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Quantized spacetime and Special Relativity

  1. Nov 17, 2008 #1
    How does special relativity affect a quantized spacetime? Specifically, how can time and space be quantized the same way for all observers?
     
  2. jcsd
  3. Nov 18, 2008 #2
    Well, if you quantize space, you break special relativity in general---discretizing space-time usually just ends up giving you a mess. I guess the loop quantum gravity people have some way around this, but I've never heard a good explanation as to why it's not a problem for them.
     
  4. Nov 18, 2008 #3

    Fra

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    I guess there is no mainstream answer to this.

    Some personal thoughts.

    First, I think it's not at all obvious to me at least what "beeing the same to all observers" means? Rovelli asks in his Relational QM - how can two observers even compare their measurements?? Ie how is the notion of comparasion defined? He argues that the answer is communication/interaction between observers. I think this is the best way of thinking of this.

    I think the essence of general covariance is that observations from all observers, while sort of different, should at some level be consistent. IE. the laws pf physics deform along with the observer, so that the equivalence class of all descriptions and observers is the fundamental description. Indespite of it's deep beauty it's not hard to see that there is something strongly realist-minded about this. The point beeing that no real observer see the entire equivalence class. The whole reasoning IMHO rests on a realist vision of symmetry. I think this won't do when it comes to incorporating QM with this. I think probably both QM and GR needs change.

    But the question is how to merge this into the big picture, which not only spacetime but also all other forces. One possibility IMO that any inconsistencies between observer view, manifest themselves as NEW forces. The mutual force is simply an mutual inconsistency, which is resolved as the interaction progresses, which results in a mutual relation which recovers the consistency of the differing views.

    The question then is if one can continously resolve all inconsistencies that appear by adding new interactions? So that you end up with a total fundamental "symmetry". I don't think it's a prior obvious that such perfect symmetry exists. I personally don't think so, but that doesn't mean we are toast. It just means that perhaps the focus should be on the evolution of the symmetries? IE. to consider a true relative symemtry, rather than aiming for a universal fixed master symmetry, that may not even exist.

    If this is so, then perhaps one might not expect perfect consistency, but OTOH the inconsistency may show us the way into the future.

    /Fredrik
     
  5. Nov 18, 2008 #4
    There IS a mainstream answer, and it's that it's not consistent with special relativity.

    A minimum length implies a breakdown of Lorentz Invariance at that scale. The reason is easy to see---in who's frame are you quantizing space? If I quantize space in my frame, and you come running by me really fast, then you see that my little chunk of quantized space is smaller by a factor of gamma. So "smallest chunk of spacetime" is a relative notion in special relativity.
     
  6. Nov 18, 2008 #5
    I always wondered about quantized space, but if you have a square area with sides which are one quantized unit long, what is the length of the diagonal?
     
  7. Nov 18, 2008 #6

    marcus

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    That would depend on what you mean by a quantized spacetime. For instance, talking about LQG, it is just a popular misconception that LQG divides spacetime up into little chunks :biggrin:.

    LQG and allied approaches quantize the gravitational field. That is, they quantize geometrical measurements. Quantizing a measurement means to represent it as an observable, an operator on the state space. In some versions of LQG and related QG, the spectrum of the area operator is discrete, in other versions it is not discrete.

    In neither case is there an implication here that space or spacetime has been divided up into little chunks. Just that if observers measure a physical area, like the are of your desk, the operation of measuring will, in some versions, have discrete spectrum. The expectation value of a measurement will in that case be an average of a countable set that includes zero, and the smallest positive eigenvalue, and the next smallest ....etc.
    If one of the observers is flying very fast past your desktop, the expectation value of his measurement of the area could be very small----smaller than the smallest positive eigenvalue.

    If you want, think of it this way. In some of the most active lines of QG research there is no minimal length (no minimum eigenvalue of the length measurement operator) or minimimal area or any such thing. A prominent example is Loll's approach, recently written up in the Scientific American. I have a link in my sig to that Loll SciAm article. Check it out. The size of the triangles goes to zero and there is no minimal length---they are explicit about this in their papers.

    Loll's approach has a quantum spacetime, but they don't call it a quantized spacetime. That might mislead people and start them talking about dividing spacetime up into little chunks. What is quantized is the geometry. Gravity is geometry and QG stands for both quantum geometry and quantum gravity.

