bluuebunny/crossref_metadata_2025 huggingface dataset:
Propagators from integral representations of Green’s functions for the N-dimensional free-particle, harmonic oscillator and Coulomb problems
Authors: Blinder S. M.
Published: March 2003

The radial Green’s functions for the N-dimensional free-particle, isotropic harmonic oscillator and Coulomb problems all contain a product of two Bessel or Whittaker functions. After integral representations for these respective products are introduced, each Green’s function exhibits the structure of a Fourier transform. One obtains thereby the Feynman propagators K(r1,r2,t) for the free particle and harmonic oscillator. In the Coulomb case, the Fourier transform involves the quantum number variable and leads instead to the recently defined Sturmian propagator. The well-known connection between Coulomb and oscillator eigenstates of various dimensionality is manifested in a new way by the structure of the propagators derived here.
Analytic treatment of the αx2+βx4 oscillator
Authors: Flessas George P. , Whitehead R. R. , Rigas A.
Published: March 2003

We carry out an investigation of the Schrödinger equation with the αx2+βx4 interaction for 0<Re( x)<∞, and β=‖β‖ exp(i(π+λ)),−π<λ<π. In the sectors 0≤arg( x)<π/3−λ/6,  0≤λ<π, and −λ/6<arg( x)<π/3−λ/6, −π<λ<0, a subdominant solution is constructed essentially in terms of the Laplace transform of a function f (s), 0≤s<∞, which is expressible as a converging power series in s. The solution, y( x), thus obtained, is compared and contrasted with the one valid in the usual case where Re( x)≤0 is included and it is argued that y( x) may not be accessible to perturbative approaches.
The generalized function solution of the Fokker–Planck equation for monoenergetic charged particle transport
Authors: Ganapol B. D. , Parlette E. B.
Published: March 2003

The solution of the Fokker–Planck equation describing the motion of an isotropic burst of ions originating at the center of an infinite medium and undergoing small angle deflections is obtained in terms of generalized functions using the multiple collision approach. The resulting solution can be shown to define a functional on the test function space Sβα for α≥ 1/2 and β≤ 1/2 and the space of polynomials specifying the dynamic moments.
On inequivalent classes of unique-mass–spin relativistic wave equations involving repeated irreducible representations with arbitrary multiplicities
Authors: Mathews P. M. , Vijayalakshmi B.
Published: March 2003

Considering all representations S(Λ) of the proper Lorentz group which are equivalent to the direct sum of three unspecified inequivalent irreducible representations, each occurring with arbitrary multiplicity, the question is investigated as to what representations of the above class can support first-order relativistic wave equations for unique-spin, unique-mass particles. An important requirement made from the outset is that the equation shall not be equivalent to any simpler one in the presence of arbitrary interactions. Strong restrictions which result on the irreducible representation content of S(Λ) are identified by a consideration of the Jordan canonical form of the matrix β0 entering such equations. With parity invariance as an additional requirement, it is shown that only 0 and 1 can be physical spins; classes of S(Λ) which can lead to new equations for these spins are determined. Finally, the restrictions which are needed to hold down the minimal degree of β0 to low values (≤6) are also determined.
Simple calculation of Löwdin’s alpha-function
Authors: Suzuki Noboru
Published: March 2003

A function of the type f(R)YML(Θ,Φ) around a specific center B is expanded about another center A as f(R)YML(Θ,Φ) =∑∞l=‖M‖(1/r)αl(fLM‖a,r) ×Y Ml(ϑ,φ), where YML(Θ,Φ) and YMl(ϑ,φ) are the complex spherical harmonics around the respective centers, which have been assumed to be quantized in the direction of A to B. The distances from the respective centers are R and r, and the separation between the two centers is a. The expansion coefficient αl( fLM‖a,r), which is usually called Löwdin’s α-function, can be expressed in a remarkably simple form by using Silverstone and Moats’ formula. Owing to the simplicity, the properties of the coefficients bKk(LM‖l) appearing in the α-function are manifested, and the recurrence formulas for evaluating bKk(LM‖l) are derived. Furthermore, when r≊0, the asymptotic expression for the α-function is given in a compact form.
Exact invariants quadratic in the momentum for a particle in a three-dimensional electromagnetic field
Authors: Lewis H. R.
Published: March 2003

