This may help: http://arxiv.org/abs/gr-qc/0205108 Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction Carlo Rovelli, Simone Speziale 12 pages, 3 figures (Submitted on 25 May 2002) "A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary."
yes but i can't understand how i can split lorentz group. namely how is possible that so(1,3)~so(3)X Something. thanks
You are right, LQG is not Lorentz covariant. Time and space are not treated in the same way. In particular, in LQG (just as in any canonical approach to quantum gravity) there is a problem of time, while there is no problem of space. Even if you accept the solution of that problem in terms of a relational time, it is still true time is not treted in the same way as space.
No, because 1.length is a state that arises as a an observable that is bounded below by the plack scale. 2.lorentz invariance arises as an average of the observables at larger scale. Thus, length is meaningless by itself, all that exists are observables in a "nothingness", which by interacting each other, make space time appears. Demystifier is right in what he says, because in the nothingness, there is a kind of absolute QM tick tack, thus, time is a paramter in this case, not a dimension. Time as a dimension shows up as a kind of constrain between the observables.
Most importantly, LQG makes here a prediction, thus becomes testable and finally, deserves the status of a scientific theory !
Can anyone explain me how the predicted dependence on frequency for the speed of light is related to the Lorentz-invariance issue?
As far as I know, there is no agreement in the present state of the theory. On needs to construct specific low energy models. There has been some claims by Smolin and other enthusiastic people. I meant to emphasize that one should not conclude the theory is "dead" but on the contrary that it is one way it becomes "alive" ! On Loop Quantum Gravity Phenomenology and the Issue of Lorentz Invariance
So, the testability of the theory is highly model dependent? That is, there may be a number of possibilities? Is it possible that the Lorentz violating effects only occur at the Planck scale?
In the present state of development, yes, I think so. Yes, and in fact if I remember correctly, Rovelli in his book gives arguments why, beyond naive expectations, Lorentz invariance might even still hold down to the quantum of length.
So, how is this different from string theory? So, the predictions might only be testable in principle? Again, how is this different from string theory?
Then quantum field theory, or more precisely, the standard way of quantization of fields, is Lorentz invariant. Even quantum gravity can be quantized in this way, and then quantum gravity is Lorentz invariant too. However, this method of quantization of gravity has other problems (non-renormalizability, background dependence, ...), which is why one searches for other approaches, like LQG and string theory, which solve some problems but cause some new ones.