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Prospects of the canonical formalism in loop quantum gravity

  1. Oct 13, 2011 #1

    tom.stoer

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    There are still unsettled questions in loop quantum gravity, especially regarding uniqueness of the Hamiltonian constraint, constraint algebra, on-shell vs. off-shell closure, operator norm and convergence, ultra-locality, possibly quantization anomalies. These questions have been asked in Nicolai's "an outside view" paper more then five years ago, they are been adressed by Alexandrov, Thiemann ist still working on these issues, ...

    So it seems that besides the reformulation of LQG in terms of spin foamns which makes the theory more tractable for practical purposes there still seems to be the question of the consistent definition of loop quantum gravity and the relation between its different formulations. It seems that not only are these formulations considered incomplete by themselves, but that both their fundamental formulations and their relation is still unclear.

    I am currently studying the paper

    http://arxiv.org/abs/1110.2157v1
    Lessons from toy-models for the dynamics of loop quantum gravity
    Authors: Valentin Bonzom, Alok Laddha
    (Submitted on 10 Oct 2011)
    Abstract: We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.

    Even if the toy models considered in this paper do not teach us anything new, its worth reading the first sections b/c the authors summarize the issues listed above, they present a rather comprehensive overview plus relevant references.

    I do not want to disparage the new SF perspective developed over the last few years, but it should be stressed that there is more than one perspective ...
     
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  3. Oct 13, 2011 #2
    Perhaps I'm naive, but I find it hard to morally believe in the canonical formulation. Again, perhaps because I'm not well-versed in it, I don't see any reason to believe that such a method can work in principle. "Quantization" is, after all, not well-defined, and one must put in more information to complete the procedure; the only benefit is that after completing the program one can trivially see that the relevant *-bracket structures are preserved --- so the reduction to classical GR is should then be obvious.

    The covariant/SF approach seem to have trouble showing for sure that GR is recovered (though I'm optimistically hopeful that it will work out), but at least the definition of the theory as a quantum theory seems clear. There is a well-defined sequence of calculations for various things, and like lattice QCD the "only" difficulty is to actually do them.

    I guess my personal flavour is to prefer well-defined, computable (though not really effectively) things over ill-defined but potentially very elegant models.
     
  4. Oct 13, 2011 #3

    atyy

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    How about LQC? That seems to be working out, even though LQG is going nowhere, and I am skeptical of the Rovellian view of spin foams.
     
  5. Oct 13, 2011 #4
    Actually there _is_ a major difference as compared to lattice QCD. As said before, the latter is UV complete, ie renormalizable and unitary, and there are reasons of universality that, roughly speaking, no matter where you start in the right universality class, you end up with the same theory in the IR.

    As for gravity, which is not UV complete (at least in the traditional sense if we neglect a possible self-unitarization by classicalization or something like that), there is no reason why any such notion of universality should hold and this may well be the inherent reason why there seems to be an infinite amount of ambiguities to even define such theories. No matter what, the discussion is bound to always come back to this or related points.

    Many people in this field have the feeling that starting with some classical gravity theory and then canonically quantizing it, is a very wrong starting point in the first place.
     
  6. Oct 13, 2011 #5

    marcus

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    There certainly are a lot of open questions to be worked on in QG! The field is in active ferment and going through a creative period of growth.

    I want to note that Eugenio Bianchi has promoted a third perspective to stand beside the two main others (abstract SF and canonical).
    http://pirsa.org/11090125/

    For what seems a long time we have been hearing suggestions about this---but I have the impression always as a side remark or footnote or lowerdimension toy illustration. I never saw it so clearly developed as in Eugenio's talk. So I think of it as his project.