For a Hamiltonian of the form H= (1)/(2) ( p−A(r,t))2+V(r,t), the conditions on the scalar and vector potentials V(r,t) and A(r,t) have been found such that there exists an exact invariant that is quadratic in each component of the canonical momentum p. The invariant is written in the form I(r,p,t)= (1)/(2)  (f2(r,t)⋅p−f1(r,t))2+f0(r,t), where f2 is a nonsingular symmetric real dyad, f1 is a real vector, and f0 is a real scalar. For the cases in which f2 is proportional to the unit dyad, or the vector potential is identically zero and f2 commutes with the time derivative of f2, all of the potentials that satisfy the conditions for such invariants to exist have been found explicitly and the invariants have been found in terms of the potentials. The derivation is a generalization of a method reported by Lewis and Leach for the corresponding one-dimensional problem. For the case in which the vector potential vanishes identically, the result is a generalization of a result found in three dimensions by Chandrasekhar. The results presented are applicable in plasma physics and stellar dynamics.
The relationship between finite groups and Clifford algebras
Authors: Salingaros Nikos
Published: March 2003

Clifford algebras are traditionally realized in terms of a specific set of representation matrices. This paper provides a more effective alternative by giving the finite group associated with each Clifford algebra. All the representation-independent algebraic results, which are really direct consequences of the underlying group structure, can thus be derived in an easier and more general manner. There are five related but distinct classes of finite groups associated with the Clifford algebras. These groups are constructed from the complex, cyclic, quaternion, and dihedral groups in a way which is discussed here in detail. Of particular utility is a table which lists the order structure of each group: this permits the immediate identification of any Clifford algebra in any dimension.
Formal linearization of nonlinear massive representations of the connected Poincaré group
Authors: Taflin E.
Published: March 2003

Let U1 be an arbitrary finite direct sum of unitary irreducible representations, each of positive (mass)2, of the connected Poincaré group P0=R4⊗SL(2,C). It is proved that each nonlinear representation of P0 with linear part U1, on the space of differentiable vectors of U1, is formally linearizable. Further, it is remarked that, commonly used, nonlinear massless representations of R2⊗SO0(1,1) are nonlinearizable and that the corresponding evolution equations do not have covariant wave operators.
On bilinear and trilinear extensions to the form of the Einstein tensor
Authors: McNamara Paul , Shepanski John
Published: March 2003

As is well known, the Einstein tensor G≡G[1] is constructed from Riemann and the metric so that it is symmetric, divergence-free, linear in Riemann, second rank, and vanishes when space-time is flat. In this paper the condition of linearity in Riemann is extended to bilinearity and trilinearity in Riemann and the resulting tensors G[2] and G[3], respectively, are found by a direct calculation. These are then compared with parts of the general tensor expression derived by Lovelock (who started with a different set of assumptions) and shown to be equal to it to within an arbitrary multiplicative constant. From this comparison it follows that G[2] and G[3] both vanish in the case where the dimension of the space is 4 (as occurs in the usual space-time).
Time-asymmetric initial data for the N black hole problem in general relativity
Authors: Kulkarni Anil D.
Published: March 2003

The time-asymmetric initial-value problem for N-body systems consisting of N black holes is described. The spacelike hypersurface for setting up the initial data is chosen to be the Einstein–Rosen manifold with N bridges. An explicit procedure in the form of an infinite series is given to construct solutions of the momentum constraints on the Einstein–Rosen manifold, starting from any solution of the momentum constraints in three-dimensional Euclidean space. A sufficient condition for the absolute convergence of a wide range of these solutions is derived. In particular, the solution representing N black holes with arbitrary momenta and spins is presented. The formulation of the Hamiltonian constraint as a boundary-value problem is discussed briefly.
A complete multipole expansion for a test particle in geometric fields
Authors: Toller M. , Vaia R.
Published: March 2003

We study the motion of a test particle interacting with a class of external fields including torsion, curvature, electromagnetic and Yang–Mills fields, using the definition of total energy momentum, angular momentum, charge and isotopic spin introduced in a preceding paper. We define a set of reduced multipole moments and we show that they give a complete description of the density and the flow of the quantities mentioned above. We write explicitly the exact evolution equations in terms of the reduced multipole moments and we show that they are the only consequences of the balance equations.
Backscattering and localization of high-frequency waves in a one-dimensional random medium
Authors: Frisch Uriel , Gautero Jean-Luc
Published: March 2003

Total reflection of waves by a one-dimensional random medium is superficially inconsistent with the decoupling of forward and backward propagating waves in the WKB limit (when no real turning points are present). Actually, complex turning points yield exponentially small reflection terms; in the random case, their cumulative effect over large distances can be evaluated using Fokker–Planck techniques, and total reflection is recovered. We also calculate the localization length as a function of wavenumber and strength of the random fluctuations.
On the representation of two-dimensional scalar wave fields in the complex plane
Authors: Nieto-Vesperinas M.
Published: March 2003

