Dynamical Triangulations in Quantum Gravity

• Dynamical Triangulations is a discrete regularization of quantum gravity that replaces the continuum path integral with a sum over piecewise-linear, simplicial manifolds.
• It enables the use of numerical Monte Carlo simulations to investigate nonperturbative phase structures and the emergence of semiclassical de Sitter geometry in Causal Dynamical Triangulations.
• The approach reveals scale-dependent spectral dimensions, illustrating a transition from classical four-dimensional behavior at large scales to reduced dimensions near the Planck scale.

Dynamical Triangulations (DT) is a lattice regularization scheme for quantum gravity—most prominently realized in its Lorentzian form as Causal Dynamical Triangulations (CDT)—that replaces the formal continuum path integral over metrics with a sum over equivalence classes of discrete, piecewise-linear manifolds constructed from simplicial building blocks. This approach enables the rigorous and nonperturbative exploration of strongly fluctuating gravitational sectors, especially in spacetime dimensions $d\geq 2$, and provides a framework to investigate the emergence of classical geometry, phase structure, and nontrivial Planck-scale phenomena.

1. Discretization of the Gravitational Path Integral

Dynamical Triangulations replaces the functional integral over $(M, g_{\mu\nu})$ with a sum over equivalence classes $[T]$ of triangulated manifolds, each specified by combinatorial data (simplicial adjacencies and labeling) and, in some variants, additional discrete geometric information (e.g., edge lengths). The path integral is regularized as

$$Z = \sum_{T \in \mathcal{G}_{a,N}} \frac{1}{C_T} \exp\big(i S^\mathrm{Regge}[T]\big)$$

where $C_T$ is the symmetry factor of triangulation $T$, $S^\mathrm{Regge}[T]$ is the Regge discretization of the Einstein–Hilbert action, and the sum runs over triangulated manifolds of fixed volume (number $N$ of simplices) built out of building blocks of edge length $a$ (Ambjorn et al., 2011). As $a \to 0$ and $N \to \infty$ with total volume $V = a^d N$ fixed, one seeks to recover the continuum physics.

This discretization implements several critical features:

• The continuum functional measure over metrics $g_{\mu\nu}(x)$ is replaced by a countable space of piecewise-linear metrics parametrized by simplicial data, drastically reducing the degrees of freedom.
• The approach allows use of numerical Monte Carlo techniques to explore the strongly nonperturbative regime, for both Euclidean and Lorentzian signatures.
• Topology of the manifold can be fixed or, in generalized settings, controlled (see generalized CDT).

2. Euclidean versus Lorentzian (Causal) Dynamical Triangulations

Euclidean DT

In the original (Euclidean) DT formulation, the sum includes all geometries with positive-definite metric signature. While powerful in low dimensions—successfully reproducing features of Liouville quantum gravity and string theory—the unrestricted path integral exhibits pathological behavior in higher dimensions, with universes dominated either by crumpled (highly connected) or branched polymer–like configurations and no robust emergence of extended semiclassical geometry (Ambjorn et al., 2011).

Causal Dynamical Triangulations (CDT)

CDT introduces a causality condition by restricting the class of triangulations to those that admit a global foliation into slices of constant (proper) time, each of fixed spatial topology, and only allows gluing rules that preserve this foliation. Each triangulation is constructed from $(d+1)$-simplices with assigned spacelike and timelike edges, ensuring the Lorentzian signature is well-defined locally before possible Wick rotation to the Euclidean sector for simulations.

Key features:

• Absence of spatial topology change and no branching in “time”; causality is strictly preserved.
• The partition function is regularized as a sum over causal triangulations:

$$Z^\mathrm{CDT}_{a, N} = \sum_{T \in \mathcal{G}^\mathrm{causal}_{a, N}} \frac{1}{C_T} \exp\big(i S^\mathrm{Regge}[T]\big)$$

• Exhibits radically different nonperturbative behavior compared to the Euclidean case, with nontrivial quantum geometry at the Planck scale but clear recovery of semiclassical features at large scales.

