bluuebunny/crossref_metadata_2025 huggingface dataset:
On q-deformed supersymmetric classical mechanical models
Authors: Colatto L. P. , Matheus-Valle J. L.
Published: July 2002

Based on the idea of quantum groups and para-Grassmannian variables, we present a generalization of supersymmetric classical mechanics with a deformation parameter q=exp(2πi/k) dealing with the k=3 case. The coordinates of the q-superspace are a commuting parameter t and a para-Grassmannian variable θ, where θ3=0. The generator and covariant derivative are obtained, as well as the action for some possible superfields.
Long-time approximation to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics
Authors: Sáenz A. W.
Published: July 2002

A simple normal-form approach is used to obtain uniform long-time approximations to the evolution of a resonant or nonresonant anharmonic oscillator system governed by a Hamiltonian Hε which is a self-adjoint operator acting in the Hilbert space ℋ=L2(Rν) (ν≥2) and is given formally by H0+εV. Here H0 denotes the Hamiltonian of ν one-dimensional harmonic oscillators whose coupling is represented by εV, where V is an operator of multiplication by a smooth function of at most polynomial growth at infinity and ε≥0 is a small parameter. We consider the general situation in which ρ≥1 of the frequencies of these oscillators are rationally independent, imposing a standard diophantine condition on the independent frequencies if ρ≥2. Under these assumptions, which are stated in a mathematically precise way in the paper, an Nth-order approximant ψN(t,ε) to the exact solution ψ(t,ε) of the Schrödinger equation id ψ(t,ε)/dt=Hεψ(t,ε) satisfying the initial condition ψ(0,ε)=ψ0 is constructed inductively, ψ0 being an arbitrary ε-independent member of a suitable family of smooth functions dense in ℋ. Our main result is that ψN(t,ε) differs from ψ(t,ε) in ℋ-norm by ≤const εN+1(‖t‖+1) for all t∈R and all ε in an arbitrary compact interval [0, ε0].
Ergodic properties of quantized toral automorphisms
Authors: Klimek Sławomir , Leśniewski Andrzej , Maitra Neepa , Rubin Ron
Published: July 2002

We study the ergodic properties for a class of quantized toral automorphisms, namely the cat and Kronecker maps. The present work uses and extends the results of Klimek and Leśniewski [Ann. Phys. 244, 173–198 (1996)]. We show that quantized cat maps are strongly mixing, while Kronecker maps are ergodic and nonmixing. We also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus. This allows us to exhibit explicit relations between our Toeplitz quantization and the semiclassical quantization of cat maps proposed by Hannay and Berry [Physica D 1, 267–290 (1980)].
Displacement-operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras
Authors: Nieto Michael Martin , Truax D. Rodney
Published: July 2002

In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials V(x,τ)=g(2)(τ)x2+g(1)(τ)x+g(0)(τ). We describe a set of number-operator eigenstates states, {Ψn(x,τ)}, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, Ψ0, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, τ. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of V(x,τ) and having isomorphic Lie algebras, will be dealt with in the following paper (II).
Displacement-operator squeezed states. II. Examples of time-dependent systems having isomorphic symmetry algebras
Authors: Nieto Michael Martin , Truax D. Rodney
Published: July 2002

In this paper, results from the previous paper (I) are applied to calculations of squeezed states for such well-known systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the (α,z)- and in the (z,α)-representations described in I. The dependence of the squeezed-state uncertainty products on the time and on the squeezing parameters is determined for each system.
Period lengths of cellular automata cam-90 with memory
Authors: Kawahara Yasuo , Lee Hyen Yeal
Published: July 2002

Cellular automata ca-90 have states 0 and 1, and their dynamics, driven by the local transition rule 90, can be simply represented with Laurent polynomials over a finite field F2={0,1}. Cellular automata cam-90 with memory, whose configurations are pairs of those of ca-90, are introduced as a useful machinery to solve certain equations on configurations, in particular, to compute fixed or kernel configurations of ca-90. This paper defines a notion of linear dynamical systems with memory, states their basic properties, and then studies some period lengths of one-dimensional and two-dimensional cellular automata cam-90 with memory.
Localization of relativistic particles
Authors: Omnès Roland
Published: July 2002

In order to discuss localization experiments and also to extend the consistent history interpretation of quantum mechanics to relativistic properties, the techniques introduced in a previous paper [J. Math. Phys. 38, 697 (1997)] are applied to the localization of a photon in a given region of space. An essential requirement is to exclude arbitrarily large wavelengths. The method is valid for a particle with any mass and spin. Though there is no proper position operator for a photon, one never needs one in practice. Causality is valid up to exponentially small corrections.
Twisted boundary conditions and transport in disordered systems
Authors: Akkermans E.
Published: July 2002

