bluuebunny/crossref_metadata_2025 huggingface dataset:
Solvable (nonrelativistic, classical) n-body problems in multidimensions. I
Authors: Calogero F. , Xiaoda Ji
Published: July 2002

Several solvable n-body problems are exhibited. They are characterized by equations of motion of Newtonian type, mjr↘̈j = F↘j, j=1,...,n, with the ‘‘forces’’ F↘j given as explicit functions of the ‘‘particle coordinates’’ r↘k and their velocities r↘̇k, k=1,...,n; the forces F↘j generally also depend on some free parameters, so that in each case there is actually a class of solvable models. The particle coordinates r↘j, and the forces F↘j, are vectors in N-dimensional space, with N=1,2,3. In this article we focus mainly on few-body problems (n=1,2,3,4) in two- and three-dimensional space (N=2,3); all these models are rotation-invariant, and some are also translation-invariant, but they are generally not Galilei-invariant. The solution of the initial-value problem is given in each case. We also exhibit one-dimensional problems (N=1), including cases with n≥4; these models are generally translation-invariant, and some are also Galilei-invariant.
The classification of decoherence functionals: An analog of Gleason’s theorem
Authors: Isham C. J. , Linden N. , Schreckenberg S.
Published: July 2002

Gell-Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where 𝒰𝒫 is the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where 𝒱 is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈𝒫(𝒱) can be written in the form d(α,β)=tr𝒱⊗𝒱(α⊗βX) for some operator X on the tensor-product space 𝒱⊗𝒱.
A renormalization method for the evaluation of lattice sums
Authors: Berman C. Leonard , Greengard Leslie
Published: July 2002

A number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums (sums over all sites of an infinite perfect lattice of some potential energy function). One classical example, the calculation of lattice sums of circular and spherical harmonics, dates back to the last century, to Lord Rayleigh’s work on computing the effective conductivity of a simple composite. While Lord Rayleigh presented an efficient asymptotic method for two-dimensional problems, he resorted to direct evaluation of the lattice sums in the three-dimensional case. More recent methods, based on Ewald summation, have been developed by Nijboer and De Wette, Schmidt and Lee, and others. In this article, a fast method for evaluating lattice sums which is based on a new renormalization identity is described.
The Chern–Simons action in noncommutative geometry
Authors: Chamseddine A. H. , Fröhlich J.
Published: July 2002

A general definition of Chern–Simons actions in noncommutative geometry is proposed and illustrated in several examples. These examples are based on ‘‘space–times’’ which are products of even-dimensional, Riemannian spin manifolds by a discrete (two-point) set. If the * algebras of operators describing the noncommutative spaces are generated by functions over such ‘‘space–times’’ with values in certain Clifford algebras the Chern–Simons actions turn out to be the actions of topological gravity on the even-dimensional spin manifolds. By constraining the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from Chern–Simons actions.
Random walks in noninteger dimension
Authors: Bender Carl M. , Boettcher Stefan , Mead Lawrence R.
Published: July 2002

One can define a random walk on a hypercubic lattice in a space of integer dimension D. For such a process formulas can be derived that express the probability of certain events, such as the chance of returning to the origin after a given number of time steps. These formulas are physically meaningful for integer values of D. However, these formulas are unacceptable as probabilities when continued to noninteger D because they give values that can be greater than 1 or less than 0. In this paper a different kind of random walk is proposed which gives acceptable probabilities for all real values of D. This D-dimensional random walk is defined on a rotationally symmetric geometry consisting of concentric spheres. The exact result is given for the probability of returning to the origin for all values of D in terms of the Riemann zeta function. This result has a number-theoretic interpretation.
Discrete differential manifolds and dynamics on networks
Authors: Dimakis Aristophanes , Müller-Hoissen Folkert , Vanderseypen Francois
Published: July 2002

A discrete differential manifold is a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. Several examples are presented and a notion of differentiability of maps between discrete differential manifolds is introduced. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity.
Gravitational collapse of vacuum gravitational field configurations
Authors: Eardley Douglas M.
Published: July 2002

This paper proves a theorem about the existence of an apparent horizon in general relativity, which applies equally well to vacuum configurations and matter configurations. The theorem uses the reciprocal of the surface-to-volume ratio of a region on a space slice to measure the radius of the region, and uses the minimum value Kmin of certain components of the extrinsic curvature to measure the strength of the gravitational field in the region. The theorem proves that, if the product of the radius times Kmin is larger than unity, then an apparent horizon must form, signaling the formation of a black hole.
On spin-(3/2) systems in Ricci flat space–times
Authors: Frauendiener Jörg
Published: July 2002

