Scaling Algebras for Charged Fields and Short-Distance Analysis for Localizable and Topological Charges

Abstract

The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behavior of quantum field theories in the setting of the model-independent, operator algebraic approach, is extended to the case of fields carrying superselection charges. In doing so, consideration will be given to strictly localizable charges ("DHR-type" superselection charges) as well as to charges which can only be localized in regions extending to spacelike infinity ("BF-type" superselection charges). A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and for charges of DHR-type, the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors.

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

“…4 When N = G F , A := B is the local net of observables associated to F, and we recover the existence of a conditional expectation from F 0,ι to the observable scaling limit net A 0,ι used in the proof of lemma 5.1 in [14].…”

“…It would be interesting to know conditions on B (and on ω 0,ι ) which guarantee that F(B 0,ι ) satisfies the assumptions in definition 4.6. It follows from the discussion in [14] that, since we already assumed that B 0,ι satisfies Haag duality, assumptions (ii) and (iii) for F(B 0,ι ) can be deduced from analogous assumptions on B. It is also reasonable to expect that suitable nuclearity requirements on B imply assumption (iv) for F(B 0,ι ), see also the paragraph following proposition 4.13.…”

“…Thanks to the results in the appendix of [23] (see also the remark in sec. 4 of [14]), the main results in [17] can be applied to A and B, and we further assume that for the corresponding DR canonical covariant field nets one has F(A) = F(B). Therefore for the canonical DR gauge groups one has that G(B) is a closed subgroup of G(A).…”

“…Let us fix a scaling limit state ω B 0,ι of B. According to the results in [14], there exists a scaling limit F(B) 0,ι of F(B) and a quotient G(B) 0,ι of G(B) by a normal closed subgroup N (B) 0,ι defined in analogy to (9), such that B 0,ι = Thanks to what was shown in section 2, using the corresponding scaling limit state ω A 0,ι := ω B 0,ι ↾ A of A we get similar relations for the nets A 0,ι , F(A) 0,ι and F(A 0,ι ). Summarizing, we get the following result.…”

“…Also, the mathematical tools that are available in this setting are well suited for providing a detailed analysis of subsystems, an issue that is central in order to obtain an intrinsic description of the observable system one starts with [12,10,11]. In another direction, the recently proposed algebraic approach to the renormalization group [8] (see also section 2) has opened the possibility of studying the short distance limit in the local quantum physics framework, and has started to convey new insight into our understanding of physically relevant issues such as confinement of colour charges and renormalization of pointlike fields [3,14,1].…”