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LQG discreteness no conflict with Lorentz contraction

  1. Aug 21, 2007 #1

    marcus

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    My impression was that only misinformed people argue that LQG discreteness is in conflict with Lorentz invariance. This was resolved a long time ago, i thought.

    Does anyone disagree? Want to talk about it? Explain something to me that I am missing?

    http://arxiv.org/abs/gr-qc/0205108
    Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
    Carlo Rovelli, Simone Speziale
    12 pages, 3 figures
    (Submitted on 25 May 2002)

    "A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary."

    Discreteness in LQG is a serious issue. For various reasons I've always thought of it as a bit of a liability for the theory. FOR ONE THING THE LOOP COMMUNITY IS ALMOST ALL WORKING ON things like spinfoam, groupfieldtheory, simplicial (e.g. Loll CDT), renorm'ble quantum metric (e.g. Reuter) and these are THINGS WHICH DON'T HAVE MINIMAL LENGTH.

    It is a strange anomaly to have a community called Loop and have almost nobody working in canonical LQG proper, and to have canonical Loop the only approach with e.g. an area operator with discrete spectrum. It makes it seem as if the community connects to discreteness in geometrical measurement only by its historical accident name.

    That is bad enough but it is worse when we get people who don't know what they are talking come in and say that LQG can't possibly be right BECAUSE discrete spectrum of a geometric observable breaks LORENTZ. It simply does not. That is just something badmouth people say who don't know any better.
     
    Last edited: Aug 21, 2007
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  3. Aug 21, 2007 #2

    marcus

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    I can sympathise with Dittrich Thiemann wanting to break LQG out of the discrete geometry bind.

    Not because of Lorentz invariance which is a hyped-up NON-ISSUE. But because it would bring a lot of change and maybe convergence with other approaches.

    But they failed to bring it off. Rovelli demolished Dittrich Thiemann argument in a gentle respectful way that leave lots of room to appreciate their valuable contribution.
    And it is definitely valuable, but my reading of what Rovelli posted today is that he still demolished their case. If you want, check it out:
    http://arxiv.org/abs/0708.2481
    Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann

    Nevertheless the discreteness issue is still a hot one. Set aside Lorentz and set aside Dittrich Thiemann, and it is still demanding attention.

    Let's consider two recent papers by Brunnemann and Rideout that show that the LQG volume spectrum approaches zero arbitrarily close. these are numbercrunch papers.

    http://arxiv.org/abs/0706.0469
    Properties of the Volume Operator in Loop Quantum Gravity I: Results
    Johannes Brunnemann, David Rideout
    37 pages, 7 figures
    (Submitted on 4 Jun 2007)

    "We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a 'volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5--7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum on 4-valent vertices are included, for which the presence of a volume gap is proved. This paper presents our main results; details are provided by a companion paper arXiv:0706.0382v1.

    http://arxiv.org/abs/0706.0382
    Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation
    Johannes Brunnemann, David Rideout
    95 pages, 65 figures
    (Submitted on 4 Jun 2007)

    "The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in gr-qc/0405060, making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4-valent vertices are included. This is a companion paper to arXiv:0706.0469, providing details of the analysis presented there."

    And here is Rideout's invited plenary talk at Loops 07
    http://www.matmor.unam.mx/eventos/loops07/talks/PL4/Rideout.pdf
    http://www.matmor.unam.mx/eventos/loops07/talks/PL4/Rideout.mp3

    They found there is no smallest nonzero volume in the spectrum of the LQG volume operator---define, according to what i understand, on the kinematical states as usual. this bothers me. I find it at first a surprising idea, and then a nagging one. The positive eigenvalues might well be a DISCRETE set and yet they get arbitrarily close to zero. Like the numbers 1/n are a discrete set but there is no smallest one. there is no volume "gap". does anyone want to explain this or put it in context?

    did anybody here attend Rideout's talk at Loops 07 in June?
     
