bluuebunny/crossref_metadata_2025 huggingface dataset:
Geometric derivation of the kinetic energy in collective models
Authors: Rowe D. J. , Rosensteel G.
Published: March 2003

A separation of the many-particle kinetic energy into collective and intrinsic components is shown to result simply from a general form of the Laplace–Beltrami operator. The geometric structure of the decomposition is thereby clearly exhibited and the intricate computations previously necessary are eliminated.
Szegö’s theorem for Hankel determinants
Authors: Geronimo J. S.
Published: March 2003

An analog of Szegö’s formula for asymptotic Töeplitz determinants is proved for Hankel determinants. The proof uses a set of recurrence formulas developed for polynomials orthogonal on a finite segment of the real line and some properties of the Jost function associated with those polynomials. The techniques of inverse scattering theory are used to calculate the correction terms to the asymptotic formula. The results are valid for weight functions that have a finite number of jump points and an absolutely continuous part that is unbounded.
The master analytic function and the Lorentz group. III. Coupling of continuous representations of O(2,1)
Authors: Basu Debabrata , Majumdar S. Datta
Published: March 2003

The Clebsch–Gordan problem for continuous representations belonging to the principal series of O(2,1) is treated by a method developed previously for the coupling of a discrete and a continuous representation. The values of the complex variable x occurring in the fundamental differential equation of the problem are restricted to lie on the unit circle, and the Clebsh–Gordan coefficients are identified with the Fourier coefficients of solutions of this equation. If j belongs to the discrete class there is only one acceptable solution of the second order equation. But, if j1,j2,j all belong to the continuous class any two independent solutions of the equation give a possible Clebsch–Gordan series. The problem of orthogonalizing the solutions in the latter case is solved and the normalization factor is determined using the Sturm–Liouville theory of differential equations. The Clebsch–Gordan coefficients generated by an orthogonal pair of solutions become automatically orthogonal. To determine the j values appearing in the reduction, a product state xm2 is expanded in a series of the coupled states gjm(x) by means of the Burchnall–Chaundy formula followed by the Sommerfeld–Watson transformation.
Tensor harmonics in the helicity basis
Authors: Akyeampong D. A.
Published: March 2003

Instead of the spherical basis vectors used to study tensor harmonics, we propose the use of the helicity basis. The rank-j tensor harmonics in this basis are explicitly worked out and their properties exhibited.
Resonances in one-dimensional Stark effect and continued fractions
Authors: Graffi S. , Grecchi V. , Levoni S. , Maioli M.
Published: March 2003

The Stieltjes type continued fraction (i.e., any diagonal Padé approximants sequence) of the perturbation series for the resonances of the so-called one-dimensional Stark effect converges to the resonances.
Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements
Authors: Maekawa Takayoshi
Published: March 2003

The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.
Relations among generalized Korteweg–deVries systems
Authors: Symes W.
Published: March 2003

This report presents certain relations among the completely integrable Hamiltonian systems introduced by Gel’fand and Dikii. These relations generalize a formula of A. Lenard linking the higher-order Korteweg–deVries equations, of which the Gel’fand–Dikii systems are a generalization. The general form of the relations, which connect the various isospectral deformations of linear differential operators, is described, and two examples are given explicitly.
The spectral properties of many-electron atomic Hamiltonians and the method of configuration interaction. I. Proof of convergence of the configuration interaction method
Authors: Choudhury M. H. , Pearson D. B.
Published: March 2003

In the method of configuration interaction for atomic structures, the Schrödinger equation for a many electron atom is reduced to an infinite system of linear equations. The eigenvalue problem associated with the finite system of equations obtained by truncating the infinite system is solved to obtain energy eigenvalues and eigenvectors. It is generally assumed that the truncation procedure is a convergent one in the sense that as one increases the size of the truncated equations, the number of eigenvalues and eigenvectors will increase and tend to those of the original infinite set. It is shown that the method of configuration interaction is a convergent procedure in the sense that given any point E in the spectrum of the Schrödinger Hamiltonian of the many-electron system, there exists an eigenvalue of the truncated matrix which is arbitrarily close to E for a sufficiently large size of the truncated matrix. Further, it is shown that the convergence referred to cannot be uniform.
Gribov degeneracies: Coulomb gauge conditions and initial value constraints
Authors: Moncrief Vincent
Published: March 2003

