The Schwinger model on $S^1$: Hamiltonian formulation, vacuum and anomaly

TL;DR

This paper develops a Hamiltonian framework for the Schwinger model on a circle, elucidating the vacuum structure, anomaly, and large gauge transformations, with implications for quantum field theory on compact spaces.

Contribution

It introduces a Hamiltonian formulation on $S^1$ that explicitly incorporates large gauge transformations and describes the vacuum and anomaly structure.

Findings

Hamiltonian is semi-bounded and self-adjoint.

Large gauge transformations act nontrivially on the vacuum.

Explicit identification of the interacting vacuum.

Abstract

We present a Hamiltonian formulation of the Schwinger model on the circle in Coulomb gauge as a semi-bounded self-adjoint operator which is invariant under the modular group ${\cal M}=\Z$ of large gauge transformations. There is a nontrivial action of $\cm$ on fermionic Fock space $\mcH_0$ and its vacuum which plays a role analogous to that of the spectral flow in the formalism involving the infinite Dirac sea. The formulation allows (i) a description of the anomaly and its relation to this group action, and (ii) an explicit identification of the interacting vacuum which arises after the destabilization of the non-interacting vacuum in $\mcH_0$.