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Causal dynamical triangulation (abbreviated as "CDT") invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz 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. The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in much greater depth, and reveal it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d 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. A brief article appeared in the February 2007 issue of Scientific American, which gives an overview of the theory, explains why some physicists are excited about it, and puts it in historical perspective. Renate Loll Renate Loll is a physicist who works at the Institute for Theoretical Physics of Utrecht University, The Netherlands. ...
This article or section does not adequately cite its references or sources. ...
Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...
Background independence is a condition in theoretical physics, especially in quantum gravity, that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. ...
In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...
In physics, Planck units are physical units of measurement originally proposed by Max Planck. ...
The boundary of the Mandelbrot set is a famous example of a fractal. ...
Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ...
Explanation
It is widely accepted that, at the very smallest scales space is not static, but is instead dynamically-varying. Near the Planck scale, the structure of spacetime itself is constantly changing, due to quantum fluctuations. This theory uses a triangulation process which is also dynamically-varying, or dynamical, to map out how this can evolve into dimensional spaces similar to that of our universe. The results of researchers suggests 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 generalized form of a triangle, in various dimensions. A 3-simplex is usually called a tetrahedron, and the 4-simplex, which is the basic building block in this theory, is also known as the pentatope, or 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 where cause precedes any event. Triangulation can be used to find the distance from the shore to the ship. ...
In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. ...
A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ...
Derivation 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, which makes this a relatively successful theory, 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. In theoretical physics, Regge calculus is a simplified form of general relativity, introduced by the Italian theoretician Tullio Regge in the early 1960s. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
A d-dimensional simplicial manifold is a simplicial complex where the neighborhood of each point, i. ...
A geodesic dome is an almost spherical structure based on a network of struts arranged on great circles (geodesics) lying approximately on the surface of a sphere. ...
Causality describes the relationship between causes and effects, and is fundamental to all natural science, especially physics. ...
This article or section is in need of attention from an expert on the subject. ...
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. ...
Disadvantages The disadvantageous aspect of this theory is that it relies heavily on computer simulations, to generate results or evidence. Some feel that this makes it a less 'elegant' solution to the problem of creating a completely successful quantum gravity theory. Also; it has been argued 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. Ergo; many physicists regard this line of reasoning as promising. Fotini Markopoulou-Kalamara is a theoretical physicist interested in foundational mathematics and quantum mechanics. ...
Lee Smolin at Harvard. ...
Related theories CDT has some similarities with loop quantum gravity, especially with its spin foam formulations. For example, 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. Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...
In physics, a spin foam is a four-dimensional graph made out of two-dimensional faces that represents one of the configurations that must be summed to obtain Feynmans path integral (functional integration) describing the alternative formulation of quantum gravity known as loop gravity or loop quantum gravity. ...
Another approach to quantum gravity that is closely related to Causal dynamic 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 very general, whereas CDT assumes a 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 Causal Set theory is not nearly as completely developed at this point. The causal sets programme is an approach to quantum gravity. ...
See also Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...
The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ...
In physics, Planck units are physical units of measurement originally proposed by Max Planck. ...
The causal sets programme is an approach to quantum gravity. ...
This article or section does not adequately cite its references or sources. ...
In theoretical physics, Regge calculus is a simplified form of general relativity, introduced by the Italian theoretician Tullio Regge in the early 1960s. ...
In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. ...
A d-dimensional simplicial manifold is a simplicial complex where the neighborhood of each point, i. ...
In physics, a spin foam is a four-dimensional graph made out of two-dimensional faces that represents one of the configurations that must be summed to obtain Feynmans path integral (functional integration) describing the alternative formulation of quantum gravity known as loop gravity or loop quantum gravity. ...
References - 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
- Markopoulo, Fotini; Smolin, Lee - Gauge Fixing in Causal Dynamical Triangulations - shows that varying the time-slice gives similar results
Early papers on the subject: - 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 arxiv.org
External links - Dynamically Triangulating Lorentzian Quantum Gravity
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