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Journal of Physics A Mathematical and General

Boson representation of Fermi systems: nature and properties of subsidiary conditions

H. Barentzen

Published Jan 11, 1987• 2 citations
Publisher

Abstract

Based on Girardeau's composite-particle theory (1971) the PPP (extended Hubbard) Hamiltonian is represented in terms of ideal-boson operators. A major unsolved problem was to find (approximate) eigenstate of the Hamiltonian which, as the same time, satisfy certain subsidiary conditions. The later are necessary to ensure that the mapping of the fermion space into the space of boson states is unique. In the present paper a partial solution to the above mentioned problem is presented by using some arguments from the representation theory of the symmetric group. The results are of immediate practical relevance for all calculation dealing with the eigenvalue problem of the Hamiltonian in ideal-boson space.

DOI
10.1088/0305-4470/20/1/021

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Based on Girardeau's composite-particle theory (1971) the PPP (extended Hubbard) Hamiltonian is represented in terms of ideal-boson operators. A major unsolved problem was to find (approximate) eigenstate of the Hamiltonian which, as the same time, satisfy certain subsidiary conditions. The later are necessary to ensure that the mapping of the fermion space into the space of boson states is unique. In the present paper a partial solution to the above mentioned problem is presented by using some arguments from the representation theory of the symmetric group. The results are of immediate practical relevance for all calculation dealing with the eigenvalue problem of the Hamiltonian in ideal-boson space.
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.
A second-quantized representation for nonrelativistic systems with composite particles is given in which composite particles of two-fermion constituents are described as elementary bosons. The Tani transformation of Girardeau is used to introduce elementary composite fields. An equivalent linear equation of motion method—introducing a ’’d matrix’’—is used in place of the usual nonlinear equation of motion method. This and a diagram formalism are developed to calculate a Hamiltonian and subsidiary condition for this model system and to derive some properties of the Tani transformation.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
The formalism derived for the boson–fermion and the baryon mapping in Paper I of this series is applied to the single-orbit quark shell model developed by Petry and co-workers for the description of nuclear properties. The multiquark space is reduced to the collective subspace of multinucleon states. A collective representation of states and operators in terms of bosons and ideal fermions or in terms of ideal baryons, respectively, is constructed. The results of the original model in the multinucleon space are exactly reproduced. Finally, the two transformations are compared with a related mapping technique recently published by Pittel, Engel, Dukelsky, and Ring.