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Dec 1997

Volume 38, Issue 12, pp. 6045-6691

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The twisted Heisenberg algebra Uh,w(H(4))

Boucif Abdesselam

J. Math. Phys. 38, 6045 (1997); http://dx.doi.org/10.1063/1.532201 (16 pages) | Cited 1 time

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A two parametric deformation of the enveloping Heisenberg algebra H(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. © 1997 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.Sv Lie algebras of Lie groups

Upper limit of the discrete hydrogen-like wave functions: Expansion in the inverse principal quantum number n−1

Bruno Blaive and Michel Cadilhac

J. Math. Phys. 38, 6061 (1997); http://dx.doi.org/10.1063/1.532202 (11 pages) | Cited 1 time

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We have expanded the Schrödinger hydrogen-like wave functions ψnlm of the discrete spectrum, with respect to the inverse principal quantum number n−1, for fixed values of the quantum numbers l,m. The Laguerre polynomials Lnα(x/n) are expanded with respect to n−1 into a sum of Bessel functions multiplied by powers rk of the distance from the origin. The coefficients of the expansion are a family of polynomials sq,k(l) of the variable l, which can be computed with a recursion formula. The development, which converges rapidly, can be truncated after a few terms, even for low levels n. © 1997 American Institute of Physics.
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02.10.De Algebraic structures and number theory
03.65.Ge Solutions of wave equations: bound states
31.10.+z Theory of electronic structure, electronic transitions, and chemical binding

Gamow-Jordan vectors and non-reducible density operators from higher-order S-matrix poles

A. Bohm, Mark Loewe, S. Maxson, P. Patuleanu, C. Püntmann, and M. Gadella

J. Math. Phys. 38, 6072 (1997); http://dx.doi.org/10.1063/1.532203 (29 pages) | Cited 18 times

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In analogy to Gamow vectors that are obtained from first-order resonance poles of the S-matrix, one can also define higher-order Gamow vectors which are derived from higher-order poles of the S-matrix. An S-matrix pole of r-th order at zR = ER−iΓ/2 leads to r generalized eigenvectors of order k = 0,1,…,r−1, which are also Jordan vectors of degree (k+1) with generalized eigenvalue (ER−iΓ/2). The Gamow-Jordan 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 higher-order pole. This microphysical state is a mixture of non-reducible components. In spite of the fact that the k-th order Gamow-Jordan vectors has the polynomial time-dependence which one always associates with higher-order poles, the microphysical state obeys a purely exponential decay law. © 1997 American Institute of Physics.
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11.55.Bq Analytic properties of S matrix
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
02.30.Tb Operator theory

Integrability of the quantum adiabatic evolution and geometric phases

G. Cassinelli, E. De Vito, and A. Levrero

J. Math. Phys. 38, 6101 (1997); http://dx.doi.org/10.1063/1.532204 (18 pages)

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We 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. © 1997 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.40.-k Geometry, differential geometry, and topology

Condition on the symmetry-breaking solution of the Schwinger–Dyson equation

G. Cheng and T. K. Kuo

J. Math. Phys. 38, 6119 (1997); http://dx.doi.org/10.1063/1.532205 (7 pages) | Cited 2 times

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We 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. © 1997 American Institute of Physics.
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12.38.Lg Other nonperturbative calculations
11.30.Qc Spontaneous and radiative symmetry breaking
11.10.St Bound and unstable states; Bethe-Salpeter equations
11.30.Rd Chiral symmetries
14.80.Va Axions and other Nambu-Goldstone bosons (Majorons, familons, etc.)

