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Regge with curved blocks; diffeo symmetry; the Dittrich Bahr papers

  1. Sep 13, 2009 #1

    marcus

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    There's an interesting series of papers by Bahr and Dittrich, and a related video talk by Dittrich at the Planck Scale site. The papers bear on issues discussed in the Lewandowski thread. The first in the series is a good introduction to the rest. Here are some links and abstracts to show what is talked about:

    http://arxiv.org/abs/0810.3594
    Diffeomorphism symmetry in quantum gravity models
    Bianca Dittrich
    Invited constribution to a special issue of Advanced Science Letters, 16 pages
    (Submitted on 20 Oct 2008)
    "We review and discuss the role of diffeomorphism symmetry in quantum gravity models. Such models often involve a discretization of the space-time manifold as a regularization method. Generically this leads to a breaking of the symmetries to approximate ones, however there are incidences in which the symmetries are exactly preserved. Both kind of symmetries have to be taken into account in covariant and canonical theories in order to ensure the correct continuum limit. We will sketch how to identify exact and approximate symmetries in the action and how to define a corresponding canonical theory in which such symmetries are reflected as exact and approximate constraints."

    http://arxiv.org/abs/0905.1670
    (Broken) Gauge Symmetries and Constraints in Regge Calculus
    Benjamin Bahr, Bianca Dittrich
    32 pages, 15 figures
    (Submitted on 11 May 2009)
    "We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so--called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken."

    http://arxiv.org/abs/0907.4323
    Improved and Perfect Actions in Discrete Gravity
    Benjamin Bahr, Bianca Dittrich
    28 pages, 2 figures
    (Submitted on 24 Jul 2009)
    "We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they - although being defined on a triangulation - reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context."

    http://arxiv.org/abs/0907.4325
    Regge calculus from a new angle
    Benjamin Bahr, Bianca Dittrich
    8 pages
    (Submitted on 24 Jul 2009)
    "In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show such a formulation allows to replace the length variables by 3d or 4d dihedral angles as basic variables. Moreover we will introduce a first order formulation, which in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant."
     
    Last edited: Sep 13, 2009
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  3. Sep 13, 2009 #2

    atyy

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    Wow, thanks, exactly what I was wondering about!
     
  4. Sep 13, 2009 #3

    marcus

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    Atyy, I just saw your reply. I'm very happy to know this turns out to mesh with what you've been thinking about!

    Here is Dittrich's talk at the Planck Scale conference, 4 July 2009. This too is a useful introduction leading into this series of papers. The presentation style is clear and deliberate---I found it helpful. And there was a strong audience response, with questions and comment, which adds to one's understanding.
    http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php [Broken]
    Dittrich pdf "In search for diffeomorphism symmetry":
    http://www.ift.uni.wroc.pl/~planckscale/lectures/4-Thursday/4-Dittrich.pdf
    Dittrich video:
    http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php?plik=http://panoramix.ift.uni.wroc.pl/~planckscale/video/Day4/4-4.flv&tytul=4.4%20Dittrich [Broken]

    Dittrich was an invited speaker at Loops 2009 in Beijing. I will give the link to the programme in case some media (e.g. slides, audio) might be posted later by the organizers of the conference.
    http://www.mighty-security.com/loop/timetable1.htm [Broken]
    Dittrich's talk was 3 August, in plenary session the morning of the first day of the conference. The title was "Diffeomorphism Symmetry and Discreteness in Quantum Gravity."

    She also presented a talk 28 August at the Emergent Gravity IV conference. Here's the abstract:
    http://www.emergentgravity.org/index.php?main=main_EGIV_programme.php&banner=banner_default.php&sooter=footer_default.php&test=end%22#Dittrich [Broken]
    It's expected that some media will eventually be posted---e.g. pdf, audio, possibly video---but as with Loops 2009 I am not sure about this.
    The title was "The fate of diffeomorphism symmetry in quantum gravity models".

    I guess I will check these links every week or so to see if media get posted.
     
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  5. Sep 13, 2009 #4

    atyy

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    From the third paper you posted http://arxiv.org/abs/0907.4323 : In this work we did not obtain explicitly an improved action which takes into account propagating degrees of freedom. This would correspond to integrating out higher frequency gravitons and their interactions and finding an effective action. We expect this to be a very complicated task leading to a non–local action. However it is a promising one with possible contacts to other quantum gravity approaches [22].

    Asymptotic Safety! (Sorry, I can't help liking particle physicist approaches :smile:)
     
  6. Sep 13, 2009 #5

    marcus

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    That stands out for me too, and I'm not sorry about it! :smile:
    Some connection there had better happen.
    In fact, as you saw and/or guessed, the reference [22] is to Reuter.

