Continuous or discrete?

Just a few of the questions running through my head today as I head out to buy propane, gas for the tractor, and to go fishing:

At the most fundamental level, is the universe continuous or discrete?  Or maybe it’s continuous but looks discrete to us?  Could we ever even know for sure?  (And, on a related note, what the hell is spacetime anyway?  Is it just a convenient way to keep track of the location of ‘events’ or does it have some real, physical nature, i.e. is it ontological?)


27 Responses to “Continuous or discrete?”

  1. BlackGriffen Says:

    Discreteness seems to be something that we could potentially observe, but never rule out. The reasoning being that you can always imagine that there’s some smaller length scale just beyond what you’re presently able to probe where the discreteness effects turn on.

    Although I would guess that the more meaningful question would be is there some fundamental length scale smaller than which nothing meaningful happens? Or are we living in Feynman’s onion, with ever more layers to peal back ad infinitum? Regardless, I seriously doubt that we’ll have the answers to that question sorted in our lifetime.

  2. quantummoxie Says:

    Ah, well, presumably that length scale is the Planck length – or, rather, that appears to be the limit to which we can actually observe.

    The reason this has been on my mind is because I’ve been thinking about field theory and whether it’s really the right approach (heresy, I know, and I’ll get crucified for it, but whatever).

  3. serafino Says:

    Doesn’t “discrete” mean there is only a limited quantity of (available) information, carried by each quantum? On the contrary, what does it mean “continuous”? Does it mean there are unlimited possibilities of information about the system under investigation?

  4. quantummoxie Says:

    I guess I was thinking in terms of spacetime rather than in terms of information. So is spacetime continuous or discrete? Or does it even have an ontological status to begin with?

  5. serafino Says:

    We can imagine a universe only made of a bi-particle, say two momentum/position entangled particles. In this case space-time (whatever it means) would be strange enough, and possibly non-local, or with non-local effects (the particles can be massive). In other words I don’t think space-time has an ontological status, rather a ‘relational’ status.

  6. quantummoxie Says:

    Well, I suppose, in some sense, everything is relational. I guess I’m thinking in more concrete, physical terms. But then again, what does that mean? I have no idea. 🙂

  7. Ian,

    I know from reading other things that you’ve written that this will be an unsatisfactory answer, but I think that David Deutsch generally has the right approach to this question. In short, he thinks that “within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse.”

  8. quantummoxie Says:


    While I’m not a big fan of the multiverse idea, it’s certainly an intriguing answer – and, at the very least, doesn’t brush the problem under the rug, as it were. 🙂

  9. Ian,

    That’s one of the things I like about the MWI. It provides a number of answers to these types of questions. 🙂

  10. Maybe you could say, well, doesn’t a maximally simplified experimental trial of Zeno’s paradoxes — you run toward a line painted on the sidewalk and to your amazement actually manage to cross to the other side — disprove physical infinitesimality and by extension the continuum? (Zeno never said his paradoxes were real. He appeared to be wondering why the hell they aren’t.)

    Okay, so I’m saying that.

  11. Rick,

    So when the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity you’re saying that the equations are wrong?

  12. Brukner and Zeilinger (for instance) say pretty much that. The continuum is a mathematical construct …

    “Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

    “This leads to the second question, namely, how to derive the Schrödinger equation. ….”

    This continues, although not ad infinitum (pp.58-9, aka pp.12-13) and do be patient — we’re dealing here with a big photocopied document which takes what might seem like an infinite time to load:

  13. quantummoxie Says:

    I vaguely recall having had this discussion with Çaslav at one time or another in relation to his “coarse-graining” stuff…

  14. Coarse-graining relates to Leggettian macrorealism, I thought, and it’s complementary to decoherence. Per Brukner we’re just crappy measuring instruments and that’s why we live in this tiresome, boring old macroworld, missing out on just about everything interesting.

    But … I’ve never talked with the guy.

