Dyson Realization Research Articles

Single boson realizations of the deformed angular momentum algebra of Witten’s types

By generalizing the Holstein-Primakoff realization and the Dyson realization of the Lie algebra SU(2), various realizations of the deformed angular momentum algebra $$ \mathcal{R}_{q,s}^{c_0 ,c_1 ,c_2 } $$ , a five-parameter deformed SU(2) by combining Witten’s two deformation schemes, are investigated in terms of the single boson and the single inversion boson respectively. For each kind, the unitary realization, the non-unitary realization, and their connection by the corresponding similarity transformation are respectively discussed.

Indecomposable representations and boson realizations of the nonlinear deformed angular momentum algebra of Witten’s first type

In this paper indecomposable representations and boson realizations of the nonlinear angular momentum algebra \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) of Witten’s first type are investigated in a purely algebraic manner. Explicit form of the master representation of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) on the space of its universal enveloping algebra is given. Then, from this master representation, other indecomposable representations are obtained in explicit form. Various kinds of single-boson, single inverse boson, and double-boson realizations of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) are respectively obtained by generalizing the Holstein–Primakoff realization, the Dyson realization, and the Jordan–Schwinger realization of the Lie algebras SU(2) and SU(1,1). For each kind, the unitary realization, the nonunitary realization, and their connection by the corresponding similarity transformation are respectively discussed. Using a kind of double-boson realizations, the irreducible representation of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) in the angular momentum basis is given.

A Holstein-Primakoff and a Dyson realization for the quantum algebra

The known Holstein-Primakoff and Dyson realizations of the Lie algebra $sl(n+1), n=1,2,...$ in terms of Bose operators (Okubo S 1975 J. Math. Phys. 16 528) are generalized to the class of the quantum algebras $U_q[sl(n+1)]$ for any $n$. It is shown how the elements of $U_q[sl(n+1)]$ can be expressed via $n$ pairs of Bose creation and annihilation operators.

The Holstein - Primakoff and Dyson realizations for the Lie superalgebragl(m/n+ 1)

The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are the same as for $gl(m+n+1)$, however both Bose and Fermi operators are involved.

Holstein–Primakoff Realizations on Coadjoint Orbits

We derive the Holstein–Primakoff oscillator realization on the coadjoint orbits of the SU (N+1) and SU (1,N) group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein–Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed.

An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1), however in the latter, Bose operators have to be replaced with Fermi operators.

Three-level Lipkin model in the context of the suq(3) algebra.

In this work we study the quantum deformation of the three-level Lipkin model. The exact spectra are obtained. The reliability of the mean-field approximation is discussed in the framework of the model. Using adequate coherent states, the mean-field ground-state and the random-phase-approximation modes are calculated. The Dyson realization of ${\mathrm{su}}_{\mathit{q}}$(3) in the Bargmann representation is derived. The effect of the quantum deformation on the ground-state energy and on the phase transitions of the model is discussed. The simulation of temperature effects by the quantum deformation is investigated.

The Dyson Realization of the Quantum Group SU(1,1)q and Its Coherent state11The project supported by Natural Science Foundation of Henan Normal University for Young Researchers and Institute of Fundamental Science and Application of Henan Province.

The Dyson realization of the quantum group SU(1,1)q is constructed and the transformation matrix, which turns the Dyson realization into the Holstein–Primakoff realization, is given. The coherent state for the Dyson realization is discussed and is found to be the q-deformed Weyl–Heisenberg coherent state.

Some realization of the quantum algebra Uq(su(2))

For the quantum algebra Uq(su(2)), a q-analogue of the usual spin coherent state is constructed. With help of coherent states the q-deformed Dyson and Holstein-Primakoff realizations of the quantum algebra Uq(su(2)) are given. A transformation matrix, which turns the Dyson mapping onto the Holstein-Primakoff, is presented.

