  Massless QED(1+1) - the Schwinger model - is studied in a covariant gauge.
The main new ingredient is an operator solution of the Dirac equation expressed
directly in terms of the fields present in the Lagrangian. This allows us to
study in detail the residual symmetry of the covariant gauge. For comparison,
we analyze first an analogous solution in the Thirring-Wess model and its
implication for the axial anomaly arising from the necessity to correctly
define products of fermion operators via point-splitting. In the Schwinger
model, one has to define the currents in a gauge invariant (GI) way. Certain
problems with their usual derivation are identified that obscure the origin of
the massive vector boson. We show how to define the truly GI interacting
currents, reformulate the theory in a finite volume and clarify role of the
gauge zero mode in the axial anomaly and in the Schwinger mechanism. A
transformation to the Coulomb gauge representation is suggested along with
ideas about how to correctly obtain other properties of the model.
