Maps Between Deformed and Ordinary Gauge Fields

In this paper, we introduce a map between the q-deformed gauge fields defined on the GL_q(N) -covariant quantum hyperplane and the ordinary gauge fields. Perturbative analysis of the q-deformed QED at the classical level is presented and gauge fixing à la BRST is discussed. Another star product defined on the hybrid (q,h) -plane is explicitly constructed.     

q-deformed probability and binomial distribution

We introduce the concept ofq-deformed probability and discuss theq-deformed binomial distribution.     

On the dynamics of the q-deformed logistic map

We analyze the q-deformed logistic map, where the q-deformation follows the scheme inspired in the Tsallis q-exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior.     

Gravitationally induced particle creation in aq-scalar field

Tom and Goodison [5] have shown that for generic values ofq, gravitationally induced particle creation is impossible in the ordinary vacuum state. Here we consider the evolution of aq-deformed scalar field in a curved spacetime and observe that if the field is either represented by a coherent state or a squeezed state, there is a change in the energy density of the field indicating the possibility of particle creation.     

Differences Among Three Deformed Oscillators

The differences among quon operators, q _a-math oscillator operators and q-deformed oscillator operators are pointed out. The q-deformed ocsillator and q _a-math oscillator are constructed in terms of q _q = 0 quon.     

Spectral inverse problem for q-deformed harmonic oscillator

The supersymmetric quantization condition is used to study the wave functions of SWKB equivalent q-deformed harmonic oscillator which are obtained by using only the knowledge of bound-state spectra of q-deformed harmonic oscillator. We have also studied the nonuniqueness of the obtained interactions by this spectral inverse method.     

Contraction rule ofq-deformed Levi-Civita symbol

In this paper we obtain theq-analog of the contraction rule of theq-deformed Levi-Civita symbol and prove it. We use this to present the simplest example ofq-vector formula for theq-outer product.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

A new extended q-deformed KP hierarchy

Abstract

A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy with self-consistent sources and the constrained q-deformed KP hierarchy, which include two types of q-deformed KdV equation with sources and two types of q-deformed Boussinesq equation with sources. All of these results reduce to the classical ones when q goes to 1. This provides a general way to construct (2+1)- and (1+1)-dimensional q-deformed soliton equations with sources and their Lax representations.     

Irreducibility and Compositeness in q-Deformed Harmonic Oscillator Algebras

A study of the reducibility of the Fock space representation of the q-deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is carried out by using the properties of the Gauss polynomials. When the deformation parameter is a root of unity, an interesting result comes out in the form of a reducibility scheme for the space representation which is based on the classification of the primitive or nonprimitive character of the deformation parameter. An application is carried out for a q-deformed harmonic oscillator Hamiltonian, to which the reducibility scheme is explicitly applied.On leave from     

Perturbative aspects of q-deformed dynamics

Within the framework of the q-deformed Heisenberg algebra a dynamical equation of q-deformed quantum mechanics is discussed. The perturbative aspects of the q-deformed Schr?dinger equation are analyzed. General representations of the additional momentum-dependent interaction originating from the q-deformed effects are presented in two approaches. As examples, such additional interactions related to the harmonic-oscillator potential and the Morse potential are demonstrated. Received: 26 February 2001 / Published online: 11 May 2001     

Polynomial relations in the centre ofU q (sl(N))

When the parameter of deformationq is a root of unity, the centre ofU _q (sl(N)) contains, besides the usualq-deformed Casimirs, a set of new generators, which are basically themth powers of all the Cartan generators ofU _q (sl(N)). All these central elements are, however, not independent. In this Letter, generalizing the well-known case ofU _q (sl(2)), we explicitly write polynomial relations satisfied by the generators of the centre. Application to the parametrization of irreducible representations and to fusion rules are sketched.On leave from SPht, CE Saclay, 91191 Gif-sur-Yvette Cedex, France.     

Coherent states and squeezed states of realq-deformed quantum oscillators

A detailed physical characterisation of the coherent states and squeezed states of a realq-deformed oscillator is attempted. The squeezing andq-squeezing behaviours are illustrated by three different model Hamiltonians, namely i) Batemann Hamiltonian ii) harmonic oscillator with time dependent mass and frequency and iii) a system with constant mass and time-dependent frequency.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Properties of the q-Analogue of a Squeezed Vacuum State

We constructed a normalizable q-analogue of squeezed vacuum state using the technique of integration within an ordered product (IWOP) of operators and the properties of the inverses of q-deformed creation and annihilation operatots. We also study its nonclassical properties and phase probability distribution.     

Miura and auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies

The Miura and anti-Miura transformations between the q-deformed KP and the q-deformed modified KP hierarchies are investigated in this paper. Then the auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies are constructed through the combinations of the Miura and anti-Miura transformations. And the corresponding results are also generalized to the constrained cases. At last, some examples of Miura and auto-Backlund transformations are given.     

Metric on quantum spaces

We introduce the analogue of the metric tensor in the case ofq-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail examples of the Manin plane and theq-deformation of SU(2). Finally we touch the topic of relations with Connes' approach.Partially supported by KBN grant 2P 302 168 4.     

On q-Deformed {\mathfrak{gl}_{\ell+1}} -Whittaker Function I

We propose a new explicit form of q-deformed Whittaker functions solving q-deformed ${\mathfrak{gl}_{\ell+1}}A representation of a specialization of a q-deformed class one lattice \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(\mathbbP^l){\mathcal{QM}_d(\mathbb{P}^{\ell})} of quasi-maps \mathbbP¹ ? \mathbbP^l{\mathbb{P}^1 \to \mathbb{P}^{\ell}} of degree d is proposed. For ℓ = 1, this provides an interpretation of the non-specialized q-deformed \mathfrakgl₂{\mathfrak{gl}_{2}}-Whittaker function in terms of QM_d(\mathbbP¹){\mathcal{QM}_d(\mathbb{P}^1)}. In particular the (q-version of the) Mellin-Barnes representation of the \mathfrakgl₂{\mathfrak{gl}_2}-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed \mathfrakgl₂{\mathfrak{gl}_2}-Toda chain is also discussed.     

Reflection equations andq-Minkowski space algebras

We express the defining relations of theq-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect to the quantum Lorentz group action in a straightforward way. Different equivalences ofq-Minkowski algebras are pointed out.On leave of absence from the St. Petersburg's Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, Spain.     

<Emphasis Type="Italic">q</Emphasis>-plane-wave solutions of <Emphasis Type="Italic">q</Emphasis>-Weyl gravity

The solutions of the q-deformed equations of quantum conformal Weyl gravity in terms of q-deformed plane waves are given. The text was submitted by the authors in English.