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Are there evidences of a discrete space-time

  1. Apr 10, 2009 #1
    Many people believe that spacetime is discrete but what evidences have they ??

    in one of issues of 'Scientific of American' explained that if space time was discrete then depending of the energy of light and since spacetime would be discrete the 'effective' space of light would not the same ,since in a discrete spacetime for short wavelength (high energy) the particle would note that space is discrete

    another possiblity is that for a discrete space time lorentz invariance would not follow, so the fundamental group of Special relativity would not be the Lorentz group, but what more evidences are there ?
  2. jcsd
  3. Apr 10, 2009 #2


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    This is a common misconception. Rovelli, one of the main developers of Loop Quantum Gravity showed several years ago that LQG was consistent with Lorentz invariance.

    One can have discrete spectra at the level of measurement---a smallest nonzero area measurement, for example----without space being "divided into little lumps". Discrete spectra of the geometric operators does not imply Lorentz violation.

    I think the paper was 2003. Look it up on arxiv if you want. This is a common confusion that comes up here at PF a couple of times a year.


    There is disagreement on the issue of vacuum dispersion. There has been some talk in both the string community and the LQG community about the possibility that very high energy photons might be slowed down----by an effect that is suppressed by a planck-scale coefficient.
    If this effect is real, it will be detectable as soon as Fermi satellite has collected several years of data on Gammaray Bursts. And if it is false then by the same token it will be ruled out. Fermi is collecting data, so this issue in in the early stages of being decided.

    The latest word on vacuum dispersion is a talk given by Amelino-Camelia at the Perimeter Institute last month.
    You should watch the talk. It is online video and quite interesting.
    You can also choose just to download the PDF file of the slides

    The conjecture (by some string and LQG people) is that there is an energy EQG which is of the order of magnitude of Planck energy (1019 GeV) such that the speed of a photon with energy E is fractionally reduced by E/EQG.

    In other words to take an unrealistic example, if the photon energy E is one billionth of EQG, or roughly speaking one billionth of the Planck energy, then the speed will be reduced by about one billionth. Actually a real gammaray photon might have an energy that is a billion-billionth of Planck (10-18 )
    So it's speed instead of being c might be (1 - 10-18) c.
    After a GRB has traveled on the order of a billion years this small slowdown would have resulted in a delay in the arrival time (of the higher as compared with lower energy photons) which Fermi can detect.

    Amelino-Camelia discusses the results from one recent GRB which Fermi has observed.

    It will take many such observations to establish any reliable result (either positive or negative) as he points out. The news is that Fermi is sensitive enough to see the effect if it is there.
    Last edited: Apr 10, 2009
  4. Apr 10, 2009 #3


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    Here is some more regarding vacuum dispersion (the conjectured slight slowing of extremely high energy photons).

    Charles Dermer of the Fermi collaboration recently reported a lower bound estimate on a Planck scale QG parameter derived from Gammaray Burst (GRB) observation by the Fermi satellite gammaray observatory launched in 2008.

    GRB 080916 C was a recordbreaking burst observed September 2008. The light had been traveling 12 billion years. Some high energy photons came in a few seconds later than average. One actually came 16 seconds after Fermi was first alerted by the initial arrival of lower energy gamma. This doesn't prove anything, they need to replicate it and confirm that the delay is proportional to the travel time.

    For bursts that traveled less than 12 billion years the delay should be proportionately less, otherwise it is not a dispersion effect.

    It is conjectured that microscopic geometry of space may depart from the apparent macroscopic norm that we are used to. There are various Quantum Geometry/Gravity models of what the micro-geometry might look like. These need to be tested. Some of the models suggest that microscopic foam-like or other irregular geometric structure at quantum level might slightly slow down very short wavelength (high energy) photons. Light of more common wavelengths would not be noticeably affected--only very high energy photons.

    The conjectured dispersion is suppressed by a parameter which Dermer calls the "quantum gravity mass MQG."

    This just the mass-equivalent of what i was calling EQG

    The present result is that if there is such dispersion then this parameter must be at least 1.5 x 1018 GeV, energy equivalent.
    For simplicity I'll continue using the energy version of the parameter as EQG

    The QG folks expect that if this dispersion effect is there the parameter will be on the order of 1.2 x 1019 GeV.
    So they seem quite happy with Dermer's report of the Fermi satellite result.

