Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent.

This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.

There is evidence [1] that at large scales CDT approximates the familiar 4-dimensional spacetime, but shows spacetime to be 2-dimensional near the Planck scale, and reveals a fractal structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work.


Near the Planck scale, the structure of spacetime itself is supposed to be constantly changing due to quantum fluctuations and topological fluctuations. CDT theory uses a triangulation process which varies dynamically and follows deterministic rules, to map out how this can evolve into dimensional spaces similar to that of our universe.

The results of researchers suggest that this is a good way to model the early universe , and describe its evolution. Using a structure called a simplex, it divides spacetime into tiny triangular sections. A simplex is the multidimensional analogue of a triangle [2-simplex]; a 3-simplex is usually called a tetrahedron, while the 4-simplex, which is the basic building block in this theory, is also known as the pentachoron. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes, where previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few.

CDT avoids this problem by allowing only those configurations in which the timelines of all joined edges of simplices agree.

CDT is a modification of quantum Regge calculus where spacetime is discretized by approximating it with a piecewise linear manifold in a process called triangulation. In this process, a d-dimensional spacetime is considered as formed by space slices that are labeled by a discrete time variable t. Each space slice is approximated by a simplicial manifold composed by regular (d − 1)-dimensional simplices and the connection between these slices is made by a piecewise linear manifold of d-simplices. In place of a smooth manifold there is a network of triangulation nodes, where space is locally flat (within each simplex) but globally curved, as with the individual faces and the overall surface of a geodesic dome. The line segments which make up each triangle can represent either a space-like or time-like extent, depending on whether they lie on a given time slice, or connect a vertex at time t with one at time t + 1. The crucial development is that the network of simplices is constrained to evolve in a way that preserves causality. This allows a path integral to be calculated non-perturbatively, by summation of all possible (allowed) configurations of the simplices, and correspondingly, of all possible spatial geometries.

Simply put, each individual simplex is like a building block of spacetime, but the edges that have a time arrow must agree in direction, wherever the edges are joined. This rule preserves causality, a feature missing from previous "triangulation" theories. When simplexes are joined in this way, the complex evolves in an orderly[how?] fashion, and eventually creates the observed framework of dimensions. CDT builds upon the earlier work of Barrett, Crane, and Baez, but by introducing the causality constraint as a fundamental rule (influencing the process from the very start), Loll, Ambjørn, and Jurkiewicz created something different.
Advantages and disadvantages

CDT derives the observed nature and properties of spacetime from a small set of assumptions, without adjusting factors. The idea of deriving what is observed from first principles is very attractive to physicists. CDT models the character of spacetime both in the ultra-microscopic realm near the Planck scale, and at the scale of the cosmos, so CDT may provide insights into the nature of reality.

Evaluation of the observable implications of CDT relies heavily on Monte Carlo simulation by computer. Some feel that this makes CDT an inelegant quantum gravity theory. Also, it has been argued[according to whom?] that discrete time-slicing may not accurately reproduce all possible modes of a dynamical system. However, research by Markopoulou and Smolin demonstrates that the cause for those concerns may be limited[how?]. Therefore, many physicists still regard this line of reasoning as promising .
Related theories

CDT has some similarities with loop quantum gravity, especially with its spin foam formulations. For example, the Lorentzian Barrett–Crane model is essentially a non-perturbative prescription for computing path integrals, just like CDT. There are important differences, however. Spin foam formulations of quantum gravity use different degrees of freedom and different Lagrangians. For example, in CDT, the distance, or "the interval", between any two points in a given triangulation can be calculated exactly (triangulations are eigenstates of the distance operator). This is not true for spin foams or loop quantum gravity in general.

Another approach to quantum gravity that is closely related to causal dynamical triangulation is called causal sets. Both CDT and causal sets attempt to model the spacetime with a discrete causal structure. The main difference between the two is that the causal set approach is relatively general, whereas CDT assumes a more specific relationship between the lattice of spacetime events and geometry. Consequently, the Lagrangian of CDT is constrained by the initial assumptions to the extent that it can be written down explicitly and analyzed (see, for example, hep-th/0505154, page 5), whereas there is more freedom in how one might write down an action for causal-set theory.
See also

Asymptotic safety in quantum gravity
Causal sets
Fractal cosmology
Loop quantum gravity
Planck scale
Quantum gravity
Regge calculus
Simplicial manifold
Spin foam


This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (April 2020) (Learn how and when to remove this template message)

Loll, Renate (2019). "Quantum gravity from causal dynamical triangulations: a review". Classical and Quantum Gravity. 37 (1): 013002. arXiv:1905.08669. doi:10.1088/1361-6382/ab57c7. S2CID 160009859.

Quantum gravity: progress from an unexpected direction
Jan Ambjørn, Jerzy Jurkiewicz, and Renate Loll - "The Self-Organizing Quantum Universe", Scientific American, July 2008
Alpert, Mark "The Triangular Universe" Scientific American page 24, February 2007
Ambjørn, J.; Jurkiewicz, J.; Loll, R. - Quantum Gravity or the Art of Building Spacetime
Loll, R.; Ambjørn, J.; Jurkiewicz, J. - The Universe from Scratch - a less technical recent overview
Loll, R.; Ambjørn, J.; Jurkiewicz, J. - Reconstructing the Universe - a technically detailed overview
Markopoulou, Fotini; Smolin, Lee - Gauge Fixing in Causal Dynamical Triangulations - shows that varying the time-slice gives similar results

Early papers on the subject:

R. Loll, Discrete Lorentzian Quantum Gravity, arXiv:hep-th/0011194v1 21 Nov 2000
J Ambjørn, A. Dasgupta, J. Jurkiewicz, and R. Loll, A Lorentzian cure for Euclidean troubles, arXiv:hep-th/0201104 v1 14 Jan 2002
Causal dynamical triangulation on

External links

Renate Loll's talk at Loops '05
John Baez' talk at Loops '05
Pentatope: from MathWorld
Simplex: from MathWorld
Tetrahedron: from MathWorld
(Re-)Constructing the Universe from Renate Loll's homepage
Renate Loll on the Quantum Origins of Space and Time as broadcast by TVO


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