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Are we digital or analog?

  1. Jul 12, 2009 #1
    Two recent posts/threads asking are spacetime and frequency continuous got me wondering: Do we live in a continuous or a discrete universe? Or a mix of both?
    Your thoughts appreciated.

    (Do we have experimental evidence anything is continuous...I can't think of anything..also, is this question essentially butting up against quantum vis relativity contradictions and so has no decent answer yet? )

    (threads are:
    Is Spacetime smooth, in the quantum physics forum:

    and Is there a limit to frequency,in the general physics forum:

    a few inputs for consideration:

    Seems like big bang and cyclic models are discontinuous affairs starting the universe,
    Seems like a lot of experimental evidence suggest discrete: all those particles in the standard model;
    Relativity sure seems continuous itself;
    the quantum nature of light (photons) strongly suggests EM waves are discrete;
    electron "orbits" are discrete,
    Pauli exclusion principle seems discrete,
    and Planck minimums and maximums don't seem a continuous concept;
    Quantum field theory in the standard model reflects lots of particles...all discrete, both matter and forces,
    "it from bit" (John Wheeler) seems discrete,
    Holographic theory seems discrete (bit per plack area),
    electric charge is discrete (1/6.25 x 1018 coulombs),
    all the nuclear constituents are particles, hence discrete;
    dimensions we live it appear discrete (3+1) but some emergent theoretical models posit fractal and partial dimensional origins;
    gravitons if they exist seem discrete;
    Confinement of a particle in a box leads to a quantization of its allowed energy eigenvalues.

    "...increasing the energy of an electromagnetic wave by one quantum energy step is the same as adding a single photon" Leonard Susskind
  2. jcsd
  3. Jul 12, 2009 #2
    In the two PF threads you linked, I have belabored the point that all the successful theories of 20th century physics are based on a continuous smooth spacetime. I won't bother repeating that point in this thread, instead I will bring in new information, and won't argue dogmatically for smooth physics: in fact, if I have time I will go into detail about why discreteness is important in physics for many observables like energy, angular momentum, and the existence of quanta (particles), although I will not necessarily say anything favorable about discrete spacetime, except that it is remotely conceivable.

    It will always be impossible to show that the mathematical definition of continuity directly applies to spacetime, since mathematical continuity requires smoothness on arbitrary small scales. This means that if we have verified smoothness up to istances as small as [itex]10^{-n} m[/itex] meters, then we still have not verified it at [itex]10^{-(n+1)} m[/itex] meters, and so the mathematical definition is not satisfied.

    In this mathematical sense you are right that we have "no evidence", but keep in mind that mathematical definitions can never be shown to apply to the world with absolute certainty, and so most physicist think that verifing continuity down to [itex]10^{-6} m[/itex] is better than verifying continuity down to [itex]10^{-5} m[/itex], even if a mathematician or philosopher could easily be skeptical and say that this proves nothing. Although the philosopher's skepticism is difficult to attack logically, I can defeat it, but that's another thread, so let's keep the spirit of physics in this thread and say that [itex]10^{-6} m[/itex] accuracy is better than [itex]10^{-5} m[/itex] accuracy.

    One relatively direct way to test spacetime on small scales is by measuring micro-gravity and looking for deviations. So far no deviations have been found down to the micrometer scale: http://www.npl.washington.edu/eotwash/experiments/shortRange/sr.html" [Broken] I consider these kinds of test to be the most direct measurements of spacetime, although electron microscopes etc routinely probe smaller distances.

    An important theoretical constraint on discrete spacetime has to do with a notion that I will intuitively describe as 'compounding' (technically I mean relevant operators in the renormalization group). Initially we may think of small scale discreteness as only causing small scale effects. But in physics the opposite often occurs, as we decrease the scale the effects become even more visible. To understand this it helps to think of chaos theory, where complex systems are extremely sensitive to initial conditions and small perurbations, and after running for a long enough time to initially identical systems can become as disparate as night and day. The bottom-line about 'compounding' is that it strongly constrains discrete theories based on physics we have already observed.

    Well, we know there is at least one consistent theory of quantum gravity in which spacetime is continuous: superstring theory. So far there aren't any consistent theories of quantum gravity with discrete spacetime, but that could be a historical accident.

