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Another Constraint on Quantum Gravity

  1. Jan 9, 2005 #1


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    This paper by Mohammed Ansari and Lee Smolin, http://arxiv.org/abs/hep-th/0412307, seems to have been overlooked here. The authors argue that for any quantum theory of gravity (they mention LQG and causal triangulations) to produce macroscopic physics - which would be necessary for it to have GR as a limit - it has to obey a certain criterion on its renormalization group flow that is of measure zero in that configuration space. That is, the odds against this criterion being met randomly are infinity to 1. Then Ansari and Smolin try to set up an evolutionary path within complex systems theory to bring the constraint about naturally. They aren't able to prove it happens, but they show a happy outcome in a very simple example.
  2. jcsd
  3. Jan 9, 2005 #2


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    let me check it out;

    Self-organized criticality in quantum gravity
    Mohammad H. Ansari, Lee Smolin
    9 pages, 9 figures

    "We study a simple model of spin network evolution motivated by the hypothesis that the emergence of classical space-time from a discrete microscopic dynamics may be a self-organized critical process. Self organized critical systems are statistical systems that naturally evolve without fine tuning to critical states in which correlation functions are scale invariant. We study several rules for evolution of frozen spin networks in which the spins labelling the edges evolve on a fixed graph. We find evidence for a set of rules which behaves analogously to sand pile models in which a critical state emerges without fine tuning, in which some correlation functions become scale invariant."
  4. Jan 9, 2005 #3


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    ---exerpt from introduction arXiv:hep-th/0412307 v3---

    Since the work of Wilson and others [1] it has been understood that the existence of a quantum field theory requires a critical phenomena, so that there are strong correlations on scales of the Compton wavelength of the lightest particle. If this scaleis to remain fixed as the ultraviolet cutof flength is taken to zero, the couplings must be tuned to a critical point, so that the ratio of the cutoff to the scale of the physical correlation length diverges. This requires asymptotic scale invariance of the kind found in second order phase transitions.

    Similar considerations apply to quantum gravity in a background independent formulation such as loop quantum gravity, or causal set models. The problem is not alleviated if the theory is shown to be finite due to there being a physical ultraviolet cutoff, as in loop quantum gravity[2]. Instead, the need for a critical phenomena is even more serious as there is no background geometry.

    This means that away from a critical point the system may not have any phenomena that can be characterized by scales much longer than the Planck length. That is to say, the volume, measured for example, by the number of events, may become large, but there may still be no pairs of events further than a few Planck times or lengths from each other.
    [my comment: this is the crumpling infinite hausdorff dimension case often found by the DT people in the 90s]

    This is seen in detail in models whose critical phenomena has been well studied, such as dynamical triangulation models[3] and Regge calculus[4]. Away from possible critical points, the average distance between two nodes or points need not grow as the number of events (or the total space-time volume) grows. Instead, one sees that for typical couplings, statistical measures of the dimension, such as the hausdorff dimension, can go to infinity or zero.

    ---end quote---

    the problems described here are familiar from DT papers in the 90s. This description could almost be taken verbatim from an Ambjorn and Loll paper some years back

    " the average distance between two nodes or points need not grow as the number of events (or the total space-time volume) grows. Instead, one sees that for typical couplings, statistical measures of the dimension, such as the hausdorff dimension, can go to infinity or zero."

    Ambjorn and Loll in almost every paper for some years would always draw pictures of the two kinds of pathological behavior (this was before Lorentzian DT): either the thing would CRUMPLE and have hausdorff dimension infinity with some nodes that almost every other node was directly connected with
    or it would get all FEATHERY fractally and have hausdorff dimension less than what it was supposed to----Smolin says zero but it could also be, like, two when it was supposed to be four.

    It was only in 2004 that Ambjorn et al showed that they were now getting hausdorff dimension 4 in their Monte Carlo runs, that is, what Smolin talks about was NOT happening.

    before 2004 they had gotten nice behavior in lower dimension toy models, where the dimensionality came out the way it was supposed to and this
    generic crumply/feathery behavior didnt happen. but last year was the first time they succeeded with a full 4D model.

