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Physical Review C

Comment on ‘‘Skeleton boson realizations of collective subalgebras’’

R. Kuchta

Published Dec 1, 1989• 2 citations
Publisher

Abstract

The skeleton boson realizations of fermion collective subalgebras proposed by Kim and Vincent are discussed. The Dyson boson mapping is found to play an essential role in the possibility of such realizations. A simple but instructive example is given to show that a straightforward application of the ``skeletonization procedure'' to the Holstein-Primakoff boson mapping may lead to serious errors in reproducing the fermion matrix elements. A possible way of constructing the modified skeleton Holstein-Primakoff realization of the collective subalgebra is suggested.

DOI
10.1103/physrevc.40.2930

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Skeleton boson realizations of collective subalgebras

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For an even number of nucleons, we consider collective motion based on any dynamical symmetry subalgebra \^A of the valence-shell bifermion algebra; the generators of \^A are coherent linear combinations of bifermion operators of the types ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, and aa. A boson algebra is obtained by Dyson mapping the whole valence-shell bifermion algebra, and then ``skeletonizing'' the resulting expressions by deleting all noncollective boson operators from them. This skeleton boson algebra is shown to have the same commutation relations as the collective bifermion algebra, provided that the bifermion algebra is self-conjugate. The resulting boson Hamiltonian is not Hermitian, but does resemble the Hamiltonian of the interacting boson model in containing only one- and two-body terms. Moreover, the skeleton boson algebra (when restricted to a collective boson subspace that is free of spurious states) is shown to be physically equivalent to the collective bifermion algebra. It is shown that by working solely with collective bosons one can identify spurious boson states unambiguously, although the procedure lacks convenience. The similarity transformation that connects conjugate versions of the skeleton boson mapping is discussed, and a general condition for hermiticity of the boson image of a Hermitian operator is given.

Physical Review C•May 1, 1988•9 citations

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Recently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson–fermion and the baryon mapping. In the present paper, the case of a restriction to a subset of three-fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson–fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three-fermion subsystems carry collective indices. Examples of simple applications are presented in the two subsequent papers of this series.

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Osvaldo Civitarese, M. Reboiro

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Physical Review C•Jun 1, 1998•11 citations
The skeleton boson realizations of fermion collective subalgebras proposed by Kim and Vincent are discussed. The Dyson boson mapping is found to play an essential role in the possibility of such realizations. A simple but instructive example is given to show that a straightforward application of the ``skeletonization procedure'' to the Holstein-Primakoff boson mapping may lead to serious errors in reproducing the fermion matrix elements. A possible way of constructing the modified skeleton Holstein-Primakoff realization of the collective subalgebra is suggested.
For an even number of nucleons, we consider collective motion based on any dynamical symmetry subalgebra \^A of the valence-shell bifermion algebra; the generators of \^A are coherent linear combinations of bifermion operators of the types ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, and aa. A boson algebra is obtained by Dyson mapping the whole valence-shell bifermion algebra, and then ``skeletonizing'' the resulting expressions by deleting all noncollective boson operators from them. This skeleton boson algebra is shown to have the same commutation relations as the collective bifermion algebra, provided that the bifermion algebra is self-conjugate. The resulting boson Hamiltonian is not Hermitian, but does resemble the Hamiltonian of the interacting boson model in containing only one- and two-body terms. Moreover, the skeleton boson algebra (when restricted to a collective boson subspace that is free of spurious states) is shown to be physically equivalent to the collective bifermion algebra. It is shown that by working solely with collective bosons one can identify spurious boson states unambiguously, although the procedure lacks convenience. The similarity transformation that connects conjugate versions of the skeleton boson mapping is discussed, and a general condition for hermiticity of the boson image of a Hermitian operator is given.
Recently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson–fermion and the baryon mapping. In the present paper, the case of a restriction to a subset of three-fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson–fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three-fermion subsystems carry collective indices. Examples of simple applications are presented in the two subsequent papers of this series.
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 dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.