Condensed matter systems are usually characterized by their response to external fields. The Kubo linear response theory is the standard theoretical tool used to analyze it. In order to obtain transport quantities with this method, we generally need to know both the eigenvalues and the eigenfunctions. This article discusses another methodological approach which in some cases allows one to obtain some characteristics of the dc dissipative transport from the response to a gauge field in a multiply connected geometry. This avoids the need of the eigenstates to characterize the transport properties which are directly read off the behavior of the energy spectrum. This method is applied to the problem of transport in metals and Anderson insulators.
Ground states and low-temperature phases of itinerant electrons interacting with classical fields: A review of rigorous results
Authors: Macris N. , Lebowitz J. L.
Published: July 2002

We review, from a unified point of view, a general class of models of itinerant electrons interacting with classical fields. Applications to the static Holstein, Kondo, and Hubbard models are discussed. The ground state structure of the classical field is investigated when the electron band is half-filled. Some of the results are also valid when there is a Hubbard interaction between spin up and spin down electrons. It is found that the ground states are either homogeneous or period two Néel configurations, depending on the geometry of the lattice and on the magnetic fluxes present in the system. In the specific models, Néel configurations correspond to Peierls, magnetic or superconducting instabilities of the homogeneous state. The effect of small thermal and quantum fluctuations of the classical fields are reviewed in the context of the Holstein model. Many of the results described here originate from the work of Elliott Lieb and collaborators.
On Einstein’s equations for space–times admitting a non-null Killing field
Authors: Rácz István
Published: July 2002

We consider the three-dimensional formulation of Einstein’s theory for space–times possessing a non-null Killing field ξa. It is known that for the vacuum case some of the basic field equations are deducible from the others. It will be shown here how this result can be generalized for the case of essentially arbitrary matter fields. The systematic study of the structure of the fundamental field equations is carried out. In particular, the existence of geometrically preferred reference systems is shown. Using local coordinates of this type two approaches are presented resulting in resolvent systems of partial differential equations for the basic field variables. Finally, the above results are applied for perfect fluid space–times describing possible equilibrium configurations of relativistic dissipative fluids.
Algebraic proof of a sum rule occurring in Stark broadening of hydrogen lines
Authors: Casini Roberto
Published: July 2002

We present the algebraic proof of a sum rule that is relevant in the theory of Stark broadening of hydrogen and hydrogenic lines. This is accomplished by applying Gauss’ recursion formulas for the hypergeometric function to the analytical expression of the dipole matrix elements involved in the summation.
Integrable discretizations of the spin Ruijsenaars–Schneider models
Authors: Ragnisco Orlando , Suris Yuri B.
Published: July 2002

Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars–Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous-time counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, including the discrete time ones.
Vandermonde-like determinants and N-fold Darboux/Bäcklund transformations
Authors: Steudel H. , Meinel R. , Neugebauer G.
Published: July 2002

We define Vandermonde-like determinants and analyze their structure. The resulting scheme is well-suited to achieve a remarkable compactness and transparency in N-soliton formulas or, more generally, in formulas for N-fold Darboux or Bäcklund transformations.
Cosmological Einstein–Yang–Mills equations
Authors: Darian B. K. , Künzle H. P.
Published: July 2002

We use a systematic construction method for invariant connections on homogeneous spaces to find the Einstein–SU(n)–Yang–Mills equations for Friedmann–Robertson–Walker and locally rotationally symmetric homogeneous cosmologies. These connections depend on the choice of a homomorphism from the isotropy group into the gauge group. We consider here the cases of the gauge group SU(n) and SO(n), where these homomorphisms correspond to unitary or orthogonal representations of the isotropy group. For some of the simpler cases the full system of the evolution equations are derived, for others we only determine the number of dynamical variables that remain after some mild fixing of the gauge.
A rejoinder on quaternionic projective representations
Authors: Adler S. L. , Emch G. G.
Published: July 2002

In a series of papers published in this journal, a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. In the present paper we give what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues.
New universality classes in one-dimensional O(N)-invariant spin-models with an n-parametric action
Authors: Seiler Erhard , Yildirim Karim
Published: July 2002

An action with n parameters, which generalizes the O(N)−RPN−1 model, is considered in one dimension for general N. We use asymptotic expansion techniques to determine where the model becomes critical and show that for the actions considered there exists a family of hypersurfaces whose asymptotic behavior determines a one-parameter family of new universality classes. They interpolate between the O(N) vector model class and the RPN−1 model class. Furthermore continuum limits are discussed, including the exceptional case N=2.
On a first integral of the Kepler problem
Authors: Vivarelli Maria Dina
Published: July 2002