    I believe this may be a sign that now LQG can successfully tackle a similar problem----so it is an opportune moment for Smolin and others to try. Unless they want to all hop the fence and work on DT instead.

    He also mentions Fotini Markopoulou's "Causal Sets" approach in the same breath as LQG. he says:
    "Similar considerations apply to quantum gravity in a background independent formulation such as loop quantum gravity, or causal set models."
    Causal Sets is not to be confused with Ambjorn Loll "Causal Dynamical Triangulations". Causal Sets is not a simplicial QG model but a kind of graph theory where you have Hilbert spaces at the nodes and Hilbertspace mappings from node to node along the links-----very abstract algebra. Causal Sets seems at the opposite end of the spectrum from Ambjorn Loll CDT stuff. It is not a primitive simplicial model that one could simulate in a computer.
    However Smolin may be right to include a mention of it here. There may be some way in which the recent progress in CDT opens up the possibility of some new results in Causal Sets as well.

    Another outcome is that this paper by Smolin Ansari will help generate a move of people from LQG to Dynamical Triangulations. But as I have suggested I think it is just as likely that they will do some re-building in LQG in order to match CDT.

    obviously too early to be speculating like this but how else are we going to voice our responses to the paper?
    Last edited: Jan 9, 2005
  5. Jan 9, 2005 #4


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    Uh oh!
    it looks more and more (reading the paper) like what you said about the merger of Sears and Kmart
    smolin is doing triangles (which would be tetrahedra in the next iteration)
  6. Jan 9, 2005 #5
    triangulated gravity

    Marcus, you cna have tringualre polyhedra that are not tetrahedrons.

    Triangulated gravity is not news to me. My Rybonic Extrpolations of Synergetics 1 and 2 has propose this for a few years now.
    http://home.usit.net/~rybo6/rybo/index.html [Broken]

    Fuller has suggested that gravity is icosahedral for much longer time than I have.

    Whats is new is my overlapping of the5-fold icoshadra to define 4-fold tetrrahedra and octahedra.

    The other news is two fold.
    1) that of Jim Lehmans only documented discovery of the 3D graphical depiction of the overlapping hexagonal "flower of life pattern" is also the curved version of Fullers Euclidean-only Isotropic Vector Matrix.

    2) and that the 4-fold triangulated hexahedrons(Pods) may be the correct goemtrical representation of either a graviton(spin-2 boson) or a Higgs boson.

    I think the former #1 is the correct and that the Higgs particle can be reprasented as a clustering of these triangulated pods as 4-fold tetrahedra or octahedra. Maybe both. I dunno.

    Fuller often stated taht it was teh artist-mathematicians hwo made discoverys before the phyiscla science.

    Fuller stated that he was using the term "spin" with great circle creation years before the particle scientist did so for subatomic particles.

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  7. Jan 9, 2005 #6
    I have been reading the Dec 27TH print-paper, I have downloaded the present print, but cannot see any revision at all?

    The spin network embedding has an interesting aspect for 'ROTATIONS' caused by the overlapping triangulated network!

    Quote: Einstein called the warping of spacetime to be 2-D.

    A third rotation (one of the line's) from the solid lines of trianglated triangle, produces a 'double spin' of whatever it encompasses?

    Another dimensional aspect for the quantum field(2-D) to be the outer energy of all 3-Dimensional spacetimes?..maybe?

    There is an error contained in the paper on the 'critical behavior' of self orginization, the authors neglect the fact that all equilibriated systems are discrete, and any dynamical evolution will destroy 'equilibrium'?

    Any system in equilibrium is as Einstein stated, in total isolation.
  8. Jan 10, 2005 #7


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    By "dynamical equilibrium state" they surely mean the kind of state they have been discussing, analogous to the sandpile. An initial equilibrium state, which changes by discrete dynamic events to a different equilibrium.
  9. May 21, 2005 #8
    I have seen a 'self-organized' experiment where a wheel containing sand is angled and slowly rotated. Over Time the grains organize themselves into distinct patterns, I wonder if anyone knows of a good site that has a similar experiment available?

    I would be interested on the geometric make-up of specific 'mixed' structured states, ie not just one geometric componant such as sand, but one that can be compared to Triangular and Hexagonal structures?

