The Schwinger Model and Its Axial Anomaly

The Schwinger model (quantum electrodynamics with massless fermions in one spatial dimension) is solved, supposing that space is a circle. This clarifies aspects of the usual version of the model, where space is a line, without changing the physics. The Hamiltonian formalism is used. On a circle, an abelian gauge field has one physical degree of freedom, and the gauge covariant Dirac operator, which couples the fermions to this degree of freedom, exhibits spectral flow. The relationship between the spectral flow and the axial anomaly is explained. Some variants of the Schwinger model are also discussed.