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Insights Struggles With The Continuum - Part 2 - Comments

  1. Sep 8, 2015 #1

    john baez

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    Last edited: Sep 6, 2016
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  3. Sep 8, 2015 #2
    Is there any previous work, any attempt on trying to discretize space, or even space time and trying to work the laws of physics, dynamics, classical mechanics, quantum mechanics, and relativity in such a discretized space? I supposse that in the limit of the grid spacing tending to zero one could get the usual mechanics in continuum space or space-time, but I would like to know what kind of physical predictions the discretized space would make, and the possibility to observe experimentally some of those predictions.

    Wouldn't the "nice" properties of space be lost, like homogenity, isotropy, etc?
     
  4. Sep 8, 2015 #3
    "At any time, we want ψ to lie in the Hilbert space consisting of square-integrable functions of all the particle’s positions. We can then formally solve Schrödinger’s equation as
    ψ(t)=exp(−itH)ψ(0)

    where ψ(t) is the solution at time t. But for this to really work, we need H to be a self-adjoint operator on the chosen Hilbert space. The correct definition of ‘self-adjoint’ is a bit subtler than what most physicists learn in a first course on quantum mechanics. In particular, an operator can be superficially self-adjoint—the actual term for this is ‘symmetric’—but not truly self-adjoint."



    Is this because we want the movement of a system in that space to be perfectly reversible? Is Reimannian continuity (if that's the right term) really about wanting to assume things are equally able or likely to go in any direction in their phase-space from any point?

    Basically I am confused here by similar but different terms "Continuity", "Reversibility", "Symmetry", "Commutativity", "Hermitian-ness" and "Self-Adjoint-ness". I think the last two may be synonymous, but are these terms just stronger/weaker versions of the same idea, precluding any preferred "direction of the grain" in the phase space?

    Thanks for the awesome articles by the way.
     
    Last edited: Sep 8, 2015
  5. Sep 8, 2015 #4

    john baez

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    Thanks!

    I suggest that you look up "Continuity", "Reversibility", "Symmetry", "Commutativity", "Hermitianness" and "Self-Adjointness" on Wikipedia. They all mean very different things - except for the last two, which are closely related. In my post I was using "symmetric" in a specific technical sense, closely akin to "hermitian", which is quite different from the general concept of "symmetry" in physics. That's why I included a link to the definition.

    Learning the precise definitions of technical terms is crucial to learning physics. You're saying a lot of things that don't make sense, I'm afraid, so I can't really comment on most of them. That sounds rude, but I'm really hoping a bit of honesty may help here.

    Anyway: we need the Hamiltonian H to be self-adjoint for the time evolution operator exp(-itH) to be unitary. And we need time evolution to be unitary for probabilities to add up to 1, as they should.
     
    Last edited: Sep 8, 2015
  6. Sep 8, 2015 #5

    john baez

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    Sure, lots.

    Yes, for example when they compute the mass of the proton, they discretize spacetime and use lattice gauge theory to calculate the answer - nobody knows any other practical way. But you try to make the grid spacing small to get a good answer.

    You'd mainly tend to lose isotropy - that is, rotation symmetry. Homogeneity - that is, translation symmetry - will still hold for discrete translations that map your grid to itself. People have looked for violations of isotropy, but perhaps more at large scales than microscopic scales.

    What seems cool to me is how cleverly chosen lattice models of fluid dynamics can actually do a darn good job of getting approximate rotation symmetry. For example, in 2 dimensions a square lattice is no good, but a hexagonal lattice is good, thanks to some nice math facts.
     
  7. Sep 9, 2015 #6
    Thanks for your answer John!

    What happens with for example angular momentum conservation when isotropy is lost? what would Newtonian physics looks like in this discretized space? Is there any need of a reformulation of Newtons laws or you can recover our macroscopic physics even with a discretized space and the usual, for example Newton's second law of physics with discrete variables (with out the need of taking the grid spacing tending to zero)? I guess that inertia would hold, as it is related to space homogenity. And I suppose that people working in numerical methods of physics actually use this kind of discretizations every time they make for example a Riemann integral, or use finite difference methods. Is this equivalent to discretizing space in Newtonian physics? and there is any departure from the predictions in continuum space or everything works in the same way?

    Perhaps I'm being too incisive about this, after all the post is about the struggles with the continuum and not about the possibility of a discretized space itself. But I thought of it as complementary. If space is not a continuum it has to be discrete, right? or there are other possibilities here?
     
