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Topic: Causal Dynamical Triangulations

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	Stefan Zohren's Homepage \| Research \| Causal Dynamical Triangulations
		Causal Dynamical Triangulations, for short CDT, is a new approach to the nonperturbative quantization of gravity.
		Another attempt is Causal Dynamical Triangulations (CDT), a covariant path integral formulation, in which Lorentzian quantum gravity is obtained as a continuum limit of a superposition of simplicial space-time geometries [2].
		In the first the dynamics is encoded in the variation of the edge lengths whereas in dynamical triangulations the edge lengths are fixed but the dynamics is encoded in the gluing of the simplicial building blocks.
	www.phys.uu.nl /~0404985/index.php?id=4,11,0,0,1,0 (1999 words)

	List of quantum gravity researchers - Wikipedia, the free encyclopedia
		Jan Ambjørn -- Expert on dynamical triangulations who helped develop the causal dynamical triangulations approach to quantum gravity.
		Abhay Ashtekar -- Inventor of the Ashtekar variables, one of the founders of loop quantum gravity.
		Lee Smolin -- One of the founders and major contributors to loop quantum gravity.
	en.wikipedia.org /wiki/List_of_quantum_gravity_researchers (444 words)

	CDT \| Musings
		The precursor to CDT was the Dynamically Triangulated Random Surface model, a model of Euclidean 2d quantum gravity (coupled to matter).
		The symptom, in the lattice model, is that the triangulations which dominate the functional integral represent a branched polymer or a crumpled phase, rather than anything resembling a smooth manifold.
		The restriction on triangulations in CDT is not, strictly, a local one.
	golem.ph.utexas.edu /~distler/blog/archives/000713.html (3481 words)

	Guide to Reality: Caution: Universe under Construction
		Causal Dynamical Triangulations (CDT) is another research program in quantum gravity which features causality at the fundamental level.
		CDT is similar to Loop Quantum Gravity (LQG) in a couple of respects.
		CDT's method in the context of spacetime geometry is to use “piecewise flat geometries” which are flat except for local subspaces where curvature is concentrated.
	guidetoreality.blogspot.com /2006/10/caution-universe-under-construction.html (742 words)

	Time Travel Portal :: View topic - Quantum Causal Histories (Site not responding. Last check: 2007-10-26)
		The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes.
		The causal set C is transformed to a description in terms of the causal pasts of the events in C.
		The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators.
	timetravelportal.com /viewtopic.php?t=1419 (1708 words)

	Time Travel Portal :: View topic - Lorentzian Quantum Gravity, Causality & Curing Sickness (Site not responding. Last check: 2007-10-26)
		Dynamically Triangulating Lorentzian Quantum Gravity by J. Ambjørn, J. Jurkiewicz and R. Loll.
		It is shown that the combinatorics involved in evaluating the one-step propagator (the transfer matrix) is that of a set of vicious walkers on a two-dimensional lattice with periodic boundary conditions and that the entropy of the model scales exponentially with the volume.
		We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically.
	timetravelportal.com /viewtopic.php?t=1748 (3521 words)

	Dario Benedetti: Quantum Gravity from Simplices: Analytical Investigations of Causal Dynamical Triangulations
		After reviewing in Chapter 1 the methods and the results of causal dynamical triangulations, we proceed in the next two chapters to illustrate in detail two original pieces of work which both represent a small step away from the special case of pure gravity in two dimensions.
		We introduce a particular model of three-dimensional causal dynamical triangulations and show that it can be solved in an asymptotic limit and a continuum quantum Hamiltonian is found.
		This is an important step in the understanding of these kinds of models, because this is the first case in which, for a dimension larger than two, a continuum limit has been obtained analytically, although many things remain to be understood and some approximations still removed.
	igitur-archive.library.uu.nl /dissertations/2007-0621-202502/index.htm (430 words)

