Operators and Observables in Isotopic Spin Space

Abstract

The usefulness of the conventional isotopic spin formalism is somewhat marred by the fact that it contains superfluous formal elements which have no physical counterpart. It is demonstrated that application of the "superselection principle" for total electric charge, recently suggested by Wick, Wightman, and Wigner, is sufficient to eliminate these superfluous elements and thus reduce the content of the theory to that of the more conventional formalism in which neutrons and protons are distinguished ab initio. This superselection principle applied to the conventional state-function representation of quantum mechanics requires that admissable state functions be eigenfunctions of the total charge and that operators representing observables commute with the total charge. The disjointness of the various subspaces in Hilbert space corresponding to different total charge eigenvalues, demanded by the superselection principle, leads to the result that many observables which are outwardly different in form are actually essentially equivalent. The construction of all inequivalent observables compounded from nucleonic isotopic spin operators only is carried out and it is shown that all such observables are simply functions of the square of the total isotopic spin and its $z$ component. The essentially uniqueness of the charge parity operator introduced by Kroll and Foldy is established.