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These two related questions cover the potential effects of quantized spacetime:

Does the Planck scale imply that spacetime is discrete?

Is spacetime discrete or continuous?

This question extends the above discussions with questions about the potential effects of discretized spacetime on conservation laws. (The reason this is important to me is that I'm writing a book that doesn't have a lot to do with quantum physics, but one of the arguments I'm considering for inclusion is related to the degree by which layers of cellular automata could "directly" simulate this universe, meaning each possible branch from quantum decoherence would have its own layer, with each layer being a physical approximation of particles including the "stringy" dimensions. Suggestions on this topic are also welcome in the comment section.)

The Planck distance is incredibly tiny, but its existence does not imply quantized spacetime. Ignoring the uncertainty principle for a moment, there's nothing that suggests one couldn't arrange "elsewhere" (noncausal) events that are a non-natural number of Planck distances apart, just so long as they're at least 1 Planck distance apart. This implies that noncausal events could be packed more tightly if arranged in a triangular tiling as opposed to a square tiling—again, without quantization of spacetime. But it does not imply that there is an underlying "tiling" of spacetime.

However, if spacetime were quantized, my intuition tells me this would mean otherwise-identical experiments could produce ever-so-slightly different results depending on whether the experiment is oriented in line with or skew with the underlying "grid" of spacetime (even if it is an irregular tiling). There would never be a way to detect this in practice, but if experiments were absolutely perfect and you did unimaginably many of them, over time the tiniest probabilistic difference might appear.

Emmy Noether proved that every experimental invariant implies a corresponding conservation law, and based on her theorem, we can (and do) infer that experimental invariance to "direction" implies that angular momentum is conserved. However, if spacetime were discretized, this invariant would not truly exist.

Questions

0) Does the uncertainty principle "blur" any possible effects from discretized spacetime to such a degree that the effects would not show up under any circumstance, thus "accidentally saving" the conservation of angular momentum?

1) If not, does the loss of the directional invariant suggest angular momentum might not be perfectly conserved, or is there another invariant which also implies the conservation law?

2) If conservation of momentum is not conserved, where might the nigh-infinitesimal energy go (or come from)? Put another way, does a loss of conservation of angular momentum also imply a loss of any other conservation laws, for example, conservation of energy?

3) Might any other physical laws be affected by loss of the directional invariant?

