Massless scalar free Field in 1+1 dimensions I: Weyl algebras Products and Superselection Sectors

TL;DR

This paper classifies superselection sectors of a 1+1 dimensional massless scalar free field model using Weyl algebra products, revealing a gauge group structure and superselection sector properties in a conformal quantum field theory context.

Contribution

It provides a detailed classification of sectors via automorphisms, introduces a gauge group framework, and analyzes the superselection structure in a conformal model with non-regular representations.

Findings

Superselection sectors are classified as solitonic and non-solitonic automorphisms.

Existence of a global compact Abelian gauge group is established.

A non-trivial center appears in the fixed-point net, trivial in the physical representation.

Abstract

This is the first of two papers on the superselection sectors of the conformal model in the title, in a time zero formulation. A classification of the sectors of the net of observables as restrictions of solitonic (twisted) and non-solitonic (untwisted) sector automorphisms of proper extensions of the observable net is given. All of them are implemented by the elements of a field net in a non-regular vacuum representation and the existence of a global compact Abelian gauge group is proved. A non-trivial center in the fixed-point net of this gauge group appears, but in an unphysical representation and reducing to the identity in the physical one. The completeness of the described superselection structure, to which the second paper is devoted, is shown in terms of Roberts' net cohomology. Some general features of physical field models defined by twisted cross products of Weyl algebras in non-regular representations are also presented.