Curvature and scaling in 4D dynamical triangulation

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 $κ_2$ (the inverse bare Newton constant) to slightly positive values around the transition $κ_2^c$. 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 $κ_2$ to $d=2$ at high $κ_2$. 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.