Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

Abstract

Given an inclusion B ⊂ F of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets B 0 ⊂ F 0 , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of F implies that of the scaling limit of B. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion A ⊂ B of local nets with the same canonical field net F, we find sufficient conditions which entail the equality of the canonical field nets of A 0 and B 0 .

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

“…, both assertions follow from the fact that, for n large enough (namely, n such thatŌ is in the left spacelike complement of the point (0, na)), [20,Thm. 4.6], also…”

“…(4.5) for an arbitrary B ∈ A (m) . Moreover, (φ λ ) is not an asymptotic isomorphism of A (m) in the sense of [20,Def. 5.4] and therefore we can not establish the analogue of Eq.…”

“…is the unitary phase of the polar decomposition of ϕ(f ), with f ∈ D R (O) such that its Fourier transform is not identically zero on the mass m hyperboloid. On the other hand, according to [22,20], the outer regularized scaling limit nets F can be identified with the corresponding nets F (0) , A (0) through the net isomorphism φ : F…”

“…where W (0) r is the net of C * -algebras generated by the operators in (4.3). For reference's sake, we note that if W ∈ W (m) r is an element of the form (4.1), then m) in the sense of [20,Def. 5.4] and therefore we can not establish the analogue of Eq.…”

The principle of locality: Effectiveness, fate, and challenges

“…Important and successful lines are, furthermore, the study of the scale limit, the analysis of the phenomenon of confinement and of renormalisation in terms of local algebras [36,37,38,39].…”