	Foliations and 2+1 Causal Dynamical Triangulation Models
		Causal Dynamical Triangulations are an approach that has progressed quite a bit in recent years producing some interesting results regarding precisely the evolution of space.
		Given a particular choice of rules, simulations of the triangulation models are known to produce good results for evolution of space.
		Whether or not the known causal dynamical triangulation models can be extended in the way I consider is a question too complex to resolve using the available analytical techniques.
	astro.uwaterloo.ca /~tkonopka/phys/p_foliations.htm (506 words)

	Spin networks, spin foams and loop quantum gravity
		Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School), 2003.
		Implementing causality in the spin foam quantum geometry, 2002.
		Causality in spin foam models for quantum gravity, 2003.
	jdc.math.uwo.ca /spin-foams (2360 words)

	Shtetl-Optimized: Insert "string" pun here (Site not responding. Last check: 2007-10-26)
		The CDT appraoch is explicitly a discretization (triangulation) of spacetime, the same spacetime that one wants to recover at long distances.
		One of the issues with dynamical triangulation is the simple fact that it has no classical continuum limit.
		This means the sum S over all triangulations (number of simplices N, link lengths a) does not approximate the integral I over all geometries in the sense that S would contain all geometries in some sense as N goes to infinity and a to zero.
	www.scottaaronson.com /blog/2005/10/insert-string-pun-here.html (9627 words)

	CNUFOS.com UFO Pictures Videos and Stories
		Twistors are by definition four-dimensional objects that locate not a position in space and time but rather a network of possible causal relationships between space-time events.
		CDT breaks down tiny units of volume and area -- the crucial stuff that makes up any spacetime -- into tiny tetrahedra, a little like a computer graphics chip renders complex surfaces by decomposing them into many itsy bitsy squares and triangles.
		CDT can be seen, says Smolin, "as a very simplified form of Loop Quantum Gravity." And even if it is not The Ultimate Theory, CDT's practitioners have developed clever solutions and approximation methods that could be used for the real thing.
	www.cnufos.com /pages/newsandstories/stringtheory.htm (1347 words)

	Qgravity.org: Technical Summary of Loop Quantum Gravity (Site not responding. Last check: 2007-10-26)
		The basic papers on causally evolving spin networks are: F. Markopoulou, {\it Dual formulation of spin network evolution} preprint, March 1997, gr-qc/9704013; F. Markopoulou and L. Smolin {\it Quantum geometry with intrinsic local causality} preprint, Dec. 1997, gr-qc/9712067, Phys.Rev. D58 (1998) 084032.
		Unfortunately, as presently understood, this is the situation with Regge calculus and dynamical triangulations in d=4.
		There are also new results in causal set dynamics (by Rideout and Sorkin, and Criscuolo and Waelbroeck) that confirm our discovery of the analogy between the classical limit of theories with fluctuating causal structure and non-equilibrium critical phenomena such as directed percolation models.
	www.qgravity.org /loop (7364 words)

	Homepage of Martin Weigel
		Dynamical triangulations provide a discrete, non-perturbative approach to quantum gravity alternative to the various further directions pursued, including string theory, loop quantum gravity, non-commutative geometry or causal sets [
		To alleviate the severe critical slowing down of the dynamics resulting from the fractal structure of the graphs, a non-local, cluster-type update in the form of the "baby-universe surgery" method is generalized to the quartic case [
		For the Poissonian Voronoï-Delaunay triangulations, while the Ising case q = 2 with α = 0 is marginal and a change of universal properties cannot normally be expected, the q = 3 Potts model, exhibiting α = 1/3, should be shifted to a new universality class.
	www.physik.uni-leipzig.de /~weigel/english/interests.html (2254 words)

	Relation to other approaches
		Arguably the most promising variant of the latter is the causal dynamical triangulations approach by Ambjørn, Jurkiewicz, and Loll [5].
		Since in dynamical triangulations there is no naive (classical) continuum limit, one cannot directly compare the discretized action used with a microscopic action in the previous sense.
		A microscopic metric operator does not exist in a dynamical triangulations approach but if one were to define coarse grained variants, their expectation value would have to be unique.
	relativity.livingreviews.org /Articles/lrr-2006-5/articlesu4.html (2081 words)

