Point In The Article, 1Jump To The Next Citation Point In The Article, 11Jump To The Next Citation Point In The Article], 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, 8Jump To The Next Citation Point In The Article, 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, c.f. (21Popup Equation).

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 tex2html_wrap_inline2837).



4.2 Path integral for dynamical 4 Dynamical triangulations4 Dynamical triangulations

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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