We discuss, using suitable function spaces, several features of the Gribov degeneracies of non-Abelian gauge theory. We show that the set of degenerate transverse potentials can be expected to fill entire neighborhoods in the space of transverse potentials. Specifically we show that if a transverse potential Ā1 sufficiently near Ā=0 has a Gribov copy Ā0 then in fact there is a whole neighborhood of Ā0 (in the transverse subspace) filled with Gribov copies of transverse potentials near Ā1. This means that degenerate potentials can be expected to have nonvanishing measure in path integral quantization. We also show how the breakdown of the canonical technique for solving the initial value constraint equations can be circumvented by using a covariant, noncanonical decomposition of the space of electric fields. We prove that the constraint subset of phase space is in fact a submanifold and establish a potentially useful orthogonal decomposition of its tangent space at any point.
A note on the iteration of the Chandrasekhar nonlinear H-equation
Authors: Bowden R. L.
Published: March 2003

An iteration scheme to solve the Chandrasekhar H equation in the form H (μ) ={1−μ ℱ 10[Ψ (s) H (s)]/(s +μ) ds}−1 is shown to converge monotonically and uniformly.
On the quantization of spin systems and Fermi systems
Authors: Combe Ph. , Rodriguez R. , Sirugue-Collin M. , Sirugue M.
Published: March 2003

It is shown that spin operators and Fermi operators can be interpreted as the Weyl quantization of some functions on a ’’classical phase space’’ which is a compact group. Moreover the transition from quantum spin to Fermi operators is an isomorphism of the ’’classical phase space’’ preserving the Haar measure.
Electromagnetic quadripotential for the pure-radiation field generated by a classical charged point-particle
Authors: Miglietta Francesco
Published: March 2003

We obtain a simple expression for the electromagnetic quadripotential corresponding to the pure-radiation field generated by a classical charged point-particle. The solution does not satisfy the Lorentz condition, and has interesting properties. It propagates inside the light cone of the particle and has a discontinuity across the sheet of the light cone itself. This discontinuity is responsible for the correct propagation of the electromagnetic effects with the velocity of light. A similar result is also obtained for the velocity-dependent field.
On the complete integrability of the stationary, axially symmetric Einstein equations
Authors: Maison Dieter
Published: March 2003

A linear eigenvalue problem in the spirit of Lax is constructed for the nonlinear differential equations describing stationary, axially symmetric Einstein spaces. In suitable variables these equations yield a generalization of the well-known sine-Gordon equation. The similarity of the system to the nonlinear σ model is pointed out.
S-wave off-shell T and K matrices for the Yukawa potential by Ecker–Weizel approximations
Authors: Das U. , Chakravarty S. , Talukdar B.
Published: March 2003

Expressions for s-state off-shell wavefunctions associated with outgoing and standing wave boundary conditions are derived for the Yukawa potential by using Ecker–Weizel approximations. The results are used to relate the T and K matrix elements to tabulated transcendental functions.
Path integrals for waves in random media
Authors: Dashen Roger
Published: March 2003

The problem of wave propagation in a random medium is formulated in terms of Feynman’s path integral. It turns out to be a powerful calculational tool. The emphasis is on propagation conditions where the rms (multiple) scattering angle is small but the log-intensity fluctuations are of order unity—the so-called saturated regime. It is shown that the intensity distribution is then approximately Rayleigh with calculable corrections. In an isotropic medium, the local or Markov approximation which is commonly used to compute first and second (at arbitrary space–time separation) moments of the wave field is explicitly shown to be valid whenever the rms multiple scattering angle is small. It is then shown that in the saturated regime the third and higher moments can be obtained from the first two by the rules of Gaussian statistics. There are small calculable corrections to the Gaussian law leading to ’’coherence tails.’’ Correlations between waves of different frequencies and the physics of pulse propagation are studied in detail. Finally it is shown that the phenomenon of saturation is physically due to the appearance of many Fermat paths satisfying a perturbed ray equation. For clarity of presentation much of the paper deals with an idealized medium which is statistically homogeneous and isotropic and is characterized by fluctuations of a single typical scale size. However, the extension to inhomogneous, anisotropic, and multiple scale media is given. The main results are summarized at the beginning of the paper.
Zorn algebra in general relativity
Authors: Oliveira Colber G. , Maia Marcos D.
Published: March 2003

The covariant differential properties of the split Cayley subalgebra of local real quaternion tetrads is considered. Referred to this local quaternion tetrad several geometrical objects are given in terms of Zorn–Weyl matrices. Associated with a pair of real null vectors we define two-component spinor fields over the curved space and the associated Zorn–Weyl matrices which satisfy the Dirac equation written in terms of the Zorn algebra. The formalism is generalized by considering a field of complex tetrads defining a Hermitian second rank tensor. The real part of this tensor describes the gravitational potentials and the imaginary part the electromagnetic potentials in the Lorentz gauge. The motion of a charged spin zero test body is considered. The Zorn–Weyl algebra associated with this generalized formalism has elements belonging to the full octonion algebra.
The Yang–Mills system in compactified Minkowski space; Invariance conditions and SU(2) invariant solutions
Authors: Harnad J. , Shnider S. , Vinet Luc
Published: March 2003

