Curvature and scaling in 4D dynamical triangulation

TL;DR

This paper investigates the geometric properties and effective dimensions of 4D quantum gravity models using dynamical triangulation, revealing scale-dependent curvature and a transition to classical spacetime behavior.

Contribution

It introduces a detailed analysis of curvature and dimension scaling in 4D dynamical triangulation models, providing new insights into the emergence of classical spacetime.

Findings

Effective curvature varies with scale and coupling

Global dimension approaches 4 at the transition

Scaling dimension d_s is approximately 4 in both regimes

Abstract

We study the average number of simplices $N'(r)$ at geodesic distance $r$ in the dynamical triangulation model of euclidean quantum gravity in four dimensions. We use $N'(r)$ to explore definitions of curvature and of effective global dimension. An effective curvature $R_V$ goes from negative values for low $\kappa_2$ (the inverse bare Newton constant) to slightly positive values around the transition $\kappa_2^c$. Far above the transition $R_V$ is hard to compute. This $R_V$ depends on the distance scale involved and we therefore investigate a similar explicitly $r$ dependent `running' curvature $R_{\rm eff}(r)$. This increases from values of order $R_V$ at intermediate distances to very high values at short distances. A global dimension $d$ goes from high values in the region with low $\kappa_2$ to $d=2$ at high $\kappa_2$. At the transition $d$ is consistent with 4. We present evidence for scaling of $N'(r)$ and introduce a scaling dimension $d_s$ which turns out to be approximately 4 in both weak and strong coupling regions. We discuss possible implications of the results, the emergence of classical euclidean spacetime and a possible `triviality' of the theory.