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It is usually said that existence of discrete spacetime violates Lorentz symmetry. What quantity is used to quantify such violation? I mean could someone points a reference for a derivation that shows such analysis.

My other question is that: if, and this is a big fat if, OPERA result was true (and I believe it is not) would that prove that spacetime is discrete? (this is like the inverse to the 1st part of this post, if space-time is discrete then Lorentz symmetry is violated, what if Lorentz symmetry is violated, does that imply space-time is discrete? or not necessarily?)

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The first part is also discussed in this question physics.stackexchange.com/q/9720/2451 while the last part is a duplicate of this question physics.stackexchange.com/q/15963/2451 – Qmechanic♦ Feb 12 '12 at 0:39
-- I'm not sure about the precise definition of Lorentz violation, but here some comments: If you have a grid with discrete points $A,B$ and a metric, which quantifies their distance $d_{AB}$, then you might be able to transform a quantity by a translation $T_{\delta= d_{AB}}$, but not by $T_{0.79}$. The Poincaré group however is a Lie-Group or ten real parameters, while a lattice doesn't let you translate continously. Notice btw. how not all of the Lorentz symmetries are translational ones, so there are other ways to break it. – Nick Kidman Feb 12 '12 at 1:35
-- Related links are this and this and also this might as an example for a periodic structure. Here is the Poincare group. Lorentz transformations are technically not the ones containing the translations, but whatever. – Nick Kidman Feb 12 '12 at 1:36
Related: physics.stackexchange.com/q/3662/2451 – Qmechanic♦ Jul 25 '12 at 16:22

2 Answers

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There are nontrivial discrete subgroups of the Lorentz group. It is easy to construct an SO(3,1) matrix that has only integer entries and yet is not just a simple rotation. A rectangular lattice in Minkowski space is invariant under the group of these transformations. Different space-time lattices have different discrete subgroups of Lorentz under which they are invariant.

This might be what you are looking for, even if it does not answer the question ...

However, there is a snag: it is extremely difficult to construct any non-trivial dynamical model of nature (quantum, classical, anything) that transforms into itself under these discrete transformations, even if their lattice does. Versions of string theory, adapted to this lattice, may give you the best promises.

So my advice is: don't believe the no-go theorems.

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But not the full six dimensional Lorentz group, right? The question is then what are the phenomenological constraints on violation of the broken part of the Lorentz group? – drake Aug 17 '12 at 22:45
I don't think one can write down field theories that break the continuous Lorentz group but keep the discrete subgroup intact. Therefore my guess is: none. But it's a guess. – G. 't Hooft Aug 22 '12 at 9:21
Could you give me any reference where I can study these discrete Lorentz's subgroups and the difficulties of constructing models invariant under their actions? – drake Aug 22 '12 at 19:02
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The analysis of Lorentz violations is found in Coleman and Glashow's "High Energy Tests of Lorentz Invariance": http://arxiv.org/abs/hep-ph/9812418 .

The discreteness leads to Lorentz-violation arguments turn out to be bogus in light of holography (in this I believe I completely agree with t'Hooft), they assume that the symmetry of the space time has to be a symmetry of the space-time points. When the bulk is emergent, even if the boundary is discrete, the emergent symmetry in most of the bulk can be as good as the longest-wavelengths of the boundary theory, so that it is essentially exact.

This is the reason that discrete theories are viable, but only in light of holography. If you make a naive lattice at the Planck scale, say, you break Lorentz invariance by too much. I'll defer to the linked paper for precise bounds, I don't know them.

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