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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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  • $\begingroup$ "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. $\endgroup$ – ACuriousMind Jan 7 '17 at 16:54
  • $\begingroup$ 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. $\endgroup$ – ACuriousMind Jan 7 '17 at 16:57
  • $\begingroup$ 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. $\endgroup$ – Trixie Wolf Jan 7 '17 at 16:59
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    $\begingroup$ 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 $\endgroup$ – user140606 Jan 25 '17 at 12:36
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    $\begingroup$ @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. $\endgroup$ – Trixie Wolf Jan 25 '17 at 12:54

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