A quadratic first integral of the Kepler problem, obtained by Benenti (“Orthogonal separable dynamical systems,” 5th International Conference on Differential Geometry and its Applications, 24–28 August 1992, Silesian University at Opava) through separation in elliptic coordinates, is shown to be intimately connected with the pre-quantization of the Kepler manifold, thus acquiring a physical interpretation.
A class of homogeneous nonlinear evolution equations with stable, localized solutions in any dimension
Authors: Auberson G.
Published: July 2002

A new set of nonlinear evolution equations is introduced and studied. These equations derive from a local Lagrangian and are (i) homogeneous and (ii) invariant under the Galilei group or the Lorentz group (including time reversal). Some of them have confined solutions with a solitonlike behavior, irrespective of the space dimension. Moreover, these solutions are shown to be stable against small and localized perturbations. Another family of localized solutions is worked out, and briefly discussed with regard to the elusive integrability properties of the new equations.
The operator algebra of the quantum relativistic oscillator
Authors: Cotăescu Ion I. , Drăgănescu Gheorghe
Published: July 2002

The operator algebras of a new family of relativistic geometric models of the relativistic oscillator [I. I. Cotăescu, Int. J. Mod. Phys. A 12, 3545 (1997)] are studied. It is shown that, generally, the operator of number of quanta and the pair of shift operators of each model are the generators of a nonunitary representation of the so(1,2) algebra, except for a special case when this algebra becomes the standard of the nonrelativistic harmonic oscillator.
W
∞
-covariance of the Weyl–Wigner–Groenewold–Moyal quantization
Authors: Dereli T. , Verçin A.
Published: July 2002

The differential structure of operator bases used in various forms of the Weyl–Wigner–Groenewold–Moyal (WWGM) quantization is analyzed and a derivative-based approach, alternative to the conventional integral-based one is developed. Thus the fundamental quantum relations follow in a simpler and unified manner. An explicit formula for the ordered products of the Heisenberg–Weyl algebra is obtained. The W∞-covariance of the WWGM-quantization in its most general form is established. It is shown that the group action of W∞ that is realized in the classical phase space induces on bases operators in the corresponding Hilbert space a similarity transformation generated by the corresponding quantum W∞ which provides a projective representation of the former W∞. Explicit expressions for the algebra generators in the classical phase space and in the Hilbert space are given. It is made manifest that this W∞-covariance of the WWGM-quantization is a genuine property of the operator bases.
Geometro-stochastically quantized fields with internal spin variables
Authors: Drechsler W.
Published: July 2002

The use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a four-dimensional internal spin space S̄ is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. S̄ is a homogeneous space of the group SL(2,C) parametrized in terms of spinors α∈C2 and their complex conjugates ᾱ. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin [m,s], with m⩾0 and s=0,12,1,32,… , with spin described in terms of the usual (2s+1)-component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable α to a compact subspace of S̄, i.e., a “spin shell” Sr=2s2 of radius r=2s in C2. This formulation of single particles or single antiparticles of type [m,s] is then used to study the geometro-stochastic (i.e., quantum) propagation of amplitudes for arbitrary spin on a curved background space–time possessing a metric and axial vector torsion treated as external fields. A Poincaré gauge covariant path integral-like representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space–time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned.
N=2
structures in string theories
Authors: Figueroa-O’Farrill José M.
Published: July 2002

The BRST cohomology of any topological conformal field theory admits the structure of a Batalin–Vilkovisky algebra, and string theories are no exception. Loosely speaking, we say that two topological conformal field theories are cohomologically equivalent if their BRST cohomologies are isomorphic as Batalin–Vilkovisky algebras. In this paper we argue that any string theory (regardless of the matter background) is cohomologically equivalent to some twisted N=2 superconformal field theory. We discuss three string theories in detail: the bosonic string, the NSR string and the W3 string. In each case the way the cohomological equivalence is constructed can be understood as coupling the topological conformal field theory to topological gravity. These results lend further supporting evidence to the conjecture that any topological conformal field theory is cohomologically equivalent to some topologically twisted N=2 superconformal field theory. We end the paper with some speculative comments on Massey products in topological conformal field theories.
Quantized Kronecker flows and almost periodic quantum field theory
Authors: Klimek Sławomir , Leśniewski Andrzej
Published: July 2002

We define and study the properties of the infinite-dimensional quantized Kronecker flow. This ℂ*-dynamical system arises as a quantization of the corresponding flow on an infinite-dimensional torus. We prove an ergodic theorem for a class of quantized Kronecker flows. We also study the closely related, almost periodic quantum field theory of bosonic, fermionic, and supersymmetric particles. We prove the existence and uniqueness of KMS and super-KMS states for the ℂ*-algebras of observables arising in these theories.
Axially symmetric metrics from Laplace’s seed by inverse scattering method
Authors: Chaudhuri S. , Das K. C.
Published: July 2002

