Locally Causal CDT and Matter Coupling
The original CDT setup hardcodes a global time direction. The Locally Causal Dynamical Triangulation extension enforces causality block by block instead. Most of the modern 2020-onward CDT literature lives here, alongside matter-coupled extensions that probe how the geometry responds to scalar and gauge fields. Most published work is in lower dimensions.
Placeholder for a 3D visualisation of Causal Dynamical Triangulation. The interactive scene will land in Phase 3. Causal Dynamical Triangulation defines a quantum theory of gravity by summing over piecewise-linear Lorentzian geometries built from four-dimensional simplices, the higher-dimensional analogs of triangles, glued together along a strict causal ordering. The sum is the gravitational path integral, evaluated nonperturbatively by Monte Carlo simulation. Spacetime is not assumed; it emerges from the sum. After almost three decades of development the program has produced a clear catalog of results: a four-dimensional de Sitter-like universe self-organizes in the C phase of the lattice's phase diagram, the spectral dimension of the emergent spacetime runs from approximately 4 at large scales to approximately 2 at the Planck scale, the phase diagram has been mapped including a recently identified bifurcation phase, and modern extensions enforce causality locally and couple matter fields to the discrete geometry. The Ambjorn-Loll 1998 paper is the foundational reference, the 2012 Physics Reports review and the 2020 Classical and Quantum Gravity review are the canonical modern surveys, and a 2026 review preprint frames the program as a working lattice theory of quantum gravity. CDT is the most computationally productive of the three discrete-spacetime programs in this chapter, complementing Causal Set Theory's order-and-number kinematics and Loop Quantum Gravity's canonical and spin-foam quantization with a concrete path-integral construction that produces actual numbers.
In one sentence
Jordan and Loll demonstrated in 2013 that the causality constraint that makes CDT work can be enforced locally rather than globally, removing the preferred-foliation construction that is the family's central foundational concern. The Locally Causal Dynamical Triangulation extension is the active 2020-2026 research frontier of the program, alongside matter-coupled CDT and the continuum-limit search.
The claim
The Locally Causal Dynamical Triangulation program was created to answer a specific critique of the original CDT construction. The original program imposes a global time slicing on every triangulated geometry: each simplex stacks into spatial layers that are constant-time slices, and the full geometry is a stack of such slices. This works mathematically and produces the program's headline results, but it appears to break full background independence by singling out a time direction at the microscopic level. Critics have argued that a fully quantum theory of spacetime should not require such a choice. Jordan and Loll's 2013 paper, titled exactly 'Causal Dynamical Triangulations without Preferred Foliation,' shows that the causal constraint can be enforced locally instead, block by block, requiring only that each individual simplex have a well-defined causal interior without imposing a global synchronization between simplices. The construction is more permissive than the original and recovers it as a special case, but it is also much more computationally expensive: the simulations have to handle configurations the original construction excluded by design.
Most of the post-2013 LCDT work has been carried out in two dimensions, where the simulations are computationally tractable enough to study the new construction in detail. The Loll-Ruijl 2015 paper 'Locally Causal Dynamical Triangulations in Two Dimensions' is the foundational 2D LCDT result. Two-dimensional gravity is a useful testing ground because it is simple enough to allow exact analytical results that the simulations can be checked against, and the LCDT results agree with the analytical expectations in cases where comparison is possible. Three-dimensional LCDT has been investigated more recently. Full four-dimensional LCDT is the long-term goal of the program but remains computationally beyond current resources for the size and statistics needed to produce reliable phase-structure results.
Alongside LCDT, the modern CDT frontier includes matter coupling and the continuum-limit search. Matter coupling means adding scalar fields, gauge fields, or fermions to the discrete geometry and observing how the phase diagram changes and how the gravitational degrees of freedom respond. The motivation is twofold: realistic phenomenology requires matter, and the gravitational coupling's flow under the renormalization group in the presence of matter is the most direct point of contact between CDT and the Asymptotic Safety program in Chapter 4. CDT functions in this sense as a lattice laboratory for testing Asymptotic Safety's conjectured ultraviolet fixed point from the discrete side. The continuum-limit search, separately, looks for a second-order phase transition in the CDT phase diagram (typically at the C-to-C_b boundary identified in the Emergent Universe variant) that would mark the existence of a continuum quantum field theory of gravity. The 2016 Ambjorn-Coumbe-Gizbert-Studnicki-Jurkiewicz paper 'Searching for a continuum limit in causal dynamical triangulation quantum gravity' is one of the canonical references for this thread.
The family stance
Spacetime is built from discrete geometric blocks glued along a strict causal ordering. Quantum gravity is defined nonperturbatively by summing over all such gluings via lattice Monte Carlo simulation, and the smooth four-dimensional spacetime of general relativity emerges from this sum in a specific phase of the lattice's phase diagram. The macroscale geometry that emerges in the C phase matches a de Sitter universe, and the microscale geometry shows a dimensional reduction from four dimensions at large scales to roughly two at the Planck scale. The same dimensional reduction appears independently in Asymptotic Safety and other quantum-gravity programs, which is taken in the literature as a possible clue that something real is happening to spacetime at the smallest scales. After three decades of development the program has produced concrete numerical results that no other quantum-gravity approach has produced at comparable detail, while remaining empirically unconfirmed and unresolved on its central technical question of whether a true continuum limit exists.
