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Argument for discreteness of spacetime

  1. Apr 3, 2010 #1


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    The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity. The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.

    I gather that this argument is far from being universally accepted. Where are the holes in the argument?

    I would appreciate nontechnical answers. My background is that I did my PhD in low-energy nuclear physics, and I have a pretty good understanding of general relativity, but I don't have any technical expertise in quantum gravity. My only knowledge of quantum gravity comes from popular-level books like Smolin's and Susskind's.
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  3. Apr 3, 2010 #2


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    I think this is the argument that gravity cannot be a "normal" quantum field theory, and that some sort of "holography" should hold, not that spacetime is discrete. eg. sec 2 & 3 of http://arxiv.org/abs/gr-qc/9508064 , which includes a short critique at the end, noting that the hoop conjecture is used - I think till this day there isn't a formal statement of the hoop conjecture.
  4. Apr 3, 2010 #3


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    Let's start from a totally different problem: In continuous spacetime (up to arbitrary small length scales) quantum fluctuations of fields could form virtual black holes; these processes would spoil any quantum field theory.

    Think about quantum fluctuations of size L where L can become arbitrary small. Let L(E) be the Compton wave length of an object of energy E; L(E) can be considered as the "typical size" of this object. If the object's energy E increases its size L(E) decreases and the object eventually becomes smaller than its own Schwarzschild radius; according to GR it eventually collapses and forms a black hole. If one equates Schwarzschild radius with Compton wavelength one observes that this process will happen at the Planck energy.

    This argument does not mean that space necessarily becomes discrete below the Planck length, but that the usual formalism of quantum field theory and GR must be replaced by something different.
  5. Apr 3, 2010 #4


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    But what does "spacetime is discrete" mean? One way to show the difficulties with that argument is to look at the example of LQG (but not at a popular level, popular exposition often misleads and confuses since we are talking about math models, not verbal models).

    In LQG one starts with a continuum---a differential manifold---representing spacetime.
    As usual it is connected. You can run a continuous path between any two points. It is not discrete---does not have discrete topology. Just the usual continuum that mathematicians have been using for over 150 years.

    On a spatial slice of that continuum one constructs quantum states of geometry.

    A Hilbert space of states of geometry. Operators corresponding to making geometric measurements. Observables.

    It turns out that the area and volume operators have discrete spectra. One proves as a theorem that there is a smallest measurable area---essentially the Planck area.

    This does not mean that space is topologically discrete. It does not consist (in the LQG context) of separate points.

    And one can prove the entropy bound in the LQG context. Indeed Ashtekar recently published a proof of the Bousso covariant entropy bound. This is something that fails as one approaches a singularity in classical GR. So Ashtekar went Bousso one better :biggrin: He proved the covariant entropy bound more generally---extending it to places where it classically fails.

    And still, in LQG, space and spacetime are not divided up into little isolated bits. We do not have simpleminded discreteness. There is a discreteness in the operator spectra---at the level of what we can know, and measure, and meaningfully talk about. We cannot measure an area smaller than Planck area.

    Notice I'm not claiming LQG is right. These are just rigorous mathematical theorems. You set up a continuum, you define quantum states of geometry in a certain seemingly natural way, you find certain operators have discrete spectra. It turns out there is a limit, for some unknown reason, on what one can measure (in the LQG context.) It is somewhat analogous to the Heisenberg limitation on how accurately one can know position and momentum---limits on knowledge, limits on what it is meaningful to talk about, on measurement. Except that these are limits in the realm of geometry itself, rather than merely in the realm of fields or particles defined on some fixed geometry.

    LQG people, as a kind of careless shorthand, especially in a popular wide-audience discussion, will talk about spacetime discreteness when what they mean is this kind of discreteness at the level of geometric information.
    Last edited: Apr 3, 2010
  6. Apr 3, 2010 #5


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    Thanks, folks, for your interesting responses!