Scalar wave fields satisfying the Helmholtz equation in two dimensions are represented by means of a complex variable associated with the two-dimensional physical plane. This characterizes the wave functions as generalizations of analytic functions, which allows the existence of a generalized Cauchy integral formula constituting the nucleus of well-known theorems of optics such as the theorem of Helmholtz and Kirchhoff and the Ewald–Oseen extinction theorem. It also seems useful in the interpretation of inverse diffraction and scattering problems.
On the Cauchy problem for the coupled Schrödinger–Klein–Gordon equations in one space dimension
Authors: Rabsztyn Szymon
Published: March 2003

The existence, uniqueness, and continuity with respect to initial data of global solutions of the Cauchy problem is proved for the Schrödinger and Klein–Gordon equations with Yukawa coupling in one space dimension. The proof is based on the standard tools for handling abstract nonlinear wave equations.
Fractional approximations for the spherically symmetric Coulomb scattering wave functions
Authors: Chalbaud Eduardo , Martín Pablo
Published: March 2003

An improved method is presented for obtaining fractional approximations. The fractional parameters are now solutions of a set of linear equations, and no nonlinear equations are involved as in the previous procedure. Excellent fractional approximations are presented for the Coulomb functions for η=0.5, 1, 2, and 5. The accuracy is sufficient for most of the computations where this function is used. The straightforward extension to higher orders is indicated.
Fractals and nonstandard analysis
Authors: Nottale L. , Schneider J.
Published: March 2003

We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future.
Maupertuis’ principle of least action in stochastic calculus of variations
Authors: Zambrini J.-C.
Published: March 2003

Within the framework of stochastic calculus of variations for time-symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time-symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.
Scattering by impurities in a solvable model of a three-dimensional crystal
Authors: Albeverio S. , Hoegh-Krohn R. , Mebkhout M.
Published: March 2003

We compute explicitly the scattering amplitude for a quantum mechanical particle scattered by a discrete set of impurities in a three-dimensional crystal with point interactions.
Analytic expressions for the matrix elements of generators of Sp(6) in an Sp(6)⊇U(3) basis
Authors: Castaños O. , Chacón E. , Moshinsky M.
Published: March 2003

Work done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group Sp(6) in an Sp(6)⊇U(3) basis. Rosensteel has derived recursion relations for these matrix elements while Filippov has determined them using generating function technique, but it would also be convenient to have explicit and analytic formulas for them. This is what we do in this paper for the case of closed shells, i.e., when the irreducible representation (irrep) of Sp(6) is characterized by equal values for the three weight generators in the lowest weight state. We also indicate how our results can be extended to the case of arbitrary irreps of Sp(6), i.e., when we have open shells.
A pair of commuting missing label operators for G⊇[SU(2)]n
Authors: Van der Jeugt J.
Published: March 2003

A general procedure, which can lead to a pair of commuting subgroup scalars in the enveloping algebra of a Lie group G, decomposed in [SU(2)]n, is discussed. The technique is illustrated by means of three explicit examples.
Structure constants for Lie algebras
Authors: Feldman G. , Fulton T. , Matthews P. T.
Published: March 2003

An orthogonal basis in root space, related to the weights of the smallest representation, is used to provide a list of the algebraic conditions which the structure constants Nαβ must satisfy for all simple Lie algebras. A particular explicit set of solutions for all the Nαβ is given.
A unified description of the representations of the graded Lie algebra Gsl(2)
Authors: Gruber Bruno , Santhanam T. S. , Wilson Raj
Published: March 2003

Indecomposable representations of the graded Lie algebra Gsl(2) are analyzed in detail. It is further shown that the study of the irreducible representations (finite- and infinite-dimensional) is intimately related to the study of these indecomposable representations.
A KdV soliton propagating with varying velocity
Authors: Au C. , Fung P. C. W.
Published: March 2003

Using one solution to the KdV equation found previously by Au and Fung [Phys. Rev. B 25, 6460 (1982)] we apply the Bäcklund transformation again to obtain a new set of solutions which are divergent at certain points in the special case where the vacuum parameter is zero. While one set of solutions is static, the other set is an asymptotic one-soliton solution propagating with a varying velocity in a ‘‘transient’’ domain of space and time. To demonstrate the main features of our discovery, we have carried out a detailed numerical analysis of our analytical solutions.
Exact solution of some nonlinear evolution equations
Authors: Tagami Yukiko
Published: March 2003

A method is developed and subsequently used to study periodic solutions of (1) the one-dimensional time-dependent Ginzburg–Landau equation, (2) an analog of the sine–Gordon equation, and (3) a special system of nonlinear equations associated with a problem of interface.
Functional integrals as integrals on locally noncompact groups with generalized measures
Authors: Meiman N. N.
Published: March 2003

