There is a variety of models of quantum field theory, where discrete spacetime is used as technical support, or even suggested as physical reality. As far as I know, all of such models faced serious problems with restoring Lorentz Symmetry in continuous limit. Is this general property of discrete models they have problems with restore continuous symmetries in general or it is certain property of Lorentz Symmetry Group and alike, in 4 dimensions?
$\begingroup$ Perhaps a discrete grid be deformed whereas a point would just move? $\endgroup$– EmilJan 2, 2018 at 11:16
1$\begingroup$ For QFTs regularized via a hypercubic space-time lattice, (euclidean) Lorentz invariance is automatically restored in the continuum limit (corresponding to the removal of the UV cutoff), thanks to the discrete hypercubic rotation subgroup preserved in the discrete space-time. So this problem is not a "general property". Saying much more may require considering a specific example, perhaps something like arXiv:0804.1145. $\endgroup$– David SchaichJan 10, 2018 at 2:29
$\begingroup$ @David - is it realistic field theory or toy model? Does it contains physical couplings or produce artificial ones as well? $\endgroup$– kakazJan 10, 2018 at 7:34
$\begingroup$ It's a fundamental definition of quantum chromodynamics (or any other vector-like gauge theory), not a toy model. There are no artificial couplings. $\endgroup$– David SchaichJan 10, 2018 at 13:42
There is absolutely no issue with restoring continuous $O(4)$ symmetry using lattice regularization. The idea is well understood at this point. The lattice spacing is simply a cut-off, and one looks for regions in the phase diagram of the model where the correlation length is much larger than the lattice spacing, making the cut-off effects disappear. In a model with a continuous phase transition one can tune a physical quantity to be constant as one takes the correlation length larger and larger, thus one is able to make predictions about quantities in physical units in the continuum limit.
The lattice QCD and lattice gauge theory communities have been doing this for decades being some of the only theorist to progress alongside with experimental high energy physics with pre- and post-dictions of the standard model.
visit [hep-lat] to see recent papers in lattice field theory, but lattice literature runs for a long time.