    Hmm...you lost me at the "but." Why do the ideas you've described require that "the usual formalism of quantum field theory and GR must be replaced by something different"?

    In the context of the original argument given in #1, it seems to me that this shows that LQG has a natural mechanism for complying with the Bekenstein bound -- and I believe this is essentially what Smolin was claiming in Three Roads. Since the discrete spectra of the area and volume operators appears to be a specific property of LQG, what happens to other models like string theory? How do they comply with the Bekenstein bound? Or is it that they don't comply with the Bekenstein bound, and that's okay because the Bekenstein bound is not firmly established on model-independent grounds?
  7. Apr 3, 2010 #6


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    The holography arguments against gravity being a normal QFT suggest that Asymptotic Safety is wrong - however, they are just heuristic, and not rigourous arguments against a non-trivial UV fixed point. Niedermeier and Reuter comment "In the context of the asymptotic safety scenario, on the other hand, the presumed reduction to effectively two-dimensional propagating degrees of freedom is a consequence of the renormalization group dynamics, which in this case acts like a ‘holographic map." http://relativity.livingreviews.org/Articles/lrr-2006-5/ [Broken]

    However, if AS doesn't work, and gravity is not a normal QFT, then a priori, a discrete spacetime cannot be ruled out. However, that doesn't mean that spacetime must be discrete. In fact in AdS/CFT, we seem to have a quantum theory of gravity (for some universe, not ours) that respects holography and has a smooth "background" with emergent spacetime.

    Area laws for the entanglement entropy - a review
    J. Eisert, M. Cramer, M.B. Plenio
    Last edited by a moderator: May 4, 2017
  8. Apr 3, 2010 #7


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    I'll try to clarify: The ideas I described are based on a rather well-understood formalism in quantum field theory. It is known that quantum field theories a plagued with infinites that have to be removed via renormalization. If one tries to apply this renormalization procedure to general relativity it fails for several reasons, so usually it is claimed that general relativity is not (perturbatively) renormalizable.

    To summarize this means that applying QFT methods to GR is inconsistent.

    Now we shouldn't jump to the conclusion that quantum gravity requires spacetime to be discrete; the only logical conclusion is that the failure of QFT methods requires a change of formalism if applied to quantum gravity. There are a couple of ideas, not all of them leading to "discrete" spacetime [in addition it is not clear if one should require spacetime to be discrete, or if one should let the formalism decide if it eventually becomes discrete; compare it to the harmonic oscillator: the discreteness of the spectrum is not an input but an output of the theory].

    Some approaches:
    - LQG
    - string theory
    - supergravity (which can be seen as low-energy limit of string theory or as a theory on its own)

    I am not completely satisfied with Marcus' statements. In LQG one starts with a continuous spacetime and applies a slightly modified formalism of quantization. The result is a theory in which area operators have discrete spectrum. BUT: these area operators are not physical (= gauge invariant) operators, so one must not conclude that "spacetime itself is discrete". In addition in the final formalism there is no spacetime anymore! One ends up with a space of so-called spin networks from which spacetime (as we know it from GR) and GR should emerge in a semiclassical limit and as low-energy effective theory, respecively.

    In certain supergravity theories spacetime stays continuous. There is still some hope that SUGRA could be a perturbatively renormalizable theory.The main difference to ordinary GR is that one uses so-called on-shell methods, both for the proof of the closure of certain algebras (which do not close off-shell) and for the proof of the finiteness of Greens functions (which should have certain symmetries valid on-shell) or physical amplitudes.
  9. Apr 3, 2010 #8


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    Everytime continuum models are used, they are part of the input to the model - there is no physical measurement process, and physical representation that is one-2-one to the continuum. So in that sense the continuum does not seem obsercavable, even in the form of in index space, as it would correspond to an infinite amount of information.

