The axial anomaly and the lattice Dirac sea

In the hamiltonian formulation of fermions coupled to external gauge fields, the axial anomaly has a simple physical interpretation in terms of level shifting at the top of the Dirac sea. We apply this formalism to lattice QED in 1 + 1 and 3 + 1 dimensions, in order to study how the lattice regulation and small fermion mass affect the picture. For a simple choice of the E and B fields, it is possible to accurately follow the time evolution of the (lattice) Dirac sea of Wilson fermions. We find that the Wilson r term plays a role, with respect to the anomaly, which is closely analogous to point-splitting in the continuum. This role is jeopardized whenever the Wilson mass scale r/a is comparable to other physical mass scales in the problem, as is the case, for example, in strong coupling calculations. Remarkably, we find that hadronic scales of only a few lattice spacings are probably sufficient to guarantee the proper anomaly structure.