FERMIONS ON THE LATTICE

As noted by Creutz, there is a problem with Yang–Mills theory in the lattice. The fact that it breaks Lorentz-invariance is something one can live with, since Lorentz symmetry is expected to be restored automatically in the continuum limit — there exist no low-dimensional effective interactions that break Lorentz-invariance so that no special measures need to be taken to restore it in the continuum limit. However, the situation is quite different in the case of chiral symmetry. Just like Pauli–Villars regulators and the dimensional regulator, also the lattice breaks chiral symmetry. The symmetry is not automatically restored in the continuum limit, since effective mass terms will be generated that survive. The reason why all known regulators have this ‘disease’ in common is the anomaly: even in the continuum theory chiral symmetry is broken down…