    Last edited: Aug 21, 2007
  4. Aug 21, 2007 #3

    Demystifier

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    Rovelli also had a nice analogy:
    Does a discrete spectra of the angular momentum in ordinary nonrelativistic QM violates rotational invariance? The answer is - of course not.

    Or even a simpler analogy:
    Does a discrete spectra of momentum of a particle in a box violates translational invariance? Again, of course not.

    One should also not forget about the Ehrenfest theorem:
    Even though the spectrum of the eigenvalues may be discrete, the average value satisfies classical equations of motion and may attain an arbitrary value (provided that the zero value is also one of the eigen-values in the spectrum).

    So, LQG is nothing special in that regard. It is just another example of QM.
     
  5. Aug 21, 2007 #4
    hum... doesn't the box itself break translation invariance??

    Yes, Carlo's analogy is great! the "minimal length" is not a problem as soon as the "length" is not simply considered as a classical object but as a quantum operator. I tried to give some Lorentz invariant toy models to illustrate this point in an old paper with Daniele Oriti, gr-qc/0405085. We use models very similar to the non-commutative geometry of the deformed special relativity type. However, it is all in flat space-time... so still quite far from LQG and the diffeomorphism invariance!

    Finally, i would like to say that we do not need to go to quantum gravity to get a notion of Lorentz invariant minimal length. We can simply look at the QFT of a massive field, there's the Compton length which is a minimal scale for all observers... The way that this minimal length is implemented does not involve gravity but simply the quantum fluctuations of the field at high energy (small scales).
    The difference with quantum gravity is that the qg minimal length is supposed to depend on the gravitational constant and not of any details of the matter fields... but the mechanism could be similar... or am i missing something?
     
  6. Aug 21, 2007 #5

    Demystifier

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    Actually yes, so maybe this analogy is not so perfect. But it is still fine, because in classical mechanics the momentum of the particle is arbitrary (not discrete), which can be obtained by boosts of the momentum. The box does not change it.
     
  7. Aug 21, 2007 #6
    Can a metric be quantized but the coordinates be continuous? Or is this a contradiction of terms? We can assign a continuous sequence of numbers to a variable; each number differs from the previous by a digit in the infinith decimal place. But perhaps the distance formula between these barely different numbers may be quite a bit larger than the previous pair of barely different numbers. Does this make sense? Thanks.
     
    Last edited: Aug 21, 2007
  8. Aug 22, 2007 #7

    Demystifier

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    Yes.
    First, if something is quantized, it does not mean that it has a discrete spectrum.
    Second, if a quantum operator has a discrete spectrum, the average values still can attain continuous values.
     
  9. Aug 22, 2007 #8

    arivero

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    What I do not understand, was this new scale suppossed to be different of Planck scale?

    Because Planck scale is simply Newton Constant in natural units, and as far as I know, Newton constant is perfectly compatible with general relativity. Of course it is not compatible with special relativity; if it were, why should be need general relativity at all?
     
  10. Aug 22, 2007 #9
    I guess I'm confused about how you can say that numbers can actually be "different" without a difference function, or metric. What property of 5 and 7 makes them different if we don't take into account 7-5=2?
     
  11. Aug 23, 2007 #10

    Demystifier

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    Think about spacetime points as elements of a set (not numbers!). Two points are either different or not. But if they are different, in general you cannot say how much they are different. To do that, you must introduce an additional structure on this set - the metric. Take some mathematics textbook to see general axioms that a metric on a set must satisfy.
     
  12. Aug 23, 2007 #11
    Spacetime has more structure than a set - it has a smooth structure, which allows us to define partial derivatives. Hence the distance between two spacetime points can have three values: zero, finite, and infinitesimal.

    This reminds me of one of my pet peeves. The anomalous dimension, i.e. the eigenvalue of the dilatation operator D, is background independent, because D = x^u d/dx^u does not depend of the metric.
     
  13. Aug 24, 2007 #12

    arivero

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  14. Jul 15, 2009 #13
    May you give examples where something is quantized not having a discrete spectrum? I don't know if you were meanning more that are many things in quantum mechanics and QFT that are conitnuus, but doesn't mean that they are quantized, as the case of momentum and position.

