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Journal of Mathematical Physics

Boson realization from quantum constraints

Mircea Iosifescu, Horia Scutaru

Published Feb 1, 1986• 1 citation
Publisher

Abstract

Generalized Holstein–Primakoff realizations deduced by Deenen, Quesne, and Papanicolaou are obtained directly from the algebraic identities satisfied in collective subspaces by the infinitesimal generators of the corresponding dynamical groups.
DOI
10.1063/1.527202

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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 Schrödinger 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 ordinary Hilbert space of collective states and the Barut Hilbert space and coherent collective states are defined in the latter. The second is carried out in terms of one boson creation and one boson annihilation operator through a generalized Holstein–Primakoff representation. The generator of a U(1) group, which is the one-dimensional analog of the U(6) group of the interacting boson model (IBM), can then be expressed in terms of the generators of 𝒮𝓅c(2,R). Finally the generalization of the preceding analysis to a d-dimensional space is outlined in the cases where d = 2 or 3. The dynamical group of collective states becomes 𝒮𝓅c(2d,R).