New operator solution of the Schwinger model in a covariant gauge and axial anomaly

TL;DR

This paper presents an operator solution to the Schwinger model in a covariant gauge, clarifying the role of gauge invariance, residual symmetries, and the axial anomaly, with implications for understanding the model's physical properties.

Contribution

It introduces a direct operator solution in a covariant gauge for the Schwinger model, improving understanding of gauge invariance and the axial anomaly compared to previous approaches.

Findings

Operator solution expressed in terms of original fields

Clarification of gauge zero mode's role in anomaly

Reformulation in finite volume and gauge-invariant currents

Abstract

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.