    The cyclic models are definitely continuous, after all the solution to the Freidman equations of cosmology for a universe with positive curvature and large positive energy density are http://mathworld.wolfram.com/Cycloid.html" [Broken], continuous curves.

    If you view the link on cycloids, you will see that the 'big crunch' events are not mathematically smooth, although they are mathematically continuous. I'm not sure if you know the difference between smoothness and continuity, I would be happy to explain, but the big bang and big crunch are continuous without being smooth. There is no mathematical sense in which they are discrete (I can explain the precise meaning of this term as well, if desired).

    Yes, families of massive particles have discrete masses, this is correct! Understanding this is called the mass gap problem, and the http://www.claymath.org/millennium/Yang-Mills_Theory/" [Broken]

    Keep in mind that quantum fields are continuous, scroll down on this page to see some of the state of the art on localizing particles observationally: http://en.wikipedia.org/wiki/Squeezed_coherent_state" [Broken]

    Yes, both special and general relativity are deeply rooted in smooth spacetime.

    The Wigner functions in the wikipedia link above literally applies to photons in an EM wave. The thing that is discrete is their energy, not spacetime itself.

    Obviously fermion spin is discrete, but the exclusion principle also applies to the spatial wavefunctions for two identical fermions with the same spin, and these spatial wave functions are continuous.

    Planck units are combinations of fundamental constants of QM, relativity, and gravity. Beyond this, everything else is theoretical, and there is no doubt that bizarre physics is going on at this scale, but at present we just don't know enough to say that spacetime is discrete on that scale.

    A better word would be 'discretization', since 'quantization' has a precise meanig in physics that is different from what you think. For example, momentum and position are quantized in QM but a particle can have its momentum and position be any real number with respect to some coordinate system, these variables are continuous.

    It is a general feature of QM that particles in stable bound states, such as in a box, or in a harmonic oscillator, or in a hydrogen atom, will have discrete energy levels, while unstable scattering and decay states will have continuous energy levels.
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  4. Jul 12, 2009 #3


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    An open string on the guitar only resonates at integer multiples of the fundamental. Is the guitar string discrete or continuous?
  5. Jul 12, 2009 #4


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    Is this the 1 000 000$ question in "Who wants to be a physicist?" :biggrin:
  6. Jul 12, 2009 #5
    Here's an interesting, if somewhat windy comment:

    good grief! (I am again reminded why I did not choose math as a major...)
    any way, it's from


    Does this apply to QCD and QED in the standard model??
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  7. Jul 12, 2009 #6


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    what does apply?
  8. Jul 12, 2009 #7
    Civilized, great input....I'll have to think about some...

    I disagree...that is what has always been thought about most theories and we have found smart ways and new technology...but I take your point...Leonard Susskind may have some "work arounds"...I'll see if I can find any hints....

    I may have gotten carried away!! I do believe that's correct.

    I don't think it's especially relevant....(See my prior post from Robert Mills, for example)

    be back later..
  9. Jul 12, 2009 #8


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    yeah but is a guitar string quantized or discrete?
  10. Jul 12, 2009 #9


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    There is another metaphysical option here that may be worth considering as a grounding to physical expectations.

    Instead of either/or it could be both - and both in a special "limits" sense. That is asymptotic. Continuous and discrete would both be extremes that would be approached infinitely and infinitesimally closely, but never actually reached.

    So on the global scale, the world would be continuous. On the most local scale, it would be discrete. Or rather it would approach both these extremes on both these scales without ever completely being either/or.

    Of course, mathematically, we can then model the world in terms of these limits states. The metaphysics can say one thing, the physics apparently another. Which is what I feel has happened with GR/QM for example.

    An analogy might be the number line. We have the apparent paradox that we can add up to a continuous quantity (infinity) by adding up in discrete steps. Well I'm with the ancient greeks in saying metaphysically, we can only approach infinity via a discrete counting process, not arrive at it. But mathematically, we can then simplify in a useful way by acting as if the limit state is indeed reached.

