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On a quantum scale the smallest unit is the Planck scale, which is a discrete measure.

There several question that come to mind:

  1. Does that mean that particles can only live in a discrete grid-like structure, i.e. have to "magically" jump from one pocket to the next? But where are they in between? Does that even give rise to the old paradox that movement as such is impossible (e.g. Zeno's paradox)?
  2. Does the same hold true for time (i.e. that it is discrete) - with all the ensuing paradoxes?
  3. Mathematically does it mean that you have to use difference equations instead of differential equations? (And sums instead of integrals?)
  4. From the point of view of the space metric do you have to use a discrete metric (e.g. the Manhattan metric) instead of good old Pythagoras?

Thank you for giving me some answers and/or references where I can turn to.

Update: I just saw this call for papers - it seems to be quite a topic after all: Is Reality Digital or Analog? FQXi Essay Contest, 2011. Call for papers (at Wayback Machine), All essays, Winners. One can find some pretty amazing papers over there.

Emilio Pisanty
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vonjd
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    Possibly related: http://physics.stackexchange.com/q/817/2451 http://physics.stackexchange.com/q/9076/2451 – Qmechanic May 11 '11 at 10:17
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    Another related: http://physics.stackexchange.com/q/4453/ – qftme May 13 '11 at 15:25
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    Regarding your point about Zeno's paradox, the existence of quanta of space would in fact disprove Zeno's paradox. The paradox was introduced by Zeno's teacher Parmenides to prove that motion was impossible, by arguing that Achilles would never catch the tortoise because as Achilles got closer to the tortoise, he would need to cover half the remaining distance, and half of that distance, and he would never reach an end of the halves. But if space were quantized, the final quanta couldn't be halved, so Achilles would cross it and catch the tortoise. – Ernie Sep 06 '15 at 17:52
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    If the speed of light is also the speed of causality, is there a limit to the shortest wavelength a photon can be which when expressed as a distance becomes or defines a quanta of time? – Kelly S. French Aug 17 '16 at 18:28
  • @Ernie Since Zeno's paradox is disproved by observation, does that imply the existence of quantized space? – Michael Oct 16 '18 at 17:25
  • @Michael : Lubos Motl's answer, below, is more insightful than my comment. I naively assumed that the Planck scale is reality, rather than just a way of measuring reality. However, there might be a ratcheting mechanism at the Planck scale that allows matter to form and that allows movement. This is a hypothesis that needs to be refined and made more specific if it is to be tested. – Ernie Oct 17 '18 at 23:37
  • @Michael There's a logical fallacy there. If A implies B and you observe B, that doesn't prove A. The fact that A implies B doesn't mean that A is the only way you can get B. – JBentley Nov 25 '20 at 10:42
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    PSE has a much more recent Q&A, "Is time emergent from quantum entanglement?", that describes an experiment bearing on the OP's sub-question #2. – Edouard Mar 26 '21 at 04:27

2 Answers2

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The answer to all questions is No. In fact, even the right reaction to the first sentence - that the Planck scale is a "discrete measure" - is No.

The Planck length is a particular value of distance which is as important as $2\pi$ times the distance or any other multiple. The fact that we can speak about the Planck scale doesn't mean that the distance becomes discrete in any way. We may also talk about the radius of the Earth which doesn't mean that all distances have to be its multiples.

In quantum gravity, geometry with the usual rules doesn't work if the (proper) distances are thought of as being shorter than the Planck scale. But this invalidity of classical geometry doesn't mean that anything about the geometry has to become discrete (although it's a favorite meme promoted by popular books). There are lots of other effects that make the sharp, point-based geometry we know invalid - and indeed, we know that in the real world, the geometry collapses near the Planck scale because of other reasons than discreteness.

Quantum mechanics got its name because according to its rules, some quantities such as energy of bound states or the angular momentum can only take "quantized" or discrete values (eigenvalues). But despite the name, that doesn't mean that all observables in quantum mechanics have to possess a discrete spectrum. Do positions or distances possess a discrete spectrum?

The proposition that distances or durations become discrete near the Planck scale is a scientific hypothesis and it is one that may be - and, in fact, has been - experimentally falsified. For example, these discrete theories inevitably predict that the time needed for photons to get from very distant places of the Universe to the Earth will measurably depend on the photons' energy.

The Fermi satellite has showed that the delay is zero within dozens of milliseconds

http://motls.blogspot.com/2009/08/fermi-kills-all-lorentz-violating.html

which proves that the violations of the Lorentz symmetry (special relativity) of the magnitude that one would inevitably get from the violations of the continuity of spacetime have to be much smaller than what a generic discrete theory predicts.

In fact, the argument used by the Fermi satellite only employs the most straightforward way to impose upper bounds on the Lorentz violation. Using the so-called birefringence,

http://arxiv.org/abs/1102.2784

one may improve the bounds by 14 orders of magnitude! This safely kills any imaginable theory that violates the Lorentz symmetry - or even continuity of the spacetime - at the Planck scale. In some sense, the birefringence method applied to gamma ray bursts allows one to "see" the continuity of spacetime at distances that are 14 orders of magnitude shorter than the Planck length.

It doesn't mean that all physics at those "distances" works just like in large flat space. It doesn't. But it surely does mean that some physics - such as the existence of photons with arbitrarily short wavelengths - has to work just like it does at long distances. And it safely rules out all hypotheses that the spacetime may be built out of discrete, LEGO-like or any qualitatively similar building blocks.

