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

Abstract

We introduced in a previous paper a general notion of asymptotic morphism of a given local net of observables, which allows to describe the sectors of a corresponding scaling limit net. Here, as an application, we illustrate the general framework by analyzing the Schwinger model, which features confined charges. In particular, we explicitly construct asymptotic morphisms for these sectors in restriction to the subnet generated by the derivatives of the field and momentum at time zero. As a consequence, the confined charges of the Schwinger model are in principle accessible to observation. We also study the obstructions, that can be traced back to the infrared singular nature of the massless free field in d = 2, to perform the same construction for the complete Schwinger model net. Finally, we exhibit asymptotic morphisms for the net generated by the massive free charged scalar field in four dimensions, where no infrared problems appear in the scaling limit.

Scaling Limits of Lattice Quantum Fields by Wavelets

“…Concerning ( 4.32 ), by the argument in the proof of [ 20 , Lemma A.5], there holds and therefore, by a similar argument as the above one, is smooth in . …”

“…Then using Lemma 4.21 and the argument of the proofs of Lemmas A.4 and A.6 of [20], one verifies that Q ± are trace-class, and this, together with Lemma 4.20, implies the quasiequivalence statement by [3]. Now, as we take K ≥ 6 and the density result (Lemma 4.6) is a local property and hence holds also for L = ∞, we have π ∞ (W ∞,L (S)) = π ∞ (W L (S)) , and the latter is a type III factor [2].…”

“…Following the steps of [ 20 , Appendix A], we first show that the two representations are quasi-equivalent by the main theorem in [ 3 ], then this is actually unitary equivalence because local algebras are type III factors. The assumptions of [ 3 ] are satisfied if one proves the following facts.…”

“…Assume . Consider the completion of the space of real functions in (thought as compactly supported functions on ) with respect to the norms , and, on these spaces, the operators defined by Then using Lemma 4.21 and the argument of the proofs of Lemmas A.4 and A.6 of [ 20 ], one verifies that are trace-class, and this, together with Lemma 4.20 , implies the quasiequivalence statement by [ 3 ].…”

Modular structure of the Weyl algebra

“…Note that, in the massless case, our notation is unconventional: Ḣ0 is the usual one particle space and H 0 has not been defined yet. See also [11,5] for related structures.…”