Exact semiclassical expansions for one-dimensional quantum oscillators

Eric Delabaere, Hervé Dillinger, and Frédéric Pham

J. Math. Phys. 38, 6126 (1997); http://dx.doi.org/10.1063/1.532206 (59 pages) | Cited 32 times

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A set of rules is given for dealing with WKB expansions in the one-dimensional 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 Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.” © 1997 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Sq Semiclassical theories and applications
02.30.Lt Sequences, series, and summability

On the geometric quantization of Jacobi manifolds

Manuel de León, Juan C. Marrero, and Edith Padrón

J. Math. Phys. 38, 6185 (1997); http://dx.doi.org/10.1063/1.532207 (29 pages) | Cited 9 times

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The 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 Poisson manifolds is naturally extended. A notion of polarization is discussed and the quantization problem is studied. The existence of prequantization representations is also considered. © 1997 American Institute of Physics.
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03.65.-w Quantum mechanics
02.40.-k Geometry, differential geometry, and topology

Self-dual solitons in N=2 supersymmetric Chern-Simons gauge theory

Wifredo García Fuertes and Juan Mateos Guilarte

J. Math. Phys. 38, 6214 (1997); http://dx.doi.org/10.1063/1.532208 (16 pages) | Cited 1 time

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The low energy effective theory of planar QED with a non-local four-fermion 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 self-duality and supersymmetry, finding in both cases a plethora of (some new) topological and non-topological solitons. The non-relativistic 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 quasi-particle and quasi-hole excitations. © 1997 American Institute of Physics.
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11.15.-q Gauge field theories
11.30.Pb Supersymmetry
11.10.Lm Nonlinear or nonlocal theories and models
73.43.-f Quantum Hall effects
12.20.Ds Specific calculations

Quantum mechanics in classical dynamics

Andrew C. Millard

J. Math. Phys. 38, 6230 (1997); http://dx.doi.org/10.1063/1.532209 (19 pages)

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Quantum 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 anti-self-adjoint 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. © 1997 American Institute of Physics.
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03.65.Sq Semiclassical theories and applications
45.05.+x General theory of classical mechanics of discrete systems
02.30.Uu Integral transforms
02.30.Vv Operational calculus

Two-dimensional boson and W-symmetry in the quantum Hall effect

Yun Soo Myung

J. Math. Phys. 38, 6249 (1997); http://dx.doi.org/10.1063/1.532210 (16 pages) | Cited 1 time

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We perform consistently the Gupta–Bleuler–Dirac quantization for a two-dimensional boson with parameter (α) on the circle, the boundary of the circular droplet. For α = 1, 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 infinite-dimensional translation (W∞) symmetry for each Fourier (edge) mode. The α = 1 chiral wave function explains the neutral edge states for integer quantum Hall effect very well. In the case of α = −1, 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 ∣α∣ ≠ 1 wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed. © 1997 American Institute of Physics.
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73.43.-f Quantum Hall effects
03.70.+k Theory of quantized fields

Justification of the zeta function renormalization in rigid string model

V. V. Nesterenko and I. G. Pirozhenko

J. Math. Phys. 38, 6265 (1997); http://dx.doi.org/10.1063/1.532211 (16 pages) | Cited 16 times

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A consistent procedure for regularization of divergences and for subsequent renormalization of the string tension is proposed in the framework of the one-loop 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. © 1997 American Institute of Physics.
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11.25.-w Strings and branes
11.10.Gh Renormalization
12.39.-x Phenomenological quark models
12.20.Ds Specific calculations

de Rham cohomology of SO(n) by supersymmetric quantum mechanics

Kazuto Oshima

J. Math. Phys. 38, 6281 (1997); http://dx.doi.org/10.1063/1.532212 (6 pages) | Cited 1 time

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We study supersymmetric quantum mechanics on SO(n) to examine Witten’s Morse theory concretely. We give a simple instanton picture of the de Rham cohomology of SO(n). 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 SO(n), at least for n ⩽ 5. © 1997 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
11.30.Ly Other internal and higher symmetries
11.30.Pb Supersymmetry

Dynamical entropy of generalized quantum Markov chains over infinite dimensional algebras

