An algorithm for projecting a microscopic nuclear Hamiltonian with a central interaction onto the enveloping algebra of the Sp(6,R) group is presented in parallel with the development of a practical procedure for constructing the corresponding part of the Sp(6,R)⊇SO(3) integrity basis. The latter is shown to consist of the invariants of the monopole–quadrupole tensor and its polynomial analogs, coupling the collective and vortex spin degrees of freedom. The algorithm uses vector coherent-state techniques and is displayed on the basis of a generic Sp(6,R) irreducible representation. The collective Hamiltonian is also shown to be expandable in terms of the enveloping algebra of the (A−1)-dimensional rotational group, where A is the total number of nucleons.
An algorithm for projecting a microscopic nuclear Hamiltonian with a central interaction onto the enveloping algebra of the Sp(6,R) group is presented in parallel with the development of a practical procedure for constructing the corresponding part of the Sp(6,R)⊇SO(3) integrity basis. The latter is shown to consist of the invariants of the monopole–quadrupole tensor and its polynomial analogs, coupling the collective and vortex spin degrees of freedom. The algorithm uses vector coherent-state techniques and is displayed on the basis of a generic Sp(6,R) irreducible representation. The collective Hamiltonian is also shown to be expandable in terms of the enveloping algebra of the (A−1)-dimensional rotational group, where A is the total number of nucleons.
We present the symplectic no-core configuration interaction (SpNCCI) framework, in which the nuclear many-body problem is solved a symmetry-adapted basis that explicitly encodes approximate symmetries associated with nuclear collectivity and deformation. In this framework, calculations are carried out in a basis organized into Sp(3,R) irreducible representations (irreps), each of which can be expressed as an infinite tower of U(3) irreps. In this framework, matrices of realistic relative two-body operators, such as the nuclear Hamiltonian, are computed directly in the Sp(3,R) many-body basis, obviating the need to expand all Sp(3,R) many-body states in, e.g., a U(3)-coupled configuration basis. Instead, many-body matrix elements are obtained via a recurrence relation that expresses a given matrix element in terms of matrix elements between basis states with fewer oscillator quanta. To use this recurrence method for computing matrix elements of relative two-body operators, we must first expand each operator into components of U(3) tensors. To this end, we present a method for decomposing arbitrary operators into U(3) tensor components.
The generalised Holstein-Primakoff boson representation of the bifermion operators is used to construct a boson image of the microscopic nuclear Hamiltonian. The microscopic analogue of the interacting boson model (IBM) Hamiltonian is obtained. It is shown how the square-root factor in the IBM Hamiltonian can be generalised so as to take into account its dependence on the angular momentum and seniority of the collective state. The method is applicable to systems where the assumption that the projection operator P is approximately equal to unity can be used. The method is proposed to include, apart from the collective quadrupole, other collective and non-collective degrees of freedom in the Hamiltonian.
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL + (3,R) and the quotient set GL + (3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL + (3,R) collective model is provided by an irreducible representation of the boson SU(6) group.
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).
The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=A−1 relative Jacobi vectors. In this paper, we further study the Barut–Girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to 𝒮𝓅c(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to Holstein–Primakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their generating function properties, and the representations they lead to for both the collective states and their dynamical group.