  8. Sep 9, 2015 #7
    Hi - I'll be a bit naive here, forgive me, but couldn't it be that our mathematics is not representing reality, but just a model of it? Nature could be not-continuous by itself. I cannot think of a "real" particle being "infinitely close" to another one... And even quantization, or discretizations used in computerized models, aren't real themselves, but just convenient models of natures, useful to make good predictions only up to a certain "extent".
    I'm afraid we still don't have mathematical models that really represent nature completely (will ever we?) and the difficulties in having converging solutions under certain circumstances might be the result of using inadequate mathematics.
    Sorry if my point is only loosely defined, but I wonder if this or similar arguments has ever been raised before (as I guess).
    Thanks John for the excellent presentation of these concepts, anyhow.
     
  9. Sep 9, 2015 #8

    john baez

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    Typically it goes away, but if you're clever you can arrange to preserve it, by choosing particle interactions that conserve it.

    The link I sent you is all about 'lattice Boltzmann gases', which have particles moving on a lattice and bouncing off each other when they collide. This is one of the earlier papers on this subject, by Steve Wolfram.

    On a square lattice in 2 dimensions, you can easily detect macroscopic deviations from isotropy in the behavior of such a gas. On a hexagonal the deviations are much subtler, because hexagonal symmetry invariance implies complete rotation invariance for a number of tensors that are important in fluid flow.

    There's a rather beautiful harmonic oscillator where the particle moves in discrete time steps on a 2d square lattice, but I'm pretty sure the inverse square law is going to be ugly. As you said, you can just think of the lattice as the discretization imposed in numerical analysis by working with numbers that have only a certain number of digits. But in general this is pretty ugly: there's nothing especially nice about physics where 'roundoff errors' gradually violate conservation laws.

    I find it more interesting to look for discrete models where you can still use a version of Noether's theorem to get exact conservation laws from symmetries. I had a grad student who wrote his PhD thesis on this:
    and we published a paper about it:
    However, while it's fun, I don't think it's the right way to go in physics.
     
  10. Sep 9, 2015 #9

    john baez

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    We use mathematics to model nature. We find ourselves in a mysterious world and try to understand it - we don't really know what it's like. But the discrete and the continuous are abstractions we've devised, to help us make sense of things. The number 473 is just as 'man-made' as the numbers π, i, j and k.

    The universe may be fundamentally mathematical, it may not be - we don't know. If it is, what kind of mathematics does it use? We don't know that either. These questions are too hard for now. People have been arguing about them at least since 500 BC when Pythagoras claimed all things are generated from numbers.

    If we ever figure out laws of physics that fully describe what we see, we'll be in a better position to tackle these hard questions. For now I find it more productive to examine the most successful theories of physics and see what issues they raise.
     
    Last edited: Sep 9, 2015
  11. Sep 10, 2015 #10

    Demystifier

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    I agree, but let me quote the famous statement by Kronecker:
    "God made the integers, all the rest is the work of man."
     
  12. Sep 12, 2015 #11
    It doesn't seem to me that anything is ever continuous; continuous is always an approximation, even in classical mechanics.

    Take a baseball. Suppose you integrate numerically the simple case where there is just gravity, no air resistance. Can you really make your timestep arbitrarily small? How about if you integrated at a femtosecond timescale? In principle you could obtain the entire motion, but in practice each step would, individually, produce no information about the system, since the change in position would be, relative to the characteristic scales of the problem, zero during each step. There is arguably a minimum time step, which is the smallest step that produces a nonzero change in state relative to the characteristic scales.

    Space is also always discretized in terms of the characteristic scales of the system.
     
  13. Sep 12, 2015 #12

    john baez

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    When you speak of numerical integration you're no longer speaking about a baseball: you're speaking about a computer program. Of course if you do numerical integration on a computer using time steps, there will be time steps.

    I have never seen any actual discretization of space, unless you're talking about man-made structures like pixels on the computer screens we're looking at now.
     
  14. Sep 13, 2015 #13
    I asked this question to my self last night, perhaps John knows the answer (great responses btw, thank you John, I had a final exam last Friday, so I didn't had time to read all the details on your responses, but I did yesterday). The question that arised to me was, if space-time is actually discrete, do you know how small the grid spacing should be? According to what we actually know of the laws of physics, is there any upper and/or lower bounding for the space-time grid?

    Many people believe that the space grid is Plancks constant. But I don't think there is really any physical reason to believe so, what is discrete at the plancks lenght is phase space because of the uncertainty principle, not space-time (actually, plancks constant have units of action, not lenght). Space-time is always treated as a continuum in physics. And I didn't know until now about the struggles with the continuum. But usually, the continuum hypothesis works, and gives reliable physical predictions in most known physics.
     