    Loll's quantum spacetime is a continuum with a new kind of geometry, quantum geometry, in which geometric observables are quantum operators. The spacetime itself is not divided up into little bits, and it does not have a minimal length.

    Now LQG is similar in some ways and different in others. It too is constructed on a continuum: so space itself is not divided up into little bits. But in some versions of LQG the area and volume operators have discrete spectrum. In other versions (spinfoam in particular) that I keep reading about, these geometric observable operators seen NOT to have discrete spectrum, at least they haven't been shown to have. I don't know how this is going to turn out. It will be interesting to see. But in either case there is no problem with Special Relativity
     
    Last edited: Nov 18, 2008
  8. Nov 18, 2008 #7

    Fra

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    Yes regarding chunks I see your point, but I guess the way I see it the notion of "quantized spacetime" is not unambigous in what I think of as "mainstream". This was what I meant. And if you suggested that there IS a mainstream sense, that in turn is inconsistent, then that still feeds the question on howto resolve it.

    Like others said, quantization doesn't equal make into chunks.

    My comment tried to address the question in a general sense (admittedly non-mainstream), that suggests that an observer-observer inconsistencies in general might possibly be resolved, by identifying these as new interactions. Ie that the APPARENT inconsistency that two observers can not agree upon their observations, does exert a kind of influence/force in between them. Once you can classified this new "force" the consistency is recovered. That's how I personally see a possible way to resolve the "subjectivity problem" and howto maintain some level of objectivity in despite of a fundamental picture where each observer can not compare their measurements with some common standard, but only compare it by interaction with it's neighbours.

    /Fredrik
     
  9. Nov 18, 2008 #8
    What I mean by 'the same to all observers' is that it should not depend on your velocity. Lets say we have two non-interacting particles moving very quickly with respect to each other over flat quantized spacetime. Since both have an equal claim to being at rest, they should observe the spacetime over which they move to be the same, despite relativistic effects.
     
  10. Nov 18, 2008 #9

    marcus

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    I'm not sure I understand your question, if you have a question. LQG is a quantization of geometry which among other things is consistent with SR as far as we know. It does not involve thinking of space as consisting of little chunks :smile: That is not what "quantizing" geometry means.

    There are other interesting QG approaches which also do not involve thinking of space as made of little chunks. Have a look at the Loll QG link in my sig. If you have some question, how about spelling it out in detail?---I'll be glad to do my best to answer.
     
    Last edited: Nov 18, 2008
  11. Nov 19, 2008 #10

    Haelfix

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    Regge calculus is sort of the historical forebear to the whole atom of space idea. You can retain lorentz invariance provided that there is enough residual symmetry left over in the marginal operators, not unlike textbook lattice gravity.

    The problem really is getting these five (not necessarily independant) conditions to mix:
    1) Lorentz invariance or at least lorentz breaking effects but only up to very small factors (which is very constraining and hard to do)
    2) Flat spacetime in at least some sort of limit (as opposed to crumpled up phases)
    3) Manageable entropy densities (as opposed to planckian entropy^4, which leads to ridiculous cosmologies)
    4) Unitarity
    5) Existence of a continuum limit.

    All treatments known to date end up sacrificing 1 or 2 conditions, depending on the context.
     
  12. Nov 19, 2008 #11
    Does string/M-theory satisfy all 5? thx
     
  13. Nov 19, 2008 #12

    Haelfix

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    ST is not a discretization theory (atom of space idea) so it doesn't apply. Incidentally, LGQ is not really a discrete theory either. CDT is though!
     
  14. Nov 19, 2008 #13
    LQG is definitely discrete
     
  15. Nov 19, 2008 #14

    marcus

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    Heh heh. In this case Savant, I would have to say that Haelfix knows a good deal more than you about it, so it seems odd for you to flatly contradict him without offering any link to peer-review literature. What technical article have you read about LQG, where it is defined? A simple one is Rovelli Upadhya LQG Primer. Look it up on arxiv. You will see that LQG is defined on a continuum representing spacetime. The operators representing geometric measurment have discrete spectrum, not the same thing as having a spacetime made of discrete bits. Please go back and read my post on this.


