Topology and quantization of abelian sigma model in (1+1) dimensions

TL;DR

This paper investigates the quantization of the abelian sigma model in (1+1) dimensions, revealing an infinite variety of inequivalent quantum representations influenced by topology and central extensions.

Contribution

It introduces a framework for understanding the algebra of the quantum field in a topologically nontrivial setting, highlighting the role of central extensions and continuous parameters.

Findings

Existence of infinitely many unitary inequivalent representations.

Characterization of representations by a central extension and a continuous parameter.

Nontrivial commutator relations for winding operator and zero-mode momentum.

Abstract

It is known that there exist an infinite number of inequivalent quantizations on a topologically nontrivial manifold even if it is a finite-dimensional manifold. In this paper we consider the abelian sigma model in (1+1) dimensions to explore a system having infinite degrees of freedom. The model has a field variable $ \phi : S^1 \to S^1 $. An algebra of the quantum field is defined respecting the topological aspect of this model. A central extension of the algebra is also introduced. It is shown that there exist an infinite number of unitary inequivalent representations, which are characterized by a central extension and a continuous parameter $ \alpha $ $ ( 0 \le \alpha < 1 ) $. When the central extension exists, the winding operator and the zero-mode momentum obey a nontrivial commutator.