On Dilation Symmetries Arising from Scaling Limits

Abstract

Quantum field theories, at short scales, can be approximated by a scaling limit theory. In this approximation, an additional symmetry is gained, namely dilation covariance. To understand the structure of this dilation symmetry, we investigate it in a nonperturbative, model independent context. To that end, it turns out to be necessary to consider nonpure vacuum states in the limit. These can be decomposed into an integral of pure states; we investigate how the symmetries and observables of the theory behave under this decomposition. In particular, we consider several natural conditions of increasing strength that yield restrictions on the decomposed dilation symmetry.

Asymptotic morphisms and superselection theory in the scaling limit

“…Consider then a double coneÔ containing O + supp h ν for all ν ∈ N. Then if W ⊂Ô ′ is a wedge, Ω 0,ι is cyclic for the algebra π 0,ι (A(W )) by a standard Reeh-Schlieder argument. Therefore, we can find a C ∈ A(W ) such that [π 0,ι (α hν B) − π 0,ι (C)]Ω 0,ι < ε/16M, and consequently we can find κ (2) ε such that, for κ ≥ κ…”

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

“…It is then clear that A(O) ⊂ A • (O), and one finds that π 0,ι extends to a morphism π • 0,ι : A • → A 0,ι . Moreover, under the mild assumption that A has a convergent scaling limit [9,Def. 4.4], there also holds A 0,ι (O) ⊂ π • 0,ι (A • (O)).…”

Scale and Möbius Covariance in Two-Dimensional Haag–Kastler Net

“…Alternatively, one might look at smaller and smaller spacetime regions in the quantum chromodynamics, and should be able to see quarks which are otherwise confined and not visible. Such a limiting theory is expected to be simpler and to obtain the dilation symmetry (yet this is not automatic, see [BDM10]). Now, a dilation-covariant theory has often an additional symmetry, the conformal symmetry.…”