  15. quantummoxie Says:

    It does, that is true. I just happen to think that the discontinuous-to-discrete transition corresponds to the coarse-graining (I’m even more vaguely recalling that that was the gist of our discussion).

  16. “Coarse-graining relates to Leggettian macrorealism, I thought, and it’s complementary to decoherence.”

    Rick, so I guess, along with Fuchs you would also take the view that decoherence has little or no bearing in any way on the quantum foundation problem. And, what of Zeilinger’s view of Shannon — see Timpson’s various critiques?

    In any event, at the end of the day I think that Woody Allen said it best: “I hate reality, but, you know, where else can you get a good steak dinner?” 🙂

  17. As an MWer you do need to take physical decoherence seriously, I realize. Zeh and Tegmark definitely do. As I read Brukner he certainly doesn’t NOT take it seriously. I don’t THINK he’s saying that if our perceptions were only fine-grained enough we’d observe coherence, superposition and entanglement all around us. I freely confess that I don’t understand precisely what he means by coarse-grainedness and decoherence being complementary or compatible. (Ian? Help.)

    (I can imagine Brukner suggesting that infinitesimality and physical continuity result from coarse-grained perception. Is that what he said?)

    I’ve gone (meaning I’ve read or skipped and jumped around in my idiosyncratic way) through just about all of Timpson’s somewhat repetitive papers, even his humungous doctoral thesis. I know the IQOQI gang drive him nuts. I sent him an email once asking something about Brukner-Zeilinger’s Weizsäcker Festschrift paper (he was cited in it) and he replied that he hadn’t read it. I also mentioned Hans C. von Baeyer’s book “Information” (which I like … including the stuff about Jan Kåhre) and got something equally sniffy and dismissive. I think that in the intellectual sense — I know nothing about his politics or social views — he’s a reactionary ideologue. I don’t understand what problem he (and Bub also, I think) has/have with the Shannon expansion other than that it reinforces the idea that reality is interactive and counterfactual definiteness may not represent actuality … oh.

    None of this relates all that much to the Woody Allen mot (which is indeed funny). Brukner and Zeilinger aren’t virtualists along the lines of Ray Kurzweil and Nick Bostrom (who is the guy Timpson really should be going after in re: the simulation fallacy … it’s not like they’d have to travel far to debate one another). Anyway, you won’t see Anton or Časlav or Marcus or anyone else from that group up there on the stage with Ray and Nick at the Singularity Summit this summer. Fairly sure of that.

    I enjoy Chris Fuchs, but haven’t encountered his views on decoherence. Unless I have and then forgot.

  18. Not directly on point, but still rather interesting:

  19. It does kind of relate to a recent discussion between Aguirre and Scott Aaronson which got a fair amount of virtual ink on Aaronson’s blog and elsewhere. There’s a basic empiricist-metaphysicist split (Tegmark might say it’s frog vs. bird … as he preens his feathers).

    You have a cosmological model based on certain mathematical assumptions. The guiding philosophy is Platonic — reality is mathematical and the mathematical is real: any coherent mathematical formulation necessarily has a physical analogue. (Tegmark has said that explicitly, as you doubtless know.) Anyway, I keep thinking about renormalization and QED. According to Dirac, at any rate, Feynman et alii defaced his beautiful mathematics to make their grubby little theory work in the physical world. They did it mainly by tossing out infinities on an ad hoc basis. QED, one might say.

    I’m a frog. Brekekekex koax koax.

  20. quantummoxie Says:

    Let me guess which side of the aisle Scott was on. Maybe he’ll surprise me, but I would bet he was on the side of Platonism.

    This is a question I have been on various sides of over the years and that has been discussed at length both here and on the FQXi blog. But the more I tinker with crap in the lab (and my car and some op amps I have lying around and…) the more I seriously doubt the Platonic ideal and the “reality” of mathematical objects.

    That said, it wouldn’t surprise me if the universe is half-Platonic (or a third or whatever), meaning it wouldn’t surprise me if some of mathematics has physical analogues while some does not, i.e. some mathematical “objects” had an ontological status while some do not.