    But what some non-observable parts are like, is hardly an interesting question. IMHO, discrete index space from the observability point of view, doesn't imply that there is a naive objective discreteness, but neither does it imply that there is a naive objective continuum. Maybe we simply can't decided and that's enough? But which seems to be the more redundant description of the two?

    I prefer to view the information bound, not as a limit of the amount of objectively hidden information behind the screen, but as a limit of the amount of inferrable(measurable) information about the other side. But then, it is not even clear to what that there is a notion of objective screen, since each observer effectively has their own screen.

    I think there is alot around this that still extremely unclear and most things are yet very much semiclassical arguments that lack coherence of thought.

  10. Apr 3, 2010 #9


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    I'd take it to mean that, if one naively pictures the discrete event-index to be embedded in a continuum all "points" in the local neigbourhood of an index node are indistiniguishable. This doesn't perhaps forbid that there actually is a continuum, no more than we can prove that god doesn't exists but it seems at least very redundant.

    But to claim that this index, is the same for all observers is of course a a different thing.

    But I fail to see the sense in that any inside observer could resolve a continuum. This I think questions these sense and use of starting with "lets consider a continuuus manifold". It almost leaves me a flavour of the physics "spherical cow" joke.

  11. Apr 3, 2010 #10
    Lets take a step back because something in your argument is very wrong. L(E) can certainly not be considered the typical size of a object mass M=E. Think of a star or any macroscopic object the Compton wavelength is inversely proportional to the mass. (I made this mistake just the other day). What causes the problems in quantum gravity is not the presence of an apparent horizon, which defines a black hole, but the presence of a central singularity.

    As far as QFT goes it is not "QFT methods" that fail with gravity rather it is that "perturbative methods" fail when applied to gravity.
  12. Apr 3, 2010 #11
    My personal take on the matter of discrete is probably closest to marcus. I think that it is the very notion of space-time that breaks down. Rather than it being replaced by discrete spacetime instead there is some "background independent" microscopic theory which has an effective description as a continuos space-time. Of coarse I have no idea of what this theory is. The length scale at which this effective description breaks down can be considering "the smallest length" but this does not imply that lengths are discrete.
  13. Apr 3, 2010 #12
    Ben, For my part I have NOT found a convincing argument against discreteness...at least not from Smolin, Susskind, Penrose,Greene, Thorne and a few others....

    here's a thread that might be of interest if you did not see it at the time....


    and there is one other if I can find it...

    Susskind takes such a "holographic principle" to mean that spacetime IS discrete...plank size areas, one bit per Planck area...Beckenstein bound, etc...which tends to support the original premise....

    On the other hand although I argued FOR discrete spacetime in the above thread, and generally favor that concept, for me the odd characteristic is that so much of quantum theory comes from quantizing an UNPROVEN continuous formalization....which seems an odd way to therefore conclude discreteness or not. I just noticed that Marcus commented on it:
    worse, we know of situations where both formalizations (Qm and relativity) so far fail.....so I like comments above in this thread implying something is missing.

    yes!!!! so right where we are likely MOST interested in discrete vs continuous we run into THAT obstacle....In another recent thread, information as a basis for theory was discussed...and IF information in a finite area or volume is finite it's hard to figure how spacetime or anything would be infinitely divisible. (another holographic argument.)

    Originally Posted by tom.stoer
    This argument does not mean that space necessarily becomes discrete below the Planck length, but that the usual formalism of quantum field theory and GR must be replaced by something different.

    Because neither applies below Planck scale....we have no theory so far to probe there...everything seems to disappear (lose its familiar characteristics) in quantum "foam"...violent irregularities.....analogous to Heisenberg uncertainty....some observables seem to have limited access...
  14. Apr 3, 2010 #13


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    I am sorry, but your argument is wrong.

    The difference is that the Compton wave length is the typical size of an quantum object. So it does not apply to classical objects like stars, but certainly to quantum fluctuations and elementary particles. My argument is non-technical. It simply shows that as soon as quantum mechanics (Compton wave length) and gravity (Schwarzsschild radius) come together, something goes wrong.

