Dirac Formalism and Symmetry Problems in Quantum Mechanics. II. Symmetry Problems

The quantum-mechanical formalism developed in a previous article and based on the use of a rigged Hilbert space Φ⊂H⊂Φ′ is here enlarged by taking into account the symmetry properties of the system. First, the compatibility of a particular symmetry with this structure is obtained by requiring Φ to be invariant under the corresponding representation U of the symmetry group in H. The symmetry is then realized by the restriction of U to Φ and its contragradient representation Ǔ in Φ′. This double manifestation of the symmetry is related to the so-called active and passive points of view commonly used for interpreting symmetry operations. Next, a general procedure is given for constructing a suitable space Φ out of the labeled observables of the system and the representation U describing its symmetry properties. This general method is then applied to the case where U is a semidirect product G=T[squared times]Δ, with T Abelian. Finally, the examples of the Euclidean, the Galilei, and the Poincaré groups are briefly studied.