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Causal dynamical triangulation (abbreviated as CDT) theorized by
Renate Loll Renate Loll (born 19 June 1962, Aachen) is a Professor in Theoretical Physics at the Institute for Mathematics, Astrophysics and Particle Physics of the Radboud University in Nijmegen, Netherlands. She previously worked at the Institute for Th ...
, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
that, like
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
, 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 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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
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.


Introduction

Near the Planck scale, the structure of spacetime itself is supposed to be constantly changing due to
quantum fluctuation In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s and topological fluctuations. CDT theory uses a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
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 The chronology of the universe describes the history and future of the universe according to Big Bang cosmology. Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, with ...
, 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 -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 In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
. 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.


Derivation

CDT is a modification of quantum
Regge calculus In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Available ( ...
where spacetime is discretized by approximating it with a piecewise linear
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
in a process called
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
. 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 Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
. This allows a path integral to be calculated
non-perturbative In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function : f(x) = e^, which does not have a Taylor series at ''x'' = 0. Every coefficient of the Tay ...
ly, 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 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.


Related theories

CDT has some similarities with
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
, especially with its
spin foam In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structu ...
formulations. For example, the Lorentzian
Barrett–Crane model The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action. The B field in the action is supposed to be a so(3, 1)-valued 2-form, i.e. taking values in the Lie algebra of a spec ...
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. Moreover, in spin foams the discreteness is thought to be fundamental, while in CDT it is viewed as a regularization of the path integral, to be removed by the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
. 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. In the continuum limit, CDT is probably related to some version of Hořava–Lifshitz gravity. In fact, both theories rely on a foliation of spacetime, and thus they can be expected to lie in the same universality class. In 1+1 dimensions they have actually been shown to be the same theory, while in higher dimensions there are only some hints, as understanding the continuum limit of CDT remains a difficult task.


See also

*
Asymptotic safety in quantum gravity Asymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontriv ...
* Causal sets *
Fractal cosmology In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal across a wide range of scales (see also: multi ...
*
Loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
*
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It ...
* Planck scale *
Quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
*
Regge calculus In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Available ( ...
* Simplex * Simplicial manifold *
Spin foam In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structu ...


References

;Notes ;Bibliography
Quantum gravity: progress from an unexpected direction
* Jan Ambjørn, Jerzy Jurkiewicz, and
Renate Loll Renate Loll (born 19 June 1962, Aachen) is a Professor in Theoretical Physics at the Institute for Mathematics, Astrophysics and Particle Physics of the Radboud University in Nijmegen, Netherlands. She previously worked at the Institute for Th ...

"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 * Loll, R
Quantum Gravity from Causal Dynamical Triangulations: A Review
A review from May 2019, focusing on results that were recent at that time 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


External links



from Renate Loll's homepage
Renate Loll on the Quantum Origins of Space and Time
as broadcast by TVO {{Standard model of physics Physical cosmology Astrophysics Quantum gravity Physics beyond the Standard Model