Curvature and scaling in 4D dynamical triangulation

Abstract

We study the average number of simplices $N^{\prime }(r)$ at geodesic distance $r$ in the dynamical triangulation model of euclidean quantum gravity in four dimensions. We use $N^{\prime }(r)$ 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 $R_{\mathrm{eff}}(r)$. 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 $N^{\prime }(r)$ 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.

27 pages, LaTeX with postscript figures using EPSF Major revision, version as accepted by Nuclear Physics B

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