    I harbor a blind hope that Krasnov's action will turn out to have a place in all this.
     
    Last edited: Sep 13, 2009
  7. Sep 13, 2009 #6

    atyy

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    You harbor the blind hope that Krasnov's action is a perfect action?

    You harbor the blind hope that Krasnov's action is asymptotically safe?

    You harbor the blind hope that Krasnov's action is a perfect, asymptotically safe action?

    This one: http://arxiv.org/abs/0811.3147 ? I feel his discussion is very close to the spirit of Asymptotic Safety: we don't insist on an action that produces the Einstein Equations exactly - we study all possible actions that keep its symmetries and degrees of freedom - hopefully some will have a continuum limit, and even more hopefully, there will be on a "small" class that have such a limit.
     
    Last edited: Sep 13, 2009
  8. Sep 13, 2009 #7

    marcus

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    Dittrich's series of papers is probably more suitable for immediate discussion. Perhaps I shouldn't have mentioned Krasnov's proposal. However for instance:
    The Krasnov action has been offered as a possible basis for a new spinfoam model, in which the effective constants like G Newton would run with scale. This would differ from usual GR at extremely small scale (or in some other extreme regime) but might still have the usual largescale behavior.

    This is discussed in a short paper posted recently:
    http://arxiv.org/abs/0907.4064
    Gravity as BF theory plus potential
    Kirill Krasnov
    7 pages, published in Proceedings of the Second Workshop on Quantum Gravity and Noncommutative Geometry (Lisbon, Portugal)
    (Submitted on 23 Jul 2009)

    "Spin foam models of quantum gravity are based on Plebanski's formulation of general relativity as a constrained BF theory. We give an alternative formulation of gravity as BF theory plus a certain potential term for the B-field. When the potential is taken to be infinitely steep one recovers general relativity. For a generic potential the theory still describes gravity in that it propagates just two graviton polarizations. The arising class of theories is of the type amenable to spin foam quantization methods, and, we argue, may allow one to come to terms with renormalization in the spin foam context."

    He offers a generalization of the Plebanski action. To me it looks like a simplification, less cluttered. He proposes to construct a class of spinfoams based on this action. He conjectures that this larger class of spinfoams is closed under refinement. That is you can divide up the cells of the complex, and divide and divide and divide.
    He makes the risky conjecture that this could converge to a UV fixed point.
    So that one would have an explanation of the running of constants with scale. The spinfoam model, or class of models, would in a sense explain why Reuter might have found a UV fixed point. The risk of this gambit makes me nervous. But I respect Krasnov's courage and independence. He takes longshots. And Ingemar Bengtsson earlier had some similar idea of an action like what Krasnov has written. Bengtsson has now posted a paper urging that this be pursued some.
    ====quote from Krasnov's paper====

    ....Indeed, its original motivation in [14] was precisely to come to terms with the renormalization in quantum gravity.

    Translated into the language of spin foams this renormalization motivation may be formulated as follows. In spin foam approach to quantum gravity one obtains an amplitude for a manifold by “gluing” together amplitudes for the individual spacetime simplices, see e.g. [3] and references therein for more details. Let us consider the “renormalization” in the context of spin foams, i.e. analyze what happens when one computes the simplex σ amplitude as the result of integration over the labels of the “smaller” simplices that are glued together to make σ (in a technical jargon this corresponds to an e.g. 5 → 1 move). When the elementary simplex amplitudes are built as dictated by the Plebanski action (5) (or its SO(4) version), the new simplex amplitude – the result of the spin foam “renormalization group flow” – is of a different type, not anymore describable as coming from the original Plebanski action. This is, we believe, how the non-renormalizability of GR manifests itself in the spin foam context. Thus, the spin foam renormalization group flow does not preserve the classical action (5) one starts from. As we have already said, we find this entirely natural, and having to do with the non-renormalizability of the underlying theory.

    It is however possible (but quite non-trivial to show) that some larger class of theories may be closed under such a renormalization group flow. In the discrete setting of spin foams this would manifest itself in the simplex amplitude given by the result of the 5 → 1 move being of the same type as one started from, but with all the coupling constants – parameters of the theory – being changed in some subtle way. Should one find the class of theories with such a property, one can then see whether its UV completion exists by determining whether there is some non-trivial UV fixed point of the flow. This fixed point, if exists, would then provide the sought UV theory. It is then clear that the first step in the direction of this program is to enlarge the class of gravity theories that is being considered. We would like to propose the class (6) as a viable and natural arena for these ideas in the spin foam context.
    ==endquote==

    The 1 → 5 move is where you take a 4-simplex, whose boundary is 5 tetrahedra, and place a vertex inside it, say at the middle. And then you connect that vertex to each of the 5 tets and get five smaller simplices. So the original 4-simplex is divided up into five smaller ones.
    Unless I'm mistaken, he is simpy thinking of the flow going in the opposited direction, to the IR. So that's why he writes the reverse move 5 → 1.