Two-soliton solutions of axially symmetric Einstein field equations are presented using two different Laplace’s solutions as seed. The derived stationary solutions reduce to some well-known solutions such as Schwarzschild, Kerr, Kerr–Newman–Unti–Tamburino, and Kerns–Wild. The surface geometry of the metrics namely, the location of singularities, event horizon, infinite red shift surface, surface area of the horizon, and Gaussian curvatures are studied. Finally, the generated two-soliton solutions are compared with our previous solutions [Pramana, J. Phys. 40, 277 (1993)], which was obtained by the method of Gutsunaev–Manko [Gen. Relativ. Gravit. 20, 327 (1988)] for the same set of Laplace’s solutions as seed. The comparison shows that the solutions generated from the Inverse Scattering Method of Belinskii–Zakharov [Sov. Phys. JETP 48, 985 (1978); 50, 1 (1979)] is more general than the solutions obtained from the method of Gutsunaev–Manko [Pramana, J. Phys. 40, 277 (1993)].
On the construction of a Dirac spinor that generates a specified tetrad
Authors: Nash Patrick L.
Published: July 2002

It is well known that a complex four-component Dirac spinor defines an orthonormal tetrad on flat Minkowski spacetime. For example, a spinor solution to the Dirac equation determines the four-velocity and Pauli-Lubanski spin vector, comprising the timelike and first spacelike members of the particle’s tetrad, as well as two other spacelike members that are rarely discussed. Also, of particular note is the complex null tetrad formalism that provides a map from a two-component SL(2, C) spinor and its complex conjugate to a tetrad. The inverse problem is studied here. Given a tetrad, a complex Dirac spinor valued function of the tetrad is explicitly defined in such a way that certain sums of bilinear products of the components of this spinor exactly reproduce the tetrad. This spinor may be used in the Feynman propagator representation of the solution to the Dirac equation to generate a Dirac wave spinor with desired initial properties. The mappings studied here represent a new class of nonlinear mappings SO(3,1)+↑→SO(3,1)¯.
Divergences in the solutions of the plasma screening equation
Authors: Das Shawpawn Kumar , Janaki Mylavarapu Sita , Dasgupta Brahmananda
Published: July 2002

Classical kinetic theory using Boltzmann statistics shows that the potential distribution φ(r) in the screening cloud surrounding a single test charge at rest within a plasma is governed by a three-dimensional spherically symmetric plasma screening equation ∇2φ(r)=A(exp(+αφ)−exp(−βφ)), r≠0, where A=4πn0ε, α=ε/Te, β=ε/Ti, ε=electronic charge, Te=electron temperature, Ti=ion temperature, and n0=electron and ion density at large distances from the charge Q. In this paper it is proved rigorously that any nontrivial solution of the screening equation must have the following property: If φ(r)=potential at a radial distance r and limr→∞ φ(r)=0, then, for any positive integer n, as r→0 either rnφ→+∞ and rnφ′→−∞ or rnφ→−∞ and rnφ′→+∞.
Gauge fixing in the partition function for generalized quantum dynamics
Authors: Adler Stephen L.
Published: July 2002

We discuss the problem of gauge fixing for the partition function in generalized quantum (or trace) dynamics, deriving analogs of the De Witt–Faddeev–Popov procedure and of the BRST invariance familiar in the functional integral context.
Classical limits for quantum maps on the torus
Authors: Lesniewski Andrzej , Rubin Ron , Salwen Nathan
Published: July 2002

We provide a rigorous canonical quantization for the following toral automorphisms: cat maps, generalized kicked maps, and generalized Harper maps. For each of these systems we construct a unitary propagator and prove the existence of a well-defined classical limit. We also provide an alternative derivation of Hannay and Berry results for the cat map propagator on the plane.
Symmetry and symmetry breaking in quantum-chromo (flavor)-dynamics
Authors: Völkel A. H.
Published: July 2002

