Lattice gauge theory under a Lorentz transformation Taking a grid of 'evenly-spaced' space-time points. e.g. at integer values of (x,y,z,t). Now do a Lorentz boost on this grid. We end up with a grid of points which are much closer together.
It is bothering me that the grid of points only looks evenly distributed in certain frames (related by spacial rotations) but looks different under Lorentz boosts. The points that looked like neighbours no longer look like neighbours.
When working out results in lattice QCD, does this mean that the results you get don't exhibit Lorentz symmetry. And how can we be sure in the limit the Lorentz symmetry is conserved?
Edit: I just came across this animation from John Baez on time crystals.

Which seems to suggest that for certain lattices like a triangular lattice, the Lorentz transformation can preserve the Lattice form. Although looking at the middle point, it's neighbours are continually changing. And Lattice QCD is based on calculations of nearest neighbours (discrete derivative). I guess one could reformulate the calculations to all neighbours a unit Minkowski-distance away. But this would increase the number of neighbours a lot for large volumes of space-time. (and does this obey the concept of `locality'??)
 A: Lattice theories are not invariant under Lorentz transformations.  They are usually defined in Euclidean  signature (imaginary time) so the symmetry woud be ${\rm SO}(N)$ rather than ${\rm SO}(N-1,1)$ ---  but apart from this the lattice breaks the rotational symmetry down to a discrete subgroup of 90 degree rotations. It's only the in the continuum limit (i.e  close to a critical point) that  full rotational invariance is restored.  
A: Yes, a Lorentz boost breaks the symmetry of the lattice. And it creates a different lattice. We in fact use this! On the lattice your momenta can only be multiple of $2\pi/L$ where $L$ is the spatial extent. When you boost, $L$ changes and so do your momenta. Looking at an interacting system from two pions you will see that the moving system has a different total energy (of course). But when you boost that back such that you are in the center of mass frame (at the cost of a distorted lattice) you will find that the energy is different from a two pion state where the total momentum was zero! This way you can create different interacting energies even though the individual momenta are quantized by the enclosing box.
The group theory will get pretty messy though, and you need to be careful with the reduced symmetries.
