Causal Dynamical Triangulation

A nonperturbative lattice-gravity construction that sums Regge-weighted, causally admissible piecewise-flat spacetime histories and searches their phase structure for a continuum quantum geometry.

Version

v2 · 2026-09-06 · History

Domain-specific #

1443

Origin domain

quantum gravity

Subdomain

lattice approaches to quantum gravity

Aliases

Causal dynamical triangulations, CDT, Lorentzian dynamical triangulations

Core Idea

Causal dynamical triangulation (CDT) is a nonperturbative construction for a gravitational path integral in which continuous Lorentzian spacetimes are regulated by piecewise-flat geometries built from simplices, only causally admissible gluings enter the configuration space, and the geometry itself is summed over. The construction is background independent in the operational sense that it does not place quantum fields on one fixed spacetime lattice: different triangulations represent different discrete spacetime geometries, and the partition function weights an ensemble of them.

The phrase names more than a research program or theory label. It names a stable chain of roles. One fixes dimension, topology or boundary data, simplex types, edge-length assignments, and causal assembly rules. These choices determine a class \(\mathcal C\) of Lorentzian triangulations. Each triangulation \(T\) carries a Regge-discretized gravitational action \(S_R[T]\) and a symmetry factor \(C_T\). Schematically, the regulated Lorentzian sum is

\[ Z_a=\sum_{T\in\mathcal C}\frac{1}{C_T}\,e^{iS_R[T]}, \]

where \(a\) denotes the lattice scale. The causal structure of standard CDT permits a triangulation-by-triangulation analytic continuation to a real statistical weight, giving the Euclideanized form

\[ Z_a^{E}=\sum_{T\in\mathcal C}\frac{1}{C_T}\,e^{-S_R^{E}[T]}. \]

This is not the replacement of Lorentzian physics by an independently postulated Euclidean ensemble. The admissible histories and their continuation are defined from the Lorentzian construction. The Euclideanized weights make Monte Carlo investigation possible in cases where the sum is not solved analytically.

The standard construction organizes spacetime into discrete proper-time slices, fixes the spatial topology during evolution, and uses spacelike and timelike edges with prescribed squared lengths. Four-dimensional simulations commonly use two types of four-simplex, distinguished by how their vertices lie in adjacent slices^[1]. Generalized locally causal models show that a preferred global foliation is not the deepest invariant: causal consistency can be retained without distinguished slicing^[2]. The durable identity is therefore a causally restricted sum over Lorentzian simplicial histories with a controlled statistical continuation, not the assertion that every possible CDT formulation must use one particular foliation.

CDT is a regulator-and-continuum-limit strategy. Its simplices are not automatically claimed to be fundamental spacetime atoms. The physical objective is to identify phases and critical behavior from which lattice spacing can be removed while suitable renormalized observables remain finite. Numerical observations such as an extended four-dimensional phase, a de Sitter-like average volume profile, or scale-dependent spectral dimension are evidence produced inside studied ensembles; they are not definitional guarantees and are not presented here as proof that CDT is the correct quantum theory of gravity.

The candidate survives at 0.99 confidence as a domain-specific abstraction. Removing the causal admissibility rule, the geometry sum, the Regge weight, or the continuum-limit program changes the construction rather than merely changing its implementation. Yet its vocabulary and diagnostics remain specific to quantum gravity and simplicial geometry, so it does not qualify as a prime.

Structural Signature

The recurring construction is:

continuum gravitational path-integral problem → Lorentzian simplicial regulator → causally admissible configuration class → Regge action and symmetry weight → controlled Wick rotation → analytic enumeration or ensemble sampling → invariant geometric observables → phase and finite-size analysis → candidate continuum limit

Sig role-phrases:

