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Discrete spacetime (Some contemplations of mine).

  1. Dec 1, 2007 #1
    Im reading 'three roads to quantum gravity', and it's written there that any theory trying to unify between QM and GR has as conclusion that space has a discrete quantity where you cannot divide space more than this.
    now how would you propose to verify this assertion?
    I mean assume you've divide space up to the planck scale 10^-33m, why would this opertaion would stop there, what would stop us from dividing space even more than that?
    and how would we know for sure that we cannot divide it even more than that?

    thanks in advance.
     
  2. jcsd
  3. Dec 1, 2007 #2
    it is an easy thing to prove that spacetime must be discrete- if it were continuous then every finite space would contain infinite information- and infinite entropy- and so infinite instability- all particles would immediately diverge in structure to become alien to each other and increasingly weak in interaction until no interactions of any kind are possible- isotropy would collapse into chaos immediately and never form any consistent structure- continuous systems are therefore unphysical
     
    Last edited: Dec 1, 2007
  4. Dec 1, 2007 #3

    dst

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    Good thinking but what stops you from going down, past the Planck scale? Is this going to end up similar to not exceeding c?
     
  5. Dec 1, 2007 #4

    f-h

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    The best that can be said (outside of thermodynamics), I believe is that there is a certain point where the size of a blackhole of a certain mass becomes equivalent to the compton wavelength of the same mass thus it implies that nothing can be localized better than that.
     
  6. Dec 2, 2007 #5
    so there's a bound on entropy?
    I thought that entropy increases with time and thus some equate the direction of time with the direction of entropy, but if entropy is bounded then obviously there would be an end to time, but doesnt it depend if the universe were to end in a big wimp or to keep expanding (or to oscillate)?
     
  7. Dec 2, 2007 #6
    btw, also without the hypothesis that space is continuous, don't we have in a finite volume of space an infinite information, I mean every particle and virtual particle carries information, or virtual particles do not carry information?
     
  8. Dec 2, 2007 #7

    marcus

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    Several people have been talking about particles. Haelfix reminded us that particles do not exist except in the flat space approximation. (or in similarly restricted contexts)

    down at planckscale geometry is normally expected to be chaotic simply due to HUP. so there would be drastic variations in curvature, or non-metric, so that curvature would not be defined.

    hence the idea of a particle is irrelevant at small scale (it is a useful emergent concept at larger scale)

    Given this, I find the estimates of information content given earlier in the thread rather dubious.
    (entropy discussion earlier in thread based on concepts which are not rigorously defined)

    BTW both Loll and Reuter approaches to quantum spacetime (Triangulations and FixedPoint approaches) let the scale go to zero. It is not clear that this invalidates either approach, it may in fact work to their advantage. Both discover a fractal-like structure of hausdorff dimension approaching 2 at small scale, an odd coincidence since formally the two approaches are radically different.
     
  9. Dec 2, 2007 #8
    you mean that the notion of classical point particle doesnt exists, this i know, but does it mean that in every QG theory we replace quantum particle with something else, i know that in string theory we see strings as the builidng blocks.
     
  10. Dec 2, 2007 #9

    Fra

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    Some reflections...

    This got me thinking and curious how most people think of information and it's association to physical structures:

    Where is say to take an example, the information one system has about a second system, stored and encoded? In the first of the second system? Or maybe also in the communication channel pipelines connecting them?

    Who is "us"? you or me, or a particle in a remote galaxy? and is there a hidden assumption here that everybody must agree?

    The way I imagine this is that, an observer stops dividing it into smaller pieces, when he simply can't related to smaller pieces, and the concept of further division is undefined, relative to this observer.

    For example, there is no such thing as half a yes or half a no. Sure we can imagine 50% yes or 50% no or so, but then it's not too hard too believe that relating to a mixture or superposition requires more information. For a human beeing, 50% yes and 50% no is no problem, but for an observer with only two distinguishable microstates, I can't see how they can *relate to* such a division?

    I think this goes beyond spacetime considerations. Wouldn't the general question apply to any structure, and the relevance of a continuum in our models in the first place.

    Or maybe it's a limit of the state of the art of human logic? I mean, starting from basic logic, the values are usually true or false. From that, and axioms, we can build mathematics. Including continuum mathematics. The other way around seems strange.

    /Fredrik
     
  11. Dec 2, 2007 #10

    marcus

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    no, Loop, I do not mean this.

    What I had to say was not limited to the classical notion.
    For example in Quantum Field Theory, if you would try to define it on such a curved space, there is no notion of a quantum particle.