Symmetries of flavored quantum-chromo(flavor)-dynamics (QCD) and their breaking are reinvestigated within the Lagrange–Hamilton frame as well as on the basis of axiomatic gauge quantum field theory aiming at the resolution of some shortcomings by which QCD is still burdened despite all its undisputable successes: The Noether energy momentum tensors arising from those general representations (substitution rules) of the translation group that leave the standard SU(3) local color gauge and global flavor invariant chiral QCD-Lagrangian (and hence the Euler–Lagrange field equations) form invariant are shown to contain, besides the standard color and flavor symmetric canonical part, additional terms breaking the internal SU(3) symmetries of the Lagrangian down to chiral SU(2)×U(1) flavor symmetry, and in case of color confinement also the color group down to U(1)×SU(2) color symmetry only. With these general representations (covariance conditions) of the translation group as an input, a strictly Poincaré covariant gauge-quantum-field theory with color confinement is formulated and flavor Goldstone states are shown to occur with finite masses due to relative Einstein causality, the spectrum condition in color neutral sectors, translational covariance, and current conservation. Identifying the finite mass Goldstone states with pseudoscalar mesons, the axial vector part of the chiral SU(3) flavor symmetry is spontaneously broken, leaving only the chiral SU(2)×U(1) subgroup as a strict flavor symmetry of QCD.
Diagrams in classical and semiclassical perturbation theory
Authors: Panayotaros Panayotis
Published: July 2002

We describe a diagrammatic method for the Poincare–Birkhoff normal forms algorithm of classical mechanics, and indicate the use of the diagrams with an example from hydrodynamics. We also present a generalization of the diagrammatic method to quantum mechanics. The quantum diagrams can be used in the semiclassical version of normal forms developed by Graffi, Paul, and others.
A reformulation of simple liquids theory—Renormalization by one-, two-, and three-particle densities
Authors: Okumura Ko
Published: July 2002

We reformulate the theory of simple liquids in a field theoretical way by taking into account the triplet potential wijk, in addition to the external potential ui and the pair potential vij. The innovation here is the inversion method and the on-shell expansion which are the building blocks of a novel use of Legendre transformation developed in field theory. By the inversion method, we renormalize the theory in terms of one-, two-, and three-particle densities, and present a diagrammatic representation for a thermodynamical functional, which is the entropy except for a trivial constant, in terms of renormalized variables. In other words, we present an expression for the entropy in terms of only one-, two-, and three-particle densities: the n-particle density where n⩾4 does not appear in the expression. The on-shell condition, which is a starting point of the on-shell expansion, of the thermodynamical functional thus obtained (the entropy) leads to a set of three self-consistent equations for one-, two-, and three-particle densities. Through one of the self-consistent equations, we can systematically improve the Kirkwood’s superposition approximation for the three-particle density. The on-shell conditions for other thermodynamical functionals, also obtained in this article, are found to be extentions of various well-known equations in the theory of simple liquids. The formulation presented here is complementary to the conventional re-summation techniques for renormalization of diagrams. In the present formulation, we do not have to care about the topological structure of diagrams, often characterized by the irreducibility of diagrams. Instead, by a perturbative calculation, we can automatically single out the diagrams with the topological structure predicted by the re-summation techniques.
Orbit spaces of compact coregular simple Lie groups with 2, 3 and 4 basic polynomial invariants: Effective tools in the analysis of invariant potentials
Authors: Sartori G. , Talamini V.
Published: July 2002

The coregular real representations of the compact simple Lie groups are classified and the equalities and inequalities defining their orbit spaces are determined in the case of 2, 3 and 4 basic polynomial invariants. The results are obtained making use of the classification of the complex coregular simple Lie groups given by G. W. Schwarz and of the determination of all the allowable metric matrices for q-dimensional (q⩽4) orbit spaces of compact coregular linear groups, recently obtained by the present authors. The results are used to determine in a rigorous way the minima of two SO(10) and E6 invariant Higgs potentials.
Evaluation of nonrelativistic one-loop Feynman integrals by distributional methods
Authors: Wagner Peter
Published: July 2002

In this paper, we use Feynman’s parameter formula and distribution theory in order to represent the integrals ∫Rn+1Πk=1l(it−(1)/(2m)|x−yk|2+μ)−1dt dx for m>0, μ>0, yk∈Rn, by definite integrals over simplices. We obtain explicit expressions in the cases of n=2 and n=l=3, respectively.
Particles in singular magnetic field
Authors: Marcinek Władysław
Published: July 2002

An algebraic formalism for description of quantum states of charged particle with spin moving in two-dimensional space under the influence of a singular magnetic field is developed in terms of graded algebras. The fundamental assumption is that the particle is transformed into a composite system which consists of quasiparticles, quasiholes, and magnetic fluxes. Such a system is endowed with generalized statistics determined by a grading group and a commutation factor on it. Composite systems corresponding to the quantum Hall effect and the electronic magnetotransport anomaly are described. The Fock space representation is also given.
Transition to criticality in circle maps at the golden mean
Authors: Dixon Tony W. , Kenny Brian G.
Published: July 2002

