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Linear Random Operator Equations in Mathematical Physics. III
Authors: Adomian G.
Published: February 2004

The problem of wave motion in a stochastic medium is treated as an application of stochastic operator theory and as a generalization of Papers I and II (and previous work by the author) to the case of partial differential equations and random fields without monochromaticity assumptions and closure approximations. Connections to the theory of partial coherence are considered. The stochastic Green's function for the two-point correlation of the solution process can be determined so the correlation can be obtained. Spectral spreading in a ``hot'' medium is easily demonstrable and can be calculated.
A Second-Order JWKB Approximation with One Turning Point and Two Singular Points: Stability of an Accelerating Liquid Sphere
Authors: Harper E. Y. , Chang I-Dee , Grube G. W.
Published: February 2004

A formal higher-order matching procedure is employed to obtain a second-order asymptotic solution, of the JWKB type, to a Legendre-like differential equation with a large parameter. The equation has two second-order poles and a first-order turning point. In addition to the usual nonuniformity, the second-order JWKB approximation exhibits a divergent integral at these points. Eigenfunctions and eigenvalues, valid to the second order of approximation, are found by simultaneously matching the latter approximation to a turning-point expansion and two boundary-layer expansions. The solutions, which heretofore have not been described, are appropriate to the neutral stable surface waves manifest by an accelerating liquid sphere.
Solution of the Schrödinger Equation in the Hardy-Lebesgue Space
Authors: Ifantis Evangelos K.
Published: February 2004

We present in this paper a uniform technique for the study of differential and difference equations in the Hardy-Lebesgue space, i.e., the Hilbert space consisting of analytic functions in the unit disk, taking the Schrödinger equation as an example. The approach followed is based on the representation of the Hardy-Lebesgue space by means of the unilateral shift operator, and reduces the problem of solving the Schrödinger equation to a perturbation problem of non-self-adjoint operators in an abstract separable Hilbert space.
Fourier Transform Methods in Linear Transport Theory
Authors: Case K. M. , Hazeltine R. D.
Published: February 2004

We generalize a three-dimensional Fourier transform method presented previously, to solve various forms of the linearized transport equation in planar geometry. Infinite-space problems are in general easily treated by our procedures. Half-space problems may also be solved analytically in, at least, the following cases: (i) separable scattering kernel (arbitrary particle speed); (ii) one-speed, anistropic scattering with rotation-invariant kernel; (iii) one-dimensional energy-dependent kernel of Kac; (iv) multigroup transport with down scattering only. The general importance of recursion relations to problems with nonseparable kernels is emphasized. A comparison is made between our methods and the two-dimensional singular eigenfunction approach, and a criterion is presented for the analytic solubility of any problem of the general form considered.
Markov Approach to Density Fluctuations Due to Transport and Scattering. II. Applications
Authors: van Vliet K. M.
Published: February 2004

Applications of a macroscopic and microscopic nature are given of the Markov formalism, developed in the previous paper, for finding the pair-correlation function, two-point covariance function, and spectral intensities from the phenomenological transport equations, transitions, and scattering. As macroscopic examples, we discuss Rayleigh diffusion, Coulomb correlations, and space-charge-limited flow in solids. The diffusion equation is shown to be not strictly Markovian, though correct Langevin diffusion sources are easily found. Dielectric relaxation is shown to be a macroscopic manifestation of Coulomb correlations in a plasma. The Λ theorem of the previous paper yields for the pair-correlation functions the original Debye-Hückel result plus a δ term. The spectra for space-charge-limited flow due to single-carrier injection are treated with the Green's-function procedure. The discrepancies between existing theories are removed since the Coulomb correlations in the steady state are shown to be not of the Debye-Hückel type. As microscopic applications, we consider ideal gases in the presence of streaming due to fields, an extension of Lax's work. The connection with the BBGK two-point equations for one- and two-body collisions is established. Finally, we integrate over the stochastic Boltzmann equation to obtain stochastic hydrodynamic equations. These provide a microscopic basis is for the noise sources associated with electrical conduction, heat conduction, and with the Navier-Stokes equation. Results for the hot-electron Boltzmann gas are compared with those of Price.
Double Laplace Transformation in Mixed Boundary-Initial Value Problems
Authors: Evans Kenneth , Jackson E. Atlee
Published: February 2004

