Algebras of unbounded operators and vacuum superselection rules in quantum field theory. II. Mathematical structure of vacuum superselection rules
The algebraic structure of quantum-field systems with vacuum superselection rules is analyzed in the framework of Wightman axiomatics on the basis of the mathematical formalism developed in Part I [ibid. 59, 28-48 (1984; Zbl 0559.47033)]. Two main theorems are obtained. The first asserts that a system with a discrete vacuum superselection rule, like systems with ordinary charge superselection rules, can always be described by a global algebra R of class P (direct sum of \(I_{\infty}\) type factors), and this property of the global algebra is equivalent to discreteness of the decomposition of the generating Wightman functional with respect to pure states, and also the existence of a discrete decomposition of the Hilbert space space into an orthogonal sum of vacuum superselection sectors. In accordance with the second theorem, there is a discrete vacuum superselection rule in all quantum-field systems for which the induction \(R'\to R'_{P_ 0}\), where \(P_ 0\) is the projection operator onto the vacuum subspace \({\mathcal H}_ 0\), has a discrete decomposition into irreducible elements (in particular, in all systems with finite-dimensional \({\mathcal H}_ 0)\). Other forms of vacuum structure in quantum field theory are analyzed.
algebraic structure of quantum-field systems with vacuum superselection rules
discreteness of the decomposition of the generating Wightman functional with respect to pure states
existence of a discrete decomposition of the Hilbert space space into an orthogonal sum of vacuum superselection sectors
scientific article; zbMATH DE number 3967218