A functional integral is considered as an integral on a locally noncompact group with a generalized measure; the general method of integration on such groups is described, and conditions under which the integral exists are formulated. In particular, the action functional is determined by means of a group-theoretical action method. The path integral for a quantum-mechanical motion of a nonrelativistic particle in the Euclidean space is considered from this point of view. The corresponding locally noncompact group is the set of all alternative trajectories where the group operation is the pointwise addition. The internal group theoretical action functional present in the generalized measure is coinciding with the classical action functional. Thus a group-theoretical definition of the classical action functional is obtained.
Construction of the tensor product for the lattices of properties of physical entities
Authors: Aerts Dirk
Published: March 2003

We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.
Two-dimensional scattering: The number of bound states from scattering data
Authors: Cheney Margaret
Published: March 2003

Relations are found between scattering data and the spectrum for the two-dimensional Schrödinger operator Δ+V(x), where V is a local noncentral potential. In particular, a two-dimensional version of the Levinson theorem is obtained; this theorem gives the number of bound states in terms of the change in phase of the determinant of the scattering operator.
Some magnetohydrostatic models of cylindrical symmetry in general relativity
Authors: Roy S. R. , Bali Raj
Published: March 2003

The purpose of this paper is to study the problem of equilibrium of a cylindrically symmetric distribution of perfect fluid under the action of an incident magnetic field. The magnetic field is due to an electric current along the axis of symmetry surrounded by static fluid of infinite electric conductivity. Two models have been derived and their physical features have been discussed.
A class of shear-free perfect fluids in general relativity. II
Authors: Collins C. B. , White A. J.
Published: March 2003

We continue our previous investigation of shear-free perfect fluids in general relativity, under the assumptions that the fluid satisfies an equation of state p=p(μ) with μ+p≠0, and that the vorticity and acceleration of the fluid are parallel (and possibly zero). We classify algebraically the set of such solutions into thirteen invariant nonempty cases. In each case, we investigate the allowed isometry groups and Petrov types, and invariantly characterize the special subcases that arise. We also show how the various subcases are related to each other and to the works of previous authors.
Phase space of general relativity revisited: A canonical choice of time and simplification of the Hamiltonian
Authors: Ashtekar Abhay , Horowitz Gary T.
Published: March 2003

The phase space of general relativity is considered in the asymptotically flat context. Using spinorial techniques introduced by Witten, a prescription is given to transport rigidly the space-time translations at infinity to the interior of the (spatial) three-manifold. This yields a preferred four-parameter family of lapses and shifts and hence reduces the infinite-dimensional freedom in the choice of ‘‘time’’ to the restricted freedom available in special relativity. The corresponding Hamiltonians are computed and are shown to have an especially simple form: the Hamiltonians are ‘‘diagonal’’ in the (spatial) derivatives of variables which define ‘‘time.’’ Furthermore, the Hamiltonians (generating timelike translations) are shown to be positive in a neighborhood of the constraint submanifold of the phase space, even at points at which the ADM energy is negative.
Symmetric space property and an inverse scattering formulation of the SAS Einstein–Maxwell field equations
Authors: Eriş Ahmet , Gürses Metin , Karasu Atalay
Published: March 2003

We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight-dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ-model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov–Mikhailov and Mikhailov–Yarimchuk.
Generalized vector products, duality, and octonionic identities in D=8 geometry
Authors: Dündarer Resit , Gürsey Feza , Tze Chia-Hsiung
Published: March 2003

In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.
Linear transport in nonhomogeneous media. III
Authors: Daniel R. , Pomraning G. C. , Larsen Edward W.
Published: March 2003

A set of continuum and discrete eigenfunctions is constructed for linear half-space transport problems with single scatter albedos which are bilinear functions of position. For Hölder continuous distributions incident upon the half-space, these eigenfunctions are proved to be half-range complete and the expansion coefficient of the continuum eigenfunctions is proved to be continuous.
Proof of the decoupling theorem of field theory in Minkowski space
Authors: Manoukian Edward B.
Published: March 2003

The decoupling theorem of quantum field theory is proved in Minkowski space. It states the vanishing property, in the distributional sense, of the renormalized Feynman amplitudes when any subset of the underlying masses are scaled to infinity. To prove the theorem we were able, in the process, to bound the corresponding integrals, in the ε → +0 limit, by similar Euclidean integrals. All subtractions of renormalization are assumed to be carried out at the origin of momentum space with the degree of divergence of a subtraction coinciding with the dimensionality of the corresponding subdiagram.
Local gauge invariant Lagrangians in classical field theories
Authors: Grigore D. R.
Published: March 2003

We investigate the most general local gauge invariant Lagrangian in the framework of classical field theory. We rederive essentially Utiyama’s result. Our proof makes clear the importance of the so-called current condition, i.e., the requirement that the Noether currents are not identically zero. This condition is of importance both in the general motivation for the introduction of the Yang–Mills fields and for the actual proof. Some comments are made about the basic mathematical structure of the problem—the gauge group.
Static solutions of the coupled Yang–Mills–Weyl equations
Authors: Magg M.
Published: March 2003

