*-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     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Equivalence of deformations and associated *-products

In this letter we study the non-trivial formal differentiable deformations of the Lie algebraN=C (W, IR) whereW is a symplectic manifold. Under some assumptions (satisfied in particular forW=IR²ⁿ ) we show that these deformations are all equivalent, up to a monomial change of the parameter, to one of them (Moyal for IR²ⁿ ). Furthermore, if there exists a differentiable *-product corresponding to one of them, each of them is induced by a *-product which is essentially unique.Aspirant du Fonds National belge de la Recherche Scientifique.     

A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

Semiclassical limit for Chern-Simons theory on compact hyperbolic spaces

The invariant integration method for Chern-Simons theory defined on compact hyperbolic spaces of the form Γℍ³ is verified in the semiclassical approximation. The semiclassical limit for the partition function is calculated. The contribution to the sum over topologies in three-dimensional quantum gravity is briefly discussed. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 65–69 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.     

Abelian faces of state spaces ofC*-algebras

LetF be a closed face of the weak* compact convex state space of a unitalC*-algebraA. The class ofF-abelian states, introduced earlier by the author, is studied further. It is shown (without any restriction onA orF) thatF is a Choquet simplex if and only if every state inF isF-abelian, and that it is sufficient for this that every pure state inF isF-abelian. As a corollary, it is deduced that an arbitraryC*-dynamical system (A, G, ) isG-abelian if and only if every ergodic state is weakly clustering. Nevertheless the set of allF-abelian (or evenG-abelian) states is not necessarily weak* compact.     

Generation of GHz repeat rate hyperbolic secant nearly transform-limited pulses

The nearly transform-limited pulses are obtained by gain-switching techniquc and spectral window method. The pulse width are 20～26ps, the products of pulse width and spectral width are 0.32y～0.42, repetition rate can be as high as 4.0 GHz.     

On the implementability of non*Bogoliubov automorphisms of CAR algebras on fock spaces

We give necessary and sufficient conditions for the implementability of non^*Bogoliubov automorphisms of CAR algebras on Fock spaces. As an application, we show that certain (projective) representations of loop groups cannot be extended to bounded representations of their complexified groups.     

Topological vertex,string amplitudes and spectral functions of hyperbolic geometry

We study radiative corrections to massless quantum electrodynamics modified by two dimension-five LV interactions $\bar{\Psi } \gamma ^{\mu } b'^{\nu } F_{\mu \nu }\Psi $ and $\bar{\Psi }\gamma ^{\mu }b^{\nu } \tilde{F}_{\mu \nu } \Psi $ in the framework of effective field theories. All divergent one-particle-irreducible Feynman diagrams are calculated at one-loop order and several related issues are discussed. It is found that massless quantum electrodynamics modified by the interaction $\bar{\Psi } \gamma ^{\mu } b'^{\nu } F_{\mu \nu }\Psi $ alone is one-loop renormalizable and the result can be understood on the grounds of symmetry. In this context the one-loop Lorentz-violating beta function is derived and the corresponding running coefficients are obtained.     

The Green’s functions of the boundaries at infinity of the hyperbolic 3-manifolds

The work is motivated by a result of Manin in [1], which relates the Arakelov Green’s function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as a conformal boundary at infinity. A natural question is to what extent the result of Manin can be generalized to cases where, instead of dealing with a single Riemann surface, one has several Riemann surfaces whose union is the boundary of a hyperbolic 3-manifold, uniformized no longer by a Schottky group, but by a Fuchsian, quasi-Fuchsian, or more general Kleinian group. We have considered this question in this work and obtained several partial results that contribute towards constructing an analog of Manin’s result in this more general context.     

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and ? function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.     

Theq-difference operator,the quantum hyperplane,Hilbert spaces of analytic functions andq-oscillators

We show that the two most frequent expressions for the anomalous commutators can be both derived from quantities associated with the WZW model.     

Hard disks on the hyperbolic plane

We examine a simple hard disk fluid with no long range interactions on the two-dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is frustrated, allowing us to construct a tractable model of disordered monodisperse hard disks. We extend free-area theory and the virial expansion to this regime, deriving the equation of state for the system, and compare its predictions with simulation near an isostatic packing in the curved space.     

Distances in spaces of physical models: partition functions versus spectra

We study the relation between convergence of partition functions (seen as general Dirichlet series) and convergence of spectra and their multiplicities. We describe applications to convergence in physical models, e.g., related to topology change and averaging in cosmology.     

Quasi-elliptic and weakly coercive systems in Sobolev spaces of vector functions

The purpose of this paper is to investigate a connection between l-quasi-ellipticity and weak coercivity of systems acting on the spaces of vector functions Π (ℝ ⁿ ). Moreover, we extend some of our earlier results to the matrix case. We also show that an l-quasielliptic system in the sense of O. V. Besov is q-quasi-elliptic in the sense of L. R. Volevich. Dedicated to the blessed memory of L. R. Volevich, an outstanding mathematician     

The spaces of trial and generalized functions of infinite number of variables

In this article the spaces of trial and generalized functions of infinite number of variables closely connected with the equipment of Fock space are introduced. The spaces introduced are described in new terminology. The properties of continuity and differentiability of trial functions are studied.