  10. May 21, 2005 #9


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    From Ansari & Smolin
    If I understand what you are saying Marcus, you are not claiming that Ambjorn et al. have solved or even directly addressed the renormalization group issues that Ansari & Smolin raise. Rather you assert that the Ambjorn et al. achievement of a stable Hausdorff dimension, suggesting a fractal structure (but not proving it, you alsoo need self-similarity for fractals), gives hope that their approach may satisfy the RG fixed point constraint. If so, I don't see your reasoniing. Could you elucidate?
  11. May 21, 2005 #10


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    the earlier pathologies that plagued simplicial gravity models until around 1998 were that the LARGESCALE dimension, instead of being 4D as it should, would turn out to be either 2 or infinity (roughly speaking)

    I have a concert and must run, but basically the spacetimes would turn out either MASSIVELY CRUMPLED or else SPREADOUT FEATHERY.

    this has nothing to do with solving the renormalization problem by having dynamical dimension going down to around 2 at small scale

    this was a pathology in the large scale dimension and a cure for it was found in 1998.

    since then they have been gradually checking the model out starting at 2D and working up to 3D and then, last year, in 4D.

    You have already discussed the renormalization business yourself. that dynamical dimension business is very new. the largescale dimension is 4D but as you noted when I first posted about the "spectral dimension" paper there is this microscopic 2D dimensionality

    no it is obviously not a usual kind of "fractal" although it can have fractional dimension and it can be related to fractal-like behavior. but it cannot strictly be a fractal because it is not selfsimilar----not the same at all scales.

    In a rush so will leave you with the job of straightening out details as needed. fun isnt it :smile: , great stuff!
  12. May 21, 2005 #11


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    Hausdorff dimension can belong to other spaces than fractal ones. Indeed it corresponds to the usual dimension where that is defined.

    My oversimplified description of of an RG fixed point is that the physics (say the action and lagrangian) is scale invariant near a given scale. If a geometry were fractal (globally self-similar at ALL scales) and the geometry necessarily detemined the action, as in most GR actions, then it would seem there was no fixed point, no PARTICULAR scale near which scale-invariance holds. But Ansari & Smolin seem to assert (I am going to reread their paper) that a fixed point is a necessary feature of any QG theory?

    As you say, the current CDT theory, with its running spectral dimension, is not fractal. They don't appear to use a cutoff, either.
  13. May 21, 2005 #12
    Hi selfAdjoint

    Not fractal because not self-similar at all scales? Scales are properties imposed by the observer. Rounding errors, again imposed by the observer, lead to a "washing out" of the fractal pattern after a few hundred magnification steps. In the other direction, entire fractal sets receed to a single point when the scale taken is very large. These charachteristics of fractal patterns appear to me to be very much like observed behaviors of spacetime phenomenology approaching the Planck scale. The fractal pattern appears to me to be clearly defined as occupying one region of a scale continuum, with an upper bound set by the scale of the entire fractal set under observation, and the lower bound set by the computational power of the observer.

    The feathering, layering, foliation, time-like volumes, bones, simplices, triangles, edges, vertices, tetrahedrons, whatever, is due to the movement of the observer and related reference frame through timespace, and it is the timespace which is patterned fractally. In a sense, the observer selects self-similar regions of the fractal pattern and experiences them sequentially.
  14. May 21, 2005 #13
    ummm...what causes triangulations to become dynamic ?
  15. May 21, 2005 #14
    Triangulation is just the minimal trajectoral encircling of and area due to mass-attraction.
    “It is the persuasion of action( motion ), to attract attention( observation ) of another action( motion ). This is gravity.” (Rybo)

    Why does mass attract?

    Obviously, to cohere all of physical Universe, ergo no ultimate entropic annihilation of radiation a.k.a “heat death” of Universe.

    Universe, as the only perpetual motion machine, is perpeutated by, a oscillating between a circumferential, postive octagonal rippling of space, to a inter-mediate hexagonal reippling of space, to a negative octagonal rippling of space, over time, and back again.