A comprehensive treatment of the application of the double Laplace transform to the combined boundary-initial value problem for systems of linear, hyperbolic, partial differential equations with constant coefficients is given. The treatment is restricted to two independent variables x and t. A discussion of the relations among the boundary and initial conditions necessary for a well-formulated problem is given, and it is shown that the solution subject to these restrictions does satisfy the boundary and initial conditions. An extensive examination of the solution in different regions of space-time is given, and the connection between this solution and the more common ``normal-mode'' solution proportional to exp[ikx − iωl(k)t] or exp[ikl(ω)x − iωt] is discussed. A generalized wave equation is used throughout the paper for illustration.
Approximate Solution of a Sturm-Liouville System Using Nonorthogonal Expansions: Application to α-α Nuclear Scattering
Authors: Chatwin R. A. , Purcell J. E.
Published: February 2004

A stationary principle for the Sturm-Liouville system with inhomogeneous boundary conditions leads to an approximate solution in which nonorthogonal expansions are used. It is shown that if the nonorthogonal functions are chosen to be the eigenfunctions of a similar Sturm-Liouville equation, then the numerical solution of the boundary value problem no longer involves an integration of a second derivative, but only the more simple integration of a function. As an illustration, formulas for calculating the nuclear phase shifts of a Schrödinger equation with a long-range Coulomb potential are presented in the notation of R-matrix theory. The numerical convergence of the method is investigated for α-α nuclear scattering from real potentials that have a repulsive core followed by a short-range attraction (Ali and Bodmer). Generalizations of the method have been applied in nuclear physics to coupled differential equations and to eigenvalue problems.
Weaker Versions of Zeeman's Conjectures on Topologies for Minkowski Space
Authors: Nanda S.
Published: February 2004

In a paper [Topology 6, 161 (1967)] Zeeman conjectured that (a) the finest topology on Minkowski space that induces the one-dimensional Euclidean topology on every timelike line and (b) the finest topology on Minkowski space that induces the three-dimensional Euclidean topology on every spacelike hyperplane have the same group of homeomorphisms G which is generated by the inhomogeneous Lorentz group and the dilatations. This paper deals with two topologies on Minkowski space which are weaker than those in (a) and (b), respectively, and have the property that they induce the one-dimensional and the three-dimensional Euclidean topology on timelike lines and spacelike hyperplanes, respectively. It is then shown that both topologies have G as their homeomorphism group. Thus, what we have shown amounts to proving the weaker versions of Zeeman's conjectures.
On an Application of the Theory of Invariants
Authors: Devi V. Syamala
Published: February 2004

The problem of finding highest-weight polynomials in certain chains of subgroups of the unitary group is shown to be related to finding semi-invariants of certain ground forms according to the theory of invariants developed by mathematicians long ago.
Radial Matrix Elements of the Radial-Angular Factorized Hydrogen Atom
Authors: Cunningham M.J.
Published: February 2004

The hydrogen atom is factorized according to the scheme O(4,2)⊃O(2,1)×O(3) and the radial group O(2, 1) studied. It is shown that rkDn/(n+q), where Da is a dilatation operator, is proportional to a tensor operator in this scheme, allowing a group theoretical study of the radial matrix elements of rk, including an explanation of the Pasternack and Sternheimer selection rule.
Green's Dyadics for Elastic and Electromagnetic Waves in a Random Medium
Authors: Chung Sun Ho , Chen Yung Ming
Published: February 2004

A method for obtaining Green's dyadics of renormalized stochastic vector wave operators is presented schematically and is applied to propagation of elastic and electromagnetic waves in statistically homogeneous and isotropic media. Formula for effective propagation constants and renormalized Green's dyadics are derived and then computed explicitly for various special cases of elastic and electromagnetic waves with a special type of correlation function. Furthermore, parameters which affect the validity of such a renormalization approximation are also derived by estimating the second nonvanishing term of the renormalized series for average waves.
Uniform Asymptotic Solution of the Radial Schrödinger Equation for an Electron Bound in a Central Field Potential
Authors: Zauderer Erich
Published: February 2004

