4.3 Existence of an exponential 4 Dynamical triangulations4.1 Introduction

4.2 Path integral for dynamical triangulations

Denoting by tex2html_wrap_inline2859 the set of all triangulations of the four-sphere, the partition function for the model is given by

  equation608

where tex2html_wrap_inline2861 and tex2html_wrap_inline2863 denote the numbers of 2- and 4-simplices contained in the simplicial manifold T, and C (T) is the order of the automorphism group of T . One may think of (21Popup Equation) as a grand canonical ensemble, with chemical potential tex2html_wrap_inline2871 . It is related to the canonical ensemble with fixed volume, tex2html_wrap_inline2873, by a Legendre transform

  equation627

The metric information is encoded in the connectivity of the simplicial decomposition, since the individual 4-simplices are assumed equilateral, with the edge length a set to 1.

To understand the simple form of the action S, recall that the curvature term in Regge calculus (c.f.(12Popup Equation) is represented by tex2html_wrap_inline2879, which for fixed edge length is proportional to tex2html_wrap_inline2881 . The constant tex2html_wrap_inline2883 is determined from the condition that a triangulation of flat space should have average vanishing curvature [2Jump To The Next Citation Point In The Article, 11Jump To The Next Citation Point In The Article]. (Because the four-simplices tex2html_wrap_inline2427 are equilateral, zero curvature can only be achieved upon averaging. This explains the absence of a conventional perturbation theory around flat space.) The cosmological term is represented by tex2html_wrap_inline2887 . It is sometimes convenient to re-express tex2html_wrap_inline2861 as a function of tex2html_wrap_inline2891, using tex2html_wrap_inline2893, valid for the tex2html_wrap_inline2837 -topology. The corresponding partition function is tex2html_wrap_inline2897 (where tex2html_wrap_inline2899).



4.3 Existence of an exponential 4 Dynamical triangulations4.1 Introduction

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
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