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

Abstract

The quantum‐mechanical formalism developed in a previous article and based on the use of a rigged Hilbert space $\mathrm{\Phi \subset }\mathcal{H}\mathrm{\subset \Phi \prime }$ 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 $\mathcal{H}.$ 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 $\mathrm{G=T[squared\; times]\Delta ,}$ with T Abelian. Finally, the examples of the Euclidean, the Galilei, and the Poincaré groups are briefly studied.