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Relations between ANY TWO of LQG, TQFT, CFT wanted

  1. Jan 27, 2012 #1
    In Tom Stoer's latest thread, he says "there is growing evidence that the incompleteness of the different approaches [LQG / canonical QGR / spin foams] has a common origin", and suprised suggests that "it may be that trying to quantize an effective theory will never work, ie., without introducing the extra degrees of freedom that are needed for unitarity."

    This got me thinking: that's a statement about UV completions, can we test it? Well, are there loop-variable quantizations of nongravitational effective field theories, and can we say anything about their UV behavior? But then I thought: loop variables for an ordinary QFT aren't quite the same as LQG; what we really need is a topological QFT. Meanwhile, we usually understand the UV and IR limits of a QFT in terms of a CFT.

    So the question resolved itself into trying to understand the relationships - the similarities and differences - between LQG, TQFT, and CFT. I haven't looked too hard, but here are two statements I already found.

    1) In http://arxiv.org/abs/1010.1939, Rovelli calls his LQG model a generalization of TQFTs as defined by Atiyah.

    2) http://arxiv.org/abs/1110.5027 claims to construct a TQFT from CFT. For some reason I don't see anything about CFTs in the paper, but maybe that's in the companion papers.

    This isn't a pressing topic for me, but I'm just curious what will turn up.
  2. jcsd
  3. Jan 27, 2012 #2


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    I see not so many similarities between LQG and CFT/TQFT. The only relation I am aware of is that BF-theory (from which SF models are constructed) is a TQFT, but BF is broken down to GR, so I see no features from TQFTs which are still present in QG.

    Afaik there are no loop quantizations of other field theories b/c loop quantization generates non-separable Hilbert spaces; only b/c of to diff.-inv. one can reduce them to separable ones. We investigated QCD in loop space 20 years ago, but could not make much sense out of it
  4. Jan 27, 2012 #3


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    As to the relation of LQG to BF theory, there might be something here:

    On the relations between gravity and BF theories
    Laurent Freidel, Simone Speziale
    (Submitted on 20 Jan 2012)
    We review, in the light of recent developments, the existing relations between gravity and topological BF theories at the classical level. We include the Plebanski action in both self-dual and non-chiral formulations, their generalizations, and the MacDowell-Mansouri action.
    16 pages. Invited review for SIGMA Special Issue "Loop Quantum Gravity and Cosmology"
  5. Jan 27, 2012 #4


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  6. Jan 27, 2012 #5


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    yes, but the SU(2) Chern-Simons gauge theory does not emerge automatically; you have to 'fix' the horizon classically
  7. Jan 27, 2012 #6


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    The most interesting empirical hint on the UV limit side in gravity is that there is nothing that has ever been observed in the universe with a materially greater matter-energy density than atomic nuclei which have about the same matter-energy density as neutron stars and the smallest observed black holes and a much greater matter-energy density that heavier black holes (if you assume that the matter in them is distributed uniformly within the event horizon).

    This could be simply a product of the fact that Hawking radiation should have dissolved all smaller primordial black holes by now, and that there is no natural process that creates smaller black holes due to the dynamics of nuclear fusion and gravitity interacting in stars that have the potential to become black holes. But, perhaps it is actually impossible to even synthetically or in a freak reaction create a smaller black hole and the theoretical possibility of such a thing in the GR equations is simple an unnatural solution in a classical approximation of true quantum gravity that is actually impossible because there is some fundamental quantum gravity bound on matter-energy density.

    An intrinsic matter-energy density limit could arise pretty naturally in a version of LQG where the Standard Model particles are basically "knots" or "clumps" of excited space-time itself as a property of space-time not terribly unlike the properties of space-time implicit in GR or existing spin foam or LQG work. And, since an intrinsic matter-energy density limit and the notion that all matter-energy manifests in point (or perhaps blurry point) particles in an fundamentally grainy space-time implies a maximum energy for a photon, which in turn implies a maximum wavelength (although relating it to a maximum frequency is more suggestive of the Noether theorem relationship between time and energy), this provides a means conceptually distinct from Planck's dimensional analysis to arrive at a fundamental minimum of time-energy graininess, which would be a UV limit for quantum gravity (and presumably the truly natural renormalization scale of all other Standard Model forces).

    This is, of course, a totally non-mathematical and intuitive way to get at the issue, but when you see "cosmic censorship" of anything empirical that could resolve theoretical issues, one has to wonder if that isn't just a coincidence.

    Along the same lines, in more conventional efforts to develop quantum field theories with spin 2 gravitons where efforts have been stymied because the equations are non-renormalizable, I sometimes wonder if that fact is a feature rather than a bug. The electromagnetic force, the weak force and the strong force do not give rise to singularities in the natural world, so their equations shouldn't blow up which is what equations tend to do at singularities. But, in the case of gravity, all of the really interesting stuff (black holes, the Big Bang, inflation, etc.) happens at or near what are singularities in classical GR. Every galaxy has at least one black hole and plenty of black holes exist outside galaxies and the universe very naturally maps back to a singularity. So, an equation that was renormalizable because it wasn't encountering singularities would almost definitionally be defective. Nature produces singularities all the time, so the equations of gravity at a quantum level should not be rigorously renormalizable - they should be renormalizable only if you devise a way to partition away the singularities from the rest of the world that solves all of the cases where the equations could theoretically blow up.
  8. Jan 27, 2012 #7


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    Yeah, that always bugged me.
  9. Jan 27, 2012 #8
    Thanks everyone for the responses so far!
    Who were "we"? Are there any records of these investigations?

    Note to self: phrases to google include "Wilson loop basis", "overcomplete basis", both alone and in conjunction with "QFT". Maybe we can start to see LQG as just a special instance of QFT in a Wilson loop basis or even just a special case of QFT in an overcomplete basis.
  10. Jan 27, 2012 #9


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  11. Jan 27, 2012 #10
    That sounds like it ought to be a good point, so let's think about this.

    The best-understood cases are conformal gauge theories with gravity duals. Also, the gravity dual becomes most tractable at large N, N the order of the gauge group. It's an example of 't Hooft's topological expansion, where you have emergent string worldsheets from very complex Feynman diagrams, except that the strings here live in the emergent extra dimensions.

    We have discussed here a few times the idea that almost any field theory has something like an AdS dual, because of constructions like MERA. But the "theories" defined in these AdS "bulk spaces" may not resemble anything familiar, in most cases. Maybe it is generically some messy Vasiliev-like gravity theory without much of a local description in terms of AdS.
  12. Jan 28, 2012 #11
  13. Jan 28, 2012 #12
    I seem to remember something in the Zakopane lecture on BF-theory as the refinement limit of spinfoams? The point being that if you keep refining, you end up with a flat bit of spacetime. So in some sense, one could see spinfoams as a perturbation around BF.
  14. Jan 29, 2012 #13


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    I wonder if spin foams can do discretized large N gauge theories?

    Or maybe its easier to start with asking whether MERA, which may be related to spin networks, can do that?

    BTW, MERA can be used to show that TQFTs give the long distance physics of some lattice models.
    Last edited: Jan 29, 2012
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