    This phemomena is incurred by, crossing points of energy --[ or quasi-energy ]-- having three differrentiations in mass that defines it geometry and is modeled using R.B.Fullers Jiiterbugging cubo-octahedron a.k.a. the vector equlibrium.

    The hexagonal ripple, has four kinds of mass point/crossings.

    1) 6 single-mass points
    2) 1 double-mass point
    3) 1 quadrubple-mass point

    The hexagonal ripple has 5 single mass points plus the one double-mass point defineing its chordal circumference of the hexagon adn 1 set of quadruple mass points as the toroidal nuclear center of the hexagon.

    The octagonal ripple hs 8 single mass points defining its chordal circumference and the 1 quadruple mass point at the toridal nuclear center. This octagonal ripple is aslo known as the “saddle shape” curvature.

    Scroll to 460.1 and 461.08 for Fullers minimal graphic depiction.

    Unfortunatly Fuller did not explore at least he did not publish a more complete set of geometric configuration attainaabe with his jbug model.

    1) the sine-wave topology ( a.k.a triangualar wave form )

    2) the saddle-shape topolgy( negative curvature )

    3) the octagonally square ripple ( postive and negative )

    4) the hexagon ( flatten torus with tetrahedral nucleus )

    http://home.usit.net/~rybo6/rybo/id8.html [Broken]

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  16. May 21, 2005 #15
    Hi Spicerack

    Good question. I'll give you a philosophical answer, if you care to read it. Your question reduces down to asking what time is. It seems clear to me that this is a crucial question. Assuming that there are spatial triangulations which behave statistically within definable limits in a time frame, such that we may interpret the flux of time as a sequence of spatial foliations separated by chronic regions. The chronic parts of the idyll are not layers, exactly, because the time-like volumes of the bones are variables. There can be a lot of time between adjacent spacial surfaces or the spatial surfaces can actually be in instantaneous contact.

    We might ask if there is a Casmir-like effect between foliations, requring a minimization of time volume, but this would have to occur over time, which involves a second time dimension. Time volume changes over time you see.

  17. May 21, 2005 #16
    Hi Rybo
    I think that Fuller had the right idea with the isomatrix, a geometry of densly packed spheres. Others on this forum might have hoped I had forgotten that idea, or at least abandoned it. Nope, sorry.
  18. May 21, 2005 #17

    Richard, The spherical clsoe-packing, isotropic vector matrix( IVM ) is a 4-fold nature( regular tets and octs ) a baryon and leptosn respectivly.

    What I try to show in my Rybonic web site is that there exists a 5-fold matrix of regular icosahedra( gravity ) that bond to each tet in such a way that that each tet is internal to 4 of the bonding icosahedra and external to 4 of the bonding icosahedra.

    That is the simple explanation, however it is of corse much more complicated than that and more complex I have explantions for at this time.

    Suffice it to say, if Loop Quantum Gravity is validated/confirmed in 2006-7 then I thnk Rybonics that is that much closer to also falling onto the quantum geometry bandwagon.

  19. May 21, 2005 #18
    Hi guys

    once again pardon my ignorance but it still sounds like we're still talking about bubbles. In an ideal situation they would be prefectly spherical but in a foam they can have a multitude of sides shared with others like connected soccerballs

    Does the search for an ideal shape/fundamental object making up spacetime require it to be an ideally fixed shape or is the dynamically changing shape due to the passage of time and the uncertainty principle meaning it is what it appears to be only when we put a light on it and observe it ?

    What is it that LQG, strings And CDT all have in common besides the mahtematics or is it too early to say ?
  20. May 21, 2005 #19
    Hi Spicerack

    I donno.

    It sounds like bubbles to me too. Marcus says they take a simplical structure and run Monte Carlo diffusion tests on it...average number of steps to return. Then they change the simplical structure and do it again. Eventually they have a representative sample, I guess, of the space of all possible geometries.

    Seems to me if you take any structure, simplical or not, even a single bone, string, edge, whathaveyou, and run it through every possible invarient transform, or even a big enough sample of the possible invarient transforms, the limits on the set of transforms will surely be spherical. Instead of doing all the statistics and repeat tests and so on, why not just assume a spherical "simplex" to begin with?

