Vector states on algebras of observables and superselection rules II. Algebraic theory of superselection rules

The methods and results of Part I are used to give a new and complete mathematical formuiation of superselectionrules. This is done first for very simple “diehotomic” superselection rules and then for arbitrary rules. Three fundamental physical conditions are formulated that are equivalent to one another and, taken together, give the abstract definition of a superselection rule. A number of new features of superselection rules is revealed. The most important are the following: 1) the superselection operators in the general case belong to the center of the global algebra of observables $R$; 2) the phenomenon of superselection rules exists and possesses the complete set of necessary physical properties only for the class of theories with a sufficient set of pure vector states (this concept was introduced and studied in Part I). It is established which forms of “continuous” superselection rules are possible and which are impossible in a physical theory. A coheren superselection sector is defined as a factorial type I representation of $R$. A generalization of the principle of superposigon is formulated and it is proved that it is satisfied in a coherent sector.