It is shown that static, spherically symmetric solutions of the massless Yang–Mills–Dirac equation which carry finite energy are trivial.
On the analytic structure of the wave function for a hydrogen atom in an analytic potential
Authors: Hill Robert Nyden
Published: March 2003

The rate of convergence of an approximate method for solving Schrödinger’s equation depends on the ability of the approximating sequence to mimic the analytic structure of the unknown exact wave function. Thus a knowledge of the analytic structure of the wave function can be of great value when approximation schemes are designed. Consider the Schrödinger equation [− 1/2 ∇2−r−1+V(r)]Ψ(r)=EΨ(r) for a hydrogen atom in a potential V(r). The general theory of elliptic partial differential equations implies that Ψ is analytic at regular points, but no general theory is available at singular points. The present paper investigates the Coulomb singular point at r=0 and shows that, if V(r)=V1(x, y, z)+rV2(x, y, z) where V1 and V2 are analytic functions of x, y, z at x=y=z=0, then the wave function has the form Ψ(r)=Ψ1(x, y, z)+rΨ2(x, y, z) where Ψ1 and Ψ2 are analytic functions of x, y, z at x=y=z=0.
A recursive generation of local higher-order sine–Gordon equations and their Bäcklund transformation
Authors: Boiti Marco , Leon Jérôme J.-P. , Pempinelli Flora
Published: March 2003

A new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. It is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine–Gordon and Liouville equations in laboratory coordinates. A generalization of the well-known Bäcklund transformation and nonlinear superposition formula for the sine–Gordon equation is also obtained.
Quantum solitons of the nonlinear Schrödinger field as Galilean particles
Authors: Martínez Alonso L.
Published: March 2003

The quantum nonlinear Schrödinger field with attractive coupling is considered through the quantum inverse scattering method. The algebra of scattering data operators is formulated in terms of an infinite family of independent boson fields. It is shown that the unitary representation induced by the Galilean invariance of the model is equivalent to a direct product of simple representations. As a consequence, the quantum solitons turn out to be associated with representations describing quantum elementary Galilean particles. The characterization of quantum solitons as stable asymptotic fragments arising in the scattering of the fundamental bosons of the model is also analyzed.
Dynamical group of microscopic collective states. III. Coherent state representations in d dimensions
Authors: Deenen J. , Quesne C.
Published: March 2003

The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=A−1 relative Jacobi vectors. In this paper, we further study the Barut–Girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to 𝒮𝓅c(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to Holstein–Primakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their generating function properties, and the representations they lead to for both the collective states and their dynamical group.
Mixed supertableaux of the superunitary groups. I. SU(n‖1)
Authors: Delduc F. , Gourdin M.
Published: March 2003

Young supertableaux of the graded unitary groups SU(n‖1) are classified and interpreted in terms of representations of the group. A distinction between typical and atypical representations arises naturally, and we propose a new type of generalized atypical supertableaux for non-fully-reducible representations.
Differentiable vectors and sharp momentum states of helicity representations of the Poincaré group
Authors: Ek Bengt , Nagel Bengt
Published: March 2003

This paper discusses some mathematical difficulties in handling sharp momentum eigenvectors for a massless helicity representation of the Poincaré group, related to the non-nuclearity of the space of differentiable vectors, and to the existence of singularities in the Lorentz group generators. A simple characterization of the nuclear space of differentiable vectors of the extension of the representation to a representation of the conformal group is given in terms of functions on the space R4− {0}. Using a fibration of this space over the forward light cone (in momentum space), the singularity in the generators is shown to be related to the fact that the standard presentation of the helicity representations should be reformulated in terms of nontrivial sections over the light cone. The problem is partly identical with the one encountered in the study of an electron in a magnetic monopole field, the generator singularity taking the place of the Dirac string.
Construction of extremal vectors for Verma submodules of Verma modules
Authors: Lorente Miguel , Gruber Bruno
Published: March 2003

In this paper we discuss certain aspects of the theory of Verma submodules of Verma modules of simple Lie algebra. We describe a simple algorithm for the complete determination of the highest weights which are associated with the Verma submodules. Moreover, we give the proof for a method which permits an explicit determination of the extremal vectors which define the Verma submodules. For the case of the simple Lie algebra Al the Verma modules are obtained in explicit form.
Weak solutions of a quasistatic model of plasmas
Authors: Yoshida Zensho , Inoue Nobuyuki , Uchida Taijiro
Published: March 2003

We study an analytical structure of a quasistatic model of magnetically confined plasmas. Applying the fixed point theorem, we construct global-in-time weak solutions.
The scattering of an obliquely incident surface wave by a submerged fixed vertical plate
Authors: Mandal Birendranath , Goswami Sudip Kumar
Published: March 2003