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Trixie Wolf
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    "Discretized spacetime" alone is not sufficient to fix a theory so that we can make statements about predictions. For instance, classical mechanics implicitly assumes continuous spacetime, so we cannot even talk about the classical notion of "angular momentum". For field theories, it is possible to put them in discrete spacetime, usually called a "lattice", and the predictions depend on what theory you are discretizing and what exactly the discretization procedure is. There are more issues, therefore I think this question is too broad/vague to be usefully answered until you narrow it down. – ACuriousMind Jan 07 '17 at 16:54
  • As a comment to a claim made in the question: Noether's theorem crucially assumes a continuous one-parameter group of symmetries to deduce a conserved quantity (and it also assumes continuous spacetime/configuration space in its proof). It is not about "experimental invariants", but about such continuous symmetries of the physical theory. Discrete symmetries do not induce conserved quantities by the standard versions of Noether's theorem. – ACuriousMind Jan 07 '17 at 16:57
  • I mean to refer to any discretization process. Are you saying that there exist theories in which angular momentum wouldn't be affected by things being forced to map to a cubic tiling? Imagine the tiling near the order of the width of an electron. – Trixie Wolf Jan 07 '17 at 16:59
  • @ACuriousMind I'm not sure I agree. Noether's Theorem doesn't prove the existence of invariants from a blank slate. It uses an invariant discovered by experimentation to deduce conserved quantities. The fact that experiments are no different depending on the "direction" of the experiment tells us that the invariant exists. If you're suggesting otherwise, then how does Noether's Theorem prove the directional invariant? – Trixie Wolf Jan 07 '17 at 17:04
  • Noether's theorem is a purely theoretical statement about the conserved quantities associated to a system with a given action functional (note that the definition of the action usually already presumes continuous spacetime, unless you are doing lattice theory, where choosing the "right" discretization of things like derivatives becomes relevant). We observe the world to be the same in all directions, so we use an action that is translationally and rotationally invariant to describe it. For this action, Noether's theorem can be applied, but it in and of itself makes no mention of "experiment". – ACuriousMind Jan 07 '17 at 17:08
  • Also, I don't make any claim about angular momentum being or not being conserved in discretized theories, because "discretized theory" is far too broad a class of theories. You need to choose a specific discrete theory for us to be able to answer the question of whether or not angular momentum is conserved in it. – ACuriousMind Jan 07 '17 at 17:10
  • @ACuriousMind I don't understand the difference between what you're saying and what I'm saying. We have to observe the world to be the same in all directions in order to get the rotational invariant, and we do that experimentally. The reason Noether's Theorem had an immediate and powerful effect is that centuries of consistent experimental observations which revealed natural invariants could be used to deduce conserved quantities. By "experimental" I am referring to these observations, because "casual observations" outside of controlled experimentation doesn't prove anything at all. – Trixie Wolf Jan 07 '17 at 17:13
  • @ACuriousMind Again, I'm asking if there are any discretized theories in which the discretization would lead to a loss of the conservation of angular momentum. I would like to know in which theories this loss may exist, and what the implications might be. – Trixie Wolf Jan 07 '17 at 17:15
  • However, if spacetime were quantized, my intuition tells me this would mean otherwise-identical experiments could produce ever-so-slightly different results depending on whether the experiment is oriented in line with or skew with the underlying "grid" of spacetime (even if it is an irregular tiling). Why? I am not being glib with you, but I can't escape the conclusions that 1. You are applying classical ideas (a "fixed" grid), given that all theories way above that scale are probabilistic and 2. even through we can't prove it, it's probably a dynamic rather than a static, "region". –  Jan 07 '17 at 18:24
  • Apologies if I have misread you though.... –  Jan 07 '17 at 18:25
  • Why is a quantization of spacetime more likely to be dynamic? Relativistic effects? – Trixie Wolf Jan 07 '17 at 18:28
  • It's a frustrating guessing game, without experimental evidence I have a feeling that, as mentioned in the links you posted, it could be a base layer that we can't even imagine. Not that you personally are advocating any particular view but we are in the same position as trying to describing the properties of a quantum world to an 1890's classical physicist. Worse actually, we have no analog of black body radiation Why is it possibly dynamic: because QFT fields are dynamic in the sense that they DO things and are they discrete too? If not why not, they should be in a discrete ST. –  Jan 07 '17 at 18:49
  • @Countto10 I'm really approaching this from the perspective of computational theory. Almost all computational theorists would say that the Universe we see could be modeled by simple determinism at the lowest level, without question. I'm trying to tap into which models physics theorists find plausible (in terms of mathematical consistency) for a lowest-level framework, but physics theorists might not be the right ones to ask. I may need to educate myself a great deal more in the math of quantum physics to answer this question, which is frustrating but not surprising. – Trixie Wolf Jan 25 '17 at 11:59
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    Quantum math itself is easy enough to learn, I really mean that, it's just that there is a lot of it. It (as you already know) is the concepts behind it and getting rid of classical ideas that is the problem. Look at the amount of questions asked here about wave particle duality for an example, they are nearly all (wrongly) based on trying to relate QM to something classical. 120 years on, we are still nowhere near a generally accepted interpretation –  Jan 25 '17 at 12:36
  • I am painfully aware that I am a newbie, waffling on about QM, compared to the knowledge and experience of most of the high rep users here, but I would be surprised if they disagreed with me. Finally, look at the number of questions that are shunted over to philosophy SE, as not experimentally testable, therefore not physics. –  Jan 25 '17 at 12:41
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    @Countto10 I agree (I do understand you can't map QM to classical analogies: I'm in computer science and news media never describe quantum computing accurately), and I'm not trying to explain QM in a classical way here. I just want to know in which ways it could consistently be modeled discretely. Again, I expect that pretty much every computational theorist (and probably most mathematicians) would agree this is possible irrespective of how weird and non-classical QM is. If most physicists disagree, I'd like to know why the ways we model physics are inconsistent with computation. – Trixie Wolf Jan 25 '17 at 12:54
  • How would you test your model, and do you model say, the electron field as a lattice? If you eventually come up with the same results as the real field, or any sort of correlation, then what physicist could argue with you?. Or argue against your demonstration that it really is a math based reality ( as I believe it is, but that's just because of my ignorance of the subject ...Confidence is what you have before you understand the problem, W. Allen). Or am I assuming you are further along than you really are? No need to reply, just thinking aloud. A link to a good basic article would be great. –  Jan 25 '17 at 14:12

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