Obstruction Results in Quantization Theory

We define the quantization structures for Poisson algebras necessary to generalise Groenewold and Van Hove's result that there is no consistent quantization for the Poisson algebra of Euclidean phase space. Recently a similar obstruction was obtained for the sphere, though surprising enough there is no obstruction to the quantization of the torus. In this paper we want to analyze the circumstances under which such obstructions appear. In this context we review the known results for the Poisson algebras of Euclidean space, the sphere and the torus.Comment: 34 pages, Latex. To appear in J. Nonlinear Scienc

Three dimensional quadratic algebras: Some realizations and representations

Four classes of three dimensional quadratic algebras of the type \lsb Q_0 , Q_\pm \rsb $=$ $\pm Q_\pm$, \lsb Q_+ , Q_- \rsb $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of $su(2)$ and $su(1,1)$. The resulting matrix representations and single variable differential operator realizations are obtained. Some remarks on the mathematical and physical relevance of such algebras are given.Comment: LaTeX2e, 23 pages, to appear in J. Phys. A: Math. Ge

New q-deformed coherent states with an explicitly known resolution of unity

We construct a new family of q-deformed coherent states $|z>_q$, where $0 < q < 1$. These states are normalizable on the whole complex plane and continuous in their label $z$. They allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem and expressed in terms of one of the two Jacksons's $q$-exponentials. They also permit exact evaluation of matrix elements of physically-relevant operators. We use this to show that the photon number statistics for the states is sub-Poissonian and that they exhibit quadrature squeezing as well as an enhanced signal-to-quantum noise ratio over the conventional coherent state value. Finally, we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics.Comment: LaTeX, 26 pages, contains 9 eps figure

Generally Deformed Oscillator, Isospectral Oscillator System and Hermitian Phase Operator

The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q->1.Comment: 21 pages, latex, no figure

Deformed oscillator algebras for two dimensional quantum superintegrable systems

Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.Comment: 22 pages, THES-TP 10/93, hep-the/yymmnn

The quantum superalgebra Uq[osp(1/2n)]U_q[osp(1/2n)]: deformed para-Bose operators and root of unity representations

We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are pointed out. Finally, using a realization in terms of ``$q$-commuting'' $q$-bosons, we construct an irreducible finite-dimensional unitary Fock representation of $U_q[osp(1/2n)]$ and its decomposition in terms of $U_q[gl(n)]$ representations when $q$ is a root of unity.Comment: 15 pages, LaTeX (latex twice), no figure

On boson algebras as Hopf algebras

Certain types of generalized undeformed and deformed boson algebras which admit a Hopf algebra structure are introduced, together with their Fock-type representations and their corresponding $R$-matrices. It is also shown that a class of generalized Heisenberg algebras including those algebras including those underlying physical models such as that of Calogero-Sutherland, is isomorphic with one of the types of boson algebra proposed, and can be formulated as a Hopf algebra.Comment: LaTex, 18 page

Boson-Fermion Models for Osp(1,2) and Uq[Osp(1,2)]

A reducible representation of the simple Lie superalgebra osp(1,2) is constructed from two pairs of boson and one pair of fermion creation and annihilation operators. The representation contains in direct sum every distinct (up to equivalence), finite-dimensional, irreducible representation of osp(1,2) exactly once and so defines a model, generalizing to osp(1,2) Schwinger's boson model of su(2). The model of osp(1,2) is further generalized to a model of the quantum superalgebra U(q)[osp(1,2)] in the case that q is not a root of unity. Matrix elements of generators and basic properties of the R matrix, known from previous studies, are rederived using the boson-fermion calculus