• Gravitational histories as the variable. The objects being summed are inequivalent spacetime geometries, not field configurations on one fixed geometry.
• Piecewise-flat simplicial regulator. Curved spacetime is represented by gluing flat \(d\)-simplices; curvature is concentrated on codimension-two hinges in the Regge description.
• Lorentzian causal admissibility. Only triangulations satisfying the construction's temporal-orientation and causal-consistency rules enter \(\mathcal C\). Standard CDT realizes this with neighboring spatial slices and fixed spatial topology.
• Fixed local building data. Spacelike and timelike edge lengths, allowed simplex types, dimension, boundary conditions, and topology specify a finite regulated ensemble at a chosen volume.
• Combinatorial dynamism. Connectivity varies across the sum. “Dynamical” refers to summing different triangulations/geometries, not to a deterministic update law for one lattice.
• Regge gravitational weight. A discretized Einstein–Hilbert action assigns relative amplitude or Euclideanized Boltzmann weight; automorphisms are accounted for by a symmetry factor.
• Controlled analytic continuation. The Lorentzian causal construction supplies a well-defined continuation of each admissible history to a real statistical weight suitable for analysis or sampling.
• Observable extraction. Diffeomorphism-invariant or relational quantities—volume profiles, dimensions, correlations, transfer-matrix elements, or curvature proxies—are measured on the ensemble rather than read from one triangulation.
• Phase and scaling search. Bare couplings and finite volume are varied to locate phases, transitions, scaling behavior, and possible renormalization-group fixed points.
• Continuum-limit obligation. A physical theory requires more than a large triangulation; the cutoff must be removable through a controlled critical limit with stable renormalized quantities.

Recognition requires the coupled package. A simplicial gravity model without Lorentzian causal restriction is dynamical triangulation but not CDT. A single Regge mesh is not a dynamical ensemble. A Monte Carlo code that samples fixed-background fields is not summing spacetime geometries. Conversely, an analytic two-dimensional CDT model remains CDT even if no Monte Carlo sampling is used.

What It Is Not

• Not any triangulation of spacetime. Triangulation is the regulator language; CDT additionally selects a causal Lorentzian configuration space, assigns gravitational weights, sums geometries, and seeks a continuum theory.
• Not Euclidean dynamical triangulation with a causal adjective. In Euclidean dynamical triangulations the configuration space is Euclidean from the outset. CDT begins with Lorentzian causal histories and uses their structure to define the continuation and suppress classes of geometry admitted in the unconstrained Euclidean sum.
• Not Regge calculus alone. Regge calculus provides piecewise-flat geometry and a discretized action. Many Regge calculations hold a triangulation fixed and vary edge lengths. CDT commonly fixes local edge lengths and sums combinatorially distinct triangulations.
• Not a fixed-background lattice field theory. In lattice QCD, for example, fields live on a prescribed lattice approximating background spacetime. In CDT the lattice connectivity represents the gravitational geometry and is itself part of the ensemble.
• Not a claim of fundamental spacetime discreteness. The regulator is discrete, but standard CDT treats physical predictions as belonging to a continuum limit if one exists.
• Not Monte Carlo simulation as such. Markov-chain sampling is a principal computational instrument in three and four dimensions, but the identity is the path-integral construction. Analytically solved or transfer-matrix formulations do not cease to be CDT.
• Not a theorem that four-dimensional classical spacetime must emerge. Extended phases and large-scale geometric behavior are empirical results of particular models, topologies, volumes, actions, and coupling regions. They motivate the construction without defining every ensemble outcome.
• Not necessarily a preferred foliation as fundamental ontology. Standard CDT uses one; locally causal variants preserve the causal construction without it. A draft that equates all CDT with immutable global slicing is too narrow.
• Not causal inference. “Causal” concerns Lorentzian spacetime order and admissible gluing, not statistical identification of cause and effect from data.

Scope of Application

CDT belongs to nonperturbative quantum gravity and its boundary fields in lattice gravity, Regge calculus, random geometry, statistical field theory, and computational physics. Its legitimate habitats are uses of the construction itself, not every paper about quantum spacetime.