Assuming Z (the nuclear charge) is large, it is shown, using a theory of McKelvey, how a uniform asymptotic solution of the radial Schrödinger equation for an electron bound in a central field potential and for a related problem may be obtained. Requiring the asymptotic solutions to be bounded leads to a determination of the energy expansion parameters.
On Canonical Commutation Relations and Infinite-Dimensional Measures
Authors: Hegerfeldt Gerhard C.
Published: February 2004

Recently, Araki has proved ray continuity for the measure-space realization of the CCR's. In this paper, stronger continuity properties of this realization are derived, from which Araki's result follows as a corollary. It is shown that V(g) is continuous on complete metric subspaces of Uπ for any metric stronger than the weak topology. Surprisingly no further assumptions on the measure are needed. If Uπ is an F- or LF-space, V(g) is continuous on the whole of Uπ. These results are equivalent to certain continuity properties of the Radon-Nikodym derivative of the measure. Via these, it is then shown that every quasi-invariant measure on a nuclear F- or LF-space, such as S or D, is a superposition of ergodic measures. In the derivation, their close connection with irreducible representations of the CCR's is exploited.
Application of the Theory of Orlicz Spaces to Statistical Mechanics. I. Integral Equations
Authors: Cheng Ing-Yih S. , Kozak John J.
Published: February 2004

In this paper it is suggested that there may exist a fundamental relationship between the variables of thermodynamics, the operators associated with certain nonlinear integral equations of statistical mechanics, and the properties of a class of convex functions, called N functions, investigated by Krasnosel'skii and Rutickii. In particular, it is pointed out that the most general theoretical framework within which all these problems can be studied is that provided by the theory of Orlicz spaces. In the first part of our study, presented here, it is shown that the existence of solutions to certain nonlinear integral equations, derived either from the BBKYG hierarchy or from the grand partition function using a variational approach, can be established with some generality. The relationship between our results and those obtained by Ruelle is discussed.
Asymptotic Behavior of Vacuum Space-Times
Authors: Couch W. E. , Torrence R. J.
Published: February 2004

Newman and Penrose have given conditions on the asymptotic form of the Weyl tensor in empty space-time that are sufficient to insure that the space-time is asymptotically flat at null infinity and has the peeling property. We give considerably weaker conditions and show them to be sufficient for asymptotic flatness. Under the weaker conditions the asymptotic behavior of the Weyl tensor is more general than the case where the peeling property holds. The asymptotic dependence on a suitably defined radial coordinate is given for the basis null tetrad, the spin coefficients, and the tetrad components of the Weyl tensor.
Non-L2 Solutions of Exactly Soluble Relativistic Wave Equations and the Lee Model
Authors: Nelson C. A. , Rajagopal A. K.
Published: February 2004

In nonrelativistic potential theory there can exist singularities of the S matrix which are associated with wavefunctions belonging to a non-L2 class. In this paper the non-L2 character of these singularities is shown to persist in several relativistic models: in (1) two schemes, including that of the Klein-Gordon equation and the coupling of a classical relativistic field to an external exponential source, in (2) the pair theory, classical and quantized, with separable interactions, and in (3) the Lee model. For (2) and (3), poles of the ``proper'' Jost function f̄_(k, 0), i.e., the Fredholm determinant for the outgoing scattering state, correspond to non-L2 class solutions of the associated dynamical field equation in coordinate space. These non-L2 solutions will be called ``shadow'' fields. They are of special importance and bear the same relation as the ``shadow'' states in potential theory because they also are of dynamical origin and do not appear in the completeness or unitarity relations by virtue of their non-L2 status.
Structure of the Wigner 9j Coefficients in the Bargmann Approach
Authors: Wu Alfred C. T.
Published: February 2004

Bargmann's treatment of the Clebsch-Gordan (3j) and Racah (6j) coefficients is here extended to the case of Wigner 9j coefficients. The generating function for the 9j coefficient is computed by the analytic method. The result is compared to the Schwinger's expression derived with the algebraic (boson operator) method. The full symmetry of the Wigner 9j coefficients is manifest and transparent in the Bargmann's formalism. A new explicit expression for the Wigner 9j coefficient is derived as a sixfold sum which may be regarded as the analog of the Racah's formula for the Racah coefficient.
A Class of Matrix Ensembles
Authors: Dyson Freeman J.
Published: February 2004

