Causal Dynamical Triangulations: Phases and Continuum Evidence

Causal dynamical triangulations (CDT) sum piecewise-flat Lorentzian geometries with a preferred discrete proper-time foliation, fixed spatial topology, and causal gluing rules. A transfer-matrix-compatible Wick rotation makes Monte Carlo possible. Its extended phase and four-dimensional finite-size scaling are important continuum evidence, but a critical regulator-removal trajectory and renormalized observables are still required.

Required background. Lines of Constant Physics and Continuum Extrapolation supplies the continuum test. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions supplies integration-domain discipline.

Helpful background. Autocorrelation Times and Effective Sample Size fixes numerical errors. Correlated Evidence, Independence, and Triangulation constrains evidential pooling.

The CDT lattice sum

Four-dimensional CDT glues $(4,1)$ and $(3,2)$ Lorentzian four-simplices between integer-time spatial triangulations. Spacelike edges have squared length $a^2$ and timelike edges $-\alpha a^2$ before Wick rotation. For fixed topology,

$$Z(\kappa_0,\kappa_4,\Delta) =\sum_T\frac{1}{C_T}\, \exp[-S_E(T)],$$

where $C_T$ is the automorphism factor and the Euclidean Regge action can be written

$$S_E(T)=-\kappa_0N_0+\kappa_4N_4+\Delta\,\Xi(T).$$

$N_0$ and $N_4$ count vertices and four-simplices; $\Xi$ is a declared linear combination distinguishing simplex types. $\kappa_0$, $\kappa_4$, and $\Delta$ encode inverse Newton coupling, cosmological tuning, and time–space asymmetry. The Wick rotation is defined simplex by simplex and is not an unrestricted Euclidean topology sum Ambjørn, Jurkiewicz, and Loll 2001.

Application: extended-volume scaling

At approximately fixed total four-volume $N_4$, measure the spatial three-volume $N_3(i)$. In the extended de Sitter-like phase its ensemble mean is well fitted by

$$\langle N_3(i)\rangle =A N_4^{3/4} \cos^3\!\left[ \frac{i-i_0}{s_0N_4^{1/4}} \right]$$

on its support. The time extent scales as $N_4^{1/4}$ and spatial volume as $N_4^{3/4}$, consistent with Hausdorff dimension four Ambjørn, Jurkiewicz, and Loll 2005.

For a phase scan, record $N_0/N_4$, simplex-type ratios, volume-profile covariance, susceptibility peaks, and hysteresis across several $N_4$. Estimate integrated autocorrelation time $\tau_{\mathrm{int}}$ and use $N_{\mathrm{eff}}\simeq N_{\mathrm{cfg}}/(2\tau_{\mathrm{int}})$ in errors. A transition suitable for a continuum limit should show finite-size scaling consistent with diverging correlation length, not only a sharp visual change.

Adversarial finite-size test

Repeat the fit at multiple volumes, boundary conditions, time periods, and volume-fixing strengths. Exclude the stalk consistently and propagate covariance. If $s_0$, effective dimension, or transition order drifts without convergence, the claimed scaling window is not controlled. A first-order line does not normally yield a conventional divergent-correlation-length continuum limit; candidate higher-order boundaries require dedicated scaling.

Scope

CDT supplies a defined regulated Lorentzian configuration space, numerical phase structure, a transfer matrix, and extended semiclassical observables. It has not thereby uniquely recovered continuum four-dimensional quantum gravity, all matter sectors, or a complete observable algebra. Compare its configuration space with Euclidean Dynamical Triangulations and Their Relation to CDT.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Ambjørn, J., J. Jurkiewicz, and R. Loll. “Dynamically Triangulating Lorentzian Quantum Gravity.” Nuclear Physics B 610 (2001): 347–382. DOI.
• Ambjørn, J., J. Jurkiewicz, and R. Loll. “Reconstructing the Universe.” Physical Review D 72 (2005): 064014. DOI.