    In fact, I wouldn’t be surprised if someone were to demonstrate that the Platonic ideal could only exist in the presence of hidden variables (now try wrapping your head around that concept!).

  21. quantummoxie Says:

    Anyway, you won’t see Anton or Časlav or Marcus or anyone else from that group up there on the stage with Ray and Nick at the Singularity Summit this summer. Fairly sure of that.

    Maybe Anton, though. He shows up at some weird things.

  22. Rick,

    Now I’m going to show my ignorance to an even greater degree than previously. Two questions:

    1. The paper says “[a]s long as we are willing to neglect a part
    of the wavefunction with vanishing Hilbert-space norm,
    then we end up with a superposition of a huge number of
    different states, each describing outcomes of an infinite
    number of widely separated identical measurements in
    our infinite space.”

    Does this relate to the renormalization technique of tossing out infinities on an ad hoc basis?

    2. I was aware that Tegmark generally takes the view that “reality is mathematical and the mathematical is real: any coherent mathematical formulation necessarily has a physical analogue.”

    But my reading of the paper was that they only went so far as to claim to have shown that “the Level I Multiverse is the same as the Level III Multiverse (and if ination instantiates more than one solution to a more
    fundamental theory of physics, then the Level II Multiverse is the same as the Level III Multiverse).” They didn’t reach the metaphysical heights of the pure Plantonic mountaintop. Perhaps this doesn’t relate to your point, but there it is anyway.

    I do appreciate your thoughtful replies.

  23. qmoxie,

    I don’t recall that Scott and Anthony got into Platonism as such. But in the past he’s said he can’t even understand what Tegmark might mean by “a mathematical object.”

    Anton seems to be sympathetic to comparatively conventional (although not overpoweringly dogmatic) religious types, e.g., the Dalai Lama, with whom he periodically hangs. You could probably say “Your Holiness” too if it got you and yours an expenses-paid holiday in Dharamsala. His affection for the Gospel of John, verse 1 (“In the beginning was the Word”) might be indicative too. Maybe he’s a closet mystic. Just as long as he keeps working.

  24. Mike,

    I recall that Tegmark bangs on a bit about contemporary Platonism and Aristotelianism and says modern Platonists are many-worlders (or many-minders) who see reality as mathematical and have a Bird overview of things, whereas Aristotelians tend to be verbal and prone to the Copenhagen Interpretation, meaning they have the sweeping overview of Frogs. Doesn’t he also say that in the Level IV multiverse you must have universes that obey entirely different physical laws from any we know or have ever conceived, or could ever conceive — that being true simply because the mathematics must exist out there somewhere?

    (Maybe that isn’t relevant to what YOU just said.)

  25. Mike,

    Dunno whether infinite superposition relates immediately to renormalization. You’d need to compare the mathematics.

    In quantum computation some people think of a qubit as an infinite superposition. By the same token people talk about “quantum information.” (That seems to be one of Timpson’s beefs IIRC, the arguably loose use of the concept “information.”) There probably is such a thing as “quantum information” in the sense that stuff is communicated at that level. But we can’t read “quantum code” so it’s all basically hand-waving as far as we’re concerned. What I’m leading into here is that when you measure a qubit all you’re going to take away is a classical bit. Is that true because Nature won’t let us “go deeper” in terms of measurement or is it because we couldn’t perceive a superposition anyway or is it some complementary combination of those considerations?

    Measuring a qubit and ending up with a bit feels a little like tossing out infinities in renormalization. But I have no idea at all whether there’s any theoretical justification for that fantasy.

  26. Just occurred to me that you’re wondering whether neglecting the part of the wave function with a vanishing Hilbert space norm amounts to renormalizing Hilbert space. That too comes from Dirac. Don’t know.

    • Yes, this was, I think, what I was trying to figure out. Obviously, I don’t know the answer, and my math skills aren’t even close to what it would take to figure this out. Thanks.

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