How do you compute the shape of spacetime when quantum theory says it has every possible shape at once?
Welcome to FreeAstroScience. Renate Loll’s group answered that question by gluing hundreds of thousands of four-dimensional triangles into digital universes, and one of those universes measured 1.80 dimensions at short distances. We read the two Physical Review Letters behind that number to see what it does and does not prove.
Causal dynamical triangulations (CDT) computes the quantum shape of spacetime by summing millions of triangulated geometries in a path integral, keeping only those in which causes precede effects. The method produced an extended four-dimensional universe from first principles in 2004, and in 2005 the same simulations measured a spectral dimension of 1.80 ± 0.25 at short distances against 4.02 ± 0.1 at large ones. Renate Loll, Jan Ambjørn, and Jerzy Jurkiewicz built the approach starting in 1998.
Updated August 2026: this article now carries the published numbers from the 2005 spectral-dimension letter, the 2011 independent replication, and the April 2026 arXiv survey. First published on 12 February 2026.
Through the 1990s, the lattice road to quantum gravity looked closed. Earlier groups had glued flat triangles into random quantum geometries in a timeless setting called Euclidean space, where the time direction behaves like just another direction of space, and their simulations kept producing wreckage. Geometries crumpled into dense balls or stretched into thin branched filaments, with nothing recognizable as a large universe in either pile. Renate Loll had spent ten years on pen-and-paper loop quantum gravity by then, and she admits she envied the groups doing concrete numerical work. Together with Jan Ambjørn she suspected the missing ingredient was time itself, and the repair they built came to be called causal dynamical triangulations.
Causal dynamical triangulations kept the arrow of time
Keeping time’s arrow is the whole difference between CDT and the failed Euclidean program. Without a genuine time direction, the older path integrals admitted geometries in which travel into the past was allowed, and those pathological configurations swamped everything else in the sum.
In 1998 Ambjørn and Loll published a first Lorentzian calculation in Nuclear Physics B showing that preserving causal structure produces a different theory outright, not a technical variant of the old one. One line of the recipe changed. Instead of cutting building blocks out of timeless Euclidean space, CDT cuts them out of ordinary spacetime, so every block carries a built-in arrow from past to future.
Readers who followed our June piece on loop quantum gravity’s matter-first rebuild will recognize both the family resemblance and the split. Both programs make spacetime discrete near the Planck scale. Loop quantum gravity quantizes geometry directly through spin networks, while CDT never quantizes geometry by hand at all: it feeds triangulated histories into a path integral and lets a computer discover what quantum spacetime does. June’s piece covered a 2026 reformulation still finding its feet; the triangle program is older and has numbers on the board.
What does the path integral actually sum?
It sums every shape spacetime could take. In Loll’s description, the path integral is a universal procedure for infusing Einstein’s gravity with a quantum essence: the universe we observe is a superposition of all possible spacetime geometries, gentle ones and violent ones blended together. Three ingredients go in, and Loll suspects nothing further is needed.
- Einstein’s geometric picture of gravity.
- The quantum rule that the shape spacetime shows us averages over every shape it could take.
- Causality, meaning causes come before effects.
No strings enter the recipe, no loops, and no extra dimensions. Triangles make the sum computable. Glue six equilateral triangles flat around a point and you get a patch of flat spacetime; remove one and reconnect its neighbors, and the sheet pops into a cone, which is curvature concentrated at a point. Adding or removing triangles across a whole mesh captures any curved geometry with no coordinates needed.
Summing every possible gluing is impossible, so the team samples. A computer generates enormous numbers of random triangulations, weighs them by the rules of Einstein’s theory, and reads off which universes dominate the blend. Loll’s group was not the first to attempt this. By her own account, they were the first to make the procedure yield a universe resembling ours.
How did four dimensions emerge in 2004?
Gradually, and only after the causal version had been rebuilt dimension by dimension. Encouraged by the 1998 calculation, the team kept going. Simulations in three dimensions using tetrahedra came next, and the full four-dimensional case arrived in 2004, published in Physical Review Letters.
Dimension is a slippery notion on a random lattice, so the team measured it by growth. They ran universes of 50,000, then 100,000, then 200,000 four-dimensional building blocks and watched how the assembled shape scaled. It grew exactly the way a three-dimensional space with one time direction should grow. Nobody had produced that behavior before. The same kind of blocks, glued without a causal arrow in the earlier Euclidean runs, had piled into crumpled balls and filament webs, so nothing about the outcome was automatic.
A large universe assembling itself from first principles answers a different question from the one in our piece on why space has three dimensions, which reads dimensionality off the inverse-square law of forces. There the number is measured in the laboratory. Here the number grows inside the machine.
The number that made spacetime look fractal
CDT’s biggest surprise came from asking how things spread. A drop of ink diffuses differently across a two-dimensional page than through a three-dimensional glass of water, so diffusion itself can measure the dimension of the space it happens in. Physicists call the result the spectral dimension.
In 2005, Ambjørn, Jurkiewicz and Loll released virtual ink into a simulated quantum universe of 181,000 four-dimensional building blocks and tracked its spread. Read at large scales, the spectral dimension came out at 4.02 ± 0.1, an ordinary spacetime. At the shortest scales the lattice could probe, it fell to 1.80 ± 0.25. Their letter compresses the whole behavior into one fitted curve.