    Last edited: Sep 13, 2015
  15. Sep 13, 2015 #14
    Wait, you think your point particle with no air resistance is a baseball? It seems to me that neither the differential equation nor the difference equations found on my computer are baseballs, just models of baseballs.

    Space is clearly discrete for the baseball model. A femtometer is zero compared to the typical flight distance, so it is below some minimal distance. Worse, your continuum approximation breaks down for distances on the scale of the atoms which make up the baseball.

    It seems to me that much of the confusion regarding the continuum arises from assuming that "infinitesimals" and "infinity" have meaningful interpretations outside of notions of characteristic scale in a problem! There is only one kind of infinity, and that's what happens when you take a number significantly larger than the largest scale of your system, and such a number is indeed finite. The distance from here to the andromeda galaxy is "infinite" compared with the typical baseball flight distance; a wavefunction of an electron prepared in a lab is zero when evaluated at the Andromeda galaxy since this is much larger than say, the width of the potential.
     
  16. Sep 13, 2015 #15

    Nugatory

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  17. Sep 13, 2015 #16

    Nugatory

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    If you can obtain the entire motion "in principle" and only the practical difficulties of getting measurements at the desired resolution stops us from doing it "in practice", then the underlying system is (pretty much by definition) continuous at that scale.
     
  18. Sep 13, 2015 #17
    Well if the universe has a finite age/size this would presumably put realistic limits on what can or cannot be computed, but I digress from this point: apart from the fact that your continuum model breaks down at the femtosecond timescale, if you are interested in the ballistics of the baseball, no information about the ballistics is stored at this scale, you'd have to run through thousands of meaningless individual steps before such information began to emerge.

    the question is, if at a certain resolution no information about the system is generated, can one really argue that any time elapsed at all? Time is related to how a system changes, which is a matter of what you want to learn about the system. If I'm interested in macroscopic details, a glass of water in thermodynamic equilibrium, while containing many changes at spatial and temporal resolutions I do not care about, is unchanging on the macroscopic scale, and is at this scale time independent.

    the whole point I'm meandering towards here is that information, I think, plays an enormous role in this problem, and was not really discussed in this post.
     
  19. Sep 14, 2015 #18

    john baez

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    Mainly, it should be small enough that we haven't seen it.

    This is a good excuse for me to update my page on distances. Quite a while ago I wrote that ##10^{-18}## meters, or an attometer; approximately the shortest distance currently probed by particle physics experiments at CERN (with energies of approximately 100 GeV)."

    But the Large Hadron Collider is now running at about 13 TeV or so. So this is about 100 times as much energy... so, using the relativistic relation between energy and momentum, and the quantum relation between momentum and inverse distance, they should be probing physics at distances of about ##10^{-20}## meters.

    They're not mainly looking for a spacetime grid, obviously! But if there a grid this big, we'd probably notice it pretty soon.

    So, I'd say smaller than ##10^{-20}## meters.

    The upper bound comes from the fact that we haven't seen a grid yet.

    There's no real lower bound. We expect quantum gravity effects to kick in at around the Planck length, namely ##10^{-35}## meters. But the argument for this is somewhat handwavy, and there's certainly no reason to think quantum gravity effects means there's a grid, either at this scale or any smaller scale.

    Really? As you note, that makes no sense, because Planck's constant is a unit of action, not length.

    The Planck length is a unit of length. Ever since Bohr there have been some handwavy arguments that quantum gravity should become important at this length scale. For my own version of these handwavy arguments, check out this thing I wrote:
     
  20. Sep 14, 2015 #19

    john baez

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    No, I never said that. Right now we were talking about baseballs, not point particles. You said:

    I replied:

    Neither of us were talking about point particles. I was just pointing out how you jumped from a baseball (an actual physical object) to numerical integration (a computer model of an object). The model is not continuous, but that doesn't prove the baseball isn't continuous.

    It depends on your model. For numerical integration, it's discrete. For analytical or symbolic integration it's continuous. They both work fine for a falling baseball. One won't learn much about the ultimate nature of reality from this.
     
    Last edited: Sep 14, 2015
  21. Sep 14, 2015 #20
    I have taken a look at it yesterday, I didn't have much time at that moment, and had to left the computer to make stuff. I was really concentrated reading at it when I was interrupted and had to leave. That post looks great too, I'll end the reading just right now, thanks!


    Thank you very much John, great insight, and great responses. Thank you for your time.

    Regards.
     
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