The Dirac formulation of massless spin-(3/2) fields is discussed. The existence and uniqueness for the solutions of the spin-(3/2) field equations in Dirac form is proven. It is shown that the system of equations can be split into a symmetric hyperbolic system of evolution equations and a set of constraint equations. The constraints are shown to propagate on a curved manifold if and only if it is an Einstein space. The gauge freedom present in the spin-(3/2) system is discussed and it is shown that the complete system ‘‘solutions modulo gauge’’ has a well posed Cauchy problem if and only if the Einstein equations hold.
Space–time defects
Authors: Letelier Patricio S. , Wang Anzhong
Published: July 2002

The theory of distributions in Riemannian spaces due to Lichnerowicz is used to obtain exact solutions to the Einstein equations for space–times that have null Riemann–Christoffel curvature tensors everywhere except on a hypersurface. The cases of spherically, cylindrically, plane, and axially symmetric space–times in which the matter content of the singular surfaces can be described by a barotropic equation of state are treated in some detail. Solutions with null curvature tensor, except on (a) concentric spheres, (b) concentric cylinders, (c) parallel planes, and (d) parallel discs, are exhibited and studied.
A note on CP
n-instantons and the family’s index theorem
Authors: Scott S. G.
Published: July 2002

For any non-negative integer p we calculate the number of CPn-instantons of charge np+n over a Riemann surface of genus np+p.
Gauge theory on a non-simply connected domain and representations of canonical commutation relations
Authors: Arai Asao
Published: July 2002

A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.
Quantum integrable systems and representations of Lie algebras
Authors: Etingof Pavel I.
Published: July 2002

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra 𝔤𝔩(N) under a certain homomorphism from the center of U(𝔤𝔩(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine 𝔤𝔩(N) at the critical level k=−N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky–Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. A new expression is also given for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of 𝔤𝔩(N).
Oscillatory integrals on Hilbert spaces and Schrödinger equation with magnetic fields
Authors: Albeverio Sergio , Brzeźniak Zdzisl/aw
Published: July 2002

The theory of oscillatory integrals on Hilbert spaces (the mathematical version of ‘‘Feynman path integrals’’) is extended to cover more general integrable functions, preserving the property of the integrals to have converging finite dimensional approximations. An application to the representation of solutions of the time-dependent Schrödinger equation with a scalar and a magnetic potential by oscillatory integrals on Hilbert spaces is given. A relation with Ramer’s functional in the corresponding probabilistic setting is found.
L functions in scattering on p-adic multiloop surfaces
Authors: Chekhov L.
Published: July 2002

The scattering processes on multiloop infinite graphs with one and the same valence p+1 of all vertices are studied herein. These graphs are discrete spaces of a constant negative curvature since they can be interpreted as quotients of the p-adic hyperbolic plane over free acting discrete subgroups of the projective group PGL(2,Qp). They are, in fact, identical to p-adic multiloop surfaces. Releasing from a graph its subgraph containing all loops, which is called the reduced graph Tred, we can get L functions corresponding to these finite closed graphs. For an infinite graph, the notion of spherical functions is introduced. These are the eigenfunctions of a discrete Laplace operator acting on the graph. s-wave scattering processes are considered and in this way they define the scattering amplitudes ci. In the discrete case, their number coincides with ‖Tred‖—the number of vertices of the reduced graph. Taking the product over all ci, we obtain the determinant of the scattering matrix which turns out to be presented as a ratio of two L functions: C∼L(α+)/L(α−). Here the L function is the Ihara–Selberg function depending only on the form of Tred. All the dependence on the initial value of p remains in arguments α± = t/2p ±√t2/4p2−1/p, t being the eigenvalue of the Laplacian. The Hashimoto–Bass theorem is used for expressing L function L(u) of any finite graph via the determinant of a local operator Δ(u) acting on this graph.
Path integrals in polar variables with spontaneously broken symmetry
Authors: Friedberg Richard
Published: July 2002

The path-integral formalism is applied to a two-dimensional complex quantum field theory with a normal-ordered Hamiltonian and spontaneously broken symmetry in the classical ground state. In most systems the spurious high-frequency divergences arising naively from the path-integral treatment can be avoided by deriving this treatment carefully from the operator formalism. But here, in order to justify the Gaussian approximation at the one-loop order, one needs to express the complex field in polar variables. To avoid having to deal with the polar form of an operator conventional holomorphic variables are used to derive a path-integral formulation entirely in terms of a c-number complex field in position space. Then the holomorphic variables are dropped, the c-number field is re-expressed in polar form, and Fourier analysis is applied to the polar variables. In the one-loop approximation the usual grand partition function is recovered with the first two ultraviolet divergences already subtracted out from the sum over zero-point energies.
(2,0) superconformal anomaly
Authors: Lhallabi T.
Published: July 2002