	Blogger: Email Post to a Friend
		You triangulate it into simplices, and you imagine that every simplex has a region of flat Minkowski spacetime in it.
		The path integral will be dominated by heavily fluctuating configurations where the topology changes a lot and where the causal relations are totally obscured - and the results of these path integrals will be non-renormalizably divergent - at least if you expand them perturbatively.
		The allowed configurations they include must be not only nice, but they must have the trivial causal diagram as well as a fixed topology - namely S3 x R in their main example.
	www.blogger.com /email-post.g?blogID=8666091&postID=110065519385426918 (2035 words)

	Deriving Dimensions
		In Einstein's theory, spacetime is a stretchy, dynamical entity.
		However, the precise state of any dynamical thing remains uncertain, according to quantum mechanics.
		In previous work, researchers imposed no such "causality." The team also took care to connect the triangles together in a way that preserves the cause-before-effect ordering.
	focus.aps.org /story/v14/st13 (671 words)

	The Quantum Pontiff » Beyond the Frinkahedron
		causal dynamical triangulations approach to quantum gravity: hep-th/0509010: “The Universe from Scratch” by J. Ambjørn, J. Jurkiewicz.
		These are the triangulations that are consistent with a discrete time coordinate (which in topology would be called a Morse function or a foliation) without any topology changes in the spatial slices (no Morse tranisitions).
		The point of that is that by construction all properties of the discretization, the fact that it is geometrical and has nice causality properties, looks x-dimensional and what not- those are exactly the things that are to be “forgotten” in a very technical sense.
	dabacon.org /pontiff/?p=1053 (2340 words)

	[No title]
		Abstract: Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant.
		Dynamical Networks for Information Processing, Information Sciences Abstract: Coupled evolution of state and topology of dynamical networks is introduced.
		Due to a well organized tensor structure, the governing equations are presented in a canonical form, and required attractors as well as their basins can be easily placed and controlled.
	www.comdig.org /issue_txt2.php?id_issue=2004.41 (6512 words)

	With all due respect, is there going to be mud wrestling? Text - Physics Forums Library
		Personally, I think that CDT and Causets are similar: a causal triangulated manifold is actually an excellent example of a certain kind of causal set----the set of simplexes causally ordered by the way they are glued.
		CDT seems to insist on the continuum, and that is fine, but it does not negate the usefulness of and need for discretion.
		It is therefore not unlikely that the mechanism of a dynamical dimensional reduction from 4 to 2 dimensions which occurs in QEG is the same phenomenon as the dimensional reduction observed in the Monte Carlo studies of causal dynamical triangulations.
	www.physicsforums.com /archive/index.php/t-84110.html (15062 words)

	Scientific Commons: R. Loll
		Loll, R. Westra, W. Zohren, S. The recent progress in the Causal Dynamical Triangulations (CDT) approach to quantum gravity indicates that gravitation is nonperturbatively renormalizable.
		Dittrich, D. Loll, R. We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two-tori.
		Dittrich, B. Loll, R. We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two-tori.
	en.scientificcommons.org /r_loll (3613 words)

	Not Even Wrong » Blog Archive » Seattle Conference
		The fact is, gravity is a force, and as such must be the property of an underlying motion, but gravitational force is one that cannot be defined in terms of a quantum field on a fixed background of three space, for reasons that are clearly related to dimensions of the motion involved.
		In all cases (causal set theory, loop quantum gravity, triangulation models), the speculations are based on the naive picture of space as being composed of infinitely sharp points - like in the classical theory - which are moreover exactly discrete.
		Secondly, after integrating out all dynamical variables but the scale factor a() in the full quantum theory, the correlation function between scale factors at different (proper) times is described by the simplest minisuperspace model used in quantum cosmology.
	www.math.columbia.edu /~woit/wordpress/?p=226 (4007 words)

