R Le Blanc

The symplectic shell-model theory of collective states

Abstract The implications of the microscopic structure of collective Lie algebras and their hydrodynamic (contraction) limits are explored. A decomposition of the nuclear Hubert space into collective subspaces and symplectic shells is proposed. The resultant symplectic shell model is shown to provide a basis for microscopic calculations which are immediately interpretable in collective-model terms.

A rotor expansion of the su(3) Lie algebra

Vector coherent state (VCS) theory is used to give a rotor expansion of the su(3) Lie algebra in a way that parallels the boson expansions that have been made for other Lie algebras. The construction provides a systematic procedure for calculating Hermitian matrix elements of su(3) in an SO(3)-coupled basis and represents a new development in VCS theory and in the theory of induced representations.

An effective shell-model theory of collective states

Abstract A simple quadrupole-quadrupole model is used to illustrate how the coupling between different symplectic shells can be eliminated to obtain effective interactions and operators that act only within shells. The physical content of the model is transparent and relates to the calculation of core-polarization effects due to extra-core particles in the unified model. But, since the symplectic shell model is based on a group, rather than an independent-particle structure, it avoids the necessity of distinguishing core and extra-core particles or of introducing redundant collective variables and thus fully respects the Pauli principle.

A coupled rotor-vibrator model as the macroscopic limit of the microscopic symplectic model

Abstract A simple coupled rotor-vibrator model is presented as the macroscopic limit of the microscopic symplectic model which is valid for medium and heavy nuclei of moderately large deformation. It is shown that this model, with distinct rotor and vibrator degrees of freedom, leads to a new perspective on the microscopic structure of collective states. Furthermore, it reveals simple relationships between the irrotational flow model, the geometrical model, the unified model and the shell model. A possible connection with the IBM model is also suggested.

Highest weight representations for gl(m/n) and gl(m+n)

It is shown how vector coherent state (VCS) theory enlightens the simultaneous discussion of representation theory for classical Lie algebras and superalgebras and provides an optimal framework for the explanation of the noted similarities and dissimilarities of their representations. Reducibility, atypicality, and the positive‐definitiveness of inner products in VCS representation spaces are discussed. The discussion is exemplified through the parallel and explicit construction of highest weight ladder representations of the Lie algebra gl(m+n) and superalgebra gl(m/n) in gl(m)⊕gl(n) bases.

New perspective on the U(n) Wigner-Racah calculus. II. Elementary reduced Wigner coefficients for U(n)

For pt.I, see ibid., vol.20, no.9, p.2241-50 (1987). Exploiting the powerful formalism of the vector coherent state theory expounded in part I, the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of Biedenharn and Louck, the authors rederive in a simple fashion all U(n): U(n-1) reduced Wigner coefficients for the elementary U(n) tensors.

Superfield and matrix realizations of highest weight representations for osp(m/2n)

A differential representation of the classical Lie superalgebra osp(m/2n) acting on superfield functions is given. This representation is used to construct matrix representations of the finite‐dimensional irreducible representations of the algebra. Inner products on the irreducible spaces are discussed and classes of star and grade‐star equivalent representations are identified.

THE COUPLING OF VALENCE SHELL AND PARTICLE-HOLE DEGREES OF FREEDOM IN A PARTIAL RANDOM PHASE APPROXIMATION

It is well known that the random phase approximation breaks down in the absence of a substantial energy gap between occupied and unoccupied single-particle states. Particle-hole excitations are then inevitably accompanied by substantial rearrangements of the particles in the neighbourhood of the Fermi surface. To accommodate this situation, a partial RPA is introduced which corresponds to replacing only the particle-hole degrees of freedom by bosons but leaving the valence space degrees of freedom intact. The PRPA is therefore a mapping of the many-fermion dynamics into the dynamics of a coupled boson-valence space. In application of the PRPA, algebraic methods, of either a fermionic or Lie algebra type, can be introduced, if desired, to facilitate the treatment of the valence space degrees of freedom. Results of applications are presented in which the valence space particles are treated in the rotational and SU(3) models, and are coupled strongly to giant dipole and quadrupole resonances.

Screening and vertex operators for u(n)

Screening and vertex operators for u(n) are given in terms of generators of the group left action on the flag manifold U(n-1)/SU(n). Their matrix elements in a basis of holomorphic sections of a vector bundle are computed.

Canonical orthonormal basis for SU(3)⊃SO(3). I. Construction of the basis

A canonical orthonormal basis is given for generic representations of the group chain SU(3) contains/implies SO(3).