    Unless of course they find that the limits on the set of transforms turns out not to be spherical. I havn't done any calculations, it just seems to me that diffusion tests as described "ought" to vary pretty equally in every direction from some average middle.

    We shall see. Marcus?

  21. May 22, 2005 #20


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    hi Richard, it sounds like you are pointing to spicerack's post
    so I could respond to it

    first of all, two extremely smart ladies name Renate Loll and Bianca Dittrich have written a CDT paper where they try DIFFERENT mix of building blox, not just simplexes. that shows you can do that in CDT. but it hasnt been tried much. the field is new. simplexes are the simplest blox.

    it just happened that Loll and Dittrich found that using a certain mix including other shape building blocks facilitated what they were trying to do in that particular paper

    they will soon post a new Loll-Dittrich CDT paper, this time about using CDT to model BLACK HOLES. that will a first (because the field is still very new and even the first black hole model has not been done yet) and so that will be very interesting. we will see if they try any changing of the mix of blox.

    that part seems to interst you all but it does not me. for me it basically does not matter the shape of the basic cells as long as they are very simple and fit together. I actually am partial to simplexes

    SPICERACK please asimilate thing one about CDT that the spacetime IS NOT MADE OF SIMPLEXES. it is a continuum. Only thing is this CDT continuum
    is not describable except by a process of finer and finer approximations, by things assembled out of building blocks. you make the size of the basic block go to zero and the approximation gets better and better.

    so what is made of simplex blox is is only an APPROXIMATE spacetime to what the theory says is the real spacetime

    so in some sense it hardly matters what the basic block is, as long it is adequate to take care of business and accomplish the approximation.

    This spicerack was not a bad question, i will repeat it:

    Maybe someone else will jump in and answer that. I do not know. right now
    I AM INTERESTED IN THE OPPOSITE QUESTION namely what is about CDT that is radically DIFFERENT from both string and LQG. Does that interest you at all? Can you understand why I would be looking at that, for any new approach? For any new approach I think it is a good idea to see what is radically different about it.

    In other approaches to quantum gravity, they tend to have the dimension of spacetime put in by hand. it is either 4 (put in by hand) or some other fool number like 11 or 17 (this not the real number but some fool number established ahead of time to get things to work right for the theorybuilder)

    I suspect that theories like this, where the dimension of spacetime has to be an integer chosen and put in by hand, are TOAST :surprised

    The reason these other approaches have a prior chosen spacetime dimension is simply because they are built on a mathematical object invented in 1850 called a smooth manifold. Smooth manifolds have coordinates (like x,y,z) and the number of coordinates is the dimension of the manifold.

    the reason CDT is different is that it is NOT built on such a manifold, it is built on a later invention called a simplicial manifold where you DONT NEED COORDINATES and you dont have a fixed dimensionality. there are different ways to define the dimension, by performing various experiements inside the spacetime, and the dimensionality DOESNT HAVE TO BE AN INTEGER (because there are no xyz coordinates that you have to count up and have that be the dimension) and the dimensionality can VARY depending on the scale you are looking. if you look with a magnifying glass it can be different. that is the wonderful thing about simplicial manifolds. to my mind.

    however you can set up Einstein gravity in a simplicial manifold. A man called Tulio Regge saw how to do that in 1950. you do it by counting the blocks that come together around the bones of the manifold (a simp manif has a web of borders of borders of the blocks and that web is called the bones and it is like a scaffolding of the manif.

    do you know who Cicero was? He was a good writer of like 100 to 50 BC IIRC. his name was TULIUS. Tulio Regge was named after Cicero. Not all romans were named Julius, some were named Tulius.

    CDT is exactly as different from LQG and string as 1950 is from 1850. heh heh some people might not like my saying that :wink:
  22. May 22, 2005 #21
    thanks Marcus, Nightcleaner and Rybo

    That's more than enough information to keep me amused, confused and enlightened for quite some time

    I like how you say Marcus, it's not whats the same it's whats different that makes it interesting, so yes I can understand your excitement

    CDT sounds like modelling the wire frame of an object before you skin it up in a 3d computer program ?
  23. May 22, 2005 #22


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    Wouldn't it be fun if this were so? But two caveats:

    1) As you have posted the dimension that varies in Ambjorn et al. is the Spectral Dimension of the flow. They identify this with spacetime dimension, but is this justified. I get the impression that a low Hausdorff dimension in the flow context is not surprisng or new per se.