    The conjectured fractional difference in speed is equal to the difference in photon energy delta E divided by the parameter EQG.

    So let's do an example. One of the photons that arrived late was 13 GeV, lets assume that there was a 10 GeV difference between that and the main batch and we will take an EQG that is right at Dermer's lower bound. Say EQG = 1.5 x 1018 GeV.

    The conjecture is that that photon would be fractionally slower by the fraction delta E/QG = 10/(1.5 x 1018) = 1/(1.5 x 1017)

    So the predicted delay, in seconds, is going to be (12 billion years)/(1.5 x 1017) = 2.5 seconds.

    That seems small compared with the 16 second delay that we heard about, but that was measured from the very first arrival detection of the burst, the trigger event. So it's probably not so unreasonable. It leaves some room for random spread.
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  5. Apr 10, 2009 #4
    There is no experimental confirmation so far.

    Its an open question but quantum theoretical considerations suggest that Planck scale is the minimum for most of what we experience with quantum foam disrupting time,space,mass, etc at the sub Planck scales....in other words, the conjecture is that Planck scale energy is so disruptive at tiny distances and times that only energy exists....
    You can check various theories under headings like Penrose Spine networks, Planck scale, spin foam, and aspects of string theory....

    I was not aware that Rovelli had shown consistency between LQG and Lorentz invariance...I'm not sure thats good news or not... In general, relativity is likewise classical/continuous and does not do well at inherent singularities (points)...by positing discrete (Planck minimums) rather than point characteristics quantum theory would appear to have a better chance of dealing with sub microscopic considerations....as might string theories....
  6. Apr 10, 2009 #5
  7. Apr 10, 2009 #6


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    I'll get the link, so you can see for yourself what the assertion is. Better not to rely on other people's paraphrase.

    Wow, that was quick! I just googled "Rovelli Lorentz" and got two articles by Rovelli about this published in Physical Review D,
    in 2002 and 2003.

    Here is one:
    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."

    The point I guess is that you can have a minimal observable length or area and still enjoy Lorentz invariance.

    LQG does not represent spacetime as divided up into little bitty lumps. (Spacetime already in 1915 General Relativity has no objectified physical existence, as Einstein pointed out on several occasions.) LQG, like GR before it, is about geometry. In the LQG case that means quantum states of geometry and geometric observables, measurements in other words. The discreteness in LQG is at the level of observables (discrete eigenvalues of measurement operators.)
    Last edited: Apr 10, 2009
  8. Apr 12, 2009 #7
    But if space time spectrum is discrete how can you even define continous operations such us infinitesimal rotations, integrals , derivatives ??
  9. Apr 12, 2009 #8


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    Do you know what a spectrum is? Maybe not.
    Think of the discrete energy levels of a hydrogen atom. Discreteness of the operation of measuring or observing---a discrete set of possible outcomes.
    The spectrum belongs to the operator, not to space or spacetime.

    We do not know that space exists, what we know is that we can measure the area of a tabletop, or the length of a broom-handle.
    The mathematical representations of space are just provisional conveniences. In LQG space is provisionally represented by a smooth manifold, a continuum. Naturally one can do integrals and infinitesimal rotations on such a thing---that is what it is meant for, after all :biggrin:. To do calculus!

    In LQG the mathematical representation of space, and of spacetime, is not discrete. But by a lovely accident the quantum observables corresponding to the measurement of physically real areas (and other geometric quantities) turn out to have a discrete set of outcomes.

    What this could mean is that in LQG the ability to do calculus only emerges at some macroscopic scale. Down at Planck scale geometry may be chaotic, foamy, choppy, irregular as to dimension etc. I wouldn't say discrete. It could be impossible to do the usual sorts of differential and integral calculus and get the right answers, at that scale. Differential geometry---manifolds with a fixed integer dimensionality good down to infinitesimal lengths, might not be applicable.

    I say might.

    In any case space and spacetime are mathematically represented by smooth differentiable manifolds, as continuous as you please, in LQG. Discreteness only appears in the observables, after one has constructed quantum states of geometry and set up a Hilbertspace of the states of geometry.
    Last edited: Apr 12, 2009
  10. Apr 12, 2009 #9
    computer simulations are discrete yet they work well enough.
  11. Apr 12, 2009 #10


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    Zeta this "many people" is just a little too vague for me to deal with. It leaves me groping, wondering who these people are and what various things each of them mean by "discrete".

