Marijan Milekovic

A UNIFIED VIEW OF MULTIMODE ALGEBRAS WITH FOCK-LIKE REPRESENTATIONS

A unified view of general multimode oscillator algebras with Fock-like representations is presented. It extends a previous analysis of the single-mode oscillator algebras. The expansion of the operators is extended to include all normally ordered terms in creation and annihilation operators, and we analyze their action on Fock-like states. We restrict ourselves to the algebras compatible with number operators. The connection between these algebras and generalized statistics is analyzed. We demonstrate our approach by considering the algebras obtainable from the generalized Jordan-Wigner transformation, the para-Bose and para-Fermi algebras, the Govorkov “paraquantization” algebra and generalized quon algebra.

A Multispecies Calogero Model

We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.

On the Clebsch-Gordan coefficients for the two-parameter quantum algebra SU(2)p,q

We show that the Clebsch-Gordan coefficients for the SU(2)p,q algebra depend on a single parameter Q= square root pq, contrary to the explicit calculation of Smirnov and Wehrhahn (1992).

EXCLUSION STATISTICS, OPERATOR ALGEBRAS AND FOCK SPACE REPRESENTATIONS

We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Greens realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio. We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space (e.g. the Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single-oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Greens realization of para-Fermi statistics). For these types of exclusions we discuss the extended Haldane statistics parameters g, recently introduced by two of us in 1996 ( Mod. Phys. Lett. A 11 3081), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.

On parastatistics defined as triple operator algebras

Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators. They are expressed as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubos and Palevs parastatistics connected to many-body Wigner quantum systems and relate them to the notion of extended Haldane statistics.We unify parastatistics, defined as triple operator algebras represented on Fock space, in a simple way using the transition number operators. We express them as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubos and Palevs parastatistics connected to many-body Wigner quantum systems. We relate them to the notion of extended Haldane statistics.

INTERPOLATION BETWEEN PARA-BOSE AND PARA-FERMI STATISTICS

Abstract Using deformed Greens oscillators and Greens ansatz, we construct a multiparameter interpolation between para-Bose and para-Fermi statistics of a given order. When the interpolating parameters q ij satisfy | q ij | q ij | = 1), the interpolation statistics is “infinite quon”-like (anyon-like). The proposed interpolation does not contain states of negative norms.

PARASTATISTICS AS EXAMPLES OF THE EXTENDED HALDANE STATISTICS

We show that for every algebra of creation and annihilation operators with a Fock-like representation, one can define extended Haldane statistical parameters in a unique way. In particular for parastatistics, we calculate extended Haldane parameters and discuss the corresponding partition functions.

Calogero Model(s) and Deformed Oscillators

We briefly review some recent results concerning algebraical (oscillator) as- pects of the N-body single-species and multispecies Calogero models in one dimension. We show how these models emerge from the matrix generalization of the harmonic oscillator Hamiltonian. We make some comments on the solvability of these models.

Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D ≥ 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.

Covariant-tensor method for quantum groups and applications I. SU(2)q

A covariant-tensor method for SU(2)q is described. This tensor method is used to calculate q-deformed Clebsch-Gordan coefficients. The connection with covariant oscillators and irreducible tensor operators is established. This approach can be extended to other quantum groups.