Boson operator realizations of su(2) and su(1,1) and unitarization

Boson operator realizations of su(2) and su(1,1) are obtained. Scalar products are introduced on ‘‘Fock spaces’’ (Verma modules) spanned by generators of the Heisenberg algebra H and by generators of su(2). These scalar products unitarize certain of the representations of H, or of su(1,1). It is shown that the Gel’fand–Dyson realization of su(1,1) implies a scalar product that unitarizes H, while the Primakoff–Holstein realizations imply a scalar product that unitarizes su(1,1). The relationship between the Gel’fand–Dyson boson operators a° and the Primakoff–Holstein boson operators b° is obtained making use of the two distinct scalar products. Generalized ‘‘vacuum states’’ are defined that are formed by polynomials in the creation–annihilation operator pairs a°a. A representation ρ of H and su(1,1) on the states a°m and an is discussed. For this representation a1=a‖0〉≠0, but rather (a°a)‖0〉 =0. The states of this representation space consist of boson states and boson-hole states. All the familiar results of H and su(1,1) representation theory are preserved within the representation ρ.

On the boson–quasifermion realization of the particle–hole SO(2Ω+1) algebra

The shell-model algebra SO(2Ω+1) generated by all bilinear and linear combinations of fermion creation and annihilation operators acting on a Fock space of Ω orbitals may be mapped into an ‘‘ideal space’’ in which particle–hole pairs are described by boson operators, while excess particles or holes are described by fermionlike degrees of freedom called ‘‘quasifermions.’’ After a review of the derivation of the nonunitary generalized Dyson realization, it is shown how this realization can be unitarized in a very simple way with the help of a recently developed technique that utilizes the Casimir invariants of a ‘‘core subalgebra.’’

DYNAMICAL SU(m)-SYMMETRIC AND SU(m/n)-SUPERSYMMETRIC LIMITS OF THE QUASIPARTICLE-PHONON NUCLEAR MODEL

It is shown that under some conditions the operators of the Quasiparticle Phonon Nuclear Model (QPM) and their commutators close the SU (m) algebra and the graded SU (m/n) algebra, respectively, for even-even and odd-even nuclei. The SU (m/n) generators and their matrix representation are then derived from the QPM operators. The generalized Dyson realization for the QPM operators is obtained. Microscopic expressions for the parameters of the IBM-1 and IBFM-1 Hamiltonians are derived.

Deductive inference of nuclear boson models (IBM, TQM, FQM) based on SU(6) dynamical symmetry

A single deductive inference of Schwinger realization (= interacting boson model—IBM), Holstein-Primakoff realization (= truncated quadrupole phonon model—TQM) and Dyson realization (= finite quadrupole phonon model—FQM) of dynamical SU(6) quadrupole collective algebra (QCA) is presented with a full scope of their isomorphism on the level of representations. Dyson realization of QCA is explicitly constructed by using holomorphically parametrized generalized coherent state and explicit form of root vectors. Utilizing appropriate orthogonalizing operators Holstein-Primakoff realization of QCA has been derived from the Dyson realization. The carrier spaces of Schwinger and Holstein-Primakoff realizations are investigated on the same footing and Marshalek's boson is rigorously derived. The intertwining operator which connects Schwinger and Holstein-Primakoff realizations is constructed and its domain and image are determined. It is shown that the intertwining operator has well-defined inverse in a definite factor space of the IBM basis space which is proved to be isomorphic to the physical subspace of the TQM basis space, meaning equivalence of IBM and TQM on level of representations.

Boson realization of sp(4). I. The matrix formulation

Holstein and Primakoff derived long ago the boson realization of a su(2) Lie algebra for an arbitrary irreducible representation (irrep) of the SU(2) group. The corresponding result for su(1,1)≅sp(2) is also well known. This raises the question of whether it is possible to obtain in an explicit, analytic, and closed form, and for any integer d, the boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the Sp(2d) group, which is a problem of considerable physical interest. The case d=2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp(4). The Dyson realization is well known, and the passage to bosons satisfying the appropriate Hermiticity conditions can be done by a similarity transformation through an operator K. What we want, though, is an explicit boson realization for sp(2d) similar to the one that exists for sp(2). In Sec. VI we show how we can get it for sp(4) if the operator K is known. Unfortunately while the matrix form of K2 can be explicitly derived from definite recursion relations, the same cannot be said of K as it involves, in general, the solution of algebraic equations of high degree. Thus the conclusion, corroborated also by a classical analysis where K does not appear, is that an explicit, analytic, and closed boson realization of sp(4), and thus also of sp(2d), is only possible for particular irreps of the corresponding groups.