    You mention string theory, but I do not know of any string theory that has been defined with the target space an irregularly curved planck-scale volume. I don't think it is relevant to talk about "string theory" here. What theory do you mean? What does it say about planck scale volumes?

    As far as I know, Loop, the idea of a particle in any welldefined theory of matter is EMERGENT at a scale larger than planck. Particles of any sort are not meaningful to talk about at very small scale.

    So people who use the idea of particles and talk about entropy which they imagine they can calculate from particles----they are necessarily talking about emergent largescale phenomena. If they are talking about very small scale, then they are arguing using meaningless words.

    In my humble opinion :smile:

    Now both Loll and Reuter let the scale go to zero and they get rather interesting theories of the quantum geometry dynamics of spacetime. These appear to be consistent well defined theories, with unique equations that you can calculate with. (Unlike some other approaches.)
    And both Loll and Reuter are very well versed in Statistical Mechanics. Especially Loll, it is something she uses very much, and her co-workers have backgrounds in. Statistical physics is an essential tool.

    I don't think any of the reasoning in this thread of why you cant let the scale go to zero makes sense, but I would also go further and say that not only do the arguments given here not make sense (because of dealing in poorly defined concepts with at best some philosophical meaning). But if there were ANY reason based on entropy considerations then probably Loll and Reuter and their co-workers would be the first to know.
     
    Last edited: Dec 2, 2007
  12. Dec 2, 2007 #11

    Fra

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    I don't have near as good overview on various research as marcus, but from skimming many papers, the concept of "entropy" are usually itself vague. This has also been pointed out by many other people, and there are various attempts to improve it. Wether it's due to my ignorance of understanding or reviewing papers I couldn't tell, but I haven't seen a satisfactory treatment yet.

    The simple entropy definition from classical mechanics is insufficient, and it seems in the QM domain the notion is at least ambigous IMO. This problem I see related to the foundational issues of probability theory in the context of physics, or the issues of selecting microstructures in the boltzmann approach.

    If you try to review this, at the same time trying to maintain the observability ideals it's unavoidable to run into the issues of information capacity. These problem appears before the notion of entropy is defined. For this reason I try to reformulate this without introducing entropy.

    I have skimmed some of the papers relating to BH entropy and so on, but they contain so much baggage at several levels, that it's hard to know where to begin.

    All the respect for everybody including my own ignorance, but I wouldn't count on that if there was an information theoretic attack to this these guys would know about it already. Very few papers I have seen, except few, question these things down to the fundamental level of probability theory and how to relate that to physics in a deeper way.

    I personally think that's what we need.

    /Fredrik
     
  13. Dec 2, 2007 #12

    jambaugh

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    This is not a proof of the discreteness of space-time but rather of the impossibility of physical continuum space-time. When you treat space-time coordinates as parametric quantities (defining degrees of transformation) rather than as ontological labels (iterating physical objects) then you neither have discrete space-time nor infinite information issues. For example, there are a continuum of angles you can rotate a spin-1/2 particle and the same continuum of angles at which you can decide to measure the components of spin but the finite (one qubit) information in that spin doesn't in the least imply that angle is quantized. It does imply that "the direction of spin" is an ill defined quantity since the components to define this direction do not mutually commute.

    I'm personally of the opinion that space-time should remain parametric in the same sense as rotation angle and other "c-numbers". I see "quantizing space-time" as meaningless in the same sense as quantizing say the complex number system we use to describe transition amplitudes.

    The issues of quantum theory and GR will not be reconciled until the equivalence principle is properly parsed as saying not that (gravitational) dynamics is "just" geometry but rather that geometry is just dynamics. Gravitation as curvature of the space-time manifold must be viewed as a Model and a Classical Model at that. It becomes ill defined in quantum theory just as does position in phase-space. What's more time has never been treated as an observable and so relativizing should dictate either that we ascribe infinite information internal clocks to all quantum systems or more properly reject spatial position as an observable in a fully relativistic treatment. So again quantizing it makes no sense.

    Born reciprocity doesn't manifest between angles and angular momenta, why should it between coordinates and linear momenta?
     