DS(σ) = 4.02 − 119 / (54 + σ)
Here DS is the spectral dimension and σ is the diffusion time, a stand-in for the distance scale being probed. In words, the dimension a random walker feels starts near two at the shortest times and relaxes to four as the walk lengthens. Feed in a large σ and the second term dies away, leaving 4.02. Set σ to zero and the curve gives 4.02 − 119/54, which works out to about 1.82, right on top of the quoted short-distance value. We ran that arithmetic ourselves, and it is worth doing once by hand to feel how steeply the dimension falls.
So is spacetime literally two-dimensional at the bottom, the way headlines still put it? Not quite. The measured value only straddles 2 through its error bar, the curve above is a fit rather than a derivation, and the letter’s own claim is more careful than most coverage of it: the authors conclude that quantum gravity may be self-renormalizing at the Planck scale, through a mechanism they name dynamical dimensional reduction. Nothing turns into a flat sheet. Space stays fully filled, while its short-range wiring behaves like a fractal in which some regions are momentarily harder to reach than others.
“This is the first genuine piece of evidence that there is nontrivial quantum structure on short scales, which I would have never thought of classically,” Loll told Quanta Magazine in a 2023 profile. She initially had to persuade her own collaborators it mattered. It became the group’s most cited paper.
One caution on the word fractal. What CDT found is Planck-scale fractality in the wiring of spacetime, a different animal from the galaxy-clustering kind we dismantled in our reality check on the fractal universe idea, which lives at survey scales and dissolves into smoothness above a few hundred megaparsecs.
One question the sources leave open, and we could not settle it: whether the short-distance value lands exactly on 2 in the continuum limit. The 1.80 comes from extrapolating a fit on a finite lattice, its error bar reaches 2.05, and neither the 2005 letter nor Loll’s interview commits to the integer.
One code ran almost every headline result
For years that sentence described the method’s quietest weakness. Both the four-dimensional emergence and the spectral drop came out of a single simulation code, however carefully written, and physics has burned itself before on numbers nobody reran independently.
Independent replication arrived when Rajesh Kommu at the University of California, Davis wrote a completely separate implementation and reproduced both signature behaviors, the emergence of classical spacetime and the smooth fall in spectral dimension, in 2+1 and in 3+1 dimensions. One replication is worth having. It remains a thin base for a claim about the fabric of reality, and no observation of any kind has yet tested CDT against nature. The distance to experiment is brutal: quantum spacetime effects should appear near the Planck length, around 10−35 m, far beyond the reach of any accelerator on Earth.
Where could anyone ever test this?
Nobody knows yet, and Loll says so plainly. Her best guess is astrophysics, where tiny effects can get amplified across cosmological distances, and her group is still working out what observable consequences CDT would leave there. No candidate signal exists today.
Freeman Dyson’s old argument hangs over the whole hunt: the gravitational analog of a photon detector would be so massive it would collapse into a black hole before registering a single graviton. Maybe single gravitons simply do not exist the way single photons do. Loll wonders aloud whether we are asking too much of nature.
She also senses a change of mood after decades dominated by strings and loops, which produced spectacular mathematics without converging on a unique quantum theory of gravity. A new humility, she calls it, and a return toward plain quantum field theory, with CDT as part of that turn. This April she and Ambjørn posted a fresh survey of the method to arXiv, twenty-eight years after their first causal calculation. We are leaving out CDT’s phase diagram here, the map of phase transitions that organizes its parameter space, because it changes neither headline result and it deserves an article of its own.
What twenty years of triangles bought
The causal move of 1998 turned a failing lattice program into a different theory. Six years later, 200,000 glued blocks grew into a four-dimensional universe on their own, and a year after that the same simulations measured a spectral dimension of 1.80 ± 0.25 at short distances, a number one independent code has confirmed and no telescope has ever tested.
We wrote this out in full at FreeAstroScience because the triangle story usually reaches you as either an arXiv abstract or a headline about 2D spacetime, and you deserve the version that carries its actual numbers. Turning hard physics into words you already own is the work we chose for ourselves here, and we want your reason wide awake while you weigh it, since a mind that stops questioning is the open door every monster prefers. Check the 1.82 yourself, argue with our reading of the error bar, and tell us where we went wrong. Come back when we open up CDT’s phase diagram, the part we set aside today. Gerd Dani, FreeAstroScience, Rimini.
Sources
- Ambjørn, J., Jurkiewicz, J., and Loll, R. (2005). Spectral Dimension of the Universe. Physical Review Letters, 95, 171301. DOI: 10.1103/PhysRevLett.95.171301. Preprint: arxiv.org/abs/hep-th/0505113
- Ambjørn, J., Jurkiewicz, J., and Loll, R. (2004). Emergence of a 4D World from Causal Quantum Gravity. Physical Review Letters, 93, 131301. Preprint: arxiv.org/abs/hep-th/0404156
- Ambjørn, J., and Loll, R. (1998). Nonperturbative Lorentzian quantum gravity, causality and topology change. Nuclear Physics B, 536, 407-434. Preprint: arxiv.org/abs/hep-th/9805108
- Kommu, R. (2012). A Validation of Causal Dynamical Triangulations. Preprint: arxiv.org/pdf/1110.6875
- Ambjørn, J., and Loll, R. (2026). Causal Dynamical Triangulations: New Lattice Theory of Quantum Gravity. Posted 7 April 2026. Preprint: arxiv.org/abs/2604.05641
- Wood, C. (2023). Renate Loll Blends Universes to Unlock Quantum Gravity. Quanta Magazine. Published 25 May 2023. quantamagazine.org
- Reccom Network (2026). Triangolazioni dinamiche causali, una teoria per la gravità quantistica. Published 12 February 2026. reccom.org