    We should not get distracted so that we forget about Dittrich's program. I have a feeling that Rovelli at some point in his 5 hours of School lectures this week is going to cite what Bahr and Dittrich are doing. It is of immediate general relevance. He almost has to. With luck we will get pdf of the slides.
     
    Last edited: Sep 13, 2009
  9. Sep 13, 2009 #8

    atyy

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    I like Krasnov's line of thinking too. I think it is related to Dittrich and Bahr's thinking also. And both I think are also related to Asymptotic Safety, if such a thing exists.

    So EPRL and FK are to Krasnov and Dittrich and Bahr as Ambjorn, Loll and Jurkiewicz are to Reuter and Percacci. It would be absolutely amazing if Krasnov guessed the result of Reuter and Percacci's calculation without actually doing the calculation.
     
  10. Sep 13, 2009 #9

    atyy

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    Does Dittrich say in her reply to Weinfurtner at the end the 3D model plus cosmological constant has (naively) too many degrees of freedom, like Horava gravity :confused:
     
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  11. Sep 13, 2009 #10

    atyy

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    http://arxiv.org/abs/0907.4323 "That perfect actions exist for asymptotically free theories follows from Wilson’s theory of renormalization group flow [12]." I wonder if the perfect actions for asymptotically safe theories also follow from renormalization flow. The quote in post #3 also from this paper would make sense if it did.

    So Dittrich and Krasnov should take the Krasnov action and do the flow and see if the effective action they get is perfect. :biggrin:

    I suppose the Dittrich and Bahr thing is not so closely tied to Asymptotic Safety as Krasnov, since it'd still be viable without Asymptotic Safety. But I think that would be inelegant, since then it'd be motivated by solving a LQG problem :tongue2:
     
    Last edited: Sep 13, 2009
  12. Sep 13, 2009 #11

    marcus

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    Atyy, you are several steps ahead. I am just waiting now, as patiently as I can, not for anything involving Krasnov, but to see how much of Dittrich's and Bahr's recent work gets into Rovelli's series of 5 talks. As I recall his time slots are all scheduled in the second half of the week, like Thurs, Fri, Sat mornings.
    I will listen again to part of Dittrich's Planck Scale talk. I remember Weinfurtner almost did not ask her question because they were out of time but then Fotini Markopoulou handed her the microphone and indicated it should be brief. I will listen again to the very end. The past I am OK with :biggrin: the future I am reluctant to talk about (when I care, like now.)
    ===============
    BTW here is Rovelli's abstract of his qg school talks. If he doesn't cite Bahr Dittrich fine OK, he's the pro. But if he does I as specatator will be extra content. See if you don't think it would fit in:

    ==quote from program==
    I present a new look on Loop Quantum Gravity, aimed at giving a better grasp on its dynamics and its low-energy limit. Following the highly succesfull model of QCD, general relativity is quantized by discretizing it on a finite lattice, quantizing, and then studying the continuous limit of expectation values. The quantization can be completed, and two remarkable theorems follow. The first gives the equivalence with the kinematics of canonical Loop Quantum Gravity. This amounts to an independent re-derivation of all well known Loop Quantum gravity kinematical results. The second the equivalence of the theory with the Feynman expansion of an auxiliary field theory. Observable quantities in the discretized theory can be identifies with general relativity n-point functions in appropriate regimes. The continuous limit turns out to be subtly different than that of QCD, for reasons that can be traced to the general covariance of the theory. I discuss this limit, the scaling properties of the theory, and I pose the problem of a renormalization group analysis of its large distance behavior.

    ==endquote==

    OK the answer to Silke starts around minute 28. I put the player on pause and let it download nearly the whole talk without playing, while I did something else, then I dragged the button almost but not quite to minute 29 and it stayed there. So I turned it on and listened. But, big deal, I can't answer very precisely. She was not talking specifically about Loop or Foam but generally about any theory of a broad class including Horava (which is Silke's special research interest and what she asked specifically about.)
    Dittrich used the example of 3D quantum gravity where discretizing (and restoring symmetry) introduces new degrees of freedom, but you find at the end that they all vanish in the infrared. She asked "how do you sure this happens in general?" Under what conditions can you expect the new DoF to vanish in the limit? It sounded to me as if she imagines that there is a theorem here which needs to be stated and proved. Which then would apply to several theories which have diffeo symmetry and which you might want to discretize, restore symmetry, and then take to the limit. I'm not sure I understand so I'll listen another time. It's quick to do when you don't have to watch the whole talk :biggrin: Just learning the ins and outs of this medium.
     