A class of random matrix ensembles is defined, with the purpose of providing a realistic statistical description of the Hamiltonian of a complicated quantum-mechanical system (such as a heavy nucleus) for which an approximate model Hamiltonian is known. An ensemble of the class is specified by the model Hamiltonian H0, an observed eigenvalue distribution-function r(E), and a parameter τ which may be considered to be a fictitious ``time.'' Each of H0, r(E), and τ may be chosen independently. The ensemble consists of matrices M which are obtained from H0 by an invariant random Brownian-motion process, lasting for a time τ and tending to pull the eigenvalues of M toward the distribution r(E). For small τ the ensemble allows only small perturbations of H0. As τ → ∞, the ensemble tends to a stationary limit independent of H0 and depending on r(E) alone. The following quantitative results are obtained. (1) It is proved that the global eigenvalue distribution in the limit τ → ∞ becomes identical with the observed distribution r(E). (2) A nonlinear partial differential equation is obtained for the global eigenvalue distribution ρ(E, τ) as a function of E and τ. Solution of this equation will show how the distribution changes from the initial form specified by H0 at τ = 0 to the final form r(E) at τ = ∞. Approximate solution shows that deviations of ρ(E, τ) from r(E) extending over an interval containing m eigenvalues will disappear exponentially as soon as τ is of the order of mD2, where D is the local mean level spacing. (3) Exact analytic expressions are obtained for the correlation functions representing the probabilities for finding n eigenvalues at assigned positions (E1, …, En) in the ensemble in the limit τ → ∞, irrespective of the positions of the remaining (N-n) eigenvalues. It is made plausible, but not proved, that these correlation functions tend to limits as N → ∞, which are universal functions independent of r(E). If proved, this statement would imply that the local statistical properties (spacing distributions, etc.) of eigenvalues in the ensemble become, when τ and N are both large, universal properties independent of the global eigenvalue distributions. In particular, the spacing distributions would be identical with those calculated for more special ensembles by Wigner, Gaudin, and Mehta.
Currents in Classical Field Theories
Authors: Hermann Robert
Published: February 2004

The current vector field associated with a one-parameter group of transformations of a classical field is defined in a coordinate-free way.
Velocity-Dominated Singularities in Irrotational Dust Cosmologies
Authors: Eardley D. , Liang E. , Sachs Rainer
Published: February 2004

We consider irrotational dust solutions of the Einstein equations. We define ``velocity-dominated'' singularities of these solutions. We show that a velocity-dominated singularity can be considered as a three-dimensional manifold with an invariantly and uniquely defined inner metric tensor, extrinsic curvature tensor, and scalar bang time function. We compute this structure for a variety of known exact models. The structure of the singularity uniquely determines the solution in a certain class of spatially inhomogeneous models. We briefly discuss the b boundary (Schmidt boundary). In an appendix we generalize conformal transformations to ``stretch'' transformations and calculate the curvature form of a stretched metric.
Kinetic Theory of Scattering by a Plasma Cylinder
Authors: Weston V. H.
Published: February 2004

The kinetic theory of scattering by a circular homogeneous isotropic plasma cyclinder is treated for plane wave incidence parallel to the axis of the cylinder. The relativistic form of the Vlasov equation is inverted, subject to the specular boundary condition, expressing the electronic distribution function in terms of the electric field intensity. After inverting Maxwell's equations and eliminating the distribution function, a set of Fredholm integral equations of the second kind are obtained for the angular Fourier components of the electric field. Since for low temperature the Neumann series converges, the low temperature solution is easily obtained. The first order temperature corrections are thus derived for the reflection coefficients associated with the Fourier components.
Lagrangian Density for Perfect Fluids in General Relativity
Authors: Ray John R.
Published: February 2004

A previously discussed variational principle for a perfect fluid in general relativity was restricted to irrotational, isentropic motions of the fluid. It is proven that these restrictions can be dropped, and the original variational principle can be generalized to general motions of the perfect fluid. The form of the basic Lagrangian density is unchanged by these generalizations. An Eulerian fluid description is used throughout. As a by-product of our variational principle, the 4-velocity is required to have the generalized Clebsch form.
Symmetry in Einstein-Maxwell Space-Time
Authors: Debney George
Published: February 2004

