Nonperturbative Quantum Gravity

TL;DR

This work presents Causal Dynamical Triangulations (CDT) as a nonperturbative, lattice-based approach to quantum gravity that preserves a Lorentzian causal structure while enabling a controlled Wick rotation for statistical analysis. By constructing discrete Lorentzian spacetimes and employing the Regge action, CDT reveals an entropically driven phase diagram with an emergent four-dimensional de Sitter universe in phase C, accompanied by calculable fluctuations that match a minisuperspace-like effective action. The study demonstrates a scale-dependent spectral dimension and consistent Hausdorff behavior, providing constructive evidence for an underlying continuum limit and potential connections to asymptotic safety and Hořava-Lifshitz gravity. These results offer a concrete, nonperturbative framework to test quantum gravity scenarios and illuminate how macroscopic spacetime geometry may emerge from quantum geometric fluctuations, with implications for UV behavior and Planck-scale physics.

Abstract

Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum. "Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Hořava-Lifshitz gravitational models.

Figures (34)

• Figure 1: Top figure: a (2,1)-simplex and a (1,2)-simplex and the way they are glued together to form a "slab" (strip). The ends of the strip should be joined to form a band with topology $S^1 \times [0,1]$. Middle figure: a (3,1)-tetrahedron (fig. (a)) and a (2,2)-tetrahedron (fig. (b))in three dimensions. Bottom figure: a (4,1)-simplex (fig. (a)) and a (3,2)-simplex (fig. (b)) in four dimensions.
• Figure 2: The asymmetry factor $\alpha$, plotted as a function of $\Delta$, for $\kappa_0=2.2$. The horizontal line is ${\tilde{\alpha}}=7/12$, the lowest kinematically allowed value of ${\tilde{\alpha}}$ (see (\ref{['5.3']})), where the (3,2)-simplices degenerate because of a saturation of a triangle inequality.
• Figure 3: Piecewise linear spacetime in (1+1)-dimensional quantum gravity, generated by computer. Proper time runs along the vertical direction.
• Figure 4: Graphical representation of relation (\ref{['2.50']}): differentiating the disc amplitude $W_\Lambda(X)$ (represented by the entire figure) with respect to the cosmological constant $\Lambda$ corresponds to marking a point somewhere inside the disc. This point has a geodesic distance $T$ from the initial loop. Associated with the point one can identify a connected curve of length $L$, all of whose points also have a geodesic distance $T$ to the initial loop. This loop can now be thought of as the curve along which the lower part of the figure (corresponding to the loop-loop propagator $G_\Lambda (X,L;T)$) is glued to the cap, which itself is the disc amplitude $W_\Lambda(L)$.
• Figure 5: In all four graphs, the geodesic distance from the final to the initial loop is given by $T$. Differentiating with respect to $T$ leads to eq. (\ref{['2.55']}). Shaded parts of graphs represent the full, $g_s$-dependent propagator and disc amplitude, and unshaded parts the CDT propagator.