3. Phase Structure and Continuum Limit

In four-dimensional CDT, numerical studies have identified three (and more recently a bifurcation subphase) distinct phases in the $(K_0, \Delta)$ coupling space:

• Phase A (branched polymer): Dominated by tree-like, highly disconnected spacetimes.
• Phase B (crumpled): Low-dimensional, highly connected geometries.
• Phase C (de Sitter or extended): Emergence of large-scale semiclassical geometry with four-dimensional volume profile $V_3(\tau) \propto \cos^3(\tau/b)$, consistent with a Euclidean de Sitter universe.

Transitions between these phases are sharp. In particular, phase C exhibits a first-order or second-order transition line, depending on the sector and precise choice of parameters (Ambjorn et al., 2011, Ambjorn et al., 2019). The order of these transitions and their properties are critical to establishing a continuum quantum gravity limit: a second-order transition may allow a divergent correlation length and a well-defined renormalization group flow.

4. Observables and Emergence of Classical Geometry

CDT realizes the dynamical emergence of semiclassical geometry directly from summation over quantum spacetimes. In phase C, the measured three-volume profile $V_3(\tau)$ matches (after rescaling) the prediction for a four-dimensional (Euclidean) de Sitter space: $V_3(\tau) \propto \cos^3(\tau/b)$ Numerical Monte Carlo simulations demonstrate nearly perfect agreement of this profile with expectations from minisuperspace cosmology (Ambjorn et al., 2011, Görlich, 2013). Higher-point correlation functions and spectral quantities extracted from the CDT ensemble also exhibit semiclassical behavior at large scales.

5. Spectral Dimension and Dynamical Dimensional Reduction

A striking quantum effect observed in DT/CDT studies is the scale-dependent spectral dimension $D_S(\sigma)$, determined via a fictitious diffusion process on the triangulated geometries. The return probability exhibits a $\sigma$-dependent scaling: $$\mathcal{R}_V(\sigma) \propto \frac{1}{\sigma^{D_S/2}}$$ For four-dimensional CDT,

• $D_S \approx 4$ at large diffusion times (infrared),
• $D_S \approx 2$ at very short scales (ultraviolet).

This "dynamical dimensional reduction" near the Planck scale highlights quantum-geometry effects absent in the classical continuum, with potential phenomenological consequences (e.g., modified dispersion relations) and links to universal behavior seen in other approaches such as Hořava–Lifshitz gravity and asymptotic safety scenarios (Ambjorn et al., 2011, Mielczarek, 2015).

6. Mathematical Foundation and Computational Techniques

DT leverages the Regge calculus discretization: $$S^\mathrm{Regge}[T] = \frac{1}{16\pi G} \sum_{hinges} V_{d-2} (\mathcal{R} - 2\Lambda)$$ The partition function's regularization translates the continuum path integral,

$$Z(G, \Lambda) = \int \frac{D[g_{\mu\nu}(x)]}{\mathrm{Diff}(M)} e^{i S^{EH}[g_{\mu\nu}]}$$

into a well-defined, numerically tractable sum. Statistical mechanics techniques—Monte Carlo sampling, finite-size scaling, and histogram analysis—are standard (Ambjorn et al., 2011, Ambjorn et al., 2019). Careful control of the continuum limit ($a \to 0, N \to \infty, V_d = a^d N$ fixed) is essential.

In the Lorentzian (CDT) context, the absence of Euclidean “spiking” pathologies (configurations dominated by isolated long simplices with infinitesimal neighbors) greatly facilitates both analytic calculations and numerical simulations (Tate et al., 2011).

7. Significance and Outlook

Dynamical Triangulations, and particularly CDT, has established itself as one of the leading nonperturbative, background-independent approaches to quantum gravity. Its successes in generating macroscopic four-dimensional semiclassical geometry, exhibiting a nontrivial quantum signature at small scales, and providing a controlled regularization of the gravitational path integral are unrivaled in other discrete or continuum quantization programs. Active research priorities include:

• Systematic mapping of phase diagrams and critical behavior.
• Inclusion of matter couplings and their influence on geometry.
• Refinement of observables (e.g., spectral functions, curvature correlators).
• Investigation of higher-order (e.g., $R^2$) corrections.
• Extension to scenarios with spatial topology change.

Dynamical Triangulations continues to provide essential physical and mathematical insights into the microstructure and emergence of spacetime (Ambjorn et al., 2011, Ambjorn et al., 2019, Duin et al., 7 Oct 2025).