special relativity - Does the discreteness of spacetime in canonical approaches imply good bye to STR?

In all the canonical approaches to the problem of quantum gravity, (eg. loop variable) spacetime is thought to have a discrete structure. One question immediately comes naively to an outsider of this approach is whether it picks a privileged frame of reference and thereby violating the key principle of the special relativity in ultra small scale. But if this violation is tolerated doesn't that imply some amount of viscosity within spacetime itself? Or am I writing complete nonsense here? Can anybody with some background in these approaches clear these issues?

Answer

Here is as I see the problem. The best way to understand it is to think what happens with rotations and quantum theory. Suppose that a certain vector quantity $V=(V_1,V_2,V_3) $ is such that its components are always multiples of a fixed quantity $d $. Then one is tempted to say that obviously rotational invariance is broken because if I take the vector $V=(d,0,0) $ and rotate it a bit, I get $V=(\cos(\phi) d, \sin(\phi) d,0) $, and $\cos(\phi) d $ is smaller than $d $. Therefore, either rotational invariance is broken, or the vector components can be smaller. Right? No, wrong. Why? Because of quantum theory. Suppose now that the quantity $V$ is the angular momentum of an atom. Then, since the atom is quantum mechanical, you cannot measure all the 3 components together. If you measure one, you can get say either $0 $, or $\hbar$, or $2\hbar$ ..., that it, precisely multiples of a fixed quantity.

Now supose you have measured that the $x$ component of the angular momentum was hbar. Rotate slowly the atom. Do you measure them something a bit smaller that $\hbar$? No! You measure again either zero, or $\hbar$ ... what changes continuously is not the eigenvalue, namely the quantity that you measure, but rather the probabilities of measuring one or the other of those eigenvalues. Same with the Planck area in LQG. If you measure an area, (and if LQG is correct, which all to be seen, of course!) you get a certain discrete number. If you boost the system, you do not meqsure the Lorentz contracted eigenvalues of the area: you measure one or the other of the same eigenvalues, with continuously changing probabilities. And, by the way, of course areas are observables. For instance any CERN experiment that measures a total scattering amplitude amplitude is measuring an area. Scattering amplitudes are in $\mathrm{cm}^2$, that is are areas.