Asymptotic morphisms and superselection theory in the scaling limit

Abstract

Given a local Haag-Kastler net of von Neumann algebras and one of its scaling limit states, we introduce a variant of the notion of asymptotic morphism by Connes and Higson, and we show that the unitary equivalence classes of (localized) morphisms of the scaling limit theory of the original net are in bijection with classes of suitable pairs of such asymptotic morphisms. In the process, we also show that the quasi-local C*-algebras of two nets are isomorphic under very general hypotheses, and we construct an extension of the scaling algebra whose representation on the scaling limit Hilbert space contains the local von Neumann algebras. We also study the relation between our asymptotic morphisms and superselection sectors preserved in the scaling limit.

Asymptotic Morphisms and Superselection Theory in the Scaling Limit II: Analysis of Some Models

“…With these definitions, the main result of [21] states that if A has convergent scaling limit and its quasi-local C*-algebra is isomorphic to A 0,ι , there is a 1-1 correspondence between unitary equivalence classes of morphisms ρ 0 :…”

“…Since the sectors of A may be described by suitable endomorphisms, following an earlier suggestion by S. Doplicher we studied the possibility to describe the superselection sectors of A 0,ι in terms of some sort of asymptotic endomorphisms of A. A general treatment of this topic appears in [21]. The emerging mathematical concept resembles the so-called asymptotic morphisms of Connes-Higson in E-theory, a variant of Kasparov KK-theory, which is a cornerstone in the formulation of Noncommutative Geometry in the sense of A. Connes [19].…”

“…In view of the above considerations, it seems desirable, both from the mathematical and the physical standpoint, to understand in some detail how the abstract concepts developed in [21] fit with the analysis of some concrete models and to present explicit examples of asymptotic morphisms associated to confined sectors. A popular toy model exhibiting the main features which are expected to characterize the confinement picture is the so-called Schwinger model, i.e., d = 2 quantum electrodynamics with massless fermions.…”

“…Hence, in the present work we discuss the Schwinger model from the point of view of our previous paper [21], and in particular we provide an explicit construction of its asymptotic morphisms.…”

Conformal Nets and KK-Theory

“…Another instance of the emergence of KK-theoretical concepts in AQFT has been recently pointed out by Conti and Morsella [CM12], where the DHR sectors of scaling limit nets (as defined by Buchholz-Verch, in order to cast the renormalization group analysis into operator algebraic terms) are described by maps of the original global quasi-local C*-algebra on four-dimensional Minkowski spacetime that are suitable modifications of the asymptotic morphisms of Connes and Higson.…”