 Causal dynamical triangulation

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Causal dynamical triangulation
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Superfluid vacuum theoryCausal dynamical triangulation (abbreviated as "CDT") invented 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 preexisting 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 great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semiclassical description. At large scales, it recreates the familiar 4dimensional spacetime, but it shows spacetime to be 2d 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, explained why some physicists are excited about it, and put it in historical perspective. The same publication gives CDT, and its primary authors, a feature article in its July 2008 issue.
Contents
Explanation
It is widely accepted that, at the very smallest scales, space is not static but is instead dynamicallyvarying. Near the Planck scale, the structure of spacetime itself is constantly changing, due to quantum fluctuations. This theory uses a triangulation process which is dynamicallyvarying and follows deterministic rules, or is 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 3simplex is usually called a tetrahedron, and the 4simplex, 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. In other words, the timelines of all joined edges of simplices must agree.
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 ddimensional 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 (d1)dimensional simplices and the connection between these slices is made by a piecewise linear manifold of dsimplices. 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 spacelike or timelike 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 nonperturbatively, 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. This is the crucial piece that this theory provides, which was missing before. When we join the pieces only in this way, the simplicial manifold evolves in a more orderly fashion than with earlier theories, and eventually creates the observed framework of dimensions. CDT builds upon the earlier work of Barrett and Crane, and Baez and Barret, which demonstrates the feasibility and utility of this approach, but by introducing the causality constraint as a fundamental rule (influencing the process from the very start) Loll, Ambjørn, and Jurkiewicz created something different. Where others had regarded causality as an emergent property, they made it one of the primary ingredients.
Advantages and Disadvantages
By far the greatest advantage of this theory is that it derives the observed nature and properties of spacetime from a minimal set of assumptions, and needs no adjusting factors. The idea of deriving what is observed from first principles is very attractive to physicists, as it often indicates a concept that is close to the truth, or offers powerful tools for investigating the nature of reality. Since it allows us to probe the character of spacetime both in the ultramicroscopic realm near the Planck scale, and at the scale of the cosmos, CDT can give us many insights into the nature of reality. This is its strength.
The disadvantageous aspect of this theory is that it relies heavily on computer simulations to generate results by Monte Carlo simulation. Some feel that this makes CDT a less "elegant" solution to the problem of creating a completely successful quantum gravity theory. Also, it has been argued that discrete timeslicing 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. 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 BarrettCrane model is essentially a nonperturbative 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 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, hepth/0505154, page 5), whereas causalset theory is not as completely developed at this point.
See also
 Causal sets
 Fractal cosmology
 Loop quantum gravity
 Pentachoron
 Planck scale
 Quantum gravity
 Regge calculus
 Simplex
 Simplicial manifold
 Spin foam
References
 Quantum gravity: progress from an unexpected direction
 Jan Ambjørn, Jerzy Jurkiewicz, and Renate Loll  "The SelfOrganizing 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 timeslice gives similar results
Early papers on the subject:
 R. Loll, Discrete Lorentzian Quantum Gravity, arXiv:hepth/0011194v1 21 Nov 2000
 J Ambjørn, A. Dasgupta, J. Jurkiewicz, and R. Loll, A Lorentzian cure for Euclidean troubles, arXiv:hepth/0201104 v1 14 Jan 2002
 Causal dynamical triangulation on arxiv.org
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
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