Volume 38, Issue 12, December 1997
Index of content:

The twisted Heisenberg algebra (H(4))
View Description Hide DescriptionA two parametric deformation of the enveloping Heisenberg algebraH(4) that appears as a combination of the standard and a nonstandard quantization given by Ballesteros and Herranz is defined and proved to be Ribbon Hopf algebra. The universal R matrix and its associated quantum group are constructed. A new solution of the Braid group is obtained. The contribution of these parameters in invariants of links and the Wess–Zumino–Witten (WZW) model are analyzed. General results for twisted Ribbon Hopf algebra are derived.

Upper limit of the discrete hydrogenlike wave functions: Expansion in the inverse principal quantum number
View Description Hide DescriptionWe have expanded the Schrödinger hydrogenlike wave functions of the discrete spectrum, with respect to the inverse principal quantum number , for fixed values of the quantum numbers . The Laguerre polynomials are expanded with respect to into a sum of Bessel functions multiplied by powers of the distance from the origin. The coefficients of the expansion are a family of polynomials of the variable , which can be computed with a recursion formula. The development, which converges rapidly, can be truncated after a few terms, even for low levels .

GamowJordan vectors and nonreducible density operators from higherorder Smatrix poles
View Description Hide DescriptionIn analogy to Gamow vectors that are obtained from firstorder resonance poles of the Smatrix, one can also define higherorder Gamow vectors which are derived from higherorder poles of the Smatrix. An Smatrix pole of th order at leads to generalized eigenvectors of order which are also Jordan vectors of degree with generalized eigenvalue The GamowJordan vectors are elements of a generalized complex eigenvector expansion, whose form suggests the definition of a state operator (density matrix) for the microphysical decaying state of this higherorder pole. This microphysical state is a mixture of nonreducible components. In spite of the fact that the th order GamowJordan vectors has the polynomial timedependence which one always associates with higherorder poles, the microphysical state obeys a purely exponential decay law.

Integrability of the quantum adiabatic evolution and geometric phases
View Description Hide DescriptionWe show that the cyclic adiabatic evolution of a quantum system is completely integrable as a classical Hamiltonian system. In this context the Berry phases arise naturally as cohomology of the invariant tori.

Condition on the symmetrybreaking solution of the Schwinger–Dyson equation
View Description Hide DescriptionWe derive a condition for a nontrivial solution of the Schwinger–Dyson equation to be accompanied by a Goldstone bound state. It implies that, for quenched planar QCD, although chiral symmetry breaking occurs when there is a cutoff, the continuum limit fails to exist.

Exact semiclassical expansions for onedimensional quantum oscillators
View Description Hide DescriptionA set of rules is given for dealing with WKB expansions in the onedimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrödinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the ZinnJustin quantization condition, and of its solution in terms of “multiinstanton expansions.”

On the geometric quantization of Jacobi manifolds
View Description Hide DescriptionThe geometric quantization of Jacobi manifolds is discussed. A natural cohomology (termed Lichnerowicz–Jacobi) on a Jacobi manifold is introduced, and using it the existence of prequantization bundles is characterized. To do this, a notion of contravariant derivatives is used, in such a way that the procedure developed by Vaisman for Poissonmanifolds is naturally extended. A notion of polarization is discussed and the quantization problem is studied. The existence of prequantization representations is also considered.

Selfdual solitons in N=2 supersymmetric ChernSimons gauge theory
View Description Hide DescriptionThe low energy effective theory of planar QED with a nonlocal fourfermion interaction including Gross–Neveu and Thirring terms is shown to be equivalent to a Chern–Simons–Higgs model of special characteristics. We study the restrictions imposed by selfduality and supersymmetry, finding in both cases a plethora of (some new) topological and nontopological solitons. The nonrelativistic limit of our model generalizes the effective Ginzburg–Landau theory for the fractional quantum Hall effect such that our solitons would be the relativistic version of quasiparticle and quasihole excitations.

Quantum mechanics in classical dynamics
View Description Hide DescriptionQuantum mechanics over an associative ring with a conjugation operation can be recast in a form familiar as a classical dynamical system. The generators of transformations on the classical phase space are the expectation values of antiselfadjoint operators and are closed under a Poisson bracket that is in direct correspondence with the quantum mechanical commutator. A prescription also exists for determining when a classical flow is equivalent to a quantum mechanical evolution.