    Also, maybe a better and much simpler example for the second is how the operators (here for the metric) having discrete spectrum have always a representation in terms of continuus variables.
     
  15. Jul 15, 2009 #14

    marcus

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    Rebel, I see you have a question for Demystifier. Sorry to butt in, I had something more to say about the general topic of the thread. But want to preserve your question:[QUOTE Demystifier;1407476]Yes.
    First, if something is quantized, it does not mean that it has a discrete spectrum.
    Second, if a quantum operator has a discrete spectrum, the average values still can attain continuous values
    ==endquote==

    [QUOTE Rebel;2272676]May you give examples where something is quantized not having a discrete spectrum? I don't know if you were meanning more that are many things in quantum mechanics and QFT that are conitnuus, but doesn't mean that they are quantized, as the case of momentum and position.

    Also, maybe a better and much simpler example for the second is how the operators (here for the metric) having discrete spectrum have always a representation in terms of continuus variables.
    ==endquote==

    I wanted to say that the thread was started in 2007 right after Dittrich and Thiemann had challenged the Loop result about discrete spectrum of the area and volume operators. I would say a few things differently now from what I said in 2007.
    At that time my intuition about that was wavering off track a little. At that time people were using Spinfoam to calculate with but Spinfoam had not yet shown that it had a place for Immirzi parameter and could confirm the earlier results about spectra.

    Since then the spinfoam approach has strongly confirmed the key results of earlier (1990s) LQG. So now there is no dissonance.

    Now when one compares Loop on the one hand and Reuter AsymSafe and Loll Triangulations on the other hand, it seems to me that on balance Loop now has more going for it. All these approaches are providing hints and clues about what is going on with geometry at very small scale, so you don't actually want to be "picking winners", but with Modesto's result Loop has the dimensional reduction at small scale which for a while it looked like only Reuter's and Loll's approaches had. Less dissonance there too. And Loop lets you do a lot more calculation. Have to see about running coupling constants. Rovelli's lectures in September may resolve that.
     
    Last edited: Jul 16, 2009
  16. Jul 16, 2009 #15
    Oh I see, I did not saw when those posts were put and I feel embarrassed.

    But now that it comes... is the question of volume operator settled by now in LQG? I don't see much about this, maybe some indicators from Ashtekar and Rovelli a decade ago, but guess they obtained different results. Sorry for don't being that updated. Also I heard once Modesto saying that this result about that paper of 2005 with Rovelli about running scale dimensions is what they wanted to do from the start, something like if Rovelli wanted to have a fractal-like structure for dimensions. And guess that was motivated exactly to relate with Loll's work, and don't know if ultimately he wanted to obtain something more for arguing more about time existence as some papers of him (an attitude about time in QG that as far as I know smolin has changed), but things did not resulted in the dimensions they wanted to obtain at first.
     
    Last edited: Jul 16, 2009
  17. Jul 16, 2009 #16

    marcus

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    I for one don't see any cause for embarrassment! I'm glad you renewed activity on this thread. It's an important point, that a lot of people seem not to realize.
    No one has ever been able to show that (in 4D) LQG leads to any sort of Lorentz violation, or even any deformation of Lorentz invariance (like DSR which has no distinguished frame of reference, all observers see the same speed c.) They've given loose arguments, and theyve thought they could show it, but it has never "stuck".

    Loop researchers have tried hard to find a way to derive some kind of bending of Lorentz and they so far can't do it. It would be great if they could, give something to test observationally. So DSR is a separate thing that has to be tested separately.

    I'm glad you brought the thread up.

    The only reason I mentioned that my earlier contribution to it was 2007 is that my perspective has changed some. Just wanted to update a bit.
     