    Different Quantum Gravity approaches have different kinds of discreteness.

    Loll's type of QG was written up in the SciAm recently. I have a link in my sig. It represents spacetime by a continuum---then it selects geometries on that continuuous spacetime which are describable using little triangle-like blocks---then it lets the size of the blocks go to zero.

    So at no time does Loll say that spacetime is discrete. But she uses a lattice-like scaffolding which approximates uncertain geometry, and then lets the size of the lattice element shrink to zero.

    Mathematically spacetime is continuuous all along. Topologically it is a cartesian product of a hypersphere and the real line. But of course people call the approach discrete.

    So "discrete" is a tricky word. Loll's approach is called discrete, and yet if you follow it down, you never get to a stage where spacetime is made of little lumps, or little marbles, or isolated points. It is always a continuum. But it is quantum, so it can be pretty chaotic and crumpled and crinkled down at very small scale. There is a kind of uncertainty or indeterminacy about the geometry at very small scale---that's the essential thing about quantum geometry/quantum gravity.

    So to get a good answer what you need to is give a link to where somebody actually says space is discrete and let us look at what they are actually doing and see what they actually mean by that.

    Could you have been thinking about Loll's article in the SciAm? It is the "signallake" link in my signature. If not, you definitely should read the article. It is one of the better quantum geometry/gravity articles that SciAm has so-far published.
  12. Apr 12, 2009 #11
  13. Apr 12, 2009 #12


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    let me add that discrete space time is a naive goal. In quantum mechanics, you have not a unit of space nor a unit of momentum, but you get a unit of "space times momentum", nameply Planck constant. Similarly, you could have a unit of area or a unit ov volume but not a unit of length.
  14. Apr 13, 2009 #13
    "Discrete" space-time does not mean that at the Planck scale you have some kind of ambient environment represented by countable, individual geometric structures ("space-time clumps"). But once you establish a mathematical space with certain properties which is spanned by a set of functions containing geometric or whatever other relevant information on quantum space-time states, you see that the action of certain operators on this space might result in a discrete spectrum for the eigenvalues of the operator. These operators represent observables of the quantum space-time states, which in certain quantum gravity models these can be, eg, the area or volume of space. (Such a procedure is just the "canonical" method of quantization, extended to the gravitational field. This is not the only possible way to address the physics at sub-Planck scales, though).

    "What is observable" has many fundamental implications that are still under debate even for quantum mechanics, so even if experiments eventually verify that area and volume are discrete quantitites -- certainly a great achievement for LQG -- I believe there will remain debates concerning the objective reality of the "quantum", interpretative issues, etc.

    A really important question in quantum gravity is how to implement the dynamics, ie, the elementary interactions between degrees of freedom at sub-Plack scales, even though I suspect the nature of the quantum gravitational degrees of freedom is not known (I'm not certain they can be inferred from the low-energy GR limit, and in any case the establishment of the existence of the graviton will have to wait experimental evidence). Regarding the underlying "substantiality" of space-time, it is a open issue, even though I interpret that classical GR regards space-time as a dynamical field. I am not certain whether some quantum gravity models (eg LQG) seek to "go away with it" (the space-time manifold) altogether.
    Last edited: Apr 13, 2009
  15. Apr 13, 2009 #14


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    There is an interesting semantic issue. You are probably familiar with the 1915 quote from Einstein
    See also page 43 of this pdf at a University of Minnesota website

    ==quote from the source material==
    ...In the introduction of the paper on the perihelion motion presented on 18 November 1915, Einstein wrote about the assumption of general covariance “[b]by which time and space are robbed of the last trace of objective reality[/b]” ([color=blue]"durch welche Zeit und Raum der letzten Spur objektiver Realität beraubt werden,"[/color])

    I would not agree with you that "GR regards space-time as a dynamical field."
    I would rather say that GR regards [B]geometry[/B] as a dynamical field. The metric represents geometry. Geometry = gravity. The metric is the gravitational field. More precisely not a single metric but a diffeo equivalence class---the mathematical object is the set of equivalent metrics representing a geometry.

    I would say that already [B]GR does away with space-time[/B].

    Now the semantic problem is how can you talk about geometry if it is not the geometry OF any substantive thing? Geometry is like the [B]smile of the cat[/B], after the cat has disappeared.