Twodimensional boson and symmetry in the quantum Hall effect
View Description Hide DescriptionWe perform consistently the Gupta–Bleuler–Dirac quantization for a twodimensional boson with parameter (α) on the circle, the boundary of the circular droplet. For we obtain the chiral (holomorphic) constraints. Using the representation of Bargmann–Fock space and the Schrödinger picture, we construct the holomorphic wave function. In order to interpret this function, we construct the coherent state representation by using the infinitedimensional translation symmetry for each Fourier (edge) mode. The chiralwave function explains the neutral edge states for integer quantum Hall effect very well. In the case of we obtain a new wave function which may describe the higher modes (radial excitations) of edge states. The charged edge states are described by the wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed.

Justification of the zeta function renormalization in rigid string model
View Description Hide DescriptionA consistent procedure for regularization of divergences and for subsequent renormalization of the string tension is proposed in the framework of the oneloop calculation of the interquark potential generated by the Polyakov–Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein–Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy.

de Rham cohomology of by supersymmetric quantum mechanics
View Description Hide DescriptionWe study supersymmetric quantum mechanics on to examine Witten’s Morse theory concretely. We give a simple instanton picture of the de Rham cohomology of We show how the reflection symmetries of the theory select the true vacuums. The number of selected vacuums agrees with the de Rham cohomology of at least for

Dynamical entropy of generalized quantum Markov chains over infinite dimensional algebras
View Description Hide DescriptionWe compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of generalized quantum Markov chain over infinite dimensional algebras. For the case in which the transition expectation is defined by a set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy. Thus we extend the result of Park [Lett. Math. Phys. 32, 63–74 (1994)] to nonAF type quantum Markov chains. We employ the main method developed in Park [Lett. Math. Phys. 32, 63–74 (1994)] with necessary modifications.

Infinite degeneracy for a Landau Hamiltonian with Poisson impurities
View Description Hide DescriptionWe consider a singleband approximation to the random Schrödinger operator in an external magnetic field. The random potential consists of delta functions of random strengths whose positions have a Poisson distribution. We prove that if the magnetic field is sufficiently high compared to the density of scatterers, then with probability one there exists an infinitely degenerate eigenenergy coinciding with the first Landau level in the absence of a random potential.

The phasedifference operator and twomode coherent states
View Description Hide DescriptionIn this paper, we introduce unitary and Hermitian phasedifference operators for the two modes of the electromagnetic field in the deformed case. The creation and annihilation operators of phasedifference quanta are given, and the algebraic properties of some operators in phase space are discussed. The phasedifference properties of twomode coherent states are investigated.

Frequency domain wavesplitting techniques for plane stratified bianisotropic media
View Description Hide DescriptionA plane harmonic electromagnetic wave obliquely incident on a plane stratified bianisotropic medium with multiple discontinuities in the parameters is considered. A covariant wavesplitting approach, based on the use of the formula of integration by parts for multiplicative integrals and the impedance concept, is presented. It encompasses various types of decomposition of the total internal field into two waves propagating in opposite directions, including the physical and vacuum wave splittings treated earlier in the literature, and provides a convenient means for both analytical investigation and numerical calculation of evolution operators (Green’s functions) and impedance tensors of split waves as well as characteristic matrices and reflection and transmission tensors of stratified bianisotropic media. The potentialities of the approach are illustrated by its application to the problems of reflection, transmission, and guided propagation, and by generalizing the method of multiple reflections to the case of stratified bianisotropic media.

An approach to the relativistic brachistochrone problem by subRiemannian geometry
View Description Hide DescriptionWe formulate a brachistochrone problem in Lorentzian geometry and we prove a variational principle valid for brachistochrones in stationary manifolds. This variational principle is stated in terms of geodesics in a suitable subRiemannian structure on M. Moreover, we prove the regularity of the solutions of our variational problem and we determine a differential equation satisfied by the brachistochrones. Some explicit examples are computed.

The ∂̄dressing method and the solutions with constant asymptotic values at infinity of DSII equation
View Description Hide DescriptionSeveral classes of exact solutions with constant asymptotic values at infinity of DSII equation are constructed via the dressing method. Among these solutions are the solutions with functional parameters, multiline solitons and breathers, and pure rational solutions.

Infinitely many Lax pairs and symmetry constraints of the KP equation
View Description Hide DescriptionStarting from a known Lax pair, one can get some infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways. In this paper, taking the well known Kadomtsev–Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in dimensions, dimensions, dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

Integrability and integrodifferential substitutions
View Description Hide DescriptionChen, Lee, and Liu presented in 1979 an algorithm for establishing integrability of twodimensional partial differential systems. It is proved here that this algorithm is invariant under the point transformations, differential substitutions, and some integrodifferential substitutions. It is also proved that canonical conserved densities of linearizable systems arising in the frameworks of the method are almost all trivial. The integrability of the nonNewtonian liquid equations is investigated and it is proved that there exist two integrable systems only. A preliminary classification of the thirdorder integrable evolution systems is presented.