The (2,0) supersymmetric Wess–Zumino–Polyakov action is constructed and the (2,0) superconformal anomaly is given. The anomalous Ward-identity in the right sector is derived and the known operator product expansion of the N=2 superstress energy tensor is recovered.
Planar coincidences for N-fold symmetry
Authors: Pleasants Peter A. B. , Baake Michael , Roth Johannes
Published: July 2002

The coincidence problem for planar patterns with N-fold symmetry is considered. For the N-fold symmetric module with N<46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case N≥46 is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.
Generalization of the Calogero–Cohn bound on the number of bound states
Authors: Chadan K. , Kobayashi R. , Martin A. , Stubbe J.
Published: July 2002

It is shown that for the Calogero–Cohn-type upper bounds on the number of bound states of a negative spherically symmetric potential V(r), in each angular momentum state, that is, bounds containing only the integral ∫∞0‖V(r)‖1/2 dr, the condition V′(r)≥0 is not necessary, and can be replaced by the less stringent condition (d/dr)[r1−2p(−V)1−p]≤0, 1/2≤p<1, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on p and l, and tend to the standard value for p=1/2.
Stationary problems for equation of the KdV type and dynamical r-matrices
Authors: Kulish P. P. , Rauch-Wojciechowski S. , Tsiganov A. V.
Published: July 2002

We study a new quite general family of dynamical r-matrices for an auxiliary loop algebra ℒ(su(2)) related to restricted flows for equations of the KdV type. This underlying r-matrix structure allows us to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems.
Identifying conical singularities
Authors: Oliveira-Neto G.
Published: July 2002

A method based upon the concept of holonomy of a metric space–time (M,g), in order to identify the presence of conical singularities in M is proposed. The validity and usefulness of this so-called holonomy method is proven by applying it to a set of four-dimensional space–times and one three-dimensional space–time. The holonomy method predictions are confirmed by the comparison with the predictions obtained after coordinate transformations which take the metrics g, to a new basis where the global properties of conical singularities are explicitly seen.
Quantization of symplectic tori in a real polarization
Authors: Manoliu Mihaela
Published: July 2002

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov–Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a four-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.
Coupling coefficients for Lie algebra representations and addition formulas for special functions
Authors: Van der Jeugt J.
Published: July 2002

Representations of the Lie algebra su(1,1) and of a generalization of the oscillator algebra, b(1), are considered. The paper then introduces polynomials which are related by the coupling (or Clebsch–Gordan) coefficients of the Lie algebra in question; by making a proper choice, these polynomials themselves are related to known special functions. The coupling of two or three representations of the Lie algebra then leads to interesting addition formulas for these special functions. The polynomials appearing here are generalized Laguerre and Jacobi polynomials for the su(1,1) case, and Hermite polynomials for the b(1) algebra.
Many-body Wigner quantum systems
Authors: Palev T. D. , Stoilova N. I.
Published: July 2002

We present examples of many-body Wigner quantum systems. The position and the momentum operators RA and PA, A=1,…,n+1, of the particles are noncanonical and are chosen so that the Heisenberg and the Hamiltonian equations are identical. The spectrum of the energy with respect to the center of mass is equidistant and has finite number of energy levels. The composite system is spread in a small volume around the center of mass and within it the geometry is noncommutative. The underlying statistics is an exclusion statistics.
The BRST operator of quantum symmetries: The quantum analogs of Donaldson invariants
Authors: Hammou A. B. , Lagraa M.
Published: July 2002

We construct the BRST operator of quantum symmetries and show that the nilpotency of this operator can either be derived from the Hopf axiom structure of the quantum group symmetries or from the Jacobi identity of their quantum Lie algebra. We extend this BRST operator to the topological transformations and we investigate the properties of invariant polynomials of curvatures from which we derive the descent equations for Donaldson invariants.
A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic
Authors: Calogero F.
Published: July 2002

We show that the dynamical system characterized by the (complex) equations of motion q̈j+iΩq̇j=∑k=1,k≠jnq̇jq̇kf(qj−qk), j=1,…,n, with f(x)=−λ℘′(λx)/[℘(λx)−℘(λμ)], is Hamiltonian and integrable, and we conjecture that all its solutions qj(t), j=1,…,n are completely periodic, with a period that is a finite integral multiple of T=2π/Ω. Here n is an arbitrary positive integer, Ω is an arbitrary (nonvanishing) real constant, ℘(y)≡℘(y|ω,ω′) is the Weierstrass function (with arbitrary semiperiods ω,ω′), and λ,μ are two arbitrary constants; special cases are f(x)=2λ coth(λx)/[1+r2 sinh2(λx)], f(x)=2λ coth(λx), f(x)=2λ/sinh(λx), f(x)=2/[x(1+λ2x2)], and of course f(x)=2/x. These findings, as well as the conjecture (which is shown to be true in some of these special cases), are based on the possibility to recast these equations of motion in the modified Lax form L̳̇+iΩL̳=[L̳,M̳] with L̳ and M̳ appropriate (n×n)-matrix functions of the n dynamical variables qj and of their time-derivatives q̇j.
Moving frames hierarchy and BF theory
Authors: Lee Jyh-Hao , Pashaev Oktay K.
Published: July 2002