• Two-dimensional quantum gravity. CDT has exactly or analytically tractable formulations in two dimensions, useful for studying transfer matrices, continuum Hamiltonians, topology change, matter coupling, and the distinction from Euclidean random geometry^[3].
• Three-dimensional model systems. Lorentzian simplicial ensembles test causal restrictions, phase behavior, and generalized formulations without preferred foliation in a lower-dimensional setting.
• Four-dimensional lattice gravity. Large-scale simulations vary bare couplings and volume, map phases and transition lines, and measure ensemble observables in the dimension relevant to phenomenological gravity.
• Emergent cosmological geometry. In the extended phase, spatial-volume profiles and their fluctuations can be compared with effective minisuperspace descriptions. “de Sitter-like” here is an ensemble observation under specified conditions, not an assumed background.
• Scale-dependent geometry. Diffusion processes, Hausdorff-type estimators, return probabilities, and related observables probe effective dimension across scales.
• Renormalization and continuum-limit searches. Phase transitions and finite-size scaling are investigated as possible routes to taking \(a\to0\) and defining renormalized observables.
• Matter-coupled and modified models. Matter fields, boundary conditions, topology choices, and modified gravitational actions can be introduced while retaining the causal triangulation architecture; each modification must be checked rather than assumed to share the same universality class.
• Comparative quantum-gravity methodology. CDT offers a discrete, sum-over-histories benchmark for comparison with asymptotic safety, causal sets, spin foams, tensor models, and Hořava–Lifshitz gravity. Similarity or a possible shared continuum class is a research question, not identity.

CDT is not directly a laboratory method, engineering workflow, or empirical cosmological fit. Its applied practice consists of constructing ensembles, implementing ergodic local moves, controlling finite-volume effects, defining observables, and testing scaling hypotheses within theoretical and computational physics.

Clarity

A proposed use is CDT when four questions have affirmative, concrete answers.

1. What geometries are summed? The dimension, topology or boundaries, building simplices, edge data, and equivalence conventions must specify a class of piecewise-flat Lorentzian histories.
2. What makes them causally admissible? Standard time-slice gluing rules or an explicit locally causal replacement must exclude the unwanted histories before weighting.
3. How are histories weighted and evaluated? A Regge-type action, measure or symmetry factor, and a justified analytic continuation or other amplitude prescription must be given.
4. How is regulator dependence tested? The work must distinguish a finite triangulation result from a phase/scaling statement and from a claimed continuum limit.

The diagnostic rejects several false positives. A paper that draws simplices to visualize curvature but does not sum triangulations is not CDT. A simulation on a random Euclidean mesh is not CDT unless a Lorentzian causal configuration space underlies it. A model can use Markov-chain moves resembling CDT moves yet fail the identity if it samples matter on fixed geometry. A generalized causal triangulation can survive the test without global slices when local causal rules and a consistent time orientation play the admissibility role.

The formula for \(Z_a\) is also a clarity device. The index set \(\mathcal C\) identifies configuration-space assumptions; \(C_T\) forces attention to overcounting; \(S_R[T]\) identifies dynamics; and the label \(a\) keeps the cutoff visible. Claims about the continuum cannot be inferred from this regulated expression alone.

Manages Complexity

The continuum gravitational path integral is difficult because its integration domain is the space of geometries modulo diffeomorphisms, its action is nonlinear, and a naive Euclidean formulation has severe measure and conformal-mode problems. CDT turns part of that problem into a controlled statistical system without pretending to solve it by definition.

First, simplicial geometry replaces coordinate fields with combinatorial incidence data and a small set of local building blocks. The Regge action turns continuum curvature integrals into functions of hinges, deficit angles, volumes, or—in fixed-edge-length implementations—counts of simplex types and vertices with bare couplings. This makes a history mechanically constructible and its weight computable.

Second, causal admissibility prunes the configuration space before summation. That restriction is not merely a speed optimization: early Lorentzian work showed that it changes the phase behavior relative to unrestricted Euclidean dynamical triangulations and supplies a nonperturbative Wick rotation^[4]. The restriction makes the scientific wager explicit: the continuum physics sought is represented within the causally constrained class.

Third, statistical-mechanics machinery decomposes the remaining problem. Local moves generate candidate triangulations while preserving manifold, topology, and causality conditions. Metropolis-type acceptance implements Euclideanized weights. Fixed or softly constrained volume allows finite-size studies. Order parameters, susceptibilities, autocorrelation times, histograms, and transfer matrices separate equilibration questions from geometric interpretation.

Finally, the phase diagram organizes otherwise disconnected simulation outputs. Measurements in one bare-coupling region cannot automatically be extrapolated to another. Candidate critical lines focus the continuum search and determine where larger-volume simulations, critical exponents, or renormalization observables are informative. The abstraction therefore compresses a formidable functional-integral problem into interacting choices about configuration space, weight, sampling, observable, and scaling.