    The paradox applies the other way too, though much less often mentioned. So we just assume the discreteness of the number 1. But really, saying 1 implies we mean 1.000.... (and not for example, 1.0000....1). So we can approach the idea of a discrete integer infinitesimally closely by constraining the continuous number line. Metaphysically, we never reach the state of perfect constraint where we can be sure we have exactly 1. But mathematically, we can chuck away the final asymptotic uncertainty and just get on with using the construct.

    So I think this is the spirit to approach the question in.

    Metaphysically, it seems obvious that reality tends towards both the continuous and the discrete. Reality encompasses both diametrically opposed and mutually contradicting (dichotomous) extremes. This then sets up the nagging question of which is the more fundamental.

    A good way out of this - metaphysically - is to say both are equally fundamental. They just actually exist in opposite directions. Opposite directions of scale. So go large and all seems continuous and connected. Go small and all breaks up into discrete and local. Reality is a "mix" in this way.

    But then mathematically, we find it more effective to separate the mixture. We jump each nearly state to a fully broken state. The continuous becomes the Continuous. The discrete becomes the Discrete. (And the mixed can then become the Mixed in more complicated modelling if we desire).

    So what I am arguing here is that too much confusion can be created by asking for evidence of "what reality really is" so we can then model it in "true fashion".

    The physics does not have to exactly reflect the metaphysics. But if the dichotomous story is "true" we should expect two complementary viewpoints on the world to emerge in our modelling.

    The two viewpoints will seem irreconcilable because each is defined by being exactly what the other is not. To be discrete is precisely to not be continuous, and vice versa. How can you map one back on to the other?

    But if we can step back and see the one world from which these two modelling extremes emerged - as moves in opposing directions of scale - then irreconcilable mathematics becomes reconciled metaphysics.

    [Ought to add that I faced exactly the same foundational issue in neuroscience - is what the brain does to produce awareness a digital or an analog process? A binary metaphysical choice that continues to create huge confusion in mind theorising. Of course, neuroscience is much further away from having effective theories in terms of either modelling choice. And indeed, a Mixed model may be what is necessary.]
    Last edited: Jul 12, 2009
  11. Jul 12, 2009 #10


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    To question the continuum models is good I think.

    Except for the two threads you mention, there is another one where I expressed some opinons before

    (1) Are there evidences of a discrete space-time

    (2) Dark energy as inertia of information-reference?
    Goto post #14

    If if you question the continuum as a scientific inquiry, then the questions becomes entangled with "is information discrete" or is evidence discrete. you might ask yourself that, and consider a finite bounded observer, what this observer can distinguish.

    I do not think it a coincidence that all the headache we see from divergences and infinities are seen in continuum models. It is as if this continuum itself contains a non-physical redundancy the we can not tame properly.

    The question is, if you remove the mathematical redundancy, what kind of structure do we get left?

  12. Jul 12, 2009 #11


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    How do we decide continuous/discrete in say electroweak theory, where the calculations all treat spacetime as continuous, but the theory does not have a formal continuum limit (because it is neither asymptotically free like QCD nor asymptotically safe)?
  13. Jul 13, 2009 #12


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    The generic way in which I imagine that some future theory will handle all interactions is that the interactions themselves, as action properties of material observers, emerge along with spacetime as evolving relations between matter.

    For me at least, the idea of reconstructing the spacetime continuum, from the physical information (in which the continum is merely a special limit) must go hand in hand with emergence of interactions.

    No spacetime, no interactions and vice versa.

    I personally think that the question of howto deal with some of the technical issues in the standard model, might not be treated just by considering the same interaction lagrangians ontop of whatever replaces spacetime. Since the lagrangians somehow loose meaning if spacetime isn't what we think, the only remedy is to reconstruct also the actions and matter.

  14. Jul 13, 2009 #13
    Given the metaphysical nature of the question, I think one would have to consider first a method of analysis in order to ascertain whether this is a false problem (or not). By itself, this is exceedingly difficult question. Given the operation of our minds, we envision models at extreme (and mutually exclusive) directions -- discrete vs. continuous. But it does not mean that those are the only possible articulations of reality.