	Guide to Reality: Geometrogenesis
		One challenge for such an approach is that the quest for finding classical dynamical space-time in the low energy limit is made difficult by the fact that the starting point is a timeless, not a dynamical theory.
		(Note though that Causal Dynamical Triangulations is an approach, discussed in this prior post, which has had some success in getting at least the right large scale dimensionality to emerge from a micro-geometric starting point).
		One is that the introduction of dynamics in the micro-theory reintroduces time in a theory that is supposed to be background independent (I personally think if local time and causality exist in the micro-theory, that’s OK).
	guidetoreality.blogspot.com /2007/04/geometrogenesis.html (872 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Date: Mon, 2 May 2005 19:26:17 GMT (19kb) Title: Foliations and 2+1 Causal Dynamical Triangulation Models Authors: Tomasz Konopka Comments: 9 pages, 3 figures, comments welcome \\ Most models of causal dynamical triangulations construct space-time by arranging a set of simplices in layers separated by a fixed time-like distance.
		The importance of the foliation structure in the 2+1 dimensional model is studied by considering variations of the model in which this property is relaxed.
		On the other hand, the introduction of new types of simpleces that puncture the foliating sheets in general leads to different low-energy behavior compared to the original model.
	www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-th:0505004 (123 words)

	Destiny Matrix (Site not responding. Last check: 2007-10-26)
		Because the measurement of lengths and distances ultimately requires an operational definition in terms of appropriate matter fields and states obeying the physical state constraints, ‘dynamical’ discreteness is expected to manifest itself in the spectra of the relevant physical observables.
		The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6].
		Once we ask about the actual local dynamics of the gravitational field, we have to adopt the viewpoint which understands general relativity to be a background independent theory within which the geometry is completely dynamical, on an equal footing with the other degrees of freedom.
	destinymatrix.blogspot.com /2006_01_01_destinymatrix_archive.html (17893 words)

	Shtetl-Optimized » Blog Archive » Insert “string” pun here
		The CDT appraoch is explicitly a discretization (triangulation) of spacetime, the same spacetime that one wants to recover at long distances.
		One of the issues with dynamical triangulation is the simple fact that it has no classical continuum limit.
		This means the sum S over all triangulations (number of simplices N, link lengths a) does not approximate the integral I over all geometries in the sense that S would contain all geometries in some sense as N goes to infinity and a to zero.
	scottaaronson.com /blog/?p=23 (10172 words)

	2d STRUCTURE/IT'S POSSIBLE EFFECTS (Site not responding. Last check: 2007-10-26)
		The relevant quantity in this connection is the degree to which the 4 configuration zones in R3 are occupied by dynamic flux elements.
		The protosimplex flow is a circulatory, periodic motion similar to an oscillation.
		The particles belonging to these invariant basic patterns are in turn combined into several families of spin isomorphisms,in which the spatial flux dynamics of the configuration zones is [b]in dynamic equilibrium[/b].
	www.geocities.com /CapeCanaveral/5473/2dspace.htm (2656 words)

	University of Toronto High-Energy Theoretical Physics: Seminar Speaker Info (Site not responding. Last check: 2007-10-26)
		To explore these web pages, use the menu located at the right (or at the bottom on some small screens).
		String theory as a theory of quantum gravity; applications to fl holes, cosmology & quantum field theories.
		Dynamical symmetry breaking, origin of fermion masses, higher derivative gravity.
	www2.physics.utoronto.ca /~heptheory (192 words)

	Steve Carlip
		An alternative is to approximate the path integral by putting the theory on a lattice, essentially breaking an infinite sum into a discrete, finite collection of "paths." For ordinary field theory on a lattice -- quantum chromodynamics, for example -- the lattice is held fixed, with fields placed at vertices or along edges.
		Called "causal dynamical triangulations" (or "Lorentzian dynamical triangulations"), this program treats time in a new way, choosing new "gluing" rules that guarantee a well-behaved direction of time and, in the process, rule out certain kinds of quantum fluctuation of topology.
		The dynamics of general relativity comes in through the Hamiltonian constraint, a not-fully-understood operator condition that determines the admissible spin networks.
	www.physics.ucdavis.edu /Text/Carlip.html (7597 words)

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