By using the complex null tetrad as basis for the tangent space, a Killing vector field (``symmetry'') is introduced into the system of Einstein's equations with Maxwell's equations. The two bivectors Fμν and Kμ;ν (the associated Killing bivector) are assumed to have a principal null direction in common. Killing's equations, Maxwell's equations, and Einstein's equations are then written down for the case where this special direction is also a principal null geodesic for the Weyl conformal tensor. A certain analog of the Goldberg-Sachs theorem is proved. The static cases, plus a sizeable class of the static algebraically special cases are examined, to wit: where the special direction is also shear-free. In particular, all such algebraically special spaces must be Petrov Type D as a result of a coupling of the principal null directions for Fμν. This algebraically special metric is derived as an example of the static classes and is a static generalization of the Reissner-Nordström metric.
Form Factors for Any Spin and Charge Coupling
Authors: Tchrakian D. H.
Published: February 2004

A prescription for decomposition into form factors of the matrix elements of totally symmetric tensor currents between (s, 0) and (s′, 0) spinor states of the incoming and outgoing particles of spins s and s′ is presented. The case of the electromagnetic current is studied in some detail, and a procedure to modify the above prescription resulting in the grouping of the form factors into Dirac-type and Pauli-type classes is given. Several examples are used for illustration.
A Geometrical Theory of the Electromagnetic Field and the Gauge Transformation
Authors: Rañada Antonio F.
Published: February 2004

A modification of Weyl's theory of the electromagnetic field is presented, such that the usual gravitational Lagrangian becomes invariant under the corresponding definition of gauge transformation. As another advantage the problem of the weight of the tensors in the construction of Lagrangians is eliminated.
Conditions for the Existence of the Generalized Wave Operators
Authors: Pearson D. B.
Published: February 2004

If E0(G) is the spectral projection operator associated with the free Hamiltonian H0, corresponding to a bounded measurable subset G of R, and E1(G) is associated with the total Hamiltonian H = H0 + V, where the operator E1(G)VE0(G) is of trace class, it is proved that the element g = E0(G)f belongs to the domain of the generalized wave operators Ω± if and only if lim lim t→±∞‖(1−E1(G))e−iH0tg‖=0. A stronger version of this result is also proved, from the theory of time-dependent scattering, and is applicable to scattering systems for which families {G1}, {G2} of measurable sets may be found such that E1(G2)VE0(G1) is of trace class.
On the Generalized Exchange Operators for SU(n)
Authors: Partensky A.
Published: February 2004

By following the work of Biedenharn we have redefined the k-particles generalized exchange operators (g.e.o.'s) and studied their properties. By a straightforward but cumbersome calculation we have derived the expression, in terms of the SU(n) Casimir operators, of the 2-, 3-, or 4-particle g.e.o. acting on the A-particle states which span an irreducible representation of the grown SU(n). A striking and interesting result is that the eigenvalues of each of these g.e.o.'s do not depend on the n in SU(n) but only on the Young pattern associated with the irreducible representation considered. For a given g.e.o., the eigenvalues corresponding to two conjugate Young patterns are the same except for the sign which depends on the parity of the g.e.o. considered. Three appendices deal with some related problems and, more specifically, Appendix C contains a method of obtaining the eigenvalues of Gel'fand invariants in a new and simple way.
Steady-State Sound Propagation in Continuous, Statistically Isotropic Media
Authors: Neubert Jerome A.
Published: February 2004

A stochastic Eulerian-Lagrangian procedure is applied to steady-state sound propagation from a small, collimated acoustic source to an omnidirectional point receiver imbedded in an infinite, continuous, statistically isotropic medium. An analytic procedure is developed for obtaining the Lagrangian measure function B(x, ξ | s) from its characteristic function φ(k, ξ | s) for stochastic-Fermat media. The results include a coefficient of intensity variation V that evinces a frequency-dependent, phase-dominance region and a frequency-independent, amplitude-dominance region. The methods employed in this study are new to the problem of sound propagation through continuous stochastic media and avoid three common difficulties: (1) range limitations due to cumulative phase effects, (2) discrete scattering assumptions, and (3) restriction to an Eulerian path.
Recurrence Formula for the Veneziano Model N-Point Functions
Authors: Mano Koichi
Published: February 2004