    I suggest a reading of Deleuze, https://www.amazon.com/Bergsonism-G...sr_1_1?ie=UTF8&s=books&qid=1247487312&sr=8-1", first chapter.
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  15. Jul 13, 2009 #14
    If we were really analog, then we would not be formally describable. It is not good enough to look at whether or not space-time is discrete. What matters is if the universe admits a formal description in terms of a finite amount of information. This amounts to saying that all processes in the universe are computable.

    It is known that classical mechanics is not formally describable in general. Classical mechanics admits the existence of a so-called "rapidly accelerating computer". This is a mechanical computer that accelerates such that the duration of a clock cycle is half of the duration of the previous clock cycle. If the first clock cycle takes one second, then in a time span of 2 seconds you can pack an infinite number of clock cycles.

    Such a computer can then "solve" the halting problem merely by running an algorithm for an infinite number of steps in a finite amount of time. It can thus be used to find the solution of mathematical problems that are not decidable. E.g. if the Riemann hypothesis is true but no proof of that exists (if it were false then a proof would exist, as you could then point out that zero that is not on the critical line), then the rapidly accelerating computer could check if all the zeroes are on the critical line one by one in a finite amount of time. It could even go through all proofs using Hilbert's proof checking's algorithm and tell you that no proof of this fact exists.
  16. Jul 13, 2009 #15

    Hans de Vries

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    The discrete state is a simplifying abstraction allowed by the semi-
    stationary nature of the continuous function.

    Quantization is per definition a simplified abstraction. Expressions
    using quantized states generally violate Special Relativity. This does
    not mean that they don't lead to the right answers but they ignore
    the underlying relativistic processes, which are often unknown.
    For example the amplitude.

    [tex]\langle \psi_1 | \psi_2 \rangle[/tex]

    violates Special Relativity. The integral over space is not allowed since
    it leads to space-like outside-the-light-cone dependencies.

    Regards, Hans
  17. Jul 13, 2009 #16
    Wow, great replies by many...Thank you so much. I must consider a number of new thoughts!!

    These are the type replies I hoped this forum would provide..... so as for all who are interested to understand better. As I may have posted in another thread, I never even thought much about discreteness of spacetime until someone asked the " is frequency continuous" question. After reading perhaps 15 or so popular contemporary books from world reknowned physicsts who repeatedly remark on it's discreteness, I never even thought there would be an "objection".

    In any case my own mind remains open....

    Chapter 16 of Roger Penrose's book, THE ROAD TO REALITY, titled The Ladder of Infinity seems to address at least some issues related to this discussion...perhaps without clear and firm resolution. 16.2 is "A Finite or Infinite Geometry for Physics?" and 16.5, "Puzzles in the Foundations of Mathematics" might even be disturbing for some... If anyone can comment on Penrose's perspectives that would be fascinating. The math discussion, including sets, classes and (16.7) "Sizes of Infinities in Physics" is rather advanced and I know enough to know what I don't know.
  18. Jul 13, 2009 #17


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    This was my point. Metaphysics always breaks down into dichotomies. We then have to ask is this just because humans are simple minded or whether this is in fact a deeper reflection of how things really are?

    This is the area I have been studying the past decade so I do feel qualified to offer some answers.

    Dichotomies are suspicious beasts because we think perhaps reality is more complex and reducing to pairs of alternatives is over-simplifying. But the key lies in the fact that dichotomies are formed by mutual exclusion. One extreme is everything the other is not. It is its antithesis. The symmetry is broken and you have instead maximal asymmetry.

    The question then is where do you go from the dichotomy. One familiar metaphysical response is to say we have two extremes, one must be valid, the other false. Or one must be fundamental, the other emergent. This is monadism - the search for a single essence. So in this case, the expectation that either continuity or discreteness will prove to be the primal view.

    An allied response is dualism. Now this accepts both extremes as fundamental. But there are two essences. So we have the dualism of substance and form, or body and mind. A double monadism.

    The alternative path (one far less frequently trodden) is instead to accept both extremes (as limit states only) and then heal the divide via their fruitful interaction. So two extremes may be produced, but the story is then in the "thirdness" (a Peircean term) of how they interact. This leads on to a hierarchy theory approach in which you have levels in dynamic interaction.