We show that the one-dimensional projection of Chern–Simons gauged nonlinear Schrödinger model is equivalent to an Abelian gauge field theory of a continuum Heisenberg spin chain. In such a theory, the matter field has geometrical meaning of coordinates in tangent plane to the spin phase space, while the U(1) gauge symmetry relates to rotation in the plane. This allows us to construct the infinite hierarchy of integrable gauge theories and related magnetic models. To each of them a U(1) invariant gauge fixing constraint of non-Abelian BF theory is derived. The corresponding moving frames hierarchy is obtained and the spectral parameter is interpreted as a constant-valued statistical gauge potential constrained by the 1-cocycle condition.
Representation properties, Racah sum rule, and Biedenharn–Elliott identity for Uq(osp(1|2))
Authors: Minnaert Pierre , Mozrzymas Marek
Published: July 2002

It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.
Proof of Serban’s conjecture
Authors: Bergère M. C.
Published: July 2002

We prove Serban’s conjecture which greatly simplifies the expression of the advanced single-particle Green function in the Calogero–Sutherland model. The importance of proving this conjecture is that it reorganizes the form factor in terms of two dimensional Coulomb gas correlators and confirms the possible existence of a bosonization procedure for this system.
The Dirac particle in the presence of the tensor wave on the background of gravitation
Authors: Shishkin G. V. , Abdel Salam W. H.
Published: July 2002

In this paper new exact solutions for the Dirac equation are presented for the case of a particle in tensor wave field on the background of gravitation. Consideration is given to different configurations introducing the tensor interaction, which result in the required separation of variables by the algebraic method and lead to the derivation of exact solutions for the Dirac equation in the corresponding field situation. At the same time, most general metrics allowing for the exact solutions are given. The cases of zero and nonzero rest mass of a particle are with treated here.
On an easy transition from operator dynamics to generating functionals by Clifford algebras
Authors: Fauser Bertfried
Published: July 2002

Clifford geometric algebras of multivectors are treated in detail. These algebras are built over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear that opposite Clifford algebras have to be used for obtaining all endomorphisms. Based on these mathematics, we give a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture, relying on geometry, does not need positivity in principle. Furthermore, we propose a transition from operator dynamics to generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerably short compared to standard ones. The transition is not injective (unique) and depends additionally on the choice of an ordering. We obtain a direct and constructive connection between orderings and the explicit form of the functional Hamiltonian. These orderings depend on the propagator of the theory and thus on the ground state. This is invisible in path integral formulations. The method is demonstrated within two examples, a nonlinear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases.
Extension of Bertrand’s theorem and factorization of the radial Schrödinger equation
Authors: Wu Zuo-Bing , Zeng Jin-Yan
Published: July 2002

The Bertrand’s theorem is extended, i.e., closed orbits still may exist for central potentials other than the power law Coulomb potential and isotropic harmonic oscillator. It is shown that for the combined potential V(r)=W(r)+b/r2(W(r)=arν), when (and only when) W(r) is the Coulomb potential or isotropic harmonic oscillator, closed orbits still exist for suitable angular momentum. The correspondence between the closeness of classical orbits and the existence of raising and lowering operators derived from the factorization of the radial Schrödinger equation is investigated.
Symmetries of asymptotically flat electrovacuum space–times and radiation
Authors: Bičák J. , Pravdová A.
Published: July 2002

Symmetries compatible with asymptotic flatness and admitting gravitational and electromagnetic radiation are studied by using the Bondi–Sachs–van der Burg formalism. It is shown that in axially symmetric electrovacuum space–times in which at least locally a smooth null infinity in the sense of Penrose exists, the only second allowable symmetry is either the translational symmetry or the boost symmetry. Translationally invariant space–times with, in general, a straight “cosmic string” along the axis of symmetry are nonradiative although they can have a nonvanishing news function. The boost-rotation symmetric space–times are radiative. They describe “uniformly accelerated charged particles” or black holes which in general may also be rotating—the axial and an additional Killing vector are not assumed to be hypersurface orthogonal. The general functional forms of both gravitational and electromagnetic news functions, and of the mass aspect and total mass of asymptotically flat boost-rotation symmetric space–times at null infinity are obtained. The expressions for the mass are new even in the case of vacuum boost-rotation symmetric space–times with hypersurface orthogonal Killing vectors. In the Appendices some errors appearing in previous works are corrected.
On infravacua and the localization of sectors
Authors: Kunhardt Walter
Published: July 2002

