Operator representations on quantum spaces

In this article we present explicit formulae for q-differentiation on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. The calculations are based on the covariant differential calculus of these quantum spaces. Furthermore, our formulae can be regarded as a generalization of Jacksons q-derivative to three and four dimensions.Received: 26 September 2002, Revised: 18 June 2003, Published online: 2 October 2003     

<Emphasis Type="Italic">q</Emphasis>-Exponentials on quantum spaces

We present explicit formulae for q-exponentials on quantum spaces which could be of particular importance in physics, i.e. the q-deformed Minkowski space and the q-deformed Euclidean space with two, three or four dimensions. Furthermore, these formulae can be viewed as 2-, 3- or 4-dimensional analogues of the well-known q-exponential function.Received: 21 January 2004, Revised: 19 May 2004, Published online: 7 September 2004     

*-products on quantum spaces

In this paper we present explicit formulas for the *-product on quantum spaces which are of particular importance in physics, i.e., the q-deformed Minkowski space and the q-deformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of up to second order, for all considered cases. Received: 6 June 2001 / Published online: 15 March 2002     

q-deformed conformal and Poincaré algebras on quantum 4-spinors

We investigate quantum deformation of conformal algebras by constructing the quantum space forsl _q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformedsu(2,2) algebra from the deformedsl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector inso _q (4,2) is constructed as a tensor product of two sets of 4-spinors. We obtain theq-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincaré algebra is derived through a contraction procedure.Work partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#030083)     

Structure of the three-dimensional quantum euclidean space

As an example of a noncommutative space we discuss the quantum 3-dimensional Euclidean space together with its symmetry structure in great detail. The algebraic structure and the representation theory are clarified and discrete spectra for the coordinates are found. The q-deformed Legendre functions play a special role. A completeness relation is derived for these functions. Received: 13 April 2000 / Published online: 18 May 2000     

<Emphasis Type="Italic">q</Emphasis>-Integration on quantum spaces

In this article we present explicit formulae for q-integration on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. Furthermore, our formulae can be regarded as a generalization of Jacksons q-integral to three or four dimensions and provide a new possibility for an integration over the whole space being invariant under translations and rotations.Received: 9 September 2003, Published online: 26 November 2003     

Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

Quantum group gauge theory on quantum spaces

We construct quantum group-valued canonical connections on quantum homogeneous spaces, including aq-deformed Dirac monopole on the quantum sphere of Podles with quantum differential structure coming from the 3D calculus of Woronowicz onSU _q (2). The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fibre, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces).Supported by St. John's College, Cambridge and KBN grant 202189101     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property.     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.     

Perturbation foundation of <Emphasis Type="Italic">q</Emphasis>-deformed dynamics

In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians; one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved. The first is Equivalence Theorem I: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. The general Equivalence Theorem II is: for any potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied. Received: 6 September 2002 / Revised version: 21 October 2002 / Published online: 14 April 2003 RID="a" ID="a" e-mail: jzzhang@physik.uni-kl.de, jzzhang@ecust.edu.cn     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed. Special representations of the translation part are investigated.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

Deformed conformal and super-Poincaré symmetries in the non- (anti-) commutative spaces

Generators of the super-Poincaré algebra in the non- (anti-) commutative superspace are represented using appropriate higher derivative operators defined in this quantum superspace. Also discussed are the analogous representations of the conformal and superconformal symmetry generators in the deformed spaces. This construction is obtained by generalizing the recent work of Wess et al. on the Poincaré generators in the θ-deformed Minkowski space, or by using the substitution rules we derived on the basis of the phase-space structures of non- (anti-) commutative-space variables. Even with the non-zero deformation parameters the algebras remain unchanged although the comultiplication rules are deformed. The transformation of the fields under deformed symmetry is also discussed. Our construction can be used for systematic development of field theories in the deformed spaces.     

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     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard su _q (2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the u _q (2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0⁺ states in the deformed and nondeformed cases.     

<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.