    I think there was even a paragraph or two about it in the Zako lectures 1102.3660. But as a side comment: the main line of development there was abstract SF (with abstract SN boundary).

    http://pirsa.org/11090125/
    Loop Gravity as the Dynamics of Topological Defects
    Eugenio Bianchi
    A charged particle can detect the presence of a magnetic field confined into a solenoid. The strength of the effect depends only on the phase shift experienced by the particle's wave function, as dictated by the Wilson loop of the Maxwell connection around the solenoid. In this seminar I'll show that Loop Gravity has a structure analogous to the one relevant in the Aharonov-Bohm effect described above: it is a quantum theory of connections with curvature vanishing everywhere, except on a 1d network of topological defects. Loop states measure the flux of the gravitational magnetic field through a defect line. A feature of this reformulation is that the space of states of Loop Gravity can be derived from an ordinary QFT quantization of a classical diffeomorphism-invariant theory defined on a manifold. I'll discuss the role quantum geometry operators play in this picture, and the prospect of formulating the Spin Foam dynamics as the local interaction of topological defects.
    21 September 2011

    Who knows if this will succeed? Progress is made by branching out and trying new ways.
     
    Last edited: Oct 13, 2011
  7. Oct 13, 2011 #6

    tom.stoer

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    In principle or in QG only? It is well-define in QM; it works well in QED and QCD; the PI formalism was derived via the canonical one, so where's the principle problem?

    It's more difficult in the canonical formalism.
     
  8. Oct 13, 2011 #7

    tom.stoer

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    Why? b/c it's too restrictive?

    It's the "shut-up-and-calculate" approach. Other's are still working on the fundamental problems.
     
  9. Oct 13, 2011 #8

    atyy

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    But it works! Isn't that a clue to make it less restrictive? (I have no idea what the clue means, but someone else like Ashtekar, Thiemann, Lewandowski or Bahr should?)

    I'm skeptical of the calculations which take the Immirzi parameter to zero and appear to be consistent with Einstein gravity. The reason is that we do expect Einstein gravity to be recovered in some limit, after all, the theory is a discretization of the Holst action. What's important is that it is recovered in the correct limit. Even more important, given that the problem of non-renormalizability is one of uniqueness, not finiteness, is that the theory is not triangulation independent, unless the Rovelli-Smerlak limit exists. Actually, that proposal is the one thing I like about Rovelli's work. I hope it exists, and that Einstein gravity is not recovered in the IP→0 limit, and that we end up with something like AdS/CFT or better.
     
  10. Oct 13, 2011 #9

    tom.stoer

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    I would say that removing all restrictions from LQC you end up with LGQ; and it works only in a very restricted sense. Most of he semiclassical limit of LQC is already present as input or as restriction. But I agree that one can learn something about LQG in general.

    This limit (or something else) should become a "dynamical" or "scaling" limit produced by renormalization, not by hand; there are some preliminary attempts ...

    To make one thing clear: I still think that LQG is a very promising approch, but this is not necessarily due to its phenomenological success (which we do not see yet) but due to mathematical rigor (which certainly plays a very important role in the deep QG regime). If the issues I listed in post #1 cannot be resolved, I am afraid the whole effort is in vain!
     
  11. Oct 13, 2011 #10

    Fra

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    I symphatize with this view.

    I think it is not the right starting point for unification either, but I think the "symptom" for applying the scheme is different in the two cases.

    What do you think about the informal suggestion that ambigousness of the hamiltonian constraint, is somehow one thing they face instead of a landscape problem? Does that make sense to you? or is it just me thinking so

    /Fredrik
     
  12. Oct 13, 2011 #11

    qsa

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    lets say LQG does manage the classical limit. and string reproduces the standard model. Are they really telling us about THE UNIFYING concept of gravity with other forces, I think even the most naive physists should know better. Let alone the million( well maybe only ten) other questions.
     
  13. Oct 13, 2011 #12
    @ Fra
    If we talk about pure gravity, any theory better reproduces the landscape of solutions to the Einstein eqs, to lowest order. If this theory is not unambigously defined, then there might be an arbitraryness on top of it, perhaps in the infinitely many counter terms added to the Einstein action, which play no big role at low energies/curvatures but become important at high energies/curvatures. Perhaps requiring the theory to be unitary would fix some or many or all such terms; perhaps unitary can never be achieved in this framework, god knows!
     
  14. Oct 13, 2011 #13

    tom.stoer

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    I agree that quantization of a classical theory seems strange b/c it tries to "invert the limit h = 0" ... and canonical quantization may be dangerous due to loss of large diffeomorphisms and things like that ... nevertheless it works in many cases. So let me ask why gravity is fundamentally different. I mean not simply more complicated in practice but really conceptionally different! In LQG it seems that it's harder than usual but that the naive approach might work. In string theory you essentially do the same (OK, you do not take gravity, but you take a classical theory and quantize it). If you want to address a REALLY fundamental issue then you should ask about an alternative to quantization i.e. a concept to define a theory w/o knowing its classical limit.
     