The problem of scattering of surface waves obliquely incident on a submerged fixed vertical plate is solved approximately for a small angle of incidence by reducing it to the solution of an integral equation. The correction to the reflection and transmission coefficients over their normal incidence values for a small angle of incidence are obtained. For different values of the incident angle these coefficients are evaluated numerically, taking particular values of the wave number and the depth of the plate, and represented graphically.
On joint lower bounds of position and momentum observables in quantum mechanics
Authors: Busch Paul
Published: March 2003

A new theorem on joint lower bounds of unsharp position and momentum observables is formulated and proven. It is developed as a straightforward formal generalization of a similar theorem for the spectral projections of the usual observables which has been employed to express the strict incompatibility of position and momentum. The relevance of our theorem to the problem of joint measurements in quantum mechanics has been discussed elsewhere.
The role of the connection in geometric quantization
Authors: McKenna I. H. , Wan K. K.
Published: March 2003

The effects on quantization of using different connections are studied. Through explicit examples it is shown that with a proper choice of connection the half-density quantization scheme with the B.W.S. condition can produce physically correct results while the complicated half-form quantization scheme with the corrected B.W.S. condition can produce physically incorrect results.
Irreducible SO(2) extended superfields
Authors: Kim Jaegu
Published: March 2003

All the irreducible SO(2) extended scalar superfields are obtained in SO(2)-invariant form. These superfields can be labelled by the eigenvalues of three Casimir operators of the enlarged algebra containing chiral transformation. They may be still more simply classified by introducing two invariant simple superspaces. Lagrangians and supersymmetry transformation rules are obtained from the explicit form of the irreducible superfields. The highest superspin multiplet is identified as the multiplet for the linearized SO(2) Weyl supergravity. The linearized SO(2) Poincaré supergravity is obtained by coupling this multiplet with the chiral superfields.
On the construction of amplitudes with Mandelstam analyticity from observable quantities
Authors: Stefanescu I. Sabba
Published: March 2003

It is shown that the problem of the construction of scattering amplitudes with Mandelstam analyticity from knowledge of their modulus in the three physical channels can be reduced, within a rather large class of functions, to the second Cousin problem of the theory of functions of two complex variables. As a consequence, it can be solved completely and explicitly. We derive conditions on the modulus function, under which at least one solution exists, as well as criteria for the correct resolution of the discrete ambiguity at fixed energy.
Nonperturbative confinement in quantum chromodynamics. III. Improved gluon propagator
Authors: Atkinson D. , Boelens H. , Hiemstra S. J. , Johnson P. W. , Schoenmaker W. J. , Stam K.
Published: March 2003

An ansatz is introduced for the three-gluon vertex that is consistent with the Slavnov–Taylor identity in Landau gauge. It is shown that the gluon has a confining infrared singularity; but there is also a tachyon, indicating an insufficiency either of quarkless QCD or at least of our approximation to it.
Inverse scattering problems: A study in terms of the zeros of entire functions
Authors: Nieto-Vesperinas M.
Published: March 2003

The study of the zeros of functions of several complex variables recently reported is applied to scalar scattered wavefields. First, an analysis of the role of the zeros of the scattering amplitude leads to an interpretation of the nonuniqueness of inverse source and scattering problems. Secondly, the analytic character of the scattered wavefunction outside the scattering volume and its zero description, as well as its role for information transmission, is discussed.
The Dirac equation and Hestenes’ geometric algebra
Authors: Hamilton J. Dwayne
Published: March 2003

Hestenes’ geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ‘‘trace’’ calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
N-body quantum scattering theory in two Hilbert spaces. IV. Approximate equations
Authors: Chandler Colston , Gibson A. G.
Published: March 2003

A rigorous mathematical theory of approximations is developed for the time-independent transition operators of N-body multichannel nonrelativistic quantum scattering theory. New basic dynamical equations are derived and shown to specify uniquely the approximate time-independent transition operators. These operator equations represent coupled integral equations with compact kernels, but it is not assumed that the equations that determine the exact transition amplitudes have compact kernels. Convergence of sequences of these approximate time-independent transition operators to the exact transition operator is established in appropriate limits. Stability of the basic dynamical equations is proved. Resolvent-type equations and their relation to the limiting absorption principle are investigated. The relation of this theory to the Petryshyn theory of A-proper operators and to the Feshbach unified theory of nuclear reactions is discussed.
Spectral properties of vaguely elliptic pseudodifferential operators with momentum-dependent long-range potentials using time-dependent scattering theory
Authors: Muthuramalingam Pl.
Published: March 2003

