Critical Behavior of Dynamically Triangulated Quantum Gravity in Four Dimensions

TL;DR

The paper investigates the four-dimensional Euclidean quantum gravity problem using dynamical triangulations to study phase structure and the emergence of a continuum limit. It employs a Grand Canonical Monte Carlo framework with observables $N_4$ and $R$ and tunable parameters $\hat{\lambda}_4$, $\hat{\lambda}_0$ to map gravity and antigravity phases and characterize the transition. The results show an asymmetric transition with a curvature-susceptibility divergence at index $-0.6$ and a geometry transition with $d_h \approx 2.3$ (Gravity), $d_h \approx 4.6$ (Antigravity), and $d_h = 4$ at criticality, indicating a nontrivial continuum limit in 4D. These findings suggest the DT framework can realize continuum four-dimensional quantum gravity and motivate further work on larger systems, matter coupling, and probes of gravitons and running couplings.

Abstract

We performed detailed study of the phase transition region in Four Dimensional Simplicial Quantum Gravity, using the dynamical triangulation approach. The phase transition between the Gravity and Antigravity phases turned out to be asymmetrical, so that we observed the scaling laws only when the Newton constant approached the critical value from perturbative side. The curvature susceptibility diverges with the scaling index $-.6$. The physical (i.e. measured with heavy particle propagation) Hausdorff dimension of the manifolds, which is 2.3 in the Gravity phase and 4.6 in the Antigravity phase, turned out to be 4 at the critical point, within the measurement accuracy. These facts indicate the existence of the continuum limit in Four Dimensional Euclidean Quantum Gravity.

Figures (6)

• Figure 1: $<R(\lambda_0)>$ for $N_4=1024-12000$.
• Figure 2: $<\lambda(\lambda_0)>$ for $N_4=1024-12000$.
• Figure 3: Direct measurements of susceptibility $\chi$. Fit of $\lambda_0$ with the polynomial of $<R>$ and susceptibility curve obtained from this fit.
• Figure 4: Finite size scaling analysis for $\chi^c$ and $\lambda_0^c$.
• Figure 5: Fit of the critical exponent $\gamma$: $\log|R^c-R| \approx |\lambda_0^c-\lambda_0|*(1-\gamma)$. The indexes on the different sides of the phase transition are clearly different.