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In Feynman's Simulating Physics with Computers, Feynman states that

"we might change the idea that space is continuous to the idea that space perhaps is a simple lattice and everything is discrete" ... "the speed of light would depend slightly on the direction, and there might be other anisotropies in the physics that we could detect experimentally"

Why does the existence of the lattice of spacetime (if any) affect the isotropy of speed of light?

The other theories which might question the validity of the Lorentz Invariance of speed of light are Loop Quantum Gravity and String Theory (a paper on this), which do not seem to directly be associated with the lattice hypothesis, although the string theory might be similar to a lattice-like spacetime.

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    $\begingroup$ He is just winging it here and not very well. The first and most important thing that a naive lattice would do would be to break relativity. This already happens in the Galilean case since we now have a preferred rest system. Neither LQG nor string theory are on lattices. We do have important examples of this in physics, of course: optics and phonons in crystals. $\endgroup$ – CuriousOne Jul 28 '16 at 18:44
  • $\begingroup$ Somewhere I've read here, that there is experimental evidence that the spacetime is continous at least until trillionths of the Planck length. $\endgroup$ – peterh Jul 28 '16 at 19:36
  • $\begingroup$ @peterh: That is a question that has raised considerable discussions among the specialists. There is no consensus on it and all experiments that indicate such a thing seem to work of rather naive assumptions. On the other hand, the Planck length is nothing but intellectual nonsense at this point. There is not one data point that indicates that anything is there there. $\endgroup$ – CuriousOne Jul 28 '16 at 20:05
  • $\begingroup$ @CuriousOne: why is he winging it by rejecting a simple lattice which you are also rejecting? By saying "simple lattice" he is actually leaving other models open in this quotation. $\endgroup$ – Jerry Schirmer Jul 28 '16 at 20:41
  • $\begingroup$ @JerrySchirmer: For one thing there is no such thing as "a simple lattice". Even in three dimensions there are 230 space groups and that's not even counting the infinite number of quasi-crystal lattices. Icosahedral systems, for instance, look almost isotropic, already. He also says nothing about what the dynamics on this lattice looks like and I am fairly sure that one could construct models that would be very hard to test unless one can come very close to the edge of the Brillouin zone. $\endgroup$ – CuriousOne Jul 29 '16 at 3:42

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