(a) Absence of singular continuous spectrum for H and unitary equivalence of h0(P) with the absolutely continuous part of H is proved for the self-adjoint operator H=h0(P)+WS(Q, P) +WL(Q, P) on L2(Rn), where (i) h0 is a smooth, nonnegative, real valued function on Rn of at most polynomial growth with h0(∞)=∞, (ii) the critical values of h0 has a countable closure, (iii) the symbol WS(x, ξ) is a smooth function of at most polynomial growth in ξ and is of short range in x, and (iv) WL(x, ξ) is a smooth real-valued function of x, ξ with at most a polynomial growth in ξ and O(‖x‖−δ) at ∞ for some δ>0. Further we have (b) essential spectra of H and h0(P) are equal and (c) eigenvalues of H can accumulate only at the critical values of h0.
The optics of null strings
Authors: Plebañski Jerzy F. , Rózga Krzysztof
Published: March 2003

The general ‘‘optical theory’’ of congruences of totally null strings is studied within complexified general relativity, employing the spinorial formalism, with the emphasis on the role of the Sommers vector. The results obtained yield a deeper geometric interpretation of the ‘‘complexified anatomy’’ of the fundamental Goldberg–Sachs theorem.
Congruences of null strings in complex space-times and some Cauchy–Kovalevski-like problems
Authors: Robinson Ivor , Rózga Krzysztof
Published: March 2003

It is shown that a problem of construction of a local congruence of null strings is equivalent to a natural Cauchy–Kovalevski-like problem, related to an equation for a spinor field kA defining the congruence. Initial data are specified on two-dimensional submanifolds. In left-conformally-flat spaces, the solution of that problem exists for arbitrary initial data.
Approximate causal solutions for a class of wave equations with backscatter
Authors: Anderson James L.
Published: March 2003

The construction of solutions of wave equations with backscatter usually requires the use of some kind of approximation scheme. A problem then arises—how can one ensure that the approximate solution is causal? In this paper we present a method of constructing approximte causal wave-zone solutions that is applicable to a class of wave equation which reduce in form, in the limit as an expansion parameter goes to zero, to the scalar wave equation in flat space-time expressed in terms of the null coordinate u0=t−r. An extension of the method which is applicable to higher spin wave equations is also given.
Scalar meson field in a conformally flat space
Authors: Sanyal Abhik Kumar , Ray Dipankar
Published: March 2003

Among several authors, who studied massive and massless scalar meson fields in general relativity, attempts to obtain a complete set of solutions for a conformally flat metric eψ(dx12 +dx22 +dx32 −dx42) were made by Ray for massive and massless mesons and Gursay for massless mesons. Both of them concluded that ψ must be a function of K0(x12 +x22 +x32 −x42) +K1x1+K2x2+K3x3+K4x4 where K0, K1, K2, K3, K4 are all constants. Both Ray and Gursay, however, overlooked an important particular case, which is studied here. As a by-product certain equations obtained by Auria and Regge in connection with ‘‘Gravitational theories with asymptotic flat instantons,’’ are solved under less restrictive assumptions.
Killing vectors in algebraically special space-times
Authors: Torres del Castillo G. F.
Published: March 2003

The form of the isometric, homothetic, and conformal Killing vectors for algebraically special metrics which admit a shear-free congruence of null geodesics is obtained by considering their complexification, using the existence of a congruence of null strings. The Killing equations are partially integrated and the reasons which permit this reduction are exhibited. In the case where the congruence of null strings has a vanishing expansion, the Killing equations are reduced to a single master equation.
Isometries and dimensional reduction
Authors: Mansouri Freydoon , Witten Louis
Published: March 2003

We study dimensional reduction in (d+n)-dimensional space-times, with or without torsion, which admit n commuting Killing vectors. The field content of theories in (d+n) dimensions reduced to d dimensions are completely worked out. The equations of motion satisfied by these fields are also derived explicitly.
Stochastic convergence of the saturation coverage of one-dimensional arrays of β-bell particles
Authors: Mellein B. , Maltz A. , Mola E. E.
Published: March 2003

This paper considers the first two moments of R( β,N)=A( β,N)/N, A( β,N) denoting the random number of unoccupied sites of a one-dimensional array of N compartments saturated by randomly placed  β-bell particles ( β≥2). It is shown that, as N → ∞, the mean of R( β,N) approaches a (non-zero) limit L( β) while its variance tends to zero thus yielding the stochastic convergence of R( β,N) to L( β).  L( β) is explicitly determined and its behavior for large β is also studied.
Geometrical properties and invariances of the generalized sigma model
Authors: Arik M. , Neyzi F.
Published: March 2003

We consider the most general conformal invariant O(3) nonlinear σ-model in two dimensions, whose Lagrangian contains only first derivatives of the field variable. This is the generalized sigma model (GSM). We point out that its integrability condition, the generalized sine–Gordon (GSG) equation, follows from the fact that the Gaussian curvature of the surface underlying the model is a constant. We show that the GSG can be formulated as an imbedding and an inverse scattering problem. Finally we examine the invariance properties of the GSG equation.
Character expansion for U(N) groups and U(N/M) supergroups
Authors: Balantekin A. B.
Published: March 2003

