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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

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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 $m$.

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In principle you don't have to preserve any symmetry at finite lattice spacing, as long as the symmetry gets restored in the continuum limit. In practice, this means that you have to preserve enough symmetry to forbid all symmetry violating operators of dimension 4 or less. This means, in particular, that you have to respect gauge invariance and at least a subgroup of the Lorentz group (hyper cubic symmetry) exactly at finite lattice spacing.

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