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Physical Review C

Vector coherent state theory in new perspective: A hybrid mapping from fermion to fermion-boson space

Jin-Quan Chen, K. T. Hecht, D.H. Feng

Published Jan 1, 1991• 2 citations

Abstract

Based on a hybrid mapping, the vector coherent state theory is revisited. The hybrid mapping of a fermion state with N coherent (collective) fermion pairs and u unpaired (noncollective) fermions leads to a hybrid state with N bosons and u fermions. The hybrid mapping is exact by which the formidable problem of finding the matrix elements of a fermion operator for a (2N+u)-fermion system is reduced to that for a u-fermion system. The Pauli effects are totally taken into account by the overlap matrix (the K matrix) for the fermion states. The application to a fermion system with the Sp(6)\ensuremath{\supset}U(3) dynamical symmetry is discussed where it is vividly seen that the occurrence of the dynamical Pauli factors is related to the fermion pair\ensuremath{\leftrightarrows}boson transformation. The intricate properties of the intrinsic operators and the K operator are discussed in detail.

DOI: 10.1103/physrevc.43.82

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Based on a hybrid mapping, the vector coherent state theory is revisited. The hybrid mapping of a fermion state with N coherent (collective) fermion pairs and u unpaired (noncollective) fermions leads to a hybrid state with N bosons and u fermions. The hybrid mapping is exact by which the formidable problem of finding the matrix elements of a fermion operator for a (2N+u)-fermion system is reduced to that for a u-fermion system. The Pauli effects are totally taken into account by the overlap matrix (the K matrix) for the fermion states. The application to a fermion system with the Sp(6)\ensuremath{\supset}U(3) dynamical symmetry is discussed where it is vividly seen that the occurrence of the dynamical Pauli factors is related to the fermion pair\ensuremath{\leftrightarrows}boson transformation. The intricate properties of the intrinsic operators and the K operator are discussed in detail.

I derive in a novel fashion exact formulas for the calculation of general matrix elements, including the overlap (norm) matrix, between states constructed from fermion pairs. Mapping the fermion pairs to bosons, I show how to construct finite and exact (in the sense of preserving matrix elements) boson representations of the norm operator and one- and two-fermion operators. This may lead to a microscopic basis for the Interacting Boson Model, as well as new truncation schemes for the nuclear shell model.

After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians, including effective operator theory to account for excluded states. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the Interacting Boson Model, but is more rigorous, general, and systematic.

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 introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include the construction of vector coherent state realizations of operators other than the group generators. The group U(3)⊇ U(2)×U(1) is used as a prototype. The construction of irreducible tensor operators with specific shift properties is illustrated with a number of examples. These show how the Wigner calculus for a higher symmetry group can be expressed solely in terms of the recoupling coefficients of the core subgroup and the simple K‐matrix elements of vector coherent state theory.

After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the interacting boson model, but is more rigorous, general, and systematic.