A certain class of superselection sectors of the free massless scalar field in three space dimensions is considered. It is shown that these sectors, which cannot be localized with respect to the vacuum, acquire a much better localization, namely in spacelike cones, when viewed in front of suitable “infravacuum” backgrounds. These background states coincide, essentially, with a class of states introduced by Kraus, Polley, and Reents as models for clouds of infrared radiation.
On the reproducing kernel of the Segal-Bargmann space
Authors: Sontz Stephen Bruce
Published: July 2002

This article revolves around the properties on the Lp scale of spaces of the integral kernel operator K whose kernel function is the reproducing kernel of the Segal-Bargmann space. We find sufficient conditions on p and q for K to be a Hille-Tamarkin (and hence compact) operator from Lp to Lq with respect to the standard Gaussian measure as well as with respect to a weighted measure on the codomain space. We also find sufficient conditions for K to be unbounded with respect to the standard Gaussian measure. Finally we give sufficent conditions for a Toeplitz operator to be Hille-Tamarkin on the Lp scale of spaces with respect to both the standard Gaussian measure and a weighted measure on the codomain space.
A braided interpretation of fractional supersymmetry in higher dimensions
Authors: Dunne R. S.
Published: July 2002

A many variable q-calculus is introduced using the formalism of braided covector algebras. Its properties are discussed in detail and related to fractional supersymmetry when certain of its deformation parameters are roots of unity. The special cases of two-dimensional supersymmetry and fractional supersymmetry are developed in detail.
Grassmann algebra and fermions at finite temperature
Authors: Charret I. C. , Corrêa Silva E. V. , de Souza S. M. , Rojas Santos O. , Thomaz M. T.
Published: July 2002

For any d-dimensional self-interacting fermionic model, all coefficients in the high-temperature expansion of its grand canonical partition function can be put in terms of multivariable Grassmann integrals. A new approach to calculate such coefficients, based on direct exploitation of the Grassmannian nature of fermionic operators, is presented. We apply the method to the soluble Hatsugai–Kohmoto model, reobtaining well-known results.
On the universality of almost every quantum logic gate
Authors: Weaver Nik
Published: July 2002

Lloyd [Phys. Rev. Lett. 75, 346 (1995)] showed that almost every quantum logic gate is universal in the sense that it can be used to approximate any unitary transformation. The argument relied on a more general fact whose proof was not given in detail. We give a complete proof of this more general fact.
Singular sector of the Kadomtsev–Petviashvili hierarchy, ∂̄ operators of nonzero index, and associated integrable systems
Authors: Konopelchenko Boris G. , Alonso Luis Martı́nez , Medina Elena
Published: July 2002

Integrable hierarchies associated with the singular sector of the Kadomtsev–Petviashvili (KP) hierarchy, or equivalently, with ∂̄ operators of nonzero index are studied. They arise as the restriction of the standard KP hierarchy to submanifolds of finite codimension in the space of independent variables. For higher ∂̄ index these hierarchies represent themselves as families of multidimensional equations with multidimensional constraints. The ∂̄-dressing method is used to construct these hierarchies. Hidden Korteweg–de Vries, Boussinesq, and hidden Gelfand–Dikii hierarchies are considered, too.
Semiclassical limit for the Binegar–Zierau quantization of the minimal nilpotent orbits of SOo(2p,2)
Authors: Mehdi Salah
Published: July 2002