The one-dimensional circle map is frequently used to model the quasiperiodic route to chaos. In this paper we study the transition to criticality for circle maps with a golden mean winding number in the limit as the order of the map inflection point N=1+ε approaches unity. The solution to the relevant pair of functional equations consists of the subcritical solution together with a perturbative correction of order ε. We derive functional relations that the perturbative correction must satisfy and produce numerical evidence that these equations are indeed satisfied. A similar analysis is performed for the golden mean eigenvalue equations. Power law conjugacy of the subcritical solution to order N=1+ε enables us to obtain a closed form solution for the perturbative correction that satisfies the constraint relations exactly. This is compared with the more interesting critical case associated with a circle map whose inflection point is infinitesimally close to unity.
The complete Schwarzschild interior and exterior solution in the harmonic coordinate system
Authors: Liu Quan-Hui
Published: July 2002

An exact solution of Schwarzschild interior and exterior solution for a star which has a uniform density and whose center pressure approaches infinite, in the harmonic coordinate system, is given. This solution enables us to demonstrate that in known literatures, a term in the Schwarzschild exterior solution is usually ignored in such a system. Neglect of the term directly leads to contradiction of the continuity of the Schwarzschild interior and exterior solution expressed in the system.
The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs
Authors: Teufel S. , Dürr D. , Münch-Berndl K.
Published: July 2002

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux-across-surfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V(x) decays like |x|−γ at infinity with γ>n∈N then the generalized eigenfunctions of the corresponding Hamiltonian −1/2Δ+V are n−2 times continuously differentiable with respect to the momentum variable.
On conserved quantities at spatial infinity
Authors: Perng Shyan-Ming
Published: July 2002

There is a well-known short list of asymptotic conserved quantities for a physical system at spatial infinity. We search for new ones. This is carried out within the asymptotic framework of Ashtekar and Romano, in which spatial infinity is represented as a smooth boundary of space–time. We first introduce, for physical fields on space–time, a characterization of their asymptotic behavior as certain fields on this boundary. Conserved quantities at spatial infinity, in turn, are constructed from these fields. We find, in Minkowski space–time, that each of a Klein–Gordon field, a Maxwell field, and a linearized gravitational field yields an entire hierarchy of conserved quantities. Only certain quantities in this hierarchy survive into curved space–time.
On the integrability of nonlinear partial differential equations
Authors: Dorren H. J. S.
Published: July 2002

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by first discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle that can be applied on nonlinearities that have a polynomial form. The method is further illustrated by finding solutions of the nonlinear Schrödinger equation and the vector nonlinear Schrödinger equation, which play an important role in optical fiber communication. Finally, it is shown that the method can also be generalized to higher dimensions.
Isospectral problem for Schrödinger operator: Evolutional viewpoint
Authors: Eleonsky V. M. , Korolev V. G.
Published: July 2002

An isospectral transform of the Schrödinger operator is considered as an evolutional problem. For a transform defined by the McKean–Trubowitz flows associated evolutional equations are derived. It is shown that for one-level and two-level flows these equations can be split into integrable Liouville equations. A relationship between the Liouville equations and the Darboux transforms is discussed; this analysis suggests that the evolutional equations can be split into the Liouville equations in the general case. A Hamiltonian formulation of the isospectral transform defined by the McKean–Trubowitz flows is presented. It is shown that this transform is performed by a canonical change of variables, which is related to the Darboux transform.
On relativistic spin network vertices
Authors: Reisenberger Michael P.
Published: July 2002

Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called “relativistic spin networks” which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the Barrett–Crane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing space-time. It is explained how this model, like the Barret–Crane model can be derived from BF theory, a simple topological field theory [G. Horowitz, Commun. Math. Phys. 125, 417 (1989)], by restricting the sum over histories to ones in which the left-handed and right-handed areas of any 2-surface are equal. It is known that the solutions of classical Euclidean general relativity form a branch of the stationary points of the BF action with respect to variations preserving this condition.
The Jacobi polynomial ensemble and the Painlevé VI equation
Authors: Haine Luc , Semengue Jean-Pierre
Published: July 2002

We consider the Jacobi polynomial ensemble of n×n random matrices. We show that the probability of finding no eigenvalues in the interval [−1,z] for a random matrix chosen from the ensemble, viewed as a function of z, satisfies a second-order differential equation. After a simple change of variable, this equation can be reduced to the Okamoto–Jimbo–Miwa form of the Painlevé VI equation. The result is achieved by a comparison of the Tracy–Widom and the Virasoro approaches to the problem, which both lead to different third-order differential equations. The Virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments, which drastically simplifies the computations.
Operator integrals and phase space observables
Authors: Lahti Pekka , Pellonpää Juha-Pekka , Ylinen Kari
Published: July 2002