In this work the SU(2) Yang–Mills equations are studied in compactified Minkowski space. The manifold is identified with that of the Lie group U(1) ×SU(2) and a classification is made of all SU(2) principal bundles over this base space in terms of homotopy classes of mappings f:S3→S3. Invariance of gauge fields under transformation groups is defined in terms of bundle mappings and the case of invariance under SU(2) translations is shown to imply a trivial bundle structure. All solutions to the field equations invariant under U(1) ×SU(2) translations are obtained as well as all (anti-) self-dual solutions invariant under SU(2) translations.
Koornwinder’s polynomials and representations of the conformal group
Authors: Ringhofer Klaus
Published: March 2003

The tensor product of two massive, spinless, positive-energy ray representations of the conformal group of spacetime SO0(4,2)/Z2 is reduced in a momentum basis. The basis vectors for the irreducible subspaces (the ’’Clebsch–Gordan coefficients’’) are found to be intimately connected with Koornwinder’s polynomials in two dimensions.
Properties of the Benjamin–Ono equation
Authors: Case K. M.
Published: March 2003

The question is raised as to whether the Benjamin–Ono equation, a nonlinear partial differential integral equation, is a completely integrable Hamiltonian system. The answer is almost certainly ’’yes.’’ Particular solutions suggest the form that general polynomial constants must have. The structure of these and an algorithm to compute them is given. Explicit formulas are given for the first six.
Thermodynamical properties of a class of solvable statistical models for hadronic matter
Authors: Miller David E.
Published: March 2003

The exact evaluation for the thermodynamical properties of relativistic quantum gas models for excited hadronic matter using a statistical picture is carried out from the analysis of the grand partition function for a general class of excitation spectra. Some special forms of these energy spectra for hadronic matter are discussed for the physical content and the relationship to the known classical limiting cases.
Applications of an algebraic quantization of the electromagnetic field
Authors: Carey A. L. , Hurst C. A.
Published: March 2003

We quantize the full component electromagnetic potential so that a version of the Haag–Kastler axioms applies. The resulting quasilocal theory is viewed as a gauge theory in the spirit of Doplicher, Haag, and Roberts. Interaction with a classical current is described and found to yield no surprises. Lacking a rigorous scattering theory for the electromagnetic field, we construct a rigorous analog of the asymptotic Hilbert spaces for quantum electrodynamics proposed by Kulish and Faddeev and Zwanziger. On this basis we are able to find the representations of the gauge group that are associated with each charge sector in the asymptotic space. These representations correspond, in a sector of charge q, to the subsidiary condition ∂μAμ(x) =−q D (x).
Formal analytic continuation of Gel’fand’s finite dimensional representations of gl(n,C)
Authors: Lemire F. , Patera J.
Published: March 2003

The article contains three results: I. It is shown that among the 2−n n! (n+1) ! discrete series of representations of the Lie algebra gl(n,C) of complex n×n matrices described in the literature, the majority are not representations at all. Thus for n=3 and 4 one has respectively 12 and 45 series of representations instead of 18 and 180. II. In addition to the p+1 discrete unitary series of representations of u(p,q) [the Lie algebra of the group U(p,q), p?q, and p+q=n] there exist other discrete series of gl(n,C) which become unitary when restricted to its real subalgebra u(p,q). For n=3 there are four such series all corresponding to the chain u(2,1) ⊆u(1,1) ⊆u(1); for n=4 there exist six such series for u(3,1) and four series for u(2,2). Furthermore, some of the gl(n,C) series whose restriction to the real case do not provide unitary representations in general, do contain (infinitely many) particular representations which are unitary. Such unitary representations are contained inside of two of the four series for n=3 and inside of seven of the 27 series for n=4. III. Some properties of indecomposable representations of the Lie algebras for the groups of inhomogeneous transformations are shown using the discrete series of gl(n,C).
Radiating Kerr–Newman metric
Authors: González C. , Herrera L. , Jiménez J.
Published: March 2003

A complete generalization of the Kerr–Newman solution to the nonstationary case is given. The possibility of associating the energy–momentum tensor with the electromagnetic field is discussed.
The semiclassical expansion of the anharmonic-oscillator propagator
Authors: Mizrahi Maurice M.
Published: March 2003

This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.
Resolvent integration techniques for generalized transport equations
Authors: Bowden R. L. , Greenberg William , Zweifel P. F.
Published: March 2003