Let n be an odd integer greater than 3 and X be the (n+1)-dimensional anti-de Sitter space–time. Binegar and Zierau have constructed in Commun. Math. Phys. 138, 245–258 (1991) a unitary representation H=H+⊕H− of the conformal group SOo(n+1,2) of X. They have shown that this particular representation is, in a certain algebraic sense, a quantization of the union, On+1min=On+1min+∪On+1min−, of the two minimal nilpotent orbits in so(n+1,2) [Commun. Math. Phys. 138, 245–258 (1991)]. (This work is part of the author’s Ph.D. thesis.) In spite of this, it is known how to obtain ℋ from On+1min using a known quantization procedure. One reason for this interest in ℋ is that, in the case n=3, H+ is the representation carried by the one-particle sector of the massless scalar field on the anti-de Sitter space–time X. In this paper, we strengthen this link between the coadjoint orbits On+1min± and H±, by studying the semiclassical limit of the latter. In this way, we clarify their appearance in the massless theory and corroborate the existing evidence that H± is the “correct” quantization of On+1min±. As a preliminary, we show that the projection onto so(n,2)* of On+1min± is the union of the two coadjoint orbits Ono± and Onmin± (Proposition IV.1); one of those is the phase space of the classical massless particle on X. We then show (Theorem VI.1 and Corollary VI.3) that the semiclassical limit of the spectral counting function of the generator of the SO(2) subgroup of SOo(n,2) in the representation H± is dominated by a Weyl term, expressed naturally in terms of the symplectic volume of a compact portion of the classical phase space Ono±. Furthermore, we show (Theorem VI.4) that the highest weight vectors of the representation coincide in the semiclassical limit with the BKW functions constructed starting from Ono±. We show in addition that, even though the orbit method applied to On+1min± does not yield H±, it nevertheless establishes a natural relation between them. Namely, the simple SO(n+1)×S0(2)-modules appearing in H+ are those we obtain if we apply the orbit method to integral SO(n+1)×S0(2)-orbits contained in the projection on (so(n+1)⊕so(2))* of On+1min± (Sec. VII). As a by-product of our analysis, we study the restriction to SOo(n,2) of ℋ (Proposition VIII.1) and we show that the unitary structure on ℋ is exactly a Klein–Gordon scalar product on X (Proposition IX.2).
Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations
Authors: Forger Michael , Sachse Sebastian
Published: July 2002

It has been proposed by Hornos and Hornos [Phys. Rev. Lett. 71, 4401–4404 (1993)] that the degeneracy of the genetic code, i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In their work, this picture was developed in the framework of simple Lie algebras. Here, we explore the possibility of explaining the degeneracy of the genetic code using basic classical Lie superalgebras, whose representation theory is sufficiently well understood, at least as far as typical representations are concerned. In the present paper, we give the complete list of all typical codon representations (typical 64-dimensional irreducible representations), whereas in the second part, we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code.
On the nonintegrability of the generalized van der Waals Hamiltonian system
Authors: Mondéjar Félix , Ferrer Sebastián
Published: July 2002

In this paper we aim to prove that, except for the three known cases, the uniparametric family of Hamiltonian systems defined by the generalized van der Waals potential is nonintegrable in the Liouville–Arnold sense. The proof is based on the theorem of Morales and Ramis about nonintegrability by differential Galois theory.
Finite energy–action solutions of p1 Yang–Mills equations on p2 Schwarzschild and deSitter backgrounds in dimensions d⩾4
Authors: Brihaye Y. , Chakrabarti A. , Tchrakian D. H.
Published: July 2002

Physically relevant gauge and gravitational theories can be seen as special members of hierarchies of more elaborate systems. The Yang–Mills (YM) system is the first member of a hierarchy of Lagrangians which we will index by p1, and the Einstein–Hilbert (EH) system of general relativity is the first member of another hierarchy which we index by p2. In this paper, we study the classical equations of the p1=1,2 YM hierarchy considered in the background of special geometries (Schwarzschild, deSitter, anti-deSitter) of the p2=1,2,3 EH hierarchy. Solutions are obtained in various dimensions and lead to several examples of nonself-dual YM fields. When p1=p2 self-dual solutions exist in addition. Their action is equal to the Chern–Pontryagin charge and can be compared with that of the nonself-dual solutions.
Critical phenomena in nonlinear sigma models
Authors: Liebling Steven L. , Hirschmann Eric W. , Isenberg James
Published: July 2002

We consider solutions to the nonlinear sigma model (wave maps) with target space S3 and base space 3+1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular solutions. For families of solutions with localized spatial support a self-similar solution is found at the boundary. For other families, we find that a static solution appears to sit at the boundary. This behavior is compared to the black hole critical phenomena found by Choptuik.
On the Lie–Poisson structure of the nonlinearized discrete eigenvalue problem
Authors: Wu Yongtang , Du Dianlou
Published: July 2002

A 3×3 discrete eigenvalue problem and corresponding discrete soliton equations are proposed. Under a constraint between the potentials and eigenfunctions, the 3×3 discrete eigenvalue problem is nonlinearized into an integrable Poisson map with a Lie–Poisson structure. Further, a reduction of the Lie–Poisson structure on the co-adjoint orbit yields the standard symplectic structure. The Poisson map is reduced to the usual symplectic map when it is restricted on the leaves of the symplectic foliation.
Relaxation and recombination channels of formation of luminescent states in a neon crystal: Evidence of electron self-trapping
Authors: Belov A. G. , Gorbulin G. M. , Fugol’ I. Ya. , Yurtaeva E. M.
Published: July 2002