A formula is presented to generate expansions over U(N) characters. The procedure to obtain the corresponding U(N/M) character expansions is also described. Several examples are worked out in both cases.
On integrability properties of SU (2) Yang–Mills fields. I. Infinitesimal part
Authors: Urbantke H.
Published: March 2003

We study the distribution of 2-plane elements on which infinitesimal parallel displacement of isovectors yields identity. This yields to an algebraic and differential classification and, in the generic case, to a quasimetric naturally associated with the field.
Dynamical invariants for two-dimensional time-dependent classical systems
Authors: Mishra S. C. , Kaushal R. S. , Tripathy K. C.
Published: March 2003

General equations are formulated to determine all potentials for two-dimensional systems of the type L= (1)/(2) ( p21 +p22) −V(q1,q2,t), which admits invariants of the form I=a0+aiξi + (1)/(2) aijξiξj, i, j=1,2, where ξ1 =ż=q̇1+iq̇2, ξ2=z̄=q̇1−iq̇2, a0, ai, aij are arbitrary functions of t, z=q1+iq2, and z̄=q1−iq2. Simplifying restrictions reduce the general equation to a tractable form. The resulting equations are solved for a special class of time-separable potentials and derive (i) the van der Waals-type long-range potential, V(r,t)=β(t)(b/r4+d) and (ii) the quark-confining logarithmic potential, V(r,t)=β(t)λ(ln r+b1/r4+d1). Invariants I for the resulting dynamical systems are found. Some observations on the present method in the context of Katzin and Levine and of Lewis and Leach analyses have also been made.
Positivity properties of phase-plane distribution functions
Authors: Janssen A. J. E. M.
Published: March 2003

The aim of this paper is to compare the members of Cohen’s class of phase-plane distributions with respect to positivity properties. It is known that certain averages (which are in a sense compatible with Heisenberg’s uncertainty principle) of the Wigner distribution over the phase-plane yield non-negative values for all states. It is shown in this paper that the Wigner distribution is unique in this respect among the members of Cohen’s class that have correct marginals or that satisfy Moyal’s formula for all states. The subset of members of Cohen’s class (not necessarily satisfying one of these two conditions) with positivity properties comparable with those for the Wigner distribution is shown to be rather small.
Signatures of finite SU( p,q) representations
Authors: Patera J. , Sharp R. T.
Published: March 2003

The signature S of a finite-dimensional representation of SU( p,q) is the difference between the number of positive and negative signs in the bilinear invariant in its diagonal form. An expression for S is derived starting from the Weyl character formula for U( p,q) representations.
The representation matrix elements of the group O+(2,2)
Authors: Syed Ansaruddin
Published: March 2003

Making use of the isomorphism of O+(2,2) with the direct product O+(2,1)×O+(2,1), the matrix elements of O+(2,2) in its unitary irreducible representations are explicitly calculated in terms of Euler angles introduced in a previous paper. The expressions so obtained consist of infinite sums of product of Clebsch–Gordan coefficients and Bargmann’s v functions, both for the group O+(2,1).
Comments on superposition rules for nonlinear coupled first-order differential equations
Authors: Winternitz P.
Published: March 2003

Some comments are made on the classification of finite-dimensional subalgebras of the Lie algebra of vector fields in n variables and of the related nonlinear ordinary differential equations with superposition principles. In particular for n=2 a very natural requirement of indecomposability implies that only two types of equations need be considered.
The summation of Bessel products
Authors: Evangelidis E. A.
Published: March 2003

We produce in what follows the closed forms for the summation of certain products of Bessel functions pertinent to a number of distinct fields of research, such as the theory of plasma waves, charged particle beam interaction with plasma, and density wave theory in galactic dynamics.
Poisson reduction and quantization for the n+1 photon
Authors: Gotay Mark J.
Published: March 2003

For a dynamical system in which the constraints are given by the vanishing of a singular momentum map J, reduction in the usual group-theoretic sense may not be possible. Nonetheless, one may still ‘‘reduce’’ J−1(0), at least on the level of Poisson algebras. An example of such a singular constrained system is the ‘‘n+1 photon,’’ that is, a massless, spinless particle in (n+1)-dimensional Minkowski space-time. We apply the generalized reduction procedure to the n+1 photon, explicitly constructing the Poisson algebra of gauge invariant observables. This technique also enables us to completely analyze the effects of the singularities in J−1(0) on the system. We then quantize, obtaining results which are in agreement with a quantization of the extended phase space and the subsequent imposition of the constraint.