Yong Moon Park and Hyun Hye Shin

J. Math. Phys. 38, 6287 (1997); http://dx.doi.org/10.1063/1.532213 (17 pages)

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We 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 non-AF type quantum Markov chains. We employ the main method developed in Park [Lett. Math. Phys. 32, 63–74 (1994)] with necessary modifications. © 1997 American Institute of Physics.
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02.50.Ga Markov processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.30.-d Quantum statistical mechanics
02.10.-v Logic, set theory, and algebra
05.70.Ce Thermodynamic functions and equations of state

Infinite degeneracy for a Landau Hamiltonian with Poisson impurities

J. V. Pulé and M. Scrowston

J. Math. Phys. 38, 6304 (1997); http://dx.doi.org/10.1063/1.532214 (11 pages) | Cited 2 times

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We consider a single-band 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. © 1997 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

The q-phase-difference operator and two-mode q-coherent states

Yaping Yang, Weiguo Feng, and Xiang Wu

J. Math. Phys. 38, 6315 (1997); http://dx.doi.org/10.1063/1.532215 (13 pages) | Cited 1 time

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In this paper, we introduce unitary and Hermitian phase-difference operators for the two modes of the electromagnetic field in the q-deformed case. The q-creation and annihilation operators of phase-difference quanta are given, and the algebraic properties of some operators in phase space are discussed. The phase-difference properties of two-mode q-coherent states are investigated. © 1997 American Institute of Physics.
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03.70.+k Theory of quantized fields
11.10.-z Field theory

Frequency domain wave-splitting techniques for plane stratified bianisotropic media

George N. Borzdov

J. Math. Phys. 38, 6328 (1997); http://dx.doi.org/10.1063/1.532216 (39 pages) | Cited 23 times

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A plane harmonic electromagnetic wave obliquely incident on a plane stratified bianisotropic medium with multiple discontinuities in the parameters is considered. A covariant wave-splitting 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. © 1997 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

An approach to the relativistic brachistochrone problem by sub-Riemannian geometry

Fabio Giannoni, Paolo Piccione, and José A. Verderesi

J. Math. Phys. 38, 6367 (1997); http://dx.doi.org/10.1063/1.532217 (15 pages) | Cited 4 times

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We 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 sub-Riemannian 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. © 1997 American Institute of Physics.
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03.30.+p Special relativity
02.40.Hw Classical differential geometry
02.30.Xx Calculus of variations
02.30.Yy Control theory

The ∂-dressing method and the solutions with constant asymptotic values at infinity of DS-II equation

V. G. Dubrovsky

J. Math. Phys. 38, 6382 (1997); http://dx.doi.org/10.1063/1.532218 (19 pages) | Cited 1 time

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Several classes of exact solutions with constant asymptotic values at infinity of DS-II equation are constructed via the math-dressing method. Among these solutions are the solutions with functional parameters, multi-line solitons and breathers, and pure rational solutions. © 1997 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

Infinitely many Lax pairs and symmetry constraints of the KP equation

Sen-Yue Lou and Xing-Biao Hu

J. Math. Phys. 38, 6401 (1997); http://dx.doi.org/10.1063/1.532219 (27 pages) | Cited 115 times

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Starting 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 (1+1)-dimensions, (2+1)-dimensions, (3+1)-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. © 1997 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

Integrability and integrodifferential substitutions

A. G. Meshkov

J. Math. Phys. 38, 6428 (1997); http://dx.doi.org/10.1063/1.532220 (16 pages) | Cited 2 times

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Chen, Lee, and Liu presented in 1979 an algorithm for establishing integrability of two-dimensional 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 non-Newtonian liquid equations is investigated and it is proved that there exist two integrable systems only. A preliminary classification of the third-order integrable evolution systems is presented. © 1997 American Institute of Physics.
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47.50.-d Non-Newtonian fluid flows
02.30.Jr Partial differential equations
02.30.Uu Integral transforms
02.30.Vv Operational calculus

Magri–Morosi–Gel’fand–Dorfman’s bi-Hamiltonian constructions in the action-angle variables

