The Lattice Schwinger Model: Confinement, Anomalies, Chiral Fermions and All That

TL;DR

This paper analytically explores the Hamiltonian formulation of the lattice Schwinger model, comparing different fermion derivatives to understand their approach to continuum physics, including confinement and anomalies.

Contribution

It provides a detailed analytical comparison of Wilson and SLAC fermion derivatives in the lattice Schwinger model, clarifying their continuum limit behavior.

Findings

Both Wilson and SLAC derivatives reproduce continuum physics accurately.

The regulated lattice charge density operator corresponds to the point-split continuum operator.

There is a connection between strong and weak coupling regimes for SLAC-type derivatives.

Abstract

In order to better understand what to expect from numerical CORE computations for two-dimensional massless QED (the Schwinger model) we wish to obtain some analytic control over the approach to the continuum limit for various choices of fermion derivative. To this end we study the Hamiltonian formulation of the lattice Schwinger model (i.e., the theory defined on the spatial lattice with continuous time) in $A_0=0$ gauge. We begin with a discussion of the solution of the Hamilton equations of motion in the continuum, we then parallel the derivation of the continuum solution within the lattice framework for a range of fermion derivatives. The equations of motion for the Fourier transform of the lattice charge density operator show explicitly why it is a regulated version of this operator which corresponds to the point-split operator of the continuum theory and the sense in which the regulated lattice operator can be treated as a Bose field. The same formulas explicitly exhibit operators whose matrix elements measure the lack of approach to the continuum physics. We show that both chirality violating Wilson-type and chirality preserving SLAC-type derivatives correctly reproduce the continuum theory and show that there is a clear connection between the strong and weak coupling limits of a theory based upon a generalized SLAC-type derivative.