Abstract Reasoning

CDT supports reasoning by interventions on roles rather than by comparison of attractive pictures.

Configuration intervention. If the causal restriction is removed while the action and simplex scale are held comparable, the admissible ensemble changes. Differences in phase behavior can then be attributed to configuration-space structure rather than to the mere use of triangles.

Weight intervention. Changing bare gravitational couplings changes relative weights within the same allowed class. Tracking observables across that change maps phases and distinguishes kinematic restrictions from dynamics.

Regulator intervention. Increasing four-volume at several couplings tests finite-size effects. A stable large-volume profile at one finite size is weaker evidence than consistent scaling across sizes.

Observable intervention. Spectral dimension, spatial volume, curvature proxies, and transfer-matrix elements answer different questions. Agreement of several invariant observables is stronger than interpreting a coordinate-dependent snapshot.

Continuum test. A continuum claim requires identifying a diverging correlation length or equivalent scaling structure in lattice units, tuning bare parameters toward criticality, and showing that selected physical ratios or renormalized quantities approach stable values. Merely setting \(a\) “very small” in notation does not perform this limit.

Variant test. If a model removes preferred foliation but preserves local causal consistency and reproduces selected long-distance behavior, that bears on which parts of standard CDT are convenient scaffolding and which are identity-bearing. It does not prove full universality; additional observables and scaling must be compared.

These inferences reveal why CDT is an abstraction rather than a list of publications. The same diagnostic questions recur whenever researchers choose topology, simplex species, move set, matter content, bare action, observable, or continuum strategy.

Knowledge Transfer

Within quantum gravity, CDT transfers literally as a construction. Techniques learned in two dimensions—transfer matrices, generating functions, controlled continuum scaling—inform higher-dimensional questions while not guaranteeing the same solution. Three-dimensional generalized models isolate the role of foliation. Four-dimensional simulations reuse the same separation among admissible histories, action weights, observables, and phase analysis. Code and move sets may change, but the role chain remains recognizable.

Transfer to neighboring fields is component-wise. Regge calculus supplies discrete curvature; lattice field theory supplies finite-size scaling and continuum-limit discipline; statistical mechanics supplies ensemble sampling and phase diagnostics; random geometry supplies combinatorial techniques. These fields can illuminate CDT because they share components, not because every lattice model becomes causal dynamical triangulation.

Beyond physics, phrases such as “build a global structure by gluing causal local pieces” are analogies. The portable skeleton is better carried by the primes Discreteness and Constraint: represent a continuous or enormous possibility space by discrete candidates, then restrict admissibility before aggregation. Calling a project schedule or causal graph “CDT” would import quantum-gravity vocabulary without the Regge action, Lorentzian simplices, geometry sum, or continuum limit. That is metaphor, not an instance.

Monte Carlo Simulation transfers as a computational method but does not parent every aspect of CDT. A two-dimensional analytic solution is still CDT, and a Monte Carlo calculation on fixed background is not. Likewise Causality is related but too general: its catalog identity concerns cause-effect structure, whereas CDT's “causal” encodes Lorentzian admissibility. Knowledge transfer is therefore strongest through shared primes and methods while the named abstraction stays domain-bound.

Examples

Canonical construction: a standard foliated ensemble

Choose a dimension \(d\), spatial topology \(\Sigma\), and discrete times \(t=0,1,\ldots,T\). Triangulate each spatial slice and fill the slab between \(t\) and \(t+1\) using allowed Lorentzian \(d\)-simplices whose vertices lie on the adjacent slices. Fix spacelike squared edges at \(a^2\) and timelike ones at a convention-dependent negative multiple before continuation. Reject gluings that violate manifold or causal constraints. For each admissible triangulation, calculate its Regge action and divide by its automorphism order. Continue the timelike length parameter so the weight becomes \(e^{-S_E}\). Enumerate analytically where possible or sample local connectivity changes that preserve the rules.

Mapped back: the example contains the piecewise-flat simplicial regulator, Lorentzian causal admissibility, fixed local building data, combinatorial dynamism, Regge gravitational weight, and controlled analytic continuation. It is CDT before any favorable emergent result is assumed.