    So everyone lands up in dichotomies (because that is the logical result of trying to divide a complex reality into its simplest possible categories or features). And then you have to chose either to make one extreme fundamental, or go triadic and model two extremes in interaction. Or take the unsatisfactory path of dualism in which you believe in two fundamentals, but see no possible causal connection between them.

    Just think of all the dichotomies we accept as useful. Symmetry-asymmetry. Algebra-geometry. Space-time. Substance-form. Local-global. Atom-void. Particle-wave. Local-nonlocal. Signal-noise.

    And recall what Bohr said about dichotomies (or complementarities as he called them). Paraphasing - when you find that the absolute negation of an obvious truth also seems obviously true, then you know you have arrived at a profound level of modelling.

    As you say, a crucial question is whether dichotomisation is just epistemology, a habit imposed on reality by inadequate human minds? Or whether it is ontic, in fact also the way reality forms itself?
  19. Jul 13, 2009 #18
    We know that the field equations of General Relativity can be derived from the least action principle of the Hilbert-Einstein action integral. And we put this into the Feynman path integral to get QG. But if we could justify the path integral from first principle and discern the Hilbert-Einstein within that, then we would know that the spacetime metric IS quantized.
  20. Jul 13, 2009 #19
    Here is a http://motls.blogspot.com/2005/11/discrete-physics.html" [Broken] that articulates a lot of points I agree with. I will give his main points in bold and my thoughts on each:

    Both discrete and continuous mathematics matter

    The classical guitar string is a great example, the string itself is can be treated as continuous, and the frequencies of the normal modes are discrete.

    Discrete and continuous concepts are related

    For example the Reimann hypothesis which connects the primes with smooth functions in the complex plane, or the entire branch of mathematics that treats the topology of manifolds, which seeks to classify smooth structures using (typically) discrete invariants.

    Crackpots are almost always discrete

    Here by crackpots I believe Motl doesn't mean the extended sense e.g. anyone whose research he doesn't like, but rather the utter and total cranks who we have all seen, e.g. work alone, barely use math, off the charts on the Baez crank scale, etc. I do agree that most really terrible cranks do use discrete concepts, but it probably mostly has to do with there not having learned calculus at all.

    Modern history of continuous dominance

    It would be difficult to argue against the statement that continuous methods have dominated fundamental physics for the last two centuries. In cases like the guitar string, the continuous description is fundamental and the discrete modes are only in the solutions to the equations; this is the same story as QM. In other words, we start with a continuous description and sometimes derive discrete solutions, this is very different than starting from a discrete description.

    I think Lubos illustrates the point well in this paragraph:

    There are considerably more points, and I suggest anyone who is interested to read the post, it is not one of his more mean-spirited rants, and it is mostly full of physics content. The latter section of his rant is dedicated to debunking the "we just invented digital computers => the universe might be a digital computer" line of thought. A good read.
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  21. Jul 13, 2009 #20


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    Continuous methods could be what works best for modelling, yet still leave the ontological question open. And a good grasp of the metaphysics would be what helps to understand the current limits of models.

    So for instance, you quote Motl on QM and the continuity of the wavefunction. Yet the wavefunction is precisely a powerful model because it (as completely as possible) excludes the discrete from its domain. The discreteness being the issue of observer collapse which happens at some point in spacetime.

    It is not that using "pure continuity" to create a model is bad. It is just that we can expect it then to exclude some matching aspect of the "purely discrete". This then has to be inserted somehow back into the picture to make it complete.

    The guitar string analogy, while very appealing, indeed useful, is also flawed in that it too must assume discreteness. A guitar string must be pegged at some distinct point. In fact at both ends. So all of the string is continuous, but it only gives well behaved harmonics due to it being discrete in its boundary.

    Imagine for example a string that is only vaguely pinned down at its ends. It can no longer be plucked crisply.

    To take this discussion in another direction, I would suggest a better analogy is fractal phenomena like critical opalescence, cantor dust, and other "is it continuous/is it discrete" examples of geometry.

    Fractals are an example of what I would call Mixed - the Continuous and the Discrete expressed in interaction over all scales.
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