    I agree with you about LQG, certainly. But sometimes I'm not sure what you mean by the words you use, Haelfix. Could you explain what you mean by saying that CDT is a "discrete theory".

    We know that CDT spacetime has no minimal length. Loll has mentioned that explicitly. It looks like a continuum (a manifold but without a metric to give it a metric geometry.)

    The CDT continuum is not in any sense discrete.

    In CDT triangulations are used to generate different possible geometries, but the size of the generalized triangles is taken to zero in the limit. This is why there is no minimal length in CDT. (By the way would you like an arxiv link for that?)

    Perhaps you should explain what you mean by "a discrete theory" before people are misled and start thinking that the CDT continuum is divided up into little bits :smile:
     
    Last edited: Nov 19, 2008
  16. Nov 19, 2008 #15
    In Three Roads to Quantum Gravity, author Lee Smolin, one of the founder of LQG explicitly states that spacetime is quantized in LQG
     
  17. Nov 19, 2008 #16

    Haelfix

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    The hope is that CDT is a continuum but this is a *hypothesis*, and its still done on a computer and very much discrete. So in other words you take the Einstein Hilbert action (or generalization), take some type of mini superspace approximation and stick it on a type of lattice (done in lorentzian signature, as opposed to euclidean, which is sort of the big success of the program), divide it up in a certain way, and see what comes out. The assumption is that at the end of the day, the details of the lattice drop out. And in practise they see that various quantities don't blow up and tend to the expected limits as they decrease the lattice spacing and increase the number of cells, the volume and so forth.

    However there is no analytic *proof* that this continuum exists and therein lies the rub.

    Lattice theories in general can give you very misleading indicators. Sometimes you can actually prove the continuum doesn't exist, even though the computer results naively looks like such a thing should. Other times a continuum theory exists, even when it shouldn't. So for instance lattice QED's UV continuum exists, but surely cannot be physical (b/c of the Landau pole or GUT unification etc). In other cases, there are several (perhaps infinite) amounts of different continuum theories possible.

    CDT's main, as yet to be resolved problem is proving the existence/uniqueness of the continuum limit, whether the inclusion of matter spoils the results, and also the loss of unitarity b/c of the particular superspace approximation they use. So theres still a lot of work to be done to really convince people that they have a final admissible TOE.
     
    Last edited: Nov 19, 2008
  18. Nov 20, 2008 #17

    marcus

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    that was a non-mathematical popularization written for wide audience.
     
  19. Nov 20, 2008 #18
    in string theory, what are its predictions for a Planck-sized string accelerating toward c? Is string theory committed to infinitely continuous sub-planckian distances and time? In string theory, is spacetime classically smooth at arbitrary small distances?
     
  20. Nov 21, 2008 #19

    Haelfix

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    I'm not the best one to ask about that sort of question, im sure a stheorist could explain it better and with more authority.

    From what I gather the question becomes illdefined. If you start probing the string (either via scattering or dumping energy into the free string) eventually past a certain point (not necessarily the planck scale, but thereabouts) you no longer are probing strings, but rather blackholes (b/c all that energy density eventually pushes the system past its Schwarschild radius). At that point, transplanckian physics is no longer well described by string theory, but becomes quasi classical again (eg GR.. scattering of blackholes and so forth).

    In a certain sense, thats kinda what you want. It makes good sense that the degrees of freedom of a QG theory when pushed to the extreme breaking point eventually lose their significance b/c you can no longer ask questions about them since they lie behind horizons.

    As for whether or not the spacetime is smooth. Well again, the question is a little fuzzy and only makes sense in certain limits. The metric is only part of the degrees of freedom of the whole (as yet to be understood) shebang, in fact its presumably not fundamental and therefore emergent. The main (string/brane) d.o.f as well as the precise nature of the moduli should in principle contribute to its dynamics, but like all emergent systems the technical details becomes really challenging. Still, since those d.o.f are decidedly quantum and fuzzy its a little hard to say with a straight face that spacetime is 'smooth'.. Its only 'smooth' when we make it so (by hand) as a sort of initial condition for calculational tractability.
     
  21. Nov 24, 2008 #20
    That does not mean that the statement was not correct on some level.
     
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