    If spacetime has no physical or objective reality (in Einstein's words) and what is more fundamental is now geometry, then this throws the ontological problem back into the lap of the observer---geometry consists of measurements and observations, of relations between events---of areas of material objects like desktops and wavelengths. Or at least it seems to me that it does. I feel the issue is unresolved.

    But i would say that LQG is not the first to [B]do away[/B] with space-time. The first steps down that road were taken in 1915. LQG is just continuing with the program.
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  16. Apr 13, 2009 #15


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    Set aside the important point of exactly what a discrete spacetime would mean and be defined, just for perspective, I would like to turn it around and ask what evidence we have that spacetime is a continuum all the way down?

    I have never seen a proof. Instead, I suspect the success of calculus is the reason why we often tend to extrapolate datapoint into a continuum context. But real numbers can be constructed from rational ones in the context of discrete systems, by considering limits.

  17. Apr 13, 2009 #16


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    a good question! But the prior question is "what evidence do we have that spacetime even exists?" More precisely, does a mathematical representation of spacetime even have to be included in a fundamental description of reality?

    This throws it back onto a process. Processes like successive approximation and taking limits. The activities of an agent who measures. Who was it who said that nature made the integers and all the rest were the work of man? Let us avoid the G-word please.
  18. Apr 13, 2009 #17


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    Does a 4D spacetime, as we know it, need to be explicit as universal in a more fundamental picture? I fully agree, that it need not, and i personally do not even expect it to. I don't even expect the topology to exist at fundamental level.

    Roughly I like this perspective. However I suspect it can be interpreted differently.

    If we think of processes as communication channels, I think these bound to
    evolve as well, an not be present in the fundamental theory.

    But I find that in any reconstruction, from the point of view of an observer, the concept of distinguishablity and counting are more easily imagined to be primitives, than is the continuum.

    I also think that the evolving emergence of topology and spacetimes, is somehow closely related to a reconstruction - by means of counting distinguishable states - of the continuum.

    Maybe it could be, that to understand the continuum, and spacetime for that matter, we NEED to question it, and understand the physical basis for it, from the point of view of a similarly emergent inside observer.

    An increasingly more complex observer, could be closer and closer to a continuum picture, than a simple observer is.

    I think this sound incredibly simple to some, as if the universe are composed of blocks. But that's not what I think. I rather think it means that the very notion of degree of freedom is emergent and relative to the observer.

    So the reason for a "possible" discrete structures such as spacetime, could IMHO be not that spacetime is actually built from blocks, but from the fact that the observers information capacity is bounded.

    So the counting, would at the fundamental level refer to countable states in the observers memory structure, which I identify with matter.

    The OT, even asked for "is there evidence of a discrete space-time"? I'd say there is some support for discrete evidence :) This is where some of the probability theory started, as a quest to "count" or rate "evidence". Jaynes, quickly jumped into the assumption that degrees of plausability is to be represented by real numbers.

    Lets assume this does not make sense. Assume that instead of thinking that discrete models approximate continuum ones, what it the continuum models instead approximates the discrete ones.

    The historical preference is not hard to see. Without computers, analysis is the tool at hand. But now the story start to change. Number crunching is not a major problem anymore, moreover, most continuum models need to be discretised anyway for any numerical solutions. If not sooner, at least at the point when yuo are to represent a real number in a computer memory.

    Just curious. What's the G-word? Geometry? or God-view? Forgive me for beeing a bit retarded concering some of the slang. It took me a while beofre to figure out what "buzz-word" meant, so this time I thought I'd ask :)

  19. Apr 14, 2009 #18


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    Penrose reasoning

    I think it's interesting to review at Penrose introductory reasoning from OLD papers...

    "My basic idea is to try and build up both space-time and quantum mechanics simultaneously|from combinatorial principles|but not (at least in the rst instance) to try and change physical theory. In the rst place it is a reformulation, though ultimately, perhaps, there will be some changes. Di erent things will suggest themselves in a reformulated theory, than in the original formulation. One scarcely wants to take every concept in existing theory and try to make it combinatorial: there are too many things which look continuous in existing theory. And to try to eliminate the continuum by approximating it by some discrete structure would be to change the theory. The idea, instead, is to concentrate only on things which, in fact, are discrete in existing theory and try and use them as primary concepts|then to build up other things using these discrete primary concepts as the basic building blocks. Continuous concepts could emerge in a limit, when we take more and more complicated systems."
    -- http://math.ucr.edu/home/baez/penrose/

    This basic idea is appealing to me as well, and has similarities with how I think.