    2) See arivero's post on the low fractal (=Hausfdorff) dimension of path integrals of [tex]\lambda \phi^4[/tex] toy quantum field theory.
  24. May 22, 2005 #23


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    I have replied to arivero's post in the other thread.

    though far from expert here, i have the impression that indeed just as you say it is not at all new or surprising for convedntional path integral PATHS to be very kinky and perhaps to have fractional dimension. But those paths are embedded in conventional 4D surroundings----there could also be very kinky surfaces in conventional settings, also embedded in conventional 4D spacetime.

    the possible difference here is that the SPACETIME ITSELF (not some embedded object) has low fractional dimension, at certain scales. so here the kinky thing is all there is. It has no surroundings. and any matterfields must be defined IN IT, not in some conventional surrounding minkowski.

    BTW as you may have noticed, they do not use the Hausdorff dimension in this recent work that we were discussing, where the dimension of spacetime is less at small scale. The Hausdorff dimension is based on VOLUME and the diffusion dimension they use is based on RETURN PROBABILITY. I was interested to see them using various different measures of dimension, which in their latest paper they point out give various different answers.

    I would be glad to know how you interpret the term "Spectral Dimension". In different branches of mathematics there are dozens or maybe scores of different spectra and spectral this and that. In this case I see a heat kernel and associated Laplace operator and i see eigenvalues and I see that what they call the spectral dimension is explicitly related to these eigenvalues.
    what do you mean when you say "spectral dimension of the Flow"?
    What Flow are you referring to? I have been unable to find a meaning for spectral dimension outside of the statistical physics of diffusion processes.

    In this paper they are not doing renormalization of any sort, they only speculate about the possibility later on, and the only flow in sight AFAIK is a simulated diffusion or random walk in 4D using an artificial time parameter. Is this diffusion (according to the heat kernel they write down) the flow you mean? Sorry if this is an obvious question, but want to make very sure.
    Last edited: May 22, 2005
  25. May 22, 2005 #24
    14 sides/faces on average

    Spicerack, 14 sides/faces is the average number that occur in clso packing of biological cells or any maleable cell-like structure.

    The cubo-octahedron/vector equilibrium/jitterbug has 14 faces/openings( 8 triangular 6 square ) and is Fullers Euclidean based Operating System of Universe that is central to the all-space filling Isotropic Vector Matrix combination of tets and octs.

    Also the tetrakaidodecahedron has 14 faces.

    Im not sure what CDT is but I think that what LQG and String theory have in common besides mathmatics is a Quantum Gravity scenario. LQG being diffrrent in there are soon ot be instruments in space to test its hypothsis, whereas Strings cannot be teste current and is not foreseen to be testable any time in the future.

  26. May 22, 2005 #25


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    Yes Rybo, quantum gravity is what we are talking about in this thread. CDT is a newer approach to quantum gravity. the first CDT papers appeared in 1998.

    LQG goes back to more like 1990, string even further back in history.

    CDT means causal dynamical triangulations
    it is a way of implementing a quantum version of Einstein 1915 classical theory of General Relativity. Other approaches tend to do this using a differentiable manifold to represent spacetime but CDT does not. It uses approximation by PL manifolds (also called simplicial manifolds)

    the einstein action is implemented without using coordinates.

    it is especially interesting because they are more able to compute, with CDT, than with other approaches, so they have been running thousands of computer simulations of spacetime generating random geometries and studying its geometry to find out statistical properties of space time, or what the different geometries have in common

    if you are interested in geometry then you may have thought some about probability distributions on the space of all geometries, that is you may have thought about random geometries. geometric probability. it can be interesting to learn about----there are some classic books on the subject.
    I guess CDT can in a very general way be said to be part of that, as well as being a quantum theory of spacetime and gravity

    here is a short CDT paper that could serve as an introduction
    http://arxiv.org/hep-th/0404156 [Broken]

    I have other CDT links in my sig at the bottom of this post
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