    You are right in the sense that this shows that perturbative methods fail - but this is how QFT is defined in most cases.

    Again: the argument is non-technical, but that was should be clear from the beginning as it was explicitly required in the first post to give non-technical arguments :-)

    Btw.: if this argument is wrong, we do you get Planck energy from it? just a coincidence? and why is used quite frequently to motivate quantum gravity?
    Last edited: Apr 3, 2010
  15. Apr 3, 2010 #14


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    How about:

    If entropy is fundamental, then something (what?) must be discrete, because entropy ~ p(x).logp(x), and if x is continuous and has units, then logp(x) will be sensitive to the choice of units, which is unphysical. So x must have some fundamental units, which would be provided by x being discrete.

    However, if the mutual information is fundamental, then the logarithmic term is log(p(x)/q(x)), which is unitless whether or not it x is discrete or continuous. Probably the "observer" point of view should take the mutual information to be fundamental, so from that point of view discreteness is not required.
  16. Apr 4, 2010 #15


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    Just a suggestion to the board. It might help if you define your terms a little bit more carefully. There are 10 different concepts floating around in this thread, and they are not at all a priori related and it makes it impossible to even begin to disambiguate the mess.

    Generally speaking, when someone says 'discrete spacetime', it means to me that you want to stick GR (or something like GR) on a lattice.

    There are many ways to do this (in fact an uncountable infinity of ways).

    Now, the canonical argument against having a fundamental theory based on this is that it breaks lorentz invariance (in the bad way), and thats quite general and not hard to prove. Lorentz invariance is only recovered in the continuum, infinite volume limit (if it all) and a theorist must go to great lenghts to ensure that both the Lorentz group as well as larger groups (like the diffeomorphism group in the case of some formulations of gravity) remain as residual symmetries upon taking this limit.

    If you do not take the continuum limit and instead keep space discretized (in the above sense), you invariably hit issues with the matter sector, b/c there is nothing that protects certain dimension 4 operators from being generated in the UV, that will then require a tremendous amount of finetuning in order to satisfy experimental bounds in the IR. There are of course many other related mathematical nogo statements to the same effect.

    Anyway, I am not aware of any formulation of quantum gravity that strictly speaking insists on spacetime being discrete. All the lattice gravity models out on the market (CDT, random triangulations, Weyl, Regge, etc) are not supposed to make sense unless you morally take the continuum limit at the end.
  17. Apr 4, 2010 #16


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    I actually don't quite know for sure what everyone means by "fundamental", but I suspect you mean that there is a measure of information (entropy) that is a timeless observer invariant? If so, it's not what I think. I think each observer and subsystem encodes their own entropy functional. The case where each subsytem encodes the same entropy functional for some common thing like space, can only be an equilibrium exception I think.

    Hmm... the way I envision things "x" here should simply a label of distinguishable events, so there is not really any units. So first we have a set of distinguishable events, a set. So far there is not yet any distance metric, and even less geometry defined. These are all higher constructs to be explained IMO.

    So in that sense I think I agree with what you say: x is discrete in the sense that it's simply a label of distinguishable events. Now wether it's more natural to mathematicall label these events by integers or real numbers is another story. Personally I find it very unnatural to think that the set of distinguishable events are uncountable simply because it seems to suggest that the observer doing the counting must be infinitely complex. The problem I have then is later when you are about to compute transition probabilities - how to you rationally compare two possibilities with infinite weight?

    Edit: Another thing we also need to reconstruct is stuff like feynmanns path integral. So far there is no proper understanding or explanation on why this is the way to evaluate transition probabilities. If we constrain ourselves by counting only physically distinguishable possibilities, I doubt that that contiuuum will be present except as some limiting case, byt but then the measure in the integral should follow from the construction, rather than beeing left unclear.