    Except I think we need to change the theory, a possibility which Penrose seems to leave open as well, in the second instance.

    But we do have to start somewhere:

    "The idea, instead, is to concentrate only on things which, in fact, are discrete in existing theory and try and use them as primary concepts|then to build up other things using these discrete primary concepts as the basic building blocks. Continuous concepts could emerge in a limit, when we take more and more complicated systems."

    I agree here too.

    So what are the primary concepts?

    Penrose's suggestion:

    "The most obvious physical concept that one has to start with, where quantum mechanics says something is discrete, and which is connected with the structure of space-time in a very intimate way, is in angular momentum. The idea here, then, is to start with the concept of angular momentum| here one has a discrete spectrum|and use the rules for combining angular momenta together and see if in some sense one can construct the concept of space from this."

    Here I think there are better starting points, that attempts to incorporate the ambition of a possible second instance from start. I think his desire to not change the theory in the first instance, probably as way to not complicate things, might prevent fidning a better starting point.

    How about forulate the theory here, not in penroses slightly realist way, but more as a more pure observer view, as a physical model for scientific inquiry.

    What else is discrete? How about evidence, or degrees of plausabilities? How do you DISTINGUISH two different degrees of plausability? Here I think it's plausible to suggest that it's a very incoherent jump, to suddently talk about real numbers. I don't think a real physical observer, can distinguish all real number from each others. The embedding of observable reality into a conntinuum seems to be an illusion.

    I expect the starting point to be that of discrete evidence, and information. And use that, in a very general senese, to build measurement operators. It would suggest that the primary concepts are living in the observable view of an observer which is herself evolving.

    The reason why I prefer to start with discrete evidence, as an ambition to construct only true inside observables, is that it seems to me to be "the most obvious physical concept that one has to start" if we want to make a theory that is to also be a proper non-realist model for scientific inquiry and measurement, which I think of as the heart of QM.

    This will itself, modify QM, to be an even MORE realistic theory of measurement, than currently. So it still keeps the heart of measurement theory, but it just aims to make it relational. Something which current QM clearly is not.

    So in that view, I still think that the concept of angular momentum is not possibly fundamental to quality as primary concepts.

    How about if we, somehow along the basic spirit of Penrose, instead where to try to reconstruct a theory of measures, and probabilities in terms of discrete information?
    And then later identify and classify things, as space, matter, forces etc as emergent measureable structures as the observer increases in complexity?

    What weirdness & clarity could that yield? That is what I desperately want to know.

  20. Apr 14, 2009 #19


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    Re: Penrose reasoning

    Noted tis potential point of confusion.
    I use the word realistic here, NOT in the sense of "realism" but in the sense of "consitency with de facto observations" in a sense of minmum speculation. It is in fact the direct oppositve to the meaning of "realism" it is usually used.

  21. Apr 14, 2009 #20

    Yes, of course I know the quotations of Einstein that you mention, and I also have a good technical knowledge of general relativity. I agree with you that I should have used a more precise term to what I referred as the "substantiality" of space-time and "space-time" as a dynamical field. So I will clarify my previous comment a bit. I believe you are making a confusion between *spacetime substantivalism* in one side, which Einstein did make an excelllent argument against by the "hole argument" and *manifold substantivalism*, in the other side, which is still under debate, at least for those with a realist philosophical position. The point is that, mathematically speaking, space-time is constructed from a manifold of events, over which more and more structure is added (metric, matter fields). General covariance implies equivalence of various distributions of metric and matter fields, making them physically equivalent, so in that sense is that Einsten's quotation regard as space-time with no objective reality. But space-time can still "exist independently" as a *manifold* of events -- this is the point of view of manifold substantivalism, and I would say that the removal of that last arena is what background independent quantum gravity theories seem look forward to, completing, in essence, the path that Einstein began in classical GR. Evidently the issue becames exceedingly difficult because you have to unify (or reformulate) the concepts of space and time that comes with quantum theory with those of GR -- you have to extend the "hole argument" accordingly -- to a "quantum hole argument".
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