Superselection Structure of Massive Quantum Field Theories in 1+1 Dimensions

TL;DR

This paper demonstrates that many massive quantum field theories in 1+1 dimensions lack locally generated superselection sectors, challenging the extension of DHR theory and impacting the understanding of soliton sectors.

Contribution

It shows that for a broad class of massive 1+1D QFTs, the DHR superselection sector framework does not apply, especially regarding localizability and Haag duality.

Findings

Superselection sectors are absent in these theories.

Haag duality holds in all locally normal representations.

The results do not extend to massless theories.

Abstract

We show that a large class of massive quantum field theories in 1+1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.