A generalized class of ’’transport type’’ equations is studied, including most of the known exactly solvable models; in particular, the transport operator K is a scalar type spectral operator. A spectral resolution for K is obtained by contour integration techniques applied to bounded functions of K. Explicit formulas are developed for the solutions of full and half range problems. The theory is applied to anisotropic neutron transport, yielding results which are proved to be equivalent to those of Mika.
Circular motion for a time-asymmetric relativistic two-body problem
Authors: Fahnline Donald E.
Published: March 2003

A formalism developed in a previous paper yields necessary and sufficient conditions and a solution for the circular motion case of the time-asymmetric relativistic two-body problem in which one particle responds to the retarded Liénard–Wiechert field of a second, while the second responds to the advanced field of the first. The necessary conditions contradict Künzle’s exceptional circular motion solution with zero angular momentum; consequently, zero angular momentum implies one-dimensional motion. The limit in which the mass of one of the particles becomes infinite commutes with the nonrelativistic limit and reduces the solution properly to the circular motion solution of the relativistic one-body Coulomb problem.
Constructive methods based on analytic characterizations and their application to nonlinear elliptic and parabolic differential equations
Authors: Barnsley M. , Bessis D.
Published: March 2003

We consider functional equations F (φ,γ) =0, in general nonlinear, whose physically meaningful solution φ[γ] depends on a real or complex parameter γ. We suppose that positivity plays an important role in the equation; for example, the physical solution φ[γ] may be forcedly positive for all γ in some range because it describes a density or a temperature. Furthermore, F may involve operators which themselves possess positivity properties; in particular we think of energy and diffusion operators of the Laplace type. This paper describes and illustrates how such positivity properties can lead one to discover the analytic nature of φ[γ], and shows how in many cases knowledge of this analytic characterization provides a method for reconstructing φ[γ] starting from the functional equation itself. First, it is shown how positivity can generate analytic characterizations with the aid of Bernsteins’ theorems. Equipped with this knowledge, we describe methods for attacking the formal perturbation series associated with F (φ,γ) =0 to yield monotone, convergent sequences of bounds on φ[γ]. Finally two classes of nonlinear problems are considered; one involves a general uniform elliptic operator while the other involves a parabolic operator; both involve a quadratic nonlinearity. In each case the key positivity properties of the operators involved are provided, and it is shown how the analyticity ’’character analysis’’ of the physically relevant solution φ[γ] is effected. In the case of the quadratically nonlinear elliptic problem a bifurcation phenomenon is associated with the solution: Here the positivity analysis enables us to describe completely the behavior of the solution when the bifurcation parameter passes through its critical value.
Modified modal theory of transient response in layered media
Authors: Tsang L. , Kong J. A.
Published: March 2003

A modified modal theory is developed for the transient response due to a line source placed in a two-layer slab medium. The double integral is evaluated by deformation on the complex wave number plane followed by a second deformation from the real frequency axis to the steepest descent path on the complex frequency plane. We first show that causality is preserved such that before the arrival time of the direct wave or the head wave, whichever takes the least time, the transient response vanishes. With the preservation of causality, we proceed to find the complete transient response at all times. The results obtained from the modal theory are then checked with those generated from direct numerical calculations with computers. Comparisons are also made with the explicit inversion technique which is applicable to the present case but appears to have severe restrictions that the modal theory does not have in generalizing to the solution of other transient problems.
A method for determining a stochastic transition
Authors: Greene John M.
Published: March 2003

A number of problems in physics can be reduced to the study of a measure-preserving mapping of a plane onto itself. One example is a Hamiltonian system with two degrees of freedom, i.e., two coupled nonlinear oscillators. These are among the simplest deterministic systems that can have chaotic solutions. According to a theorem of Kolmogorov, Arnol’d, and Moser, these systems may also have more ordered orbits lying on curves that divide the plane. The existence of each of these orbit types depends sensitively on both the parameters of the problem and on the initial conditions. The problem addressed in this paper is that of finding when given KAM orbits exist. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of nearby periodic orbits. The relation between KAM surfaces and periodic orbits has been explored extensively here by the numerical computation of a particular mapping. An important part of this procedure is the introduction of two quantities, the residue and the mean residue, that permit the stability of many orbits to be estimated from the extrapolation of results obtained for a few orbits. The results are distilled into a series of assertions. These are consistent with all that is previously known, strongly supported by numerical results, and lead to a method for deciding the existence of any given KAM surface computationally.
Asymptotic forms of radial wavefunctions and Jost functions for cutoff potentials
Authors: Romo W. J.
Published: March 2003

