Учёные записки Казанского университета: Серия Физико-математические науки (Jun 2017)
A Simple Algebraic Model for Few-Nucleon Systems in the Presence of Non-Abelian Superselection Rules
Abstract
Traditionally, the dynamics of a quantum physical system is described on the basis of the field models, where the fundamental role is played by the algebra of quantized fields and its automorphisms (forming a compact group). However, despite this important role of quantized fields, the fields are unobservable quantities. As shown by R. Haag, a quantum physical system can also be described in a dual way, where the algebra of observables and the semigroup (category) of its endomorphisms are taken as the initial object. Both approaches should provide practically the same information about the physical system, but the second approach is more natural, because it is based only on the experimentally observed information. In this case, the concept of duality reduces to the existence of a dual object to a compact group: in the case of the Abelian group, this is a group of its characters (Pontryagin duality); in the case of the non-Abelian group, this is the category of representations of the given group (Tannaka–Krein duality). From the physical point of view, the dual object describes charges (Abelian charges in the case of the Abelian compact group and non-Abelian ones in the case of the non-Abelian group) and, hence, the superselected structure of the physical system. We have developed a model for describing non-Abelian isotopic charges of nucleon systems. The category of representations of the compact isotopic rotation group describes the superselected structure by isospin, and each irreducible representation is indexed by one of the numbers {0; 1/2; 1; 3/2; 2;…}. It has been shown that a special projection operator belonging to the algebra of endomorphisms of a fixed object of the category allows to obtain a bound state of nucleons by projecting onto antisymmetric subspace. The states of such nucleons obey parastatistics of the order 2. It has been also demonstrated that the intertwining operators of objects with a vacuum sector correspond to the fields carrying isotopic charges. Since the model takes into account the conservation of isospin, it is also applicable to the study of resonances, which are under heated discussions at the present time.