Applied practice: four-dimensional phase and geometry measurement

In a four-dimensional run, researchers hold or constrain total four-volume, choose bare couplings, thermalize a Markov chain of causal triangulations, and record observables across many decorrelated configurations. Spatial three-volume as a function of discrete time can reveal an extended phase and an average profile compatible with a Euclidean de Sitter-type effective description under the studied conditions^[5]. Repeating the run at different volumes and couplings separates finite-size shape from phase behavior. A diffusion process on sampled geometries can estimate spectral dimension as a function of diffusion scale; the classic CDT result reported a value near four at large scales and a reduction toward two at short scales within that ensemble^[6]. These are measured properties, not inputs or universal guarantees.

Mapped back: the practice exercises ensemble sampling, observable extraction, phase and scaling search, and the continuum-limit obligation. The need to state topology, volume, phase, estimator, and finite-size checks prevents a numerical finding from being mistaken for a theorem about all quantum spacetime.

Structural Tensions

T1: Causal restriction versus configuration completeness. Restricting histories removes geometries associated with pathological behavior in earlier Euclidean models and enables a controlled continuation, but any restriction risks excluding configurations required by the desired continuum theory. The scientific strength and vulnerability are the same choice: CDT makes the domain of integration explicit. Diagnostic: Is the causal restriction shown to define a plausible universality class, or merely assumed harmless because its simulations behave better?

T2: Regulator tractability versus discretization dependence. Fixed simplex shapes and combinatorial moves make weights and simulations manageable, yet finite-lattice observables can reflect simplex anisotropy, topology, move-set mixing, or volume control. Diagnostic: Which reported features persist across volumes, observables, topologies, and regulator variations rather than appearing only in one discretization?

T3: Wick-rotated computability versus Lorentzian interpretation. Real Euclideanized weights make statistical sampling feasible, but physical questions concern Lorentzian spacetime. The continuation is structurally controlled, yet interpretation still requires care about what survives the rotation and continuum limit. Diagnostic: Does the claim follow from the Lorentzian construction and its defined continuation, or only from treating the Euclideanized ensemble as an independent physical model?

T4: Emergent semiclassical geometry versus quantum diversity. An extended de Sitter-like volume profile is encouraging because large-scale classical geometry is a target. Focusing only on mean shape can hide fluctuations, alternative phases, or observables that do not share the same semiclassical interpretation. Diagnostic: Is the conclusion supported by fluctuations and independent invariant observables, or only by a visually persuasive average profile?

T5: Preferred foliation versus local causal essence. Standard slicing supports transfer matrices, simple gluing rules, and efficient simulation. Foliation-free variants suggest that global slicing may be convenient rather than essential, but equivalence is not established merely by one matching large-scale observable. Diagnostic: Which results depend on the foliation, and which persist under a locally causal replacement with comparable scaling tests?

T6: Autonomous construction versus reduction to parent primes. CDT combines discreteness, admissibility constraints, weighted aggregation, and simulation, all of which travel broadly. Its autonomy lies in their specific Lorentzian-Regge path-integral organization; stripped of that physics it reduces to the parents and methods. Diagnostic: Does the case require reasoning about causal simplicial geometries and their continuum limit, or can it be stated completely as generic constrained sampling?

Structural–Framed Character

CDT is mixed-structural. Its evaluative weight is low: the construction does not define an outcome as desirable merely because it emerges. It is not human-practice-bound in the sense of a social convention, although it is a theoretical model deliberately engineered by researchers. Its institutional origin is therefore mixed: simplicial geometry, Lorentzian signature, and the Einstein–Hilbert action are mathematical-physical structures, while the exact admissibility rules, topology choices, and regulator are features of a research program.

Its operative vocabulary does not travel freely. “Simplex,” “Regge action,” “Lorentzian history,” “Wick rotation,” “bare coupling,” and “continuum limit” remain pinned to mathematical and physical frameworks. Import versus recognition is decisive: another quantum-gravity model can be recognized as CDT only if it reproduces the construction's roles, whereas use in management or information systems would be an imported analogy.

