Quantum Gravity from Causal Dynamical Triangulations: A Review

TL;DR

This review synthesizes advances in Causal Dynamical Triangulations (CDT) as a nonperturbative, Lorentzian lattice approach to quantum gravity in four dimensions. It surveys the CDT path integral, its diffeomorphism-invariant geometric encoding, and the crucial role of causal structure and Wick rotation for enabling Monte Carlo analyses. The authors detail the CDT phase diagram, the emergence of semiclassical de Sitter geometry, and the discovery of a new bifurcation phase, alongside developments in an effective transfer matrix and renormalization group strategies. They also introduce new quantum-geometric observables, including spectral and curvature probes, and discuss prospects for cosmology and UV completion, highlighting CDT’s potential as a robust framework for Planck-scale quantum gravity. Overall, CDT is presented as a mature, promising program with concrete results and clear avenues for connecting nonperturbative quantum geometry to observable cosmology and quantum gravity phenomenology.

Abstract

This topical review gives a comprehensive overview and assessment of recent results in Causal Dynamical Triangulations (CDT), a modern formulation of lattice gravity, whose aim is to obtain a theory of quantum gravity nonperturbatively from a scaling limit of the lattice-regularized theory. In this manifestly diffeomorphism-invariant approach one has direct, computational access to a Planckian spacetime regime, which is explored with the help of invariant quantum observables. During the last few years, there have been numerous new and important developments and insights concerning the theory's phase structure, the roles of time, causality, diffeomorphisms and global topology, the application of renormalization group methods and new observables. We will focus on these new results, primarily in four spacetime dimensions, and discuss some of their geometric and physical implications.

Figures (23)

• Figure 1: Regularized spacetimes in CDT quantum gravity are built from two elementary four-simplices, the (4,1)-simplex (left) and the (3,2)-simplex (right). They differ in their assignments of space- and time-like edges, drawn in blue and red respectively, see physrep for details.
• Figure 2: A "sandwich geometry" of topology $I\!\times\! {}^{(2)}\Sigma$ for CDT in $2\! +\! 1$ spacetime dimensions. The sandwich consists of a layer of three-simplices (tetrahedra) extrapolating between two adjacent spatial slices at times $t$ and $t\! +\! 1$ made up of two-dimensional triangles. The space between the lower and upper triangulation is completely filled with tetrahedra, but for simplicity only three are shown here. (Space-like edges depicted in blue, time-like ones in red.)
• Figure 3: The layered structure of a two-dimensional CDT configuration, with space-like edges drawn in blue and time-like ones in red. Left: CDT geometry consisting of a stack of three sandwiches. The spatial geometries at discrete times $t$ are open chains consisting of a variable number $l_t >0$ of space-like edges. Right: the corresponding Euclidean geometry after Wick rotation. The underlying lattice structure is unchanged, but all time-like edges have been replaced by space-like ones.
• Figure 4: Left: the two elementary Minkowskian building blocks of CDT without preferred foliation in two dimensions; in standard CDT only the lower one is present. Dotted lines indicate light rays through the corner points. Centre: gluing these triangles together can result in local causality violations, like too many light cones meeting at a vertex, as shown. Right: at a causally well-behaved vertex, one crosses exactly four light-like directions when going around the vertex.
• Figure 5: CDT phase diagram spanned by the bare couplings $\kappa_0$ and $\Delta$, consisting of the de Sitter phase $C_{dS}$, the bifurcation phase $C_{b}$, and two unphysical phases $A$ and $B$. (Fat dots and squares refer to actual measurements.)