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

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 $\varphi :S^1\to S^1$. 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.

Latex 13 pages, Nagoya University DPNU-94-35, Contributed to Yamada Conference (XXth International Colloquium on Group Theoretical Methods in Physics; July 1994 at Toyonaka in Japan)

References in corpus (3)

• Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S^1
• Quantum Mechanics on Manifolds
• Toroidal Orbifold Models with a Wess-Zumino Term

Cited by in corpus (3)

• Quantum Mechanically Induced Hopf Term in the O(3) Nonlinear Sigma Model
• Inequivalent Quantization in the Skyrme Model
• Algebra and Twisted Algebra in Toroidal Target Space