Rigorous asymptotic forms for the radial wavefunctions and the Jost functions are obtained for a broad class of cutoff potentials as the momentum k tends to infinity in the complex k plane. The asymptotic positions of the zeros of the Jost functions (S matrix poles) are also determined. It is found that the solutions of fl(−k) =0 for which ‖k‖ is very large, form a finite sequence of families of the form kN(m) ≈ αmN−iβm log N with N≈N0≫ 1 and m=1 to max with max a finite integer greater than or equal to one which is determined by the potential parameters.
Soliton solutions of nonlinear Dirac equations
Authors: Takahashi K.
Published: March 2003

Dirac equations with fourth order self-couplings are investigated in one time and three space dimensions. Both stringlike and ball-like soliton solutions carrying nontopological quantum numbers are shown to exist, depending on the symmetries taken into account. The energies (per unit length) of stringlike solutions turn out to be always smaller than those of plane wave solutions. For ball-like solutions, it is shown that the region of values of the nontopological quantum numbers exists in which the total energies of solitons are smaller than those of plane waves.
Invariant imbedding equations in general geometries: Numerical solution in spherical geometry
Authors: Shenoy S. K. , Mockel A. J.
Published: March 2003

A method for deriving the invariant imbedding equations in general geometries by the transfer matrix method is developed. Time-independent monoenergetic neutron transport in a nonmultiplying medium is considered. Specifically, invariant imbedding equations are derived for the plane, spherical, and cylindrical geometries. The reflection and transmission functions are evaluated numerically using the discrete ordinates method, for the case of a spherical shell with a perfectly absorbing core. A source of error in the results obtained is pointed out.
Note on the stability of the Schwarzschild metric
Authors: Wald Robert M.
Published: March 2003

It is shown that the standard arguments for the stability of the Schwarzschild metric can be made into a rigorous proof that the numerical values of linear perturbations of Schwarzschild must remain uniformly bounded for all time.
Group content of the Foldy–Wouthuysen transformation and the nonrelativistic limit for arbitrary spin
Authors: León J. , Quirós M. , Mittelbrunn J. Ramirez
Published: March 2003

The relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)2j. Thus for fermions we can directly take the 2(2j+1) - dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) - dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p0)1/2, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum-mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.
Positivity conditions on correlation functions that imply Debye screening
Authors: Federbush Paul
Published: March 2003

In the classical statistical mechanics setting, a set of positivity conditions on certain two-point correlation functions is exhibited that implies Debye screening for a large class of Coulomb-like models. For example, for the model treated by Brydges, for which he has rigorously proved shielding, in a range of parameters where 〈φs(x) J (y) 〉⩾ 0 for all x and y and all s odd, there is screening. (Alternative conditions require positivity for only two correlation functions.) Strong estimates are obtained for the rate of exponential falloff.
Factored irreducible symmetry operators and space groups
Authors: Folland N. O.
Published: March 2003

Induction techniques for the systematic construction of factorized irreducible symmetry operators (FISO) are applied to general space groups. Detailed results are given for the diamond structure space group and space double group. In the case of the diamond structure it is shown how the relations provided by time-reversal symmetry may be efficiently utilized to simplify calculations.
Inhomogeneous local gauge transformations in spacetimes with torsion
Authors: Flaherty Francis J. , Kerlick G. David
Published: March 2003

The formalism of bundle maps is developed, then applied to several situations of physical interest. The bundle isomorphisms are defined to be inhomogeneous gauge transformations. First, we examine the compatibility between local gauge invariance with respect to an internal symmetry and the minimal coupling of the free gauge field Lagrangian to spacetime. In both homogeneous and inhomogeneous cases, restrictions arise which limit the torsion of the spacetime covariant derivative when applied to the gauge potential. Second, we examine the group of diffeomorphisms as a spacetime symmetry in the local gauge theoretic approaches to general relativity and the Einstein–Cartan theory of gravity.
Two-body gravidynamics
Authors: Ibañez J. , Sanz J. L.
Published: March 2003

We develope a theory of gravitational interaction between two massive punctual structureless particles, from the point of view of predictive relativistic mechanics, whose main characteristics are: (i) the geometric framework is Minkowski flat space–time, M4; (ii) it is a predictive theory as opposed to the heredity which characterizes the dynamics derived from the classical field theory; (iii) the interaction occurs through light cones with a second order covariant symmetrical tensorial function on (TM4)2 which we call ’’field’’; (iv) we do not impose any ’’field equation’’ on this field, but only certain symmetries which lead to a certain indeterminacy which appears in the dynamic; (v) the dynamic is invariant with respect to the Poincaré group. From the outset, we adopt a fast motion approximation which consists of supposing that the dynamics can be developed in power series of the particle masses, and we evaluate this dynamic up to the second order of the masses. The comparison of the dynamics obtained from the Einstein, Infeld, and Hoffmann Lagrangian (which describes the gravitational interaction up to c−2 order in general relativity) helps us to fix five of the twelve arbitrary parameters which appear in our dynamics. It appears possible to explain all the gravitational experiments carried out up to now, in which neither light nor internal particle structure appear.
Variational nature of eigenvalue problems with a parameter
Authors: Schwarzmeier James L. , Symon Keith R.
Published: March 2003

