A remark on the definition of superselection rules in terms of unbounded operators

The mathematical definition of superselection rules in the case when observables are described by unbounded operators in a fixed Hilbert space (for instance, in the frame of Wightman's axioms) is examined. The additional condition (P_{H_q } D subset D) (whereD is the common domain of definition of the operators,H _q is theqth sector, and (P_{H_q } ) is the projection onH _q ) is found to be sufficient in order to preserve-as in the case of bounded observables—the one-to-one correspondence between reducing subspacesH _q and projections (P_{H_q } ) from the commutantA′ of the algebraA of observables. This additional condition is equivalent to the physical requirement that every physical vector state can be uniquely represented as a linear combination of physical states, each belonging to some sector.