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Lie groups are used to some behaviors of quantum mechanics, as well as forming a basis for Kaluza-Klein, Yang-Mills, and String theory.

But Lie groups are defined as involving a differentiable manifold. If spacetime is discrete, then no such differentiable manifold would exist.

So, most of modern theoretical physics assumes that spacetime is continuous, and without stating that this assumption is being made.

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    $\begingroup$ FWIW, there exist lattice gauge theories with continuous gauge group and discretized spacetime. $\endgroup$ – Qmechanic Apr 11 '16 at 13:25
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    $\begingroup$ I don't understand the question. The Lie group being itself a smooth manifold has nothing to do with spacetime being one. $\endgroup$ – ACuriousMind Apr 11 '16 at 13:28
  • $\begingroup$ I could be confused. How can a discrete system be (accurately) modeled with a continuous mathematics? $\endgroup$ – Jiminion Apr 11 '16 at 13:33
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    $\begingroup$ @Jiminion Because gauge transformation does not take place in spacetime. They are realized in the so-called internal space. It acts on fields, not on coordinates. The only relation among them is that the transformation can be different at any spacetime point. $\endgroup$ – Diracology Apr 11 '16 at 13:42
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    $\begingroup$ Why the downvote? This seems a reasonable question to me. It betrays a misunderstand of what a local gauge symmetry is, but isn't that the whole point of asking questions here? $\endgroup$ – John Rennie Apr 11 '16 at 16:13
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I can't comment on string theory, but in quantum field theory the U(1), SU(2), etc symmetry groups are local gauge symmetries. They are not a symmetry of the spacetime in which the symmetry is formulated. So whether the spacetime is discrete or not makes no difference to the local gauge symmetry.

As far as I know the physical significance of the local gauge symmetry is not understood. It presumably represents some internal degrees of freedom but it is not understood what these are. If string theory is a valid description then it offers us an explanation for the local gauge symmetries, but I'm afraid my grasp of string theory is too tenuous to comment further on this.

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