Lattice fermions

The problem of formulating the field theory of Dirac particles on a spatial lattice is reviewed. In one dimension we construct free massless Dirac fields and Thirring fields and show their equivalence to the $X$ - $Y$ and asymmetric Heisenberg antiferromagnetic chains. In three dimensions we find that the simplest construction describes an isodoublet of massless Dirac fields. We discuss the incorporation of gauge degrees of freedom and illustrate how chiral symmetry is spontaneously broken by the interaction of gauge and fermion fields.