The operator integral ∫f dE of a complex valued measurable function f with respect to a positive operator measure E is considered. If F is a Neumark dilation of E into a projection measure, then the “projected” operator integral pr(∫ fdF) is a restriction of the operator ∫ f dE. Necessary and sufficient conditions for the equality pr(∫f dF)=∫ f dE are obtained. The results are applied to determine the moment operators of the phase space observables generated by the number states.
Inverse problem for an inhomogeneous Schrödinger equation
Authors: Ramm A. G.
Published: July 2002

Let (l−k2)u=−u″+q(x)u−k2u=δ(x), x∈R, ∂u/∂|x|−iku→0, |x|→∞. Assume that the potential q(x) is real valued and compactly supported: q(x)=q(x)¯, q(x)=0 for |x|⩾1, ∫−11|q|dx<∞, and that q(x) produces no bound states. Let u(−1,k) and u(1,k), ∀k>0 be the data. It is shown that under the above assumptions these data determine q(x) uniquely.
Gödel-type space–times in induced matter gravity theory
Authors: Carrion H. L. , Rebouças M. J. , Teixeira A. F. F.
Published: July 2002

Five-dimensional (5D) generalized Gödel-type manifolds are examined in the light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by three essential parameters k, m2, and ω: identical triads (k,m2,ω) correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian generalized Gödel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the corresponding Lie algebra of each class are determined. It is shown that the generalized Gödel-type 5D manifolds admit maximal group of isometry Gr with r=7, r=9, or r=15 depending on the essential parameters k, m2, and ω. The breakdown of causality in all these classes of homogeneous Gödel-type manifolds are also examined. It is found that in three out of the six irreducible classes the causality can be violated. The unique generalized Gödel-type solution of the induced matter (IM) field equations is found. The question as to whether the induced matter version of general relativity is an effective therapy for these types of causal anomalies of general relativity is also discussed in connection with a recent work by Romero, Tavakol, and Zalaletdinov.
Self-duality and gravitational perturbations
Authors: Torres del Castillo G. F. , Solı́s-Rodrı́guez H. G.
Published: July 2002

It is shown that there exists a one-to-one correspondence between the real solutions of the Einstein vacuum field equations linearized about the Minkowski metric and the (complex) metric perturbations whose curvature to first order in the metric perturbation is self-dual. It is also shown that the self-duality condition of the curvature to first order in the metric perturbation is equivalent to a set of first-order equations for the metric perturbation whose solution is given by a scalar potential that obeys the wave equation.
Erratum: “Differential geometry on SU(3) with applications to three state systems” [J. Math. Phys. 39, 6125–6136 (1998)]
Authors: Byrd Mark
Published: July 2002

The first equation on page 6127, just above Eq. (2), is the equation for the right invariant vector fields (not the left invariant vector fields). This makes several sign changes in the Appendices that we present here.
Random sampling for the monomer–dimer model on a lattice
Authors: van den Berg J. , Brouwer R.
Published: July 2002

In the monomer–dimer model on a graph, each matching (collection of nonoverlapping edges) M has a probability proportional to λ|M|, where λ>0 is the model parameter, and |M| denotes the number of edges in M. An approximate random sample from the monomer–dimer distribution can be obtained by running an appropriate Markov chain (each step of which involves an elementary local change in the configuration) sufficiently long. Jerrum and Sinclair have shown (roughly speaking) that for an arbitrary graph and fixed λ and ε (the maximal allowed variational distance from the desired distribution), O(|Λ|2|E|) steps suffice, where |E| is the number of edges and |Λ| the number of vertices of the graph. For sufficiently nice subgraphs (e.g., cubes) of the d-dimensional cubic lattice we give an explicit recipe to generate approximate random samples in (asymptotically) significantly fewer steps, namely (for fixed λ and ε) O(|Λ|(ln|Λ|)2).
Structure and properties of Hughston’s stochastic extension of the Schrödinger equation
Authors: Adler Stephen L. , Horwitz Lawrence P.
Published: July 2002

Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston’s equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston’s equation to earlier work on norm-preserving stochastic equations, and show that Hughston’s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston’s, suffices to give an objective interpretation of the measurement process in quantum mechanics.
Spectral zeta functions for a cylinder and a circle
Authors: Nesterenko V. V. , Pirozhenko I. G.
Published: July 2002

Spectral zeta functions ζ(s) for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis −1⩽Re s⩽0. Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced.
Quaternionic symmetry groups and particle multiplets
Authors: Scolarici Giuseppe , Solombrino Luigi
Published: July 2002