The following theorem is proved. Let D (ω) be an operator with eigenvalues and eigenfunctions {dk(ω), νk(ω) }, where ω is a complex parameter. Given a complex number dk0, let ω0 be such that dk(ω0) =〈Ṽk(ω0) ‖D (ω0) ‖νk(ω0) 〉=dk0, where Ṽk(ω0) is the dual eigenfunction to νk(ω0). Suppose ψ and Ṽ approximate νk(ω0) and Ṽk(ω0), respectively, to order ε. Then, if D (ω) is analytic in ω in the neighborhood of ω0, and if ω′ is such that 〈Ṽ‖D (ω′) ‖ψ〉=dk0, ω′ usually will approximate ω0 to order ε2. By applying this theorem it is shown that roots of the inhomogeneous plasma dispersion relation usually will be accurate to second order if the associated normal modes and their duals are known merely to first order. The theorem can also be applied to solutions of the dispersion relation in a truncated function space.
A note on Miura’s transformation
Authors: Ablowitz Mark J. , Kruskal Martin , Segur Harvey
Published: March 2003

It has been observed by Miura that the solutions of the modified Korteweg–deVries equation can be mapped into those of the Korteweg–deVries equation. In this note we show that all of the solutions of the former, decaying sufficiently rapidly as ‖x‖→∞, map into a sparse solution set of the KdV equation. We use certain results regarding the second Painlevé transcendent to exhibit this fact.
Note on the linear representations of any dimensional Lorentz group and their matrix elements
Authors: Maekawa Takayoshi
Published: March 2003

The linear representations of the group SO(n,1) are studied in the space of functions given on the maximal compact subgroup SO(n) without making use of a special parametrization for elements of the group. The scalar products, which are invariant under the transformation by any element of SO(n,1), are introduced into the space together with the action of any representation operator of SO(n,1) on the function. The computation formula for the representation matrix elements is obtained.
Discontinuities of the Green’s functions across the characteristic surfaces of field equations
Authors: Darkhosh Teymour
Published: March 2003

A general method of calculating the retarded and advanced Green’s functions for the second order hyperbolic partial differential equations in the neighborhood of the characteristic surfaces is developed. This method is then applied to two examples: the Proca wave equation in an external electromagnetic field and the Proca wave equation in an external symmetrical tensor field. The second example is of special interest since it exhibits acausal behavior. For this case, it is shown that the field commutators do not vanish for all spacelike separations and that the commutator of the energy tensor with itself is nonzero in the region bounded by the light cone and the spacelike characteristic surface.
A remark about the duality for non-Abelian lattice fields
Authors: Bellissard J.
Published: March 2003

We give an example of a statistical mechanical model of spins with value in the non-Abelian group 𝒢3 for which the Kramers–Wannier duality holds and gives rise to self-dual interaction.
Dust distribution (cylindrically symmetric) in nonrigid rotation in Brans–Dicke theory
Authors: Bandyopadhyay N.
Published: March 2003

Solutions of the field equations of Brans–Dicke theory for the case of stationary cylindrically symmetric dust distribution in nonrigid rotation are studied. It is seen that singularities invariably occur in contrast to the singularity-free solution obtained by Maitra (1966) in the corresponding case in general relativity.
Shear-free gravitational collapse
Authors: Glass E. N.
Published: March 2003

Spherically symmetric perfect fluids are studied under the restriction of shear-free motion. All solutions of the field equations are found by solving a single second order nonlinear equation containing an arbitrary function. It is shown that this arbitrary function is a geometric invariant, E, which measures the gravitational field energy, and it is shown that E=const generates all the homogeneous density solutions. An improved proof is given for the nonexistence of any one-parameter equation of state. A number of exact solutions are presented and discussed.
Boost matrix elements of the homogeneous Lorentz group
Authors: Rashid M. A.
Published: March 2003

The boost matrix elements of the homogeneous Lorentz group are expressed in terms of analytic continuation of the Clebsch–Gordan coefficients of the three-dimensional rotation group.
New nonlinear realizations of SL(2,C)
Authors: Dalton B. J.
Published: March 2003