    Last edited: Sep 13, 2009
  13. Sep 13, 2009 #12

    atyy

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    Hmm, I don't know. If I had to guess Rovelli's abstract seems to be about EPRL, it's derivation from Holst, it's relation to LQG, and the semiclassical limit. Why do you think it's related to Dittrich and Bahr, is it the mention of LQG which Dittrich mentions? Has Rovelli mentioned QCD in any of his papers?
     
  14. Sep 13, 2009 #13

    marcus

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    I probably shouldn't get into why. I don't have sound rational reasons. Like she is the kind of horse that wins you races and if there is any justice, and if Rovelli knows what is good for his community and his line of research, damn it. How can you not feel proud of Bianca. Outstanding research, and also she writes exceptionally clear articulate English. Heh heh, in other words I don't have reasons about this, merely feelings.

    BTW in answer to your question I don't recall Rovelli mentioning QCD several times in the abstract of any earlier paper, but I could easily have missed an instance.
     
  15. Sep 13, 2009 #14

    atyy

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    Oh, I too am a Dittrich fan regardless of Rovelli :smile:
     
  16. Sep 15, 2009 #15

    atyy

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    http://relativity.livingreviews.org/Articles/lrr-2006-5/ [Broken]
    The section "A Reminder on Kadanoff–Wilson Renormalization" mentions perfect actions.
     
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  17. Sep 15, 2009 #16

    RUTA

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    I wonder why there's no more talk in Regge calculus QG about the discrete being fundamental to the continuum? E.g.,

    "The quintessence of Einstein's theory of general relativity lies in its invariance under a general coordinate transformation that leaves ds^2 unchanged. As remarked before, this invariance also holds for the discrete gravity action (1.1); the lattice can be completely arbitrary, regular or irregular, and its links can be of any lengths e, long or short. In addition, as we shall see, the lattice theory enjoys still another totally new class of symmetries which does not exist in the usual continuum theory. Aesthetically, this adds greatly to the appeal of lattice gravity. For physical applications, when e is small, our general formula (2.26) insures that all known tests of general relativity are automatically satisfied. Furthermore, by keeping e nonzero, we see that the lattice action per volume possesses only a finite degree of freedom. The normal difficulty of ultraviolet divergence that one encounters in quantum gravity disappears in the lattice theory. All these strongly suggest that the lattice theory with a nonzero e may be more fundamental. The usual continuum theory is very likely only an approximation." pp 364-365


    Nuclear Physics B245 (1984) 343-368
    LAITICE GRAVITY NEAR THE CONTINUUM LIMIT
    G. FEINBERG, R. FRIEDBERG, T.D. LEE and H.C. REN
     
  18. Sep 15, 2009 #17

    marcus

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    Because there's a page break in the discussion thread, I will copy RUTA's post:

    RUTA, I found 6 papers published in 2008 or later which cited the 1984 paper you mentioned by Richard Friedberg et al
    http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+C+NUPHA%2CB245%2C343&FORMAT=www&SEQUENCE=ds%28d%29 [Broken]
    So you picked a good paper :biggrin: It is still being cited 25 years down the line!

    I have the impression that there is a persistent interest in Regge calculus and perhaps even a recent surge. One of the invited plenary talks at the MG12 conference in Paris this July was by Herbert Hamber---I think currently the most recognized Regge expert after R. Williams.

    I don't have access here at home to the 1984 Friedberg paper. Could you explain how something that looks like a continuum, with the usual continuum symmetries at large scale, can emerge from something fundamentally discrete? I dimly comprehend how it might, but it would help if someone (you, Atyy?) could talk a bit more at length.

    And does it make any practical operational difference? If discreteness is inaccessibly fine, does it matter what sort of mathematical model one uses? I suppose yes, but cannot at the moment say why. Why should we care if the ground of geometry is continuous or grainy? Isn't it like asking if matter is waves or particles? (No it's not like that, but you get the drift of what I'm saying.)

    I'll get the link for Herbert Hamber's plenary talk at Marcel 12. For anyone not familar with it the Marcel Grossmann meeting on recent advances in General Relativity, Astrophysics, Relativistic Field Theory etc. is held every 3 years and recently drew 800-plus participants. Among the invited talks, Juan Maldacena gave the String talk, Laurent Freidel gave the Spinfoam, and Hamber gave one on "Lattice QG". That was it for QG in the plenary session. So let's see what Hamber said.
    http://www.icra.it/MG/mg12/talks_plenary/Hamber.pdf
    He introduces Regge calculus (and modern computation methods as applied to it) around slide #14. That is about 40% of the way through the slides. So you have to scroll down about 40% to skip the introduction.

