Chapter 03 · The Nature of Space and Time/Causal Dynamical Triangulation

Foundational CDT Program

1998 · Jan Ambjorn, Renate Loll

Frontier

Build spacetime out of four-dimensional simplices glued along a strict causal ordering. Sum over every such gluing on a computer. The geometry that emerges is the prediction of quantum gravity.

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In one sentence

Ambjorn and Loll proposed in 1998 that the gravitational path integral can be defined nonperturbatively by summing over discrete Lorentzian geometries built from simplices, with one structural rule: every building block must agree on which direction is the past and which is the future. The causal-foliation constraint is what rescued lattice quantum gravity from the pathological geometries that defeated earlier Euclidean approaches.

The claim

Quantum gravity is hard for a specific technical reason. The gravitational path integral, the sum over every possible spacetime weighted by its action, is a sum over an infinite space of geometries, and the techniques that work for ordinary quantum field theory (perturbation around a fixed background, lattice discretization with a fixed metric) either lose background independence or fail to converge when applied to gravity. Earlier attempts at lattice gravity in the 1980s and 1990s, called Euclidean Dynamical Triangulations, tried to make the path integral finite by chopping spacetime into discrete building blocks and summing over the gluings. The numerical experiments produced spacetimes that either crumpled into a highly connected mess with no resemblance to a four-dimensional manifold or stretched into a thin branched polymer with effective dimension barely greater than two. Neither phase looked anything like our universe. The program stalled.

Ambjorn and Loll's 1998 contribution was to change one structural rule. Instead of summing over all possible gluings of simplices, they imposed that every gluing must respect a global causal ordering: each block stacks neatly into spatial layers, and the layers are constant-time slices. This makes every term in the path integral a Lorentzian geometry rather than a Euclidean one. With the causality constraint in place the numerical pathologies disappear. Subsequent simulations show a third phase, the C phase, in which the sum over geometries produces a smooth four-dimensional universe whose large-scale shape matches a de Sitter spacetime. The program's name reflects this central design choice. Causal Dynamical Triangulation is dynamical triangulation with a causal constraint.

Modern CDT is a mature lattice quantum-gravity program. The path integral is evaluated by Monte Carlo simulation on triangulations containing on the order of one hundred thousand simplices, with running times measured in CPU-years. The program has produced a catalog of results, summarized in the 2012 Physics Reports review by Ambjorn, Gorlich, Jurkiewicz, and Loll, and again in Loll's 2020 Classical and Quantum Gravity review, that no other quantum-gravity approach has produced at comparable detail: spontaneous emergence of four-dimensional macroscale geometry, a phase diagram with at least four distinct phases including a recently identified bifurcation phase, a running spectral dimension that drops from four to roughly two at short scales, and a growing set of matter-coupled and locally-causal extensions. CDT's distinctive position in the discrete-spacetime landscape is that it does these calculations rather than only outlining the framework in which the calculations would be done.

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 gravitational path integral can be defined nonperturbatively on a lattice without picking a background metric, by summing over all causal triangulations weighted by the Regge action.
Imposing a global causal ordering on the simplicial geometries suppresses the crumpled and branched-polymer geometries that dominate the Euclidean Dynamical Triangulation path integral, allowing a smooth four-dimensional phase to exist.
The continuum limit of the lattice theory, if it exists, should approach a true renormalized quantum field theory of gravity, possibly connected to the ultraviolet fixed point of the Asymptotic Safety program.

Evidence

The framework is a working lattice theory: the path integral is well-defined at finite lattice sizes, and Monte Carlo simulations produce numerical results that other quantum-gravity approaches have not been able to compute at comparable detail.
Imposing the causal-foliation constraint demonstrably suppresses the pathological geometries that defeated Euclidean Dynamical Triangulations in the 1980s and 1990s, which is a concrete structural success of the CDT design choice.
Three decades of subsequent results (4D emergence, phase diagram, dimensional reduction, matter coupling, locally-causal extensions) build internal consistency across many independent calculations using the same path-integral construction.

Counterpoints

The causal-foliation construction picks a global time direction at the microscopic level. Critics argue this breaks full background independence and is at odds with the diffeomorphism invariance that any quantum theory of gravity should preserve. The Locally Causal Dynamical Triangulation extension is the program's response, but most published LCDT work is in lower dimensions because the local-causality construction is computationally much more expensive than the global one.
The continuum limit of the lattice theory has not been rigorously established. Without it, the program's results are statements about finite-size simulations rather than about a true continuum quantum theory of gravity. Recent work on phase transitions in the C-to-bifurcation region is encouraging but does not yet close the argument.
The bare action contains only the Einstein-Hilbert term and a cosmological constant. Whether the results survive the inclusion of higher-derivative operators or matter sectors is the subject of the program's ongoing extensions.

Variants in this family

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▸Go deeperTechnical detail with proper terminology

A causal triangulation in CDT is a piecewise-linear Lorentzian manifold built from four-simplices, each of which has four spacelike edges of fixed length and a fixed timelike edge length, glued together along their faces such that every four-simplex sits between two consecutive constant-time slices. The action on a given triangulation is the Regge action, the discrete version of the Einstein-Hilbert action with a cosmological-constant term. The path integral is the weighted sum over all such triangulations sharing a fixed topology, evaluated by Monte Carlo sampling.

The contrast with Euclidean Dynamical Triangulations is sharpened by what happens when the causal constraint is removed. In Euclidean DT every simplex can be glued to every other simplex without respecting a time direction. Numerical simulations show two failed phases, one in which the geometry is crumpled and has effectively infinite Hausdorff dimension, the other in which the geometry is a thin branched-polymer structure with Hausdorff dimension approximately two. Neither phase contains a fourth-dimensional macroscale geometry. The Ambjorn-Loll 1998 paper identifies the global causal ordering as the structural choice that opens a third phase in which the simulated spacetime is genuinely four-dimensional at large scales.

The Regge action plus the simplicial path integral leaves two free parameters in the theory, conventionally called the gravitational coupling and the cosmological constant, which together determine which of the program's phases the simulation lands in. The relevant region of parameter space, the C phase, is where the simulated spacetime self-organizes into a smooth four-dimensional manifold; outside this region the simulation produces the crumpled or branched geometries that defeated Euclidean DT. The 2020 Loll review is the canonical pedagogical introduction to the parameter structure and the modern numerical results.

A 2026 review preprint by Ambjorn and Loll, posted to arXiv as 2604.05641 with the title 'Causal Dynamical Triangulations: New Lattice Theory of Quantum Gravity,' positions the program as the lattice-quantum-gravity analog of lattice QCD: a concrete, computational definition of the gravitational path integral whose continuum limit, if it exists, would be the long-sought quantum theory of gravity. The framing as a lattice theory is editorially deliberate; it sets the empirical-status bar at the same level as lattice QCD rather than at the level of a fully axiomatized theory.

References

Last reviewed May 19, 2026

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