The portable skeleton is constrained discrete aggregation: specify discrete candidates, exclude inadmissible configurations, weight the survivors, and infer large-scale behavior. That skeleton belongs to Discreteness and Constraint, not uniquely to CDT. Its character: a highly explicit structural construction whose identity nonetheless remains framed by quantum-gravity objects, actions, observables, and validation standards.

Structural Core vs. Domain Accent

This section decides why CDT is a domain-specific abstraction rather than a prime.

What is skeletal. At maximum abstraction, CDT turns an intractable continuous sum into a discrete ensemble, treats admissibility as a first-class restriction, assigns weights, samples or enumerates configurations, measures aggregate observables, and searches for scale-stable behavior. That skeleton can recur in polymers, random graphs, lattice gases, optimization, Bayesian computation, and many other substrates. Its portability is real, but those recurrences instantiate Discreteness, Constraint, weighted aggregation, Monte Carlo Simulation, and continuum approximation—not causal dynamical triangulation.

What is domain-bound. The distinctive roles are Lorentzian piecewise-flat spacetime, spacelike and timelike simplex data, causal gluing, fixed or controlled topology, the Regge form of the gravitational action, a gravitational path integral over geometries, a history-specific analytic continuation, diffeomorphism-invariant geometric observables, and a quantum-gravity continuum limit. Remove these and the name no longer identifies the same method. Replacing spacetime triangulations with project tasks may preserve a directed gluing metaphor, but there is no curvature hinge, gravitational amplitude, sum over geometries, or cutoff-removal problem.

Why this does not clear the prime bar. A prime's recognition vocabulary and intervention logic must travel literally across unrelated domains. CDT's exact diagnostic—specify Lorentzian simplex types, causal admissibility, Regge weight, Wick rotation, invariant geometry observables, and critical continuum scaling—does not apply to biology, law, organizations, or ordinary computation without renaming almost every role. Cross-domain transfer is therefore component-wise or metaphorical. The abstraction earns autonomy inside quantum gravity because the roles constrain real modeling decisions and recur across dimensions, topologies, observables, and variants. Its broader reach belongs to its parent primes.

Instantiates / Related Primes

CDT instantiates Discreteness because a lattice scale and countable simplicial configurations replace the continuum integration domain during regularization, with an explicit bridge back toward a continuum limit. This is the strongest current live-catalog parent relation.

It also instantiates Constraint in its kinematic definition of the admissible configuration space. Causal and manifold conditions partition candidate gluings before weights are applied. Constraint is explanatory but too generic to add as a second prospective parent when Discreteness already provides the minimal current placement.

Monte Carlo Simulation is a major related prime for higher-dimensional numerical work, but it is not constitutive of every CDT formulation: analytic and transfer-matrix solutions exist. Causality is conceptually related, yet its catalog identity is generic cause-effect relationship, while CDT uses causal order in Lorentzian geometry. The catalog's Triangulation prime is not related by meaning; it denotes cross-verification with independent evidence streams, a title collision with mathematical triangulation.

Relationships to Other Abstractions

Current abstraction Causal Dynamical Triangulation Domain-specific

Parents (1) — more general patterns this builds on

• Causal Dynamical Triangulation is part of Discreteness Prime

  CDT instantiates Discreteness because a lattice scale and countable simplicial configurations replace the continuum integration domain during regularization, with an explicit bridge back toward a continuum limit.

Hierarchy paths (2) — routes to 2 parentless roots

• Causal Dynamical Triangulation → Discreteness → Boundary

Show alternative path (1)

Neighborhood in Abstraction Space

Causal Dynamical Triangulation sits in a sparse region of the domain-specific corpus (80^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Relativistic Fields & Spacetime Singularities (10 abstractions)

Nearest neighbors

• Penrose–Hawking Singularity Theorems — 0.83
• Control-Theoretic Orbit — 0.82
• Cauchy surface — 0.82
• Closed timelike curve — 0.82
• Black Hole No-Hair Theorem — 0.82