The relation between the temperature and the intensity distribution in luminescence bands in VUV and VIS spectral regions of Ne crystals is investigated. The measurements are made by using cathodoluminescent spectroscopy with steady-state excitation and are supplemented by observations of luminescence decay in perfect pure crystals, in samples with defects, and in solid solutions with electronegative (oxygen) and electropositive (xenon) impurities. The contribution of the recombination channel to the formation of the spectrum of self-trapped states in solid neon is determined. It is shown that this channel plays a decisive role in the population of 3p(3p) luminescence centers and makes a relatively small contribution to the population of the lowermost 3s(3s) excitations. The recombination channel efficiency strongly depends on temperature, increasing abruptly with the temperature value. An analysis of experimental data leads to the conclusion on the possibility of electron self-trapping in Ne cryocrystals, followed by the formation of shallow self-trapped states.
Influence of the shape of the electrodes on the tunnel current
Authors: Keijsers R. J. P. , Voets J. , Shklyarevskii O. I. , van Kempen H.
Published: July 2002

The tunnel resistance of highly stable, mechanically controlled break junctions of Al, Au, Cu, Pb, Ni, Pt, and Pt–Ir, have been recorded as a function of the electrode spacing over 6–7 decades. Clear deviations from the expected exponential behavior have been observed. Comparison with previous experimental and theoretical studies indicate that the discussed deviations in some cases are most probably due to the shape of, rather than to interactions between, the two electrodes.
The absolute-convective transition in subsonic mixing layers
Authors: Pavithran S. , Redekopp L. G.
Published: July 2002

The stability properties of subsonic mixing layers described by hyperbolic-tangent profiles of both velocity and temperature are studied. The impulse response of the flow is examined to classify the instability as being of either absolute or convective type. The boundary of the absolute-convective transition is defined as a function of the velocity ratio, the temperature ratio, and the Mach number.
Structure of Lagrangian turbulence
Authors: Viecelli James A.
Published: July 2002

A structure function describing the geometrical properties of Lagrangian turbulence is proposed. This function yields the asymptotic mixing properties of inertial range turbulence. The form of the structure function is deduced from Richardson’s particle dispersion law. Scaling relationships for the mixing volume fraction, the surface area of boundaries between materials, and the chord lengths defined by the intersection points of a ray with material boundaries are obtained. Scaling laws are also deduced for the area fraction, perimeter length, and chord lengths in a two-dimensional section, and for the trajectory of a single particle. The predictions for the two-dimensional case have been verified in several examples of mixing, using data obtained from numerical integration of the Navier–Stokes equations.
Stability of core-annular flow in a rotating pipe
Authors: Hu Howard H. , Joseph Daniel D.
Published: July 2002

The linear stability of core-annular flow in rotating pipes is analyzed. Attention is focused on the effects of rotating the pipe and the difference in density of the two fluids. Both axisymmetric and nonaxisymmetric disturbances are considered. Major effects of the viscosity ratio, interfacial tension, radius ratio, and Reynolds number are included. It is found that for two fluids of equal density the rotation of the pipe stabilizes the axisymmetric (n=0) modes of disturbances and destabilizes the nonaxisymmetric modes. Except for small ℛ, where the axisymmetric capillary instability is dominant, the first azimuthal mode of disturbance ‖n‖=1 is the most unstable. When the heavier fluid is outside centripetal acceleration of the fluid in the rotating pipe is stabilizing; there exists a critical rotating speed above which the flow is stabilized against capillary instability for certain range of small ℛ. When the lighter fluid is outside the flow is always unstable.
Comparison of closure to spectral-based large eddy simulations
Authors: Herring Jackson R.
Published: July 2002

A comparison of closure [in the guise of the test field model (TFM)] to the recent spectral large eddy simulation (LES) of Lesieur and Rogallo [Phys. Fluids A 1, 718 (1989)] is made with the primary goal of examining the closure’s verisimilitude. The spectral LES may be viewed as a direct numerical simulation (DNS) with hyperviscosity, designed to give insight into high Reynolds and Pèclet number flows. The results indicate fairly good agreement between TFM and LES—at least for a particular choice of parameters entering the theory. This is counter to the comparison of asymptotically high Reynolds number closure with LES discussed by Lesieur and Rogallo. Finally, the TFM calculations are calculated to much longer times than possible for the LES and the results are used as a basis to comment on possible long-time trends.
Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio
Authors: Finlay W. H. , Nandakumar K.
Published: July 2002

The development of streamwise-oriented, symmetric, two-dimensional vortices in curved channels of large aspect ratios is studied near the threshold of the first centrifugal instability. The nonlinear equations of motion governing the two-dimensional, stationary flows are solved numerically over a range of parameter values. The dynamical parameter is the normalized streamwise pressure gradient defined as ε=[(∂p/∂x)−(∂p/∂x)c]/(∂p/∂x)c, where (∂p/∂x)c is the critical value for the infinite geometry. The development of interior cells and the selection of the wavelength as ε is gradually increased through zero is quite different from that observed in Taylor vortex flow. For pressure gradients of 0.5% and 1.0% (ε=0.005, 0.01) above critical, the interior cells begin to grow spontaneously and are strongest in the middle of the channel. Unlike the interior cells in Taylor vortex flow, they are only weakly coupled to the end cells. The end cells (or Ekman vortices) are also found to be anomalously long.