In this paper we present a study of some of the possible realizations of SL(2,C) as a transformation group on a complex three-dimensional space S. Several inequivalent categories of nonlinear realizations are found. We demonstrate that complex vector spaces can be found which transform under certain of these nonlinear realizations and for which the Poincaré mass and spin operators are diagonal. Within these new categories we obtain one spin-0, two spin-1/2, and several spin-1 realizations. In addition we obtain one new realization for each spin greater than 1. Certain of these realizations are unusual in that for both integer and half-odd integer spins the rotational period is 4π. For each new realization we derive the invariant metric on the space S and obtain a covariant wave equation. The development in this study suggests the interesting possibility that different categories of elementary particles could correspond to different categories of SL(2,C) realizations.
Unitary declension of dynamical symmetries for the time-dependent harmonic oscillator
Authors: Günther Neil J.
Published: March 2003

Elaboration is given to a previous discussion concerning dynamical symmetries of the Lewis–Riesenfeld time-dependent harmonic oscillator. A boson operator formalism is used to define a generator of impicitly time-dependent unitary transformations. Bilinear combinations of the transformed boson operators are shown to span the symplectic algebra Sp(2n,IR) regarded as a larger noninvariance group for the Lewis–Riesenfeld Hamiltonian. The multiplet structure of the embeddings, Sp(2n) ↓SU(n), and SU (n+1) ↓SU(n), is determined for the case n=3, using established branching rules. Boson operator realizations are presented for the multiplet structures in each SU(n) declension of these dynamical groups.
Projective representations of SU(2)ΛR4
Authors: Backhouse N. B. , Hughes J. W. B.
Published: March 2003

The Mackey little group procedure is used to classify the projective irreducible unitary representations of the group SU(2)ΛR4. Whereas an arbitrary ordinary representation of this group plays the role of both spectral and dynamical group for the three-dimensional hydrogen atom, it is found that one of its projective representations acts as spectral group for the two-dimensional isotropic oscillator. The projective representations of the Lie algebra of SU(2)ΛR4 are treated as ordinary representations of the central extension of this Lie algebra, and shift operator techniques used to obtain the matrix elements of its representatives. The connection with the isotropic oscillator is shown to be related to the realizability of the extended Lie algebra by two-boson creation and annihilation operators.
Prolongation structure for Langmuir solitons
Authors: Chowdhury A. Roy , Roy T.
Published: March 2003

A systematic analysis of nonlinear partial differential equations governing the formation and evolution of Langmuir solitons has been undertaken with the help of differential forms. The technique of Wahlquist and Estabrook has been applied in conjunction with the representation theory of Lie groups to derive the 3×3 inverse scattering formalism previously derived heuristically by Yajima.
Complex conjugation of space group representations
Authors: Dirl R.
Published: March 2003

Useful relations are derived which allow us to determine for every unirrep of nonsymmorphic space groups, which contain the inversion as a point group operation, the equivalent complex conjugate representation.
Group theoretical analysis of second order phase transitions in magnetic structures
Authors: Rudra P. , Sikdar M. K.
Published: March 2003

Landau’s analysis of second order phase transitions in nonmagnetic crystals, using group theoretical results, have been extended to magnetic transitions. Corepresentation theory of magnetic space groups, having both linear and antilinear operations, has been used to obtain the criteria for second order phase transitions in magnetic structures. The possible magnetic structures after a second order phase transition from the paramagnetic phase with the space group D72h, which is the case for CuCl2⋅2H2O, have been obtained as an example.
Distributional geometry
Authors: Parker Phillip E.
Published: March 2003

Recently there has been considerable interest in geometry with nonsmooth coefficients. The general theory of such distributional geometry is begun and an application in general relativity to black holes is given.
Center of inertia and coordinate transformations in the post-Newtonian charged n-body problem in gravitation
Authors: Barker B. M. , O’Connell R. F.
Published: March 2003

We generalize the field theory propagator by finding a way to make it a function of some additional arbitrary parameters. Thus, it is now possible to obtain Lagrangians (which contain the propagator parameters) from field theory in a more general coordinate system than had previously been possible. We find the n-body (classical) Bażański Lagrangian in this more general coordinate system and we give the relationship between the various coordinate systems by an n-body coordinate transformation involving the propagator parameters. We find the center of inertia for the case of the n-body Basżański Lagrangian in the general coordinate system and find that the potential energy terms −Gmimj/rij and eiej/rij do not in general split equally between particles i and j as they do in the case of Bażański coordinates. We also find the center of inertia for the case of the n-body (unchanged) post-Newtonian Lagrangian with parameterized post-Newtonian (PPN) parameters γ and β in standard coordinates, and show that the potential energy terms do split equally between a pair of particles.
Theory of strings and membranes in an external field. I. General formulation
Authors: Aurilia A. , Christodoulou D.
Published: March 2003

