Quantum Gravity, or The Art of Building Spacetime

TL;DR

The paper presents causal dynamical triangulations (CDT) as a background-independent, Lorentzian path integral for quantum gravity implemented via a foliated set of causal, simplicial spacetimes. Using Monte Carlo simulations, it demonstrates that a macroscopic four-dimensional universe with a dynamical scale factor emerges, and its large-scale dynamics are captured by an effective action derived from integrating out all other degrees of freedom, yielding a semiclassical bounce akin to tunneling from nothing. The results reveal a scale-dependent spectral dimension, with $D_S(\sigma)$ approaching $4$ at large diffusion times and $\approx 2$ at short times, and show that the observed effective dynamics align with a minisuperspace-like model once quantum fluctuations at small scales are incorporated. These findings support CDT as a viable nonperturbative approach to quantum gravity with a controllable continuum limit and motivate further exploration of renormalization, matter coupling, and transverse gravitational degrees of freedom.

Abstract

The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence accumulated to date that a macroscopic four-dimensional world can emerge from this theory dynamically. Using computer simulations we observe in the Euclidean sector a universe whose scale factor exhibits the same dynamics as that of the simplest mini-superspace models in quantum cosmology, with the distinction that in the case of causal dynamical triangulations the effective action for the scale factor is not put in by hand but obtained by integrating out {\it in the quantum theory} the full set of dynamical degrees of freedom except for the scale factor itself.

Figures (6)

• Figure 1: The two fundamental building blocks of causal dynamically triangulated gravity. The flat four-simplex of type (4,1) on the left has four of its vertices at time $t$ and one at time $t\!+\! 1$, and analogously for the (3,2)-simplex on the right. The "gap" between two consecutive spatial slices of constant integer time is filled by copies of these simplicial building blocks and their time-reversed counterparts, the (1,4)- and the (2,3)-simplices.
• Figure 2: Snapshot of a "typical universe" consisting of approximately 91000 four-simplices as it appears in the Monte Carlo simulations at a given "computer time". We plot the three-volume at each integer step in proper time, for a total time extent of $T=40$, in units where $a_s=1$.
• Figure 3: The scaling of the volume-volume correlator, as function of the rescaled time variable $x\!=\! \delta/( N_4)^{1/4}$. Data points come from system sizes $N_4\!=\!$ 22500, 45000, 91000, 181000 and 362000 at $\kappa_0\!=\! 2.2$, $\Delta\!=\! 0.6$ and $T\!=\! 80$.
• Figure 4: The spectral dimension $D_S$ of the universe as function of the diffusion time $\sigma$, measured for $\kappa_0\!=\! 2.2$, $\Delta\!=\! 0.6$ and $t\!=\! 80$, and a spacetime volume $N_4\!=\! 181$k. The averaged measurements lie along the central curve, together with a superimposed best fit $D_S(\sigma) = 4.02\!-\! 119/(54\!+\!\sigma)$ (thin black curve). The two outer curves represent error bars.
• Figure 5: Distribution $P_{V_3}(z)$ of volume differences of adjacent spatial slices, for three-volumes $V_3=$ 10.000, 20.000, 40.000 and 80.000 tetrahedra.