Anomalies and the lattice Schwinger model: Paradigm not paradox

Abstract

This paper shows that, contrary to statements extant in the literature, it is possible to introduce fermions into a lattice gauge theory in such a way as to preserve the continuous chiral symmetries of the massless theory and the physics of the axial anomaly. The particular model discussed is the lattice Schwinger model and the methods used are based upon the nonperturbative gauge-invariant variational techniques introduced by Horn and Weinstein. It is demonstrated that the physics of the anomaly, and its relation to the angles appearing in the exact solution to the continuum model, appears in a simple and elegant way. The generalization of the model to several sets of independent fermions is discussed at the end of the paper. Some brief remarks are made about what happens if one attempts to gauge an anomalous current. These results are of interest, since the Drell-Weinstein-Yankielowicz prescription is the only known way of writing down purely lattice gauge theories with only left-handed fermions.