Motivated by recent developments in particle physics we give a mathematical formulation of the classical theory of extended objects under the influence of an external force field. Our geometric approach unifies the treatment of strings and membranes. We prove the existence and uniqueness of local solutions to the Cauchy problem associated with the object equations. We demonstrate the preservation of the topology of the object during its evolution. This allows us to extract the dynamical content of the theory. Some general properties of maximal timelike submanifolds, which represent worldtracks of extended objects when in free motion, are also deduced.
Path-integral evaluation of the space–time propagator for quadratic Hamiltonian systems
Authors: Marshall John T. , Pell John L.
Published: March 2003

Path-integral methods are used to derive an exact expression for the space–time propagator for systems with quadratic Hamiltonians. For a certain subclass of such systems, the result is reduced to a simplified closed form. The propagators for several illustrative elementary cases are exhibited in detail.
Classical predictive electrodynamics of two charges with radiation: General framework. I
Authors: Lapiedra R. , Molina A.
Published: March 2003

Outgoing radiation is introduced in the framework of the classical predictive electrodynamics using Lorentz–Dirac’s equation as a subsidiary condition. In a perturbative scheme in the charges the first radiative ’’self-terms’’ of the accelerations, momentum and angular momentum of a two charge system without external field are calculated.
Classical predictive electrodynamics of two charges with radiation: Energy and 3-momentum balance and scattering cross sections. II
Authors: Lapiedra R. , Marqués F. , Molina A.
Published: March 2003

We deal with a classical predictive mechanical system of two spinless charges where radiation is considered and there are no external fields. The terms (2,2)Paα of the expansion in the charges of the Hamilton–Jacobi momenta are calculated. Using these, together with known previous results, we can obtain the pαa up to the fourth order. Then we have calculated the ’’radiated’’ energy and the 3-momentum in a scattering process as functions of the impact parameter and the incident energy for the former and 3-momentum for the latter. Scattering cross-sections are also calculated. Good agreement with well known results, including those of quantum electrodynamics, has been found.
On stationary, axially symmetric solutions of Jordan’s five-dimensional, unified theory
Authors: Burzlaff J. , Maison D.
Published: March 2003

It is shown that the association of a linear eigenvalue problem for solutions of Einstein’s equations admitting a two-parameter Abelian group of isometries can be extended to Jordan’s five-dimensional, unified theory admitting three commuting Killing vectors. The reduction to a two-dimensional problem, the derivation of infinitely many conservation laws and the generation of one-parameter families of solutions can thereby be transcribed almost literally.
Repulsive and attractive timelike singularities in vacuum cosmologies
Authors: Miller Bonnie D.
Published: March 2003

Spherically symmetric cosmologies whose big bang is partially spacelike and partially timelike are constrained to occur only in the presence of certain types of matter, and in such cosmologies the timelike part of the big bang is a negative-mass singularity. In this paper examples are given of cylindrically symmetric cosmologies whose big bang is partially spacelike and partially timelike. These cosmologies are vacuum. In some of them, the timelike part of the big bang is clearly a (generalized) negative-mass singularity, while in others it is a (generalized) positive-mass singularity.
On angular momentum of stationary gravitating systems
Authors: Ashtekar Abhay , Streubel Michael
Published: March 2003

A theorem is proved to the effect that, for isolated gravitating systems in equilibrium, the definition of total angular momentum involving fields at null infinity agrees with that involving fields at spatial infinity.
On some representations of the Poincaré group on phase space
Authors: Ali S. Twareque
Published: March 2003

Some representations of the Poincaré group, by functions on phase space are studied both for classical as well as quantum relativistic systems. The classical representations are identified with certain canonically induced representations, and the quantum representations are then obtained on the same Hilbert space. Equations of motion on phase space are also developed.
Representations of the duals of gauge field tensors
Authors: Rund Hanno
Published: March 2003

Classical electromagnetic fields admit a perfectly symmetrical description in terms of the electromagnetic field tensors and the duals thereof, this phenomenon being a simple consequence of Maxwell’s equations for free fields. Since the Yang–Mills equations for free gauge fields possess a somewhat analogous structure, the possibility of a similar symmetrical description of gauge fields is investigated. It is shown that source-free Yang–Mills equations do, in fact, imply the existence of a new gauge in terms of which the duals of the field tensors admit a representation; however, the latter does not in general possess a structure which is identical with that of the field tensors in terms of the original gauge.