As ε is increased further to 0.04 and 0.07, the amplitude and wavelength of the interior cells are more uniform. There is, however, a complex interaction between the Ekman vortices and their neighboring interior cells, often resulting in the formation of additional cells in that region. Next, a simple Ginzburg–Landau (GL) model is tested for weakly nonlinear, two-dimensional vortices. The coefficient in the steady form of this equation is evaluated for a wide parameter range using high accuracy calculations of infinite aspect ratio neutral stability curves. (When suitably normalized, neutral stability curves are found to vary only a little with radius ratio.) For large aspect ratio curved channels, predictions from the model are compared with results from numerical simulation. The variation with ε of vortex amplitude near the center of the duct is correctly predicted by the Ginzburg–Landau equation. For given ε, however, agreement of the spanwise variation of vortex amplitude and spacing between the numerical simulation and the model is not obtained. The development of consistent amplitude equations and boundary conditions that link the interior flow to the boundary is expected to be a challenging task.
Absolute/convective instabilities and the convective Mach number in a compressible mixing layer
Authors: Jackson T. L. , Grosch C. E.
Published: July 2002

In this paper two aspects of the stability of a compressible mixing layer are considered: absolute/convective instability and the convective Mach number. It is shown that, for Mach numbers less than unity, the compressible mixing layer is convectively unstable unless there is an appreciable amount of backflow. A rigorous derivation of a convective Mach number based on linear stability theory for the flow of a multispecies gas in a mixing layer is also presented. In particular, the definition is based on the free-stream Mach number in the laboratory frame and is independent of the speed of the large-scale structures and the speed of the most unstable wave. The result is compared with the heuristic definitions of others and to selected experimental results.
Some consequences of the boundedness of scalar fluctuations
Authors: Dimotakis Paul E. , Miller Paul L.
Published: July 2002

Values of the scalar field c(x,t), if initially bounded, will always be bounded by the limits set by the initial conditions. This observation permits the maximum variance ∼(c′2) to be computed as a function of the mean value c̄. It is argued that this maximum should be expected in the limit of infinite Schmidt numbers (zero scalar species diffusivity). This suggests that c′/ c̄ on the axis of turbulent jets, for example, may not tend to a constant, i.e., independent of x/d, in the limit of very large Schmidt numbers. It also underscores a difficulty with the k−1 scalar spectrum proposed by Batchelor [J. Fluid Mech. 5, 113 (1959)].
A numerical study of bubble interactions in Rayleigh–Taylor instability for compressible fluids
Authors: Glimm J. , Li X. L. , Menikoff R. , Sharp D. H. , Zhang Q.
Published: July 2002

The late nonlinear and chaotic stage of Rayleigh–Taylor instability is characterized by the evolution of bubbles of the light fluid and spikes of the heavy fluid, each penetrating into the other phase. This paper is focused on the numerical study of bubble interactions and their effect on the statistical behavior and evolution of the bubble envelope. Compressible fluids described by the two-fluid Euler equations are considered and the front tracking method for numerical simulation of these equations is used. Two major phenomena are studied. One is the dynamics of the bubbles in a chaotic environment and the interaction among neighboring bubbles. Another one is the acceleration of the overall bubble envelope, which is a statistical consequence of the interactions of bubbles. The main result is a consistent analysis, at least in the approximately incompressible case of these two phenomena. The consistency encompasses the analysis of experiments, numerical simulation, simple theoretical models, and variation of parameters. Numerical simulation results that are in quantitative agreement with laboratory experiment for one-and-one-half (1 (1)/(2) ) generations of bubble merger are presented. To the authors’ knowledge, computations of this accuracy have not previously been obtained.
Three-dimensional instability of rotating flows with oscillating axial strain
Authors: Mansour Nagi N. , Lundgren Thomas S.
Published: July 2002

The equations of motion for perturbed uniformly rotating flows with uniform axial-time periodic strain, are derived from the Navier–Stokes equations in the low Mach number limit. The perturbation equations admit exponentially growing three-dimensional solutions for which the amplification factors per period are computed for a range of compression and swirl ratios. It is found that for a given compression ratio, the flow is stable for low swirl ratios, but at high swirl ratios the flow is unstable with the amplification factor dependent on wave angle but independent of wavelength. For an unstable swirl ratio, higher compression ratios yield larger amplification factors.