Dynamical group of microscopic collective states. I. One‐dimensional case

TLDR: In this article, it was shown that the dynamical group of collective states is then the group Spc(2,R), which is the restriction to the collective subspace of the group of linear canonical transformations in n dimensions conserving the O(n) symmetry.

Abstract: In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A‐nucleon system, defined as those A‐particle states invariant under the orthogonal group O(n) associated with the n = A−1 Jacobi vectors. The present paper discusses the case of a hypothetical one‐dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group Spc(2,R), which is the restriction to the collective subspace of the group Sp(2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well‐known realization of the dynamical group in the Schrodinger representation based upon the Dzublik–Zickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordi...