It turned out to be most difficult in the lattice approach to QCD to deal with fermions. Putting fermions on a lattice is an ambiguous procedure since the cubic symmetry of a lattice is less restrictive than the continuous Lorentz group.

The simplest chiral-invariant formulations of lattice fermions lead to a doubling of fermionic degrees of freedom, as was first noted by Wilson [Wil75], and describe from 16 to four relativistic continuum fermions, depending on the formulation. One-half of them have a positive axial charge and the other half have a negative one, so that the chiral anomaly cancels. There is a no-go theorem which says that the fermionic doubling is always present under natural assumptions concerning a lattice gauge theory.

A practical way out of this problem is to choose the fermionic lattice action to be explicitly noninvariant under the chiral transformation and to have, by tuning the mass of the lattice fermion, one relativistic fermion in the continuum and the masses of the doublers to be of the order of the inverse lattice spacing. The chiral anomaly is recovered in this way.

In this chapter we consider various formulations of lattice fermions and the doubling problem. We discuss briefly the results on spontaneous breaking of the chiral symmetry in QCD.

Chiral fermions

The quark fields are generically matter fields, the gauge transformation of which in the continuum is given by Eqs. (5.1) and (5.3), and can be put on a lattice according to Eq. (6.7).