@sb1 you've opened up a can of worms here. This is a good question but it is subject to the standard misunderstandings about the nature of LQG and quantum geometry.
There is no problem with the question of Lorentz Invariance (LI) in a discrete spacetime. If your spacetime becomes discrete then your notion of LI must change accordingly. Keep in mind that LI is nothing more than a statement that physics should be causal ref1. For a continuous $3+1$ manifold this requirement is expressed in terms of invariance of the space-time interval $ds^2 = -dt^2 + dx^2$. The corresponding symmetry group is the continuous Lie group $SO(3,1)$ or $SL(2,\mathbb{C})$.
In a discrete spacetime $ds^2$ will have to be replaced by its discrete generalization $ \hat{d} s^2 = -\hat{d}t^2 + \hat{d}x^2 $ with $\hat{d}$ being the discrete ("quantum" or "q") differential. The invariance group of this interval is $SL(2,\mathbb{Z})$.
There is a more physical route involving considerations of the transformation of punctures of black hole horizons w.r.t an external observer which leads us to see how $SL(2,\mathbb{C})$ must reduce to $SL(2,\mathbb{Z})$ in the event that the numbers of these punctures is small.
Also your intuition regarding the dissipative effects of discreteness, is correct:
But if this violation is tolerated doesn't that imply some amount of viscosity within space-time itself?
It absolutely does and this is a result that we have know from a completely different directions - primarily from AdS/CFT and the fluid-gravity correspondence tells us that there is a lower bound for the viscosity to entropy ratio of horizons - those of black holes or those experienced by accelerated observers in an otherwise flat spacetime. Eling - Hydrodynamics of spacetime and vacuum viscosity, Son and Starinets - Viscosity, Black Holes, and Quantum Field Theory. This leads to the question of information loss. Having a dissipative horizon seems to suggest that information is lost in quantum gravity. However, this is only as seen by a "local" observer - who only has access to a portion of the spacetime. For a universal observer who has access to all the regions of the spacetime, this problem will not occur.
The debate around this topic is reminiscent of that surrounding Einstein's introduction of the notion that a Lorentzian manifold, rather than a Galilean one, was the right tapestry for events in spacetime. Then people struggled mightily to adapt all the physical inconsistencies of Newtonian's theory without having to give up the cherished Galilean property of absolute space and time. Now we see a similar reluctance to abandon the safe confines of continuous manifolds for a more general framework which can also describe discrete geometries. If one is willing to take this intellectual leap then there is no problem in becoming comfortable with a notion of LI which is adapted to the discrete setting.
I expect a strong response from the usual suspect(s) ;)
This post imported from StackExchange Physics at 2014-04-01 16:18 (UCT), posted by SE-user user346
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