Luboš Motl
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    Thank you for this very exhaustive answer: +1! Yet I wonder if all of your expositions are true for all theories on the market at the moment, esp. loop quantum gravity (http://en.wikipedia.org/wiki/Loop_quantum_gravity) ? Thank you again. – vonjd May 11 '11 at 11:10
  • ...or the other way round: Does this experimentally rule out e.g. loop quantum gravity and the like? – vonjd May 11 '11 at 11:15
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    Yes, it does. Even if one could find nearly flat-space solutions in LQG, which no one can (and chances are that it's because there's no flat space in LQG), it would still be true that it would violate the Lorentz invariance much more strongly than the experimental upper bound. The spin networks or, in the path integral language, the spin foam is a (not so) modern version of the 19th century luminiferous aether. It not only violates the Lorentz invariance but also carries a huge entropy density which instantly slows objects much like a dense $10^{95},kg/m^3$ "water" slows down swimmers. – Luboš Motl May 12 '11 at 14:13
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    I've been considering posting a question about this topic. Your answer is immensely helpful - that the evidence rules out all forms of LEGO-like approaches to a GUT, but if a formal system of nodes doesn't create continuous space-time, what does?! Don't your current ascertains imply that the universe could not be fully defined by any formal system, no matter how advanced and expansive? Does that not contradict your intuition? Continuous geometry of any kind requires an assumption of infinity existing... and that seems impossible. – Alan Rominger May 25 '11 at 15:48
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    It's a great answer. I'd like to add that the Planck scale comes about from the coupling to gravity, and it should actually be energy-dependent as all other coupling constants. – WIMP Apr 05 '12 at 09:58
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    which proves that the violations of the Lorentz symmetry (special relativity) of the magnitude that one would inevitably get from the violations of the continuity of spacetime have to be much smaller than what a generic discrete theory predicts. IMO this claim is much too strong. In particular, LQG does not, as some people originally thought, make such a prediction. –  Oct 02 '13 at 15:48
  • What about good old Zeno's Paradox ? If the space were continuous, all distances would be infinite... no? – Luxspes Jul 24 '14 at 15:55
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    @AlanRominger Not really, it only says that a simulation of a universe cannot be arbitrarily accurate. And infinities aren't a problem on their own - for example, consider that although a given volume can contain an infinite amount of "points", its volume is still finite. If you have a continuous coördinate system and one particle, you have infinite possible positions for the particle, but still just one particle. – Luaan Mar 20 '17 at 12:09
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    @Luxspes Nothing to do with Zeno's Paradox - that is entirely satisfactorily explained as a sum of an infinite series. You just need to realize that while the distances are halved in each iteration, so is the time. The apparent infinity only arises because you added it - by cutting time into infinite amount of events. If we assume a constant-speed Achilles in vacuum, he simply moves linearly in space just like the tortoise; if you don't keep making the time intervals shorter, you will find one interval where Achilles is behind, and the next where he's ahead of the tortoise. – Luaan Mar 20 '17 at 12:16
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    Yes, but , what is movement but a jump between discrete positions? In a true continuum stuff simply could not move, the next coordinate would always be infinitely far away. – Luxspes Mar 20 '17 at 19:19
  • Can't quantized values be seen as a continuum of values which certain regions being forbidden? For instance, the analogy I learned was that electron orbitals have discrete transitions in the same way that only certain wavelengths can form a standing wave when in a loop. – Michael Oct 16 '18 at 17:52
  • Right, that's what the quantized or discrete values mean. My answer says that most (well, all) of the values of coordinates in space are allowed, not forbidden: the position is a quantity with a continuous spectrum (i.e. spectrum where the whole intervals, almost all numbers, are not forbidden). – Luboš Motl Oct 17 '18 at 17:38
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    Just because the wavelength of light is not constrained by quantization does not mean that a photon (particle) of light can be observed having travelled less than a Planck length. In fact no such measurement can be made directly. Similarly, the wave-function describing the momentum of any massive particle describes a smooth range of probabilities that the particle will or will not be found at a given, discrete point. I do not see how your statement that wave-functions are smooth disproves a discrete, quantized spacetime for all point-like observables. – CommaToast Nov 27 '18 at 16:47
  • If the coordinates such as x,y,z were integer-valued, then - by properties of the Fourier transform - the components of the momentum would actually be periodic. It is straightforward to rule out this possibility experimentally. It is possible to falsify any particular discrete picture, not just the integer-valued coordinates, because all such possibilities would violate the Lorentz symmetry - and they would violate other known facts. If you don't know how to disprove a hypothesis, it doesn't mean that physicists don't know how to disprove it. – Luboš Motl Nov 27 '18 at 18:47
  • @Lubos Motl --I'm not a fan of "space atoms" or "space legos", but I have to report that, on p.108 in his 2020 book titled "Helgoland", no less a light than Rovelli claims that, "There is no infinite in going toward the small: Things cannot get infinitely smaller." Is this to be taken so literally as to mean, "Nothing that's not infinitely small can become infinitely small" (which I guess would imply a causal separation between localities), or is it disagreeing with your answer? – Edouard Jul 09 '22 at 16:04
  • I realize I may be asking you to read another physicist's mind, but I lack an address for Rovelli, and I'm guessing you might be familiar enough with his work to make a guess as to which of the last two possibilities might apply. – Edouard Jul 09 '22 at 16:36
  • I know Rovelli's statements in some detail and I assure you that this one as well as most of this statements about fundamental physics are completely wrong, usually ludicrously wrong. – Luboš Motl Jul 10 '22 at 18:39
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Minimal Length Scale Scenarios for Quantum Gravity
arXiv:1203.6191
Here is a serious consideration (review paper) considering many possibilities of something similar to a discrete quantum length scale. Enjoy!
http://arxiv.org/abs/1203.6191

Jim Graber
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    not to tmention the various discrete approaches to quantum gravity (regardless of attempts to downplay them), e.g spin foams, LQG, etc.. – Nikos M. Oct 29 '14 at 15:12