  14. Dec 2, 2007 #13
    well I suppose it isn't as cut-and-dried as I put it- perhaps locally observably discrete is better? systems are defined by their interactions- it is possible that scale could be an unbounded fractal distribution of interactions in some universes- but this would make any observable space discrete with the infinte information at lower scales increasingly more weakly causally connected to the observable space and at some point in a total state of superpostion- but an observer in such a universe could in principle measure into smaller scales indefinitely- threshing out one decoherred histroy of the micro scale- however there appears to be a limit in our world [beckenstein's bound] as at some point you may have too much information in the local spacetime-singularities form and spacetime go haywire- that is the Planck scale

    so an objective fractal continuum is a possibility- sometimes- maybe- but subjectivity is discrete-
     
    Last edited: Dec 2, 2007
  15. Dec 2, 2007 #14

    marcus

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    Jim your view here strikes me as interesting and sophisticated.

    When GR is quantized the classical idea of curvature typically does get left out.
    Which I think you anticipate in what you say.
    Geometry gets quantized, say using the Einstein-Hilbert action possibly with quantum correction terms.
    And geometry takes on a new meaning----there may be no metric, where the geometry is described by new degrees of freedom and is not metrizable. The dimensionality may not be integer-valued and may become a scale dependent quantum observable.
    What you say about geometry being observed dynamically is not far off the mark, i would say!

    Anyway quantized geometry is basic, a lot of classical notions get dropped, and increasingly what one sees is a PATH INTEGRAL approach where spacetime is thought of as a sum over histories which get from an initial spatial geometry state to a final one.

    For example the spinfoam approach that Rovelli and a number of others are working on is a path integral approach. And likewise the triangulations approach pursued by Ambjorn, Loll and others.

    My impression is you never hear of metrics or of curvature in those contexts. At least I don't recall. Also individual points of spacetime have no physical meaning (but that was pointed out by Einstein in 1915 already) because these approaches are diffeomorphism invariant, like GR itself.

    If you would like to find out more of what modern nonstring Quantum Gravity is about you might look at the survey paper by Renate Loll for starters. You might enjoy it and find that modern QG fits in rather nicely with the un-naive viewpoint you have expressed in your post here

    http://arxiv.org/abs/0711.0273
    The Emergence of Spacetime, or, Quantum Gravity on Your Desktop
    R. Loll
    21 pages, 11 figures, write-up of plenary talk at GR18, Sydney, July 2007
    (Submitted on 2 Nov 2007)

    "Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up to now are far from trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales. The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated."

    This was an invited talk given at the most recent major international GR and Gravitation conference. If I remember right, there were two nonstring QG invited plenary talks and zero string QG invited plenary talks. It would help if people who want to talk about QG would get familiar with the mainstream approaches in that field. No need to exclude fringe stuff but good to be familiar with main stuff at least as a point of reference.
     
    Last edited: Dec 2, 2007
  16. Dec 3, 2007 #15

    Chronos

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    This reminds me of the 'ultraviolet catastrophe' - which was resolved by quantization. I think a similar approach would yield a similar solution [which I perceive as the essence of Vandersloot and Reuter's work]. I don't see infinite information issues. You only need as much information as necessary to encode a finite number of particles, coordinates and time slices in a finite space. If all the 'availabe space' ever 'filled', the universe would screech to a halt, IMO. Perhaps this is what fuels inflation, perhaps not. I refuse to believe the universe would paint itself into such a corner.
     
  17. Dec 3, 2007 #16

    Fra

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    It's interesting to see the different angles people view this.

    What James writes strikes me as seeing the distinction between physical reality and the mathematical model that supposedly describes reality. Certainly this is a point hard to argue on, but... if the continuum is mathematical without physical correspondence, what is there is a model where this redundancy is removed, and would allow a more compact representation? Why insist on a model which contains redundant information? As long as we can handle this, it has it's advantages, but try to apply this to a particle system.

    How does one encode a real number in a physical device? It could take infinite information to just encode a single real number? alternatively one can represent a real number as a limit, or an algorithm that one could in principle compute. But then we are transforming information vs processing time - we don't have the information, but we can compute it if we want to. That seems like a substantial difference to me?

    If mathematics is the given language for physics, I personally expect a better match than this from a fundamental theory.

    Tegemark said he think physics is mathematics. I don't adhere to Tegemark's main philosophy, but I do like some of his ideas of a tight connection to mathematics.

    /Fredrik
     
  18. Dec 7, 2007 #17

    jambaugh

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    Marcus,
    Thanks for the excellent reply and reference. I'll be sure to read it over the semester break.
     
  19. Dec 12, 2007 #18
    well iv'e chipped in also string theory cause in lee smolin's book he claims that it doesnt matter which approach you use (string or loop or the third independent) either way you get that space is discerte by planck scale, i.e this is the basic unit of measure in space, you cannot dissect space even smaller than this.

    I should get back to the book...
     
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