  15. Oct 13, 2011 #14

    marcus

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    Does "unification" always mean "unification of forces" for you?

    I think gravity is a theory of geometry (not just another force.) So trying to unify it with the three forces is the wrong goal.

    I see the program as connecting GR with QM. GR is about dynamic geometry. To connect it with QM means to find out how to do quantum geometry.

    In other words, forget about "unification of forces". Find out how to represent quantum states of geometry. A hilbert space of quantum states of the geometry of the universe. Geometric operators, corresponding to making geometric measurements (of area angle volume etc.). Quantum dynamics governing geometry.

    Then once you have a quantum geometry, put matter and forces into it.

    To me this seems like the logical unification program. "Unification of forces" does not seem logically well-founded.

    What do you think? Do you actually believe that unification means unification of forces?
    Should that be the aim of QG, then?
     
  16. Oct 13, 2011 #15

    marcus

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    It seems to me that for a couple of years now, Rovelli has been regularly making explicit that he does not think starting with classical GR and quantizing is the right way to go. He pointedly prefers the other direction. Formulate a definite quantum theory of geometry. Check that it has the right limits. See if phenomenologists can figure out practical ways to test it.

    IIRC he gives arguments for this approach in 1102.3660. Including incomplete but suggestive convergence of several GR quantization programs which all point towards this theory, increasing the likelihood of its working out.

    To believe Suprised, MANY people would agree with Rovelli that quantizing classical GR is not the way to go. Apparently many people believe it is better to do as he does, namely start with a quantum theory and check/test.
    ================================

    Tom, you ask why GR is fundamentally different. I think you have already reflected on this and have some tentative answers in mind. But I will venture an obvious one. GR is fundamentally different because it is a theory of GEOMETRY in which other things occur and other fields are located.

    Therefore it must be fundamentally different.

    A quantum theory of geometry is a theory of the framework for fields and events. A quantum state must specify such a framework---in which other things can happen. So it is quite a different problem from, say, "grand unification" of 2 or 3 particle forces.

    I am intrigued by Eugenio Bianchi's proposal because it uses a manifold---where I imagine all the usual particle stuff can be defined---and the manifold is uniformly flat except on a web of defects. These defects run through the manifold and are where the curvature lives. The idea is neither altogether novel nor certain to work, but interesting nevertheless.
     
    Last edited: Oct 13, 2011
  17. Oct 13, 2011 #16

    atyy

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    Well, here's Ashtekar and Singh's latest review. The last section does have the things I hoped they'd speculate about - lessons of LQC for canonical LQG and spin foams. http://arxiv.org/abs/1108.0893
     
  18. Oct 13, 2011 #17

    tom.stoer

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    marcus! this is not true!

    Rovelli says that quantizing a classical theory is not relevant once you have a quantum theory; therefore he concludes that it may be time to look at the quantum theories we have at hand (LQG/SF is rather a class of models, not a single theory) and to see where they take us (instead of studying their derivation). But of course these theories HAVE been created via quantization, so classical GR seems NOT to be the wrong starting point (Rovelli is not suffering from amnesia, is he?) But this program has not yet been completed. I agree that these theories seem to be consistent quantum theories, but a proof is still missing. In addition it is not clear if and how canonical LQG and SFs are related. If they are equivalent it would be nice to see why; if not it is of major importance to learn exactly why this equivalence fails!

    Rovelli is promoting one way of doing quantum gravity - and his way is certainly OK. But there are other ways to attack this problem, and these are not wrong, either. They support each other, definitly.

    I think we agree to try to falsify a theory of QG based on experiments solely could be a very long-term and therefore risky strategy. If there are doubts regarding the correctness of a physical theory one must try to find the the root of evil - and especially in QG it might very well be that you are able to find a serious flaw in the construction / quantization / consistency whereas it will take eons to find the error in the data :-)
     
  19. Oct 13, 2011 #18

    marcus

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    I did not say R. was suffering from amnesia :smile: Each time he has said this he has referred to the several past attempts to quantize, which have pointed towards but not precisely arrived at the current abstract SF formulation of LQG. These attempts at quantizing have been helpful heuristics. As he says in http://arxiv.org/abs/1102.3660 section on "Derivations".

