Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nucl-th/9303026 · Physical Review C

Boson-fermion mappings for odd systems from supercoherent states

J. Dobaczewski, Frederik G. Scholtz, H. B. Geyer

Published Nov 1, 1993• 9 citations
arXivHTMLPDF

Abstract

We extend the formalism whereby boson mappings can be derived from generalized coherent states to boson-fermion mappings for systems with an odd number of fermions. This is accomplished by constructing supercoherent states in terms of both complex and Grassmann variables. In addition to a known mapping for the full so(2N+1) algebra, we also uncover some other formal mappings, together with mappings relevant to collective subspaces.

DOI
10.1103/physrevc.48.2313

Related papers

Mapping of a classical transformation in Grassmann number space to a Fermi unitary operator

Fan Hong‐Yi

The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.

Physical Review A•Oct 1, 1989•19 citations

Generalized supercoherent states

Mohammed Daoud A, M. Kibler, Y. Hassouni

A set of operators, the so-called k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of an algebra arising from two non-commuting quon algebras. The deformation parameters q and 1/q for these quon algebras are roots of unity with q to the power k being equal to 1. The case k = 2 corresponds to fermions and the case k going to infinity to bosons. Generalized coherent states (connected to the k-fermionic states) and super-coherent states (involving a k-fermionic sector and a purely bosonic sector) are investigated.

PAN•Jan 1, 1997•0 citations

Dynamical and invariant supersymmetry in the fermion pairing problem

Pavel Cejnar, H. B. Geyer

We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersymmetries based on the dynamical algebra of the type $\mathrm{U}(K∕2\ensuremath{\Omega})$ (where $K$ and $2\ensuremath{\Omega}$ denote the number of fermion-pair and single-fermion states, respectively) naturally arise in the bosonized description of the system. Conditions for invariant supersymmetry are determined in a restricted case of bilinear supercharge operators.

arXiv:nucl-th/0306058•Physical Review C•Nov 25, 2003•3 citations

Boson-Fermion Expansions

Eugene R. Marshalek

Boson expansion theory is extended so as to allow for the coupling of several fermion-like valence particles or holes to the collective vibrations of a closed shell, described in terms of boson degrees of freedom. The result is a faithful mapping of the original many-fermion problem into a certain boson-fermion subspace without redundant variables. Generalizations of the Dyson and Holstein-Primakoff representations are derived with the aid of algebraic techniques.

Progress of Theoretical Physics Supplement•Jan 1, 1983•2 citations

Boson–fermion and baryon mapping: Construction of collective subspaces. I. Theory

Jutta Meyer

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.

Journal of Mathematical Physics•Oct 1, 1992•2 citations

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

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.

Physical Review C•Jan 1, 1991•2 citations
We extend the formalism whereby boson mappings can be derived from generalized coherent states to boson-fermion mappings for systems with an odd number of fermions. This is accomplished by constructing supercoherent states in terms of both complex and Grassmann variables. In addition to a known mapping for the full so(2N+1) algebra, we also uncover some other formal mappings, together with mappings relevant to collective subspaces.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.
A set of operators, the so-called k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of an algebra arising from two non-commuting quon algebras. The deformation parameters q and 1/q for these quon algebras are roots of unity with q to the power k being equal to 1. The case k = 2 corresponds to fermions and the case k going to infinity to bosons. Generalized coherent states (connected to the k-fermionic states) and super-coherent states (involving a k-fermionic sector and a purely bosonic sector) are investigated.
We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersymmetries based on the dynamical algebra of the type $\mathrm{U}(K∕2\ensuremath{\Omega})$ (where KKK and $2\ensuremath{\Omega}$ denote the number of fermion-pair and single-fermion states, respectively) naturally arise in the bosonized description of the system. Conditions for invariant supersymmetry are determined in a restricted case of bilinear supercharge operators.
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.
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.