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arXiv:1107.4297

Two-fermion composite quasi-bosons and deformed oscillators

Alexandre M. Gavrilik, І. І. Качурик, Yu. A. Mishchenko

Published Jul 21, 2011• 2 citations
arXivHTMLPDF

Abstract

The concept of quasi-bosons or composite bosons (like mesons, excitons etc.) has a wide range of potential physical applications. Even composed of two pure fermions, the quasi-boson creation and annihilation operators satisfy non-standard commutation relations. It is natural to try to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator, the latter constituting widely studied field of modern quantum physics. In this paper, it is proved that the deformed oscillators which realize quasi-boson operators in a consistent way really exist. The conditions for such realization are derived, and the uniqueness of the family of deformations under consideration is shown.

DOI
10.48550/arxiv.1107.4297

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The concept of quasi-bosons or composite bosons (like mesons, excitons etc.) has a wide range of potential physical applications. Even composed of two pure fermions, the quasi-boson creation and annihilation operators satisfy non-standard commutation relations. It is natural to try to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator, the latter constituting widely studied field of modern quantum physics. In this paper, it is proved that the deformed oscillators which realize quasi-boson operators in a consistent way really exist. The conditions for such realization are derived, and the uniqueness of the family of deformations under consideration is shown.
Composite bosons, here called quasibosons (e.g. mesons, excitons, etc), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the 'fermion+fermion' composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q -fermions ( q -deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is a quadratic polynomial in the number operator. In the case of two q -fermions, q ≠ 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.
Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, п-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of composite Fermi-type particles, and explore them in two major cases: (i) “boson + fermion” composite fermions (or cofermions, or CFs); (ii) “deformed boson + fermion” CFs. As we show, cofermions in both cases admit only the realization by ordinary fermions. Case (i) is solved explicitly, and admissible wavefunctions are found along with entanglement measures. Case (ii) is treated within few modes both for CFs and constituents. The entanglement entropy and purity of CFs are obtained via the relevant parameters and illustrated graphically.
Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates are computed giving a new class of deformed coherent states.
When a boson interacts with another to form a composite system with SO(3) dynamic symmetry, it is shown that there exists the q-deformed bosonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO(3) is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO(3) Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose–Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.
A new fundamental theory of composite particles in nonrelativistic many-body systems is developed. The theory is constructed making use solely of the physical Fock space for the basic elementary particles which comprise the many-body system. In this manner exchange symmetry in the elementary particles is exact. Physical composite particle creation and annihilation operators are introduced and these operators satisfy exact Bose (Fermi) commutation (anticommutati$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{o}}$n) relations depending on whether the composites correspond to bosons or to fermions. Commuting physical occupation number operators for composite particles are then constructed in the usual manner from the composite creation and annihilation operators. These number operators are highly dressed in terms of the basic elementary particle operators. Finally, the foundations of statistical mechanics for bound composite particles are formulated in terms of the appropriate occupation number operators.