    You may be very familiar with all this because of your exact specialty or general research interests. But I'm not certain and it might be new to other people.
    ==EDIT TO REPLY TO NEXT==
    About that Nuclear Physics B article, I'm a short walk from the physics building and I have to go there this afternoon to hear a talk by STEVE CARLIP! The open stacks are right there, so I will glance at it. Thanks for the offer of sharing the pdf. I'll see if I think its a keeper or just a looker. (I'm just an interested person, not an active researcher like yourself.)
    I see I made an error earlier and referred to this year's Marcel Grossmann meeting as MG13 instead of MG12. It was the twelfth in the series.
     
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  19. Sep 15, 2009 #18

    RUTA

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    Thanks, marcus. You started this thread just as my research brought me to Regge calculus QG, albeit from a very different direction.

    First, I have the 1984 paper in pdf form on my computer. How can I get it to you? I doubt it's legit to post it, since you have to pay a fee or subscribe to a citation service to get it otherwise.

    Second, if one accepts discrete theories as fundamental with the continuum theories as their approximation, then UV divergences and inequivalencies are merely artifacts of the approximation. If you believe the converse, then the avoidance of UV problems via lattice methods is ad hoc because the "real" (continuum) theories "really" have these problems. Even in the classical realm, if you consider Regge calculus as fundamental and GR as its approximation, you attack the problem of singularities (black hole, big bang, etc.) very differently.
     
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  20. Sep 16, 2009 #19

    atyy

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    I think fundamental discreteness is usually said to have the problem that the cut-off can be imposed arbitrarily, and this arbitrariness in conjunction with matter models, will lead to an infinite number of models, and no predictivity.

    Anyhow, one approach which I like that is fundamentally discrete but looks like a continuum at large distances is causal sets. They ask and answer the question "when can a Lorentzian manifold (M, g) be said to be an approximation to a causet C?" http://arxiv.org/abs/gr-qc/0601121

    Also, I think Freidel does think of spin foams as fundamentally discrete, with the continuum only emerging as an approximation. I like this better than standard LQG, because Smolin and Rovelli's "background independence" seems silly to me, and if one hypothesizes Asymptotic Safety, why not just study Asymptotic Safety, which is well motivated from the particle physicist's renormalization group? So to me a vision of spin foams providing fundamental discreteness gives LQG more identity (or maybe makes it not LQG!). Anyhow, from his remarks in the paper on the semiclassical limit, I thought he was thinking of fundamental discreteness, so looked at his other papers, as well as references he cited, and found these interesting bits:

    Freidel, http://arxiv.org/abs/hep-th/0505016 "The product involves only tree Feynman graphs. Using the correspondence between GFT Feynman graphs and discrete manifolds one sees that all the manifolds involved in the sum are of the same topology and describe a ball on the boundary of which the operators are inserted. Since this product is independent of the choice of triangulation it can be thought as a ‘continuous’ scalar product."

    Oriti, http://arxiv.org/abs/gr-qc/0607032 "we point out the connections with other approaches to quantum gravity, such as loop quantum gravity, quantum Regge calculus and dynamical triangulations, and causal sets. ................................... the spin networks appearing as boundary states or observables in the GFT framework are inherently adapted to a simplicial context in that they are always D-valent in D spacetime dimensions .................... Finally, there would be a fundamental discreteness of spacetime "
     
    Last edited: Sep 16, 2009
  21. Sep 16, 2009 #20

    Haelfix

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    It is very hard to make a truly discrete model (and not something like Regge calculus where you are supposed to morally take the continuum limit) of physics work with inflation. After ~60 efolds, the discreteness is blown up astronomically. Even if you are smart about the choice of triangulations (for instance stochastic triangulation schemes) you will still expect some statistical anisotropy to persist through many different scales.

    You could make the discreteness many orders of magnitude below the Planck scale to evade observational constraints (lorentz breaking limits, structure formation and CMB constraints, etc) after reheating, but then you would still face the divergence problem and lack of predictivity of quantum gravity (in fact it becomes uncountably worse, b/c you have the choice of an infinite variety of possible triangulations, along with the explosion of all those free parameters that must be carefully tuned). The point being, you can't appeal to fundamental discreteness to avoid the problems of quantum gravity... It makes the finetuning problems worse, not better.
     
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