    Also, from the point of view of representation of the information state vector without imaginary ensembles I think that even the probability p is discrete. The "possible" ratings might not actually cover the entire [0,1] continuum. So not only is x discrete, I think p is too. This why I'm more radical and think that the continuum probability itself needs to be replaced by a combinatorical approach, beucase I don't think the [0,1] continuuum is physical. And it really screws up the "counting" in the action computations.

    I think if you already on the input of the model, start by assuming a contiuum we have already bypassed several that I think important questions.

    So my own angle to this is more towards discrete information in general. This should apply to information of states, geomeotry and physical law. I think what we think of as the continuum spacetime, is something that will be emergent from a more abstract framework. But then even dimensionality and cardinality must be emergent. To start with a 4D continuum is not explaining anything IMO.

    Last edited: Apr 4, 2010
  18. Apr 4, 2010 #17


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    Ok now I think I see the distinciton you make. In my view, there is no observer indepdenent view at all. There are only interacting views. But there are "effective" observer independent views that are emergent locally. This should included what we usually see as symmetries of natures, but I think none of these symmetries should be seen as timless and eternal for the simple reason that it bypasses the measurement requirement, that we should only speak of what we can say/infer about nature.

    The notion of fundamental timeless observer independent symmetries is a realist type of assumption or belieft that is not the result of a scientific or measurement process.

    If we look in detail, the symmetries are ALMOST a result of human sciences, but what we inferred is a well justified expectation of such a symmetry, we have NOT deduced that these symmetry are perfect, eternal and observer independent.

    But I think we don't need that, the important thing becomes when you see theories as somehow tools for interacting with your own environment, using the imperfect tools we have is still fully rational. So that what happens during interactions is that all parties acts according to their tools, and the result is also an evolution of the "population of tools".

    In think in this hierarchy of laws and evolving laws, all interactions must fit in. Not only gravity, but also the SM-model for particlephysics. But I think that won't work unless we find an abstraction framework that works for both subsystems, such as what we study in particle physics, and cosmological theories. Clearly and something that I refuse to deny is that the notion of ensembles and repeating experiments that does make sense in particle physics when you can make preparations - something that is really a key building stone in our current "measurement theory" - just doesn't make any sense to cosmologicla models. The notion of an observer indepdenent wavefunction of the universe is just outrageous to me.

    So I think we need a "measurement theory" does makes sense also for such models. Ordinary QM, with it's fixed hilbert structures does not, that's my clear opinion. Of course there are more problems in it than just hte continuum, but I think the continuuum is a key problem, although not the only one.

  19. Apr 4, 2010 #18
    I admit that your argument is non-technical but I disagree with it and I think what you say is misleading. The Compton wavelength is the length for which quantum effects become important. To a lot of people a black hole is a collapsed star so it would seem to me that one should clarify that the "typical size" one associates with most black holes is certainly not their Compton wavelength.

    The Planck energy or Planck length is the length scale that classical gravity breaks down. This happens at the centre of a massive black hole but not at the horizon. Its here, at the centre of the black hole, if anywhere, that things "go wrong".

    I totally disagree that QFT are defined by perturbation theory. Perturbation theory is an approximation to exact QFT and will breakdown at some scale. This is true of QED, QCD and gravity.
  20. Apr 4, 2010 #19
    from atyy reference in post #2 [ http://arxiv.org/abs/gr-qc/9508064 ...a Smolin paper, 1995)

    I did not know that (!) and whereas I took this from Haelfix Post #15 to be accurate, now I am considerably less sure:

    Anyway, back to bcrowells original question, if there IS an argument against discreteness, this might be it, although I have not considered it to be personally convincing in the past and now have Smolin's view additionally....
  21. Apr 4, 2010 #20
    It's not clear to me that the holographic principle says anything about the discreteness of spacetime. Horizons block us from observing things behind it. But I don't think space all by itself is an observable thing. Only things floating about in space are observable, not space itself.
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