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Euclidean dynamical triangulations (EDT). EDT sums piecewise-flat Euclidean geometries without first restricting them as causal Lorentzian histories. CDT's controlled Lorentzian origin and admissible configuration class are the difference. Tell: Are histories causal before continuation, or is the ensemble Euclidean from the start?
• Regge calculus. Regge calculus discretizes curvature on simplicial manifolds and supplies CDT's action machinery, but it need not sum over connectivity and often varies edge lengths on one triangulation. Tell: Is geometry integrated by summing distinct causal triangulations, or is one mesh being solved?
• Lattice quantum field theory. Standard lattice field theory integrates fields on a fixed discretized background. CDT integrates gravitational geometry represented by the connectivity itself. Tell: Does the lattice remain fixed across configurations?
• Causal set theory. Causal sets begin with a locally finite partial order intended to encode spacetime causality and volume. CDT histories are piecewise-flat simplicial manifolds with a Regge action and more geometric structure. Tell: Is the basic object an order relation or a glued Lorentzian simplicial geometry?
• Spin foams and loop quantum gravity. These use different kinematic variables, amplitudes, and geometric spectra, even when their histories are combinatorial and background independent. Tell: Are the summed objects causal Regge triangulations with CDT's allowed simplex data?
• Tensor and group-field models. Their perturbative expansions can generate triangulated complexes, creating genuine contact with dynamical triangulations, but their defining fields, amplitudes, and control parameters differ. Tell: Is the triangulation ensemble primary, or generated as diagrams of another field theory?
• Hořava–Lifshitz gravity. It is a continuum field-theory framework with anisotropic scaling. Two-dimensional CDT has a demonstrated relation to a projectable Hořava–Lifshitz model, and higher-dimensional connections are studied, but possible shared universality does not make the constructions identical. Tell: Is the model defined by a continuum anisotropic action or by the CDT sum over causal simplicial histories?
• Mathematical or methodological triangulation. A mathematical triangulation decomposes a space into simplices; methodological triangulation cross-checks evidence. The encyclopedia's live prime named Triangulation uses the second sense. Tell: Does the term describe simplicial geometry, evidence convergence, or the full CDT path integral?

References

[1] Ambjørn, Jurkiewicz, and Loll. “Dynamically triangulating Lorentzian quantum gravity”. Nuclear Physics B, 2001. Ambjorn, Jurkiewicz & Loll (2001) state directly that 4D CDT simplices come in two types up to time reflection, (4,1) and (3,2), matching the article's classification. registry ↩

[2] Jordan and Loll. “Causal Dynamical Triangulations without preferred foliation”. Physics Letters B, 2013. Jordan & Loll (2013) construct a foliation-free generalization of CDT, showing causal consistency survives without a preferred global time-slicing. registry ↩

[3] Ambjørn and Loll. “Non-perturbative Lorentzian quantum gravity, causality and topology change”. Nuclear Physics B, 1998. Ambjorn & Loll (1998) establish the exactly-solvable 2D causal transfer matrix, its continuum Hamiltonian, topology change, and the contrast with Euclidean 2D gravity, but not matter coupling, which was added in later work (Ambjorn, Anagnostopoulos & Loll, c. 1999-2000). registry ↩

[4] Ambjørn, Jurkiewicz, and Loll. “Nonperturbative Lorentzian Path Integral for Gravity”. Physical Review Letters, 2000. Ambjorn, Jurkiewicz & Loll (2000) report both a triangulation-by-triangulation nonperturbative Wick rotation and the absence of the degenerate phases seen in unrestricted Euclidean dynamical triangulations, directly in the PRL abstract. registry ↩

[5] Ambjørn, et al. “Planckian Birth of a Quantum de Sitter Universe”. Physical Review Letters, 2008. Ambjorn, Gorlich, Jurkiewicz & Loll (2008) report that the measured spatial-volume profile of the extended 4D phase matches a de Sitter/minisuperspace effective description to high accuracy. registry ↩

[6] Ambjørn, Jurkiewicz, and Loll. “The Spectral Dimension of the Universe is Scale Dependent”. Physical Review Letters, 2005. Ambjorn, Jurkiewicz & Loll (2005) measure spectral dimension 4.02 ± 0.1 at large scales and 1.80 ± 0.25 at short scales – consistent with (though not exactly equal to) the two-dimensional short-scale value the article describes as a 'reduction toward two.'. registry ↩