# What symmetries does a lattice calculation need to preserve?

I've heard that it is impossible to have a properly Lorentz-invariant lattice QFT simulation, as the Lorentz invariance is spoiled by the nonzero lattice distance $a$. I've also heard that there are other symmetries that must be preserved for a lattice simulation to be realistic.

What symmetries must a lattice simulation preserve to be realistic? Why must they be preserved?

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1: Gauge symmetries, otherwise the theory is inconsistent. 2: ...? –  Michael Brown Feb 19 '13 at 1:28
Well, rotational symmetry is a proper subset of Lorentz symmetry, but it is often discussed separately (in the context of angular momentum conservation). –  dmckee Feb 19 '13 at 1:37

The only symmetry that must be preserved on the lattice is local gauge invariance. In fact the discrete lattice theory was developed by Wilson as a way to regularize quantum field theories while exactly preserving local gauge invariance. This happens because you have a lattice spacing $a$ which corresponds to a ultraviolet momentum cutoff $\Lambda$ in your path integral. In real lattice gauge theory calculations all other symmetries such as Lorentz symmetry and chiral symmetry are extrapolated in a controlled fashion by tuning a parameter in the simulations. To restore Lorentz symmetry you tune $a$ and to restore chiral symmetry you tune the quark mass $m$.