    ==quote Zako lectures page 23 and following==

    V. DERIVATIONS
    I have presented the theory without deriving it from classical general relativity. There are a number of distinct derivations that converge to* the theory. In this last section, I sketch some basic ideas in these derivations. A word of caution is however needed.
    Quantum-gravity research has often focused on setting up and following “quantization paths” from classical general relativity to a quantum theory. These are very useful to provide heuristic indications for constructing the quantum theory, but they are neither sufficient nor necessary for taking us to quantum gravity. If there was a straightforward quantization route, the quantum theory of gravity would have been found long ago. Any generalization requires a certain amount of guesswork. The “quantization paths” sketched below must be seen as nothing more than heuristics, which have given suggestions useful for construction of the theory, and shed light on aspects of the definitions.
    The theory itself should not be evaluated on the basis of whether or not quantization procedures have been “properly followed” in setting it up. It must be judged on the basis of two criteria. The first is whether it provides a coherent scheme consistent with what we know about Nature, namely with quantum mechanics and, in an appropriate limit, with classical general relativity. The second is to predict new physics that agrees with future empirical observations. This is all we demand of a quantum theory of gravity.


    Since for the moment we do not have so many useful empirical observations, it might sound that the considerations above give us far to much freedom. How then to choose between different quantum gravity theories, or different ways of constructing the theory? This question is asked often. I think it is a misleading question, for the following reason. At present, we do not have several consistent, complete and predictive theories of quantum gravity. In fact, we are near to have none at all. Most of the quantum gravity approaches lead to very incomplete theories where predictions are impossible. Therefore the scientifically sound problem, today, is whether any complete and consistent quantum theory of gravity can be set up at all. If we can solve this problem, it is already a great success, after decades of search. The issue of checking whether this is the right theory, namely the theory that agrees with experiments, comes after.
    ==endquote==

    *"to the theory" here is to be understood in the sense of "towards". None of the several quantizations arrive exactly at the present formulation---they all point towards it from different directions.
     
    Last edited: Oct 13, 2011
  20. Oct 14, 2011 #19

    Fra

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    I agree, this was what I meant too with not thinking it's right starting point for unification of the other forces either (I mean even w/o gravity).

    /Fredrik
     
  21. Oct 14, 2011 #20

    tom.stoer

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    @marcus: first of all I think we agree that Rovelli does not say that classical GR with its quantization is a wrong starting point! He talks about "useful ... heuristic indications" - but means that this is of of limited relevance. And he makes clear why it's of limited relevance, namely b/c "distinct derivations ... converge to the theory" - or perhaps they don't - we don't know yet - b/c "we do not have several consistent, complete and predictive theories of quantum gravity ... we are near to have none at all". So it's about the relevance or the weight of different approaches and different interpretations and ratings of "quantization" or "construction" - and that's that's pretty subjective.

    I don't think that everybody in the community agrees with him. Think about Ashtekar's point of view - you'll find it in the LQC review paper http://arxiv.org/abs/1108.0893 - or think about Thiemann's research program; or Nicolai's "outside view" to which - after 6 years - there still seems to be no fully satisfactory reply; or think about the overview presented in the paper I cited in the first post of this thread.

    Let me stress some of my recent statement:
    If the issues I listed ... cannot be resolved, ... the whole effort is in vain.
    If there are doubts regarding the correctness of a physical theory one must try to find the the root of evil.
    If they (SF and canonical LQG) are equivalent it would be nice to see why; if not it is of major importance to learn exactly why this equivalence fails.


    We do not know why quantization works at all, we do not know why "inverting the singular limit h=0" together with some heuristics works at all. But we know that it does work very well in many successfull theories - and therefore it is of major importance to understand whether it works in QG as well - and if not - why it fails.

    If Rovelli thinks that this is of little relevance than I totally disagree.
     
    Last edited: Oct 14, 2011
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