Predictions
- The causality constraint that makes the CDT path integral well-defined can be enforced locally rather than globally, recovering the original construction as a special case while permitting a wider class of triangulated geometries.
- The local construction is computationally much more expensive than the global one; most LCDT results to date are in two dimensions, with three-dimensional investigations active and four-dimensional results still out of reach at the precision needed for full phase-structure mapping.
- Matter-coupled CDT modifies the phase diagram in computable ways, with the modification dependent on the matter content (scalar, gauge, fermion) and the matter-matter and matter-gravity couplings. The pattern of modification is a direct probe of the gravitational degrees of freedom in the presence of matter and is the most direct CDT-side contact with Asymptotic Safety's matter-coupled fixed-point calculations.
Evidence
- The existence of the LCDT construction directly demonstrates that the preferred-foliation choice in the original CDT is not an inescapable feature of the program. Practitioners take the concern seriously enough to have built and tested a structural alternative.
- Two-dimensional LCDT results agree with analytical expectations for two-dimensional quantum gravity where comparison is possible, supporting the view that the local-causality construction is a consistent extension rather than a structurally different theory.
- Matter-coupled CDT produces phase-diagram shifts that match qualitative expectations from continuum gravity-plus-matter analyses, providing internal-consistency evidence for both the CDT phase structure and the matter-coupling construction.
Counterpoints
- Most of the post-2013 LCDT literature lives in two dimensions because the local-causality construction is computationally much more expensive than the global one. Four-dimensional LCDT, where the program's most important questions live, remains computationally limited. Critics argue that the LCDT response to the preferred-foliation concern is real in principle but not yet realized in the dimension that matters.
- Matter coupling at present requires simplified matter sectors (single scalar fields, abelian gauge fields, fermions in restricted regimes) because of the same computational cost. Realistic Standard Model matter content coupled to CDT is beyond current resources and may require substantial algorithmic innovation rather than just more compute time.
- The continuum-limit search has identified candidate second-order phase transitions but has not closed the rigorous argument that the CDT lattice theory defines a continuum quantum field theory of gravity. Without this the LCDT and matter-coupling extensions remain extensions of a lattice construction whose long-distance status is not yet settled.
- The cross-program contact with Asymptotic Safety is qualitative at present rather than quantitative. CDT and Asymptotic Safety see similar dimensional reduction at short scales and similar features of matter-coupling, but a direct numerical match between the discrete and continuum approaches is still work in progress.
Variants in this family
▸Go deeperTechnical detail with proper terminology
The Jordan-Loll 2013 construction replaces the global causality constraint with a local one by retaining the requirement that each four-simplex have a well-defined causal interior (a non-degenerate distinction between past, future, and spacelike-separated directions within the simplex) but dropping the requirement that the timelike directions of neighboring simplices agree globally. The path integral now includes configurations in which the time direction varies smoothly across the geometry, recovering the original CDT only as a special case where all such variations vanish.
Two-dimensional LCDT is a useful testing ground because two-dimensional gravity is exactly soluble in the continuum (matrix models, Liouville theory) and the discrete LCDT result can be checked against analytical expectations. Loll and Ruijl's 2015 paper carries out this comparison and finds agreement between the LCDT correlation functions and the matrix-model results in the regime where both apply, which validates the local-causality construction at the dimension where independent checks are available.
Matter coupling in CDT proceeds by adding the matter degrees of freedom on the existing simplicial geometry and supplementing the Regge action with discrete versions of the matter action. The simplest case, a single scalar field, has been studied in several CDT extensions; gauge fields and fermions are more recent and computationally more demanding. The relevant observables are the changes in the phase boundaries of the gravitational phase diagram and the effective coupling of the gravitational degrees of freedom to matter, both of which can be measured directly in the simulation.
The continuum-limit search, summarized in the 2016 Ambjorn-Coumbe-Gizbert-Studnicki-Jurkiewicz paper among others, identifies candidate second-order phase transitions in the CDT phase diagram at the boundaries between the C phase and adjacent phases (particularly C_b). A second-order transition with the right scaling properties would mark a continuum limit and would establish CDT as a true continuum quantum field theory of gravity rather than only a well-defined lattice construction. The candidate transitions have been mapped in increasing detail but the rigorous demonstration that the scaling matches a renormalizable continuum theory is still open.
References
- EstablishedJordan, S., Loll, R. (2013). 'Causal Dynamical Triangulations without Preferred Foliation.' Physics Letters B 724, 155
- EstablishedLoll, R., Ruijl, B. (2015). 'Locally Causal Dynamical Triangulations in Two Dimensions.' Physical Review D 92, 084002
- EstablishedAmbjorn, J., Coumbe, D., Gizbert-Studnicki, J., Jurkiewicz, J. (2016). 'Searching for a continuum limit in causal dynamical triangulation quantum gravity.' Physical Review D 93, 104032
Last reviewed May 19, 2026
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