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This question already has an answer here:

We may also ask: Is the assumption of continuity of spacetime required by Lorentz Invariance?

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marked as duplicate by David Z Sep 30 '16 at 5:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ I'm not sure why this got downvoted. Inquiring whether a discrete spacetime breaks Lorentz invariance seems sensible to me since the answer is that yes it does. $\endgroup$ – John Rennie Sep 28 '16 at 16:23
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    $\begingroup$ Though I think this has been asked before, for example Discreteness of Spacetime and Violation of Lorentz symmetry. A Nobel prize winner answered that question! $\endgroup$ – John Rennie Sep 28 '16 at 16:25
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    $\begingroup$ +1 Trying to balance the books, I would be interested in the answer to a good question. $\endgroup$ – user108787 Sep 28 '16 at 16:26
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/20860/2451 $\endgroup$ – Qmechanic Sep 28 '16 at 22:06
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Yes. The Lorentz group contains continuous (i.e. arbitrarily small) rotations and boosts, and any discretization of spacetime would fail to be invariant under rotations or boosts smaller than the scale of, say, the primitive unit cell (if the discretization forms a lattice).

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    $\begingroup$ But ... isn't this argument absolutely wrong?!! Consider the statement: The group SU(2) requires continuous (i.e. arbitrarily small) rotations, so any discretization of angular momentum would fail to be invariant under rotations. How can that argument fail while your answer is correct? $\endgroup$ – Peter Shor Sep 28 '16 at 17:59
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    $\begingroup$ This argument invariably comes up when somebody asks whether spacetime can be discrete, and nobody yet has answered my objection to my satisfaction. Quantum mechanics is non-intuitive. And we have no idea how to quantize space-time (if that's what the OP was asking about). If he was asking about classical discretization of space-time, your answer is indeed correct. $\endgroup$ – Peter Shor Sep 28 '16 at 18:01
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    $\begingroup$ @PeterShor You are right that if we are considering the possibility of a dynamical and quantum-mechanical spacetime, then the question becomes much more subtle, and as you say I don't think anyone knows the answer. Since neither the question body nor the tags mention quantum mechanics, I was assuming that the OP was referring to classical special relativity (or a quantum-mechanical system living on a fixed, classical spacetime background). But your clarification is very conceptually important. $\endgroup$ – tparker Sep 28 '16 at 18:58
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    $\begingroup$ Well, yes and no. You can answer this question in terms of special relativistic QFT, i.e. The standard model. Lorentz transformation like translations are continuous transformations, nobody is quantizing Minkowski spacetime. It leads to the global symmetry causing a conservation law. SU(2) is a gauge symmetry, a local symmetry, part of the SU(2) x U(1) electroweak isospin symmetry, which causes the discrete 'charge' conservation laws on isospin. Like U(1) leads to the discrete charge conservation. Quantizing General Relativity, i.e., dynamic and discrete spacetime, is not so easy, as you know $\endgroup$ – Bob Bee Sep 28 '16 at 21:14
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    $\begingroup$ @tparker, Peter Shor and Bob Bee, Thanks for the answer and interesting and conceptual comments. As you noted the question here doesn't concern a quantum spacetime. In addition, I'd like to know about so-called discrete Lorentz transformations and their relation to spacetime symmetries. In this regard I will try to ask a question, too. $\endgroup$ – user130644 Sep 28 '16 at 23:50