  18. Jul 17, 2009 #17

    tom.stoer

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    The argumentation relies heavily on the states; if we want to discuss Lorentz symmetry, the operator algebra itself has to be discussed

    A few comments are in order:
    - the angular momentum is a perfect example where a continuous symmetry leads to discrete eigenvalues
    - a symmetry is manifest after quantization if the operator algebra remains valid = if there are no anomalies
    - the area and volume operator and not diff. invariant, so I can't say if these arguments apply (this is a major difference to angular momentum!)
    - in the final theory the generator of infinitesimal diffeomorphisms does no longer exist, so comparison to to angular is not straightforward
    - I don't know if the algrabra of constraints itself closes or if it closes only weakly = within the physical Hilbert space; therefore the question of anomalies is not clear to me
     
  19. Jul 17, 2009 #18
    To which id like to add: a closed universe only permits discrete eigenvalues.

    Continuous spectra, move over.
     
  20. Jul 17, 2009 #19

    marcus

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    It sounds like you are talking about the old LQG. The canonical approach.
    I think the best survey of the current LQG situation is the talk that Rovelli gave to Strings 2008 at Geneva. He also re-wrote his Living Reviews article last year.

    Have you watched the Strings 2008 talk?

    I guess the situation could seem a bit confusing---there was a little "revolution" in 2007, when a spinfoam formulation appeared which preserved all the kinematic features of canonical (oldstyle) LQG. So one could throw out the old canonical approach---which few people have worked on in recent years---and move completely to the path integral or covariant approach.
    I haven't seen Rovelli or anyone else call it a "revolution" (which would sound a bit pompous and self-important) but it has been referred to as a "new look".
    There was a redefinition of the field/program in or around 2007. One could call it a "restyling" or a "make-over"---a bit lighter and less solemn sounding than "revolution".

    I'm not sure how much of this you already know. I would have assumed you know the whole story but I'll recap---when Rovelli gives his introduction to LQG to the strings 2008 audience he gives a brief account of oldstyle LQG including the area and volume operator results (which are kinematic) up to but not including dynamics. Then he says "now we come to dynamics, you can forget everything I just told you, what follows is a completely new derivation of LQG".

    That is, as he defines the program, LQG is no longer based on the canonical formalism---it is entirely based on path integral. This is what most of the Loop community seems to have wanted for the past 10 years, since they almost all worked on spinfoam and the related group field theory (see Oriti's papers for explanation of GFT). What enabled the desired switch was in 2007 a new version of spinfoam was found to contain the Immirzi parameter and to MATCH the oldstyle canonical results (that were purely kinematic.) They found in 2007 that one could move over to a different definition of LQG without sacrificing anything valuable of the earlier results.

    If you look at the LQG papers from 2006 onwards that study the graviton propagator and the large-scale limit you will see that they are all using spinfoam (and related group field theory) methods---the active research on the key questions is essentially all path integral. Perhaps this is partly sociological: the majority of the young researchers in the field have come out of Marseille. But I think that the more important reason is that Thiemann's heroic effort to save the canonical dynamics did not succeed, they might come back to it after the path integral approach is worked out.

    Another element to the picture is a further new look hinted at in Rovelli's course description for his lecture series for the Corfu school. If that gambit succeeds it will further re-define LQG, if it fails then they will continue with the redefinition as of 2007.
    You probably have the links, but in case anyone wants them I will list a few.
    Strings 2008 intro to LQG video:
    http://cdsweb.cern.ch/record/1121957?ln=en
    Slides:
    http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917

    Page of helpful excerpts from recent seminar talk quoted here:
    https://www.physicsforums.com/showthread.php?p=2236221#post2236221

    Abstracts for upcoming Corfu school (September 2009):
    http://www.maths.nottingham.ac.uk/qg/CorfuSS.html

    2008 Living Reviews article on LQG:
    http://relativity.livingreviews.org/Articles/lrr-2008-5/
     
    Last edited: Jul 17, 2009
  21. Jul 17, 2009 #20

    atyy

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    Yes, I am totally confused - I've often read spin foams is supposed to reproduce the results of the canonical approach. If spin foams is new, how is it motivated? Or if spin foams is equivalent to the canonical approach, am I allowed to use the old motivation?
     
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