Delicacies of the Mass Perturbation in the Schwinger Model on a Circle

TL;DR

This paper investigates the topological differences in the fermionic Hilbert bundle of the Schwinger model on a circle, demonstrating that perturbation theory remains valid in the strong coupling limit despite topological discontinuities.

Contribution

It shows that the structure of the massless fermionic Hilbert bundle can be preserved dynamically in the strong coupling limit, justifying perturbation theory in this regime.

Findings

Massless fermion bundle is topologically non-trivial

Massive fermion bundle is topologically trivial

Perturbation theory is justified in the strong coupling limit

Abstract

The Hilbert bundle for the massless fermions of the Schwinger model on a circle, over the space of gauge field configurations, is topologically non-trivial (twisted). The corresponding bundle for massive fermions is topologically trivial (periodic). Since the structure of the fermionic Hilbert bundle changes discontinuously the possibility of perturbing in the mass is thrown into doubt. In this article, we show that a direct application of the anti-adiabatic theorem of Low, allows the structure of the massless theory to be dynamically preserved in the strong coupling limit, ${e\over m}>>1$. This justifies the use of perturbation theory in the bosonized version of the model, in this limit.