Roman G. Smirnov

J. Math. Phys. 38, 6444 (1997); http://dx.doi.org/10.1063/1.532221 (11 pages) | Cited 1 time

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A constructive method of transforming a completely integrable in Liouville’s sense Hamiltonian system into Magri–Morosi–Gel’fand–Dorfman’s (MMGD) bi-Hamiltonian form is presented. The approach is carried out by making use of the action-angle coordinates. The classical Kepler problem is shown to be a MMGD bi-Hamiltonian system. Explicit plethoras of higher-order conserved quantities for the Kepler problem are derived by employing Oevel’s method based on the existence of the MMGD bi-Hamiltonian representation. © 1997 American Institute of Physics.
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95.10.Ce Celestial mechanics (including n-body problems)
45.05.+x General theory of classical mechanics of discrete systems
02.30.Uu Integral transforms
02.30.Vv Operational calculus

Darboux and binary Darboux transformations for the nonautonomous discrete KP equation

R. Willox, T. Tokihiro, and J. Satsuma

J. Math. Phys. 38, 6455 (1997); http://dx.doi.org/10.1063/1.532222 (15 pages) | Cited 12 times

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It is shown how Darboux and binary Darboux transformations for a nonautonomous discrete KP equation can be obtained from fermion analysis. This equation is obtained by considering a generalized Miwa transformation; it is also shown to be linked to the discrete KP equation by a special gauge transformation. The Darboux and binary Darboux transformations are used to discuss general classes of solutions in the form of Casorati- and Gramm-type determinants. N-soliton solutions are discussed as well. © 1997 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
02.30.Uu Integral transforms
02.30.Vv Operational calculus

Energy conditions for a spherically symmetric kink space–time

K. A. Dunn, Tina A. Harriott, and J. G. Williams

J. Math. Phys. 38, 6470 (1997); http://dx.doi.org/10.1063/1.532223 (5 pages)

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The tetrad formalism is used to study a broad class of spherically symmetric kink space–times whose Einstein tensor is computed and then diagonalized. A simple example of such a space–time is shown to satisfy the weak energy condition and to be extendible to a space–time that is geodesically complete. © 1997 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
02.40.Ky Riemannian geometries

Graphical classification of global SO(n) invariants and independent general invariants

Shoichi Ichinose and Noriaki Ikeda

J. Math. Phys. 38, 6475 (1997); http://dx.doi.org/10.1063/1.532183 (47 pages) | Cited 3 times

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This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is presented. Through the analysis, we explain the following points: (a) a graphical representation is introduced to express invariants clearly; (b) every graph of invariants is specified by a set of indices; (c) a number, called weight, is assigned to each invariant. It expresses the symmetry with respect to the suffix-permutation within an invariant. Interesting relations among the weights of invariants are given. Those relations show the consistency and the completeness of the present classification; (d) some reduction procedures are introduced in graphs for the purpose of classifying them. Secondly the above result is applied to the proof of the independence of general invariants with the mass-dimension M6 for the general geometry in a general space dimension. We take a graphical representation for general invariants too. Finally all relations depending on each space-dimension are systematically obtained for 2, 4, and 6 dimensions. © 1997 American Institute of Physics.    
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04.20.Cv Fundamental problems and general formalism
02.10.-v Logic, set theory, and algebra
02.20.-a Group theory

Reissner–Nordström-like solutions of the SU(2) Einstein–Yang/Mills equations

J. A. Smoller and A. G. Wasserman

J. Math. Phys. 38, 6522 (1997); http://dx.doi.org/10.1063/1.532224 (38 pages) | Cited 6 times

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We introduce a new class of spherically symmetric solutions of the SU(2) Einstein–Yang/Mills equations. These solutions have a Reissner–Nordström-type essential singularity at the origin, and are well behaved in the far field. These solutions are needed to classify all spherically symmetric solutions which are smooth, asymptotically flat in the far field, and have finite (ADM) mass. © 1997 American Institute of Physics.
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04.20.Jb Exact solutions
11.15.-q Gauge field theories
11.30.Hv Flavor symmetries
03.65.Pm Relativistic wave equations
11.10.Jj Asymptotic problems and properties
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