We consider the quaternionic complete symmetry group of a massive physical system, obtained extending the connected Poincaré group and the internal symmetry group by means of the CPT and the generalized parity operators. We classify the irreducible Q-representations of this group crossing the generalized Wigner and Frobenius–Schur classifications, and obtain 14 different cases. Some novelties arise in this context, such as the failure of the statement that only irreducible representations must be associated with particle multiplets, and a suggestion on the possible forms of a parity-violating Hamiltonian.
Belinskii–Zakharov formulation for Bianchi models and Painlevé III equation
Authors: Manojlović Nenad , Miković Aleksandar
Published: July 2002

We show that α<0, β>0, γ=δ=0 Painlevé III equation arises as a zero-curvature condition in the Belinskii–Zakharov inverse scattering formulation for Bianchi cosmological models. For special values of the parameters this Painlevé III equation becomes the dynamical equation for Bianchi I, II, VI0, and VII0 models.
Square-integrable imprimitivity systems
Authors: Cassinelli G. , De Vito E. , Levrero A.
Published: July 2002

We give a definition of square-integrability for imprimitivity systems and we prove that the square-integrable ones share most of the properties of square-integrable representations of groups.
Twisting invariance of link polynomials derived from ribbon quasi-Hopf algebras
Authors: Links J. R. , Gould M. D. , Zhang Y.-Z.
Published: July 2002

The construction of link polynomials associated with finite dimensional representations of ribbon quasi-Hopf algebras is discussed in terms of the formulation of an appropriate Markov trace. We then show that this Markov trace is invariant under twisting of the quasi-Hopf structure, which in turn implies twisting invariance of the associated link polynomials.
Asymptotics of minimizers of variational problems involving curl functional
Authors: Pan Xing-Bin , Qi Yuanwei
Published: July 2002

In this paper we are concerned with singularly perturbed variational problems involving the curl functional, which arise in the mathematical theory of liquid crystals. The asymptotic behavior of the minimizers in the singular limiting process is discussed, which is closely related to the variational problems for curl functional under various constraints.
Hamiltonian description of the motion of discontinuity surfaces
Authors: Kats A. V. , Kontorovich V. M.
Published: July 2002

The Hamiltonian description of the motion of arbitrary discontinuity surfaces is proposed on the basis of the variational principle, taking into account the conservation laws in terms of consecutively introduced volume potentials (of the Clebsch type) as constraints. Such a method of introduction of Hamiltonian variables makes it possible to generalize the known canonical variables for the interface between two media to the cases of shock waves and slip surfaces. The results are compared with the introduction of surface Hamiltonian variables through the canonical transformations of bulk Hamiltonian variables. The results permit a direct generalization to the case of magnetohydrodynamics, plasmas, superfluid liquids, and other media for which the bulk Hamiltonian equations are known.
Peculiarities in the electron properties of δ〈Sb〉-layers in epitaxial silicon. III. Electron–phonon relaxation
Authors: Kashirin V. Yu. , Komnik Yu. F. , Anopchenko A. S. , Mironov O. A. , Emeleus C. J. , Whall T. E.
Published: July 2002

Complex studies of weak electron localization, electron–electron interaction, and electron overheating in Si crystals containing a δ〈Sb〉-layer with various concentrations of Sb atoms are carried out in order to obtain information on the characteristic times of inelastic electron relaxation. The temperature dependence of the electron–phonon relaxation time τep derived from the electron overheating effect can be described by the dependence τep∝T−p, where p≅3.7±0.3, which corresponds to the case qTl<1(qT is the wave vector of the thermal phonon and l the electron mean free path).
Relaxation of the electric resistance of YBaCuO single crystals due to hydrostatic pressure and jumpwise temperature variation
Authors: Obolenskiı̆ M. A. , Balla D. D. , Bondarenko A. V. , Vovk R. V. , Prodan A. A. , Ivanova T. F.
Published: July 2002

The effect of hydrostatic pressure on pressure, temperature, and time dependences of the conductivity in the ab-plane of YBa2Cu3O7−x single crystals with different oxygen concentrations (0.1<x<0.5) is studied. It is shown that in oxygen-deficient samples with x>0.2, temperature and pressure variations may induce a nonequilibrium state in which effects associated with the variation of unit cell volume and redistribution of oxygen in Cu–O planes must be distinguished. Characteristic conductivity relaxation times under the effect of pressure and during annealing at room temperature are determined. It is concluded that these processes have the same origin.
Simulations of scalar mixing in grid turbulence using an eddy-damped closure model
Authors: Eswaran V. , O’Brien E. E.
Published: July 2002

A physical-space eddy-damped quasinormal Markovian model for the two-point scalar correlation function is used to simulate wind tunnel experiments on scalar mixing. The simulations closely approximate experimentally observed trends of scalar variance and length scales in both heated-grid and heated-screen experiments. The simplicity of the model should enable its effective use in modeling the analogous turbulent mixing terms in the two-point scalar probability density function equations.