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 axiomaties on the basis of the mathematical formalism developed in Part I [6]. 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 state space into an orthogonal sum of vacuum superseleetion 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 $\mathscr H_0$, has a discrete decomposition into irreducible elements (in particular, in all systems with finite-dimensional $\mathscr H_0$). Other forms of vacuum structure in quantum field theory are analyzed.