What happens to Lattice Fermion near Continuum Limit?

Abstract

A Ginsparg-Wilson Relation (GWR) is obtained in the presence of chiral symmetry breaking terms. It leads to the PCAC relation as well as an anomaly relation on the lattice. For general fermions, the deviation from the exact GWR is getting small when the block-spin transformations are performed iteratively. Based on a simple geometrical interpretation of the Dirac operator satisfying the GWR, we find some physical properties shared by the lattice fermions near the continuum limit. In two-dimensions, we explicitly construct the GW Dirac operator by using a conformal mapping.