137 research outputs found
Realizations for Kepler and Oscillator Potentials and q-Canonical Transformations
The realizations of the Lie algebra corresponding to the dynamical symmetry
group SO(2,1) of the Kepler and oscillator potentials are q-deformed. The
q-canonical transformation connecting two realizations is given and a general
definition for q-canonical transformation is deduced. q-Schr\"{o}dinger
equation for a Kepler like potential is obtained from the q-oscillator
Schr\"{o}dinger equation. Energy spectrum and the ground state wave function
are calculated.Comment: 12 pages, Latex twice, (Comparison with the other approaches and some
refs. added. The version which will appear in J. Phys. A
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
Time quantization and q-deformations
We extend to quantum mechanics the technique of stochastic subordination, by
means of which one can express any semi-martingale as a time-changed Brownian
motion. As examples, we considered two versions of the q-deformed Harmonic
oscillator in both ordinary and imaginary time and show how these various cases
can be understood as different patterns of time quantization rules.Comment: 4 pages, 2 figure
Finite Number of States, de Sitter Space and Quantum Groups at Roots of Unity
This paper explores the use of a deformation by a root of unity as a tool to
build models with a finite number of states for applications to quantum
gravity. The initial motivation for this work was cosmological breaking of
supersymmetry. We explain why the project was unsuccessful. What is left are
some observations on supersymmetry for q-bosons, an analogy between black holes
in de Sitter and properties of quantum groups, and an observation on a
noncommutative quantum mechanics model with two degrees of freedom, depending
on one parameter. When this parameter is positive, the spectrum has a finite
number of states; when it is negative or zero, the spectrum has an infinite
number of states. This exhibits a desirable feature of quantum physics in de
Sitter space, albeit in a very simple, non-gravitational context.Comment: 25 pages, 5 figure
Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters
We present a systematic technique to construct solutions to the Yang-Baxter
equation which depend not only on a spectral parameter but in addition on
further continuous parameters. These extra parameters enter the Yang-Baxter
equation in a similar way to the spectral parameter but in a non-additive form.
We exploit the fact that quantum non-compact algebras such as
and type-I quantum superalgebras such as and are
known to admit non-trivial one-parameter families of infinite-dimensional and
finite dimensional irreps, respectively, even for generic . We develop a
technique for constructing the corresponding spectral-dependent R-matrices. As
examples we work out the the -matrices for the three quantum algebras
mentioned above in certain representations.Comment: 13 page
Unified -deformation of one-parametric q-deformed oscillator algebras
We define a generalized -deformed oscillator
algebra and study the number of its characteristics. We describe the structure
function of deformation, analyze the classification of irreducible
representations and discuss the asymptotic spectrum behaviour of the
Hamiltonian. For a special choice of the deformation parameters we construct
the deformed oscillator with discrete spectrum of its "quantized coordinate"
operator. We establish its connection with the (generalized) discrete Hermite I
polynomials
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