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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.

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Another related: – qftme May 13 '11 at 15:25
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 6 '15 at 17:52
up vote 56 down vote accepted

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

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,

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.

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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 ( ? 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
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
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 5 '12 at 9:58
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. – Ben Crowell Oct 2 '13 at 15:48

Minimal Length Scale Scenarios for Quantum Gravity
Here is a serious consideration (review paper) considering many possibilities of something similar to a discrete quantum length scale. Enjoy!

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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

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