A recurrence formula is derived for a function which reduces to the Veneziano model (n + 3)-point function. It is shown that the formula is equivalent to, but is more self-contained than, the Hopkinson and Plahte formula in that it does not require the prescription for the parameters involved.
Stationary Axially Symmetric Fields and the Kerr Metric
Authors: Misra R. M. , Pandey Deo Bhushan
Published: February 2004

In this note the axially symmetric metric for stationary gravitational field, in a slightly general form, is discussed. The vacuum field equations for this metric are given. Specialization of this metric leads to a different form of field equations previously discussed in literature. In particular, the Kerr metric is given in a new form. A justification for interpreting the Kerr metric as an exterior solution corresponding to a spinning rod or a rotating spherical body is given.
Properties of Lorentz Covariant Analytic Functions
Authors: Stapp Henry P.
Published: February 2004

A theorem is proved that asserts, roughly, that a function that is real Lorentz covariant anywhere is complex Lorentz covariant everywhere in its domain of regularity. It is also shown that the analytic continuation of a scattering function from a regularity domain in the physical region of a given process along all paths generated by complex Lorentz transformations leads to a function that is single-valued in the neighborhood of all these paths. Applications are discussed. The results derived constitute necessary preliminaries to a discussion of the analytic structure of scattering functions given in other papers.
Some Remarks on Three-Waves Interaction
Authors: Casati G. , Lazzaro E.
Published: February 2004

The problem of nonlinear interaction among three waves is analyzed in the case in which one of the waves is initially absent. Clarification is made of some mathematical aspects of the interaction mechanism.
Algebraic Solution for a Dirac Electron in a Plane-Wave Electromagnetic Field
Authors: Beers Brian , Nickle H. H.
Published: February 2004

An algebraic classification is given for the solutions of the Dirac equation for an electron interacting with a classical plane-wave electromagnetic field. The solutions appear as the carrier space of the direct sum of the positive and negative energy, mass m, spin-½ representations of the restricted Poincaré group. An explicit construction is given for the generators of the representation. The explicit position space form of the solutions follows readily from the relatively simple form of these operators. Via these solutions, an expression for the propagator of the interacting electron is given.
The Scattering of Three Impenetrable Particles in One Dimension
Authors: McGuire J. B. , Hurst C. A.
Published: February 2004

A special one-dimensional quantum mechanical three-body problem with an interaction among the particles is solved exactly. The interaction is introduced in such a way that the particles retain their order along a line, and hence the particles are impenetrable. The assumption of impenetrability is consistant with the existence of bound states in all accessible channels, and the probabilities for direct and exchange scattering of one bound pair and one free particle are calculated both below and above the breakup threshold. Above the breakup threshold, the probabilities for breakup and free particle scattering are also found. The absolute squares of all elements of the scattering matrix are given explicitly, and the scattering matrix is shown to be unitary.
The Convex Structure of the Set of N-Representable Reduced 2-Matrices
Authors: Erdahl R. M.
Published: February 2004

It is shown that in the finite-dimensional case the extreme points of the convex set of all N-representable 2-matrices D2 are exposed points and that for the case N = 3 and N = 4 the preimage of an exposed point under the contraction map is unique. In addition, the convex structure of both D2 and its associated polar is given in the case where the 1-rank is equal to N + 2. A long list of exposed points of D2 is given which includes all previously published examples as well as some new ones. However, this list is shown not to be exhaustive.
Asymptotic Expansions of Fourier Integrals with Light Cone Singularities
Authors: Schwarz Fritz
Published: February 2004

Let the integral transforms φ̃(y) and φ̃(y1,y2) be defined through φ̃(y)=∫d4xeixys(x)φ(x) and φ̃(y1,y2)=∫d4x1eix1y1∫d4x2eix2y2s(x1,x2)φ(x1,x2), respectively. Here all variables x and y are 4-vectors in Minkowski space. The functions φ are elements of S, and the factors s contain certain types of light cone singularities. The integral transforms φ̃ are investigated with respect to their characteristic properties implied by these light cone singularities using the method of van der Corput's neutralizers. It turns out that the behavior of φ̃ is determined by the light cone singularities if one goes to infinity in the space of the y variables along an arbitrary straight line. All characteristically different cases are classified and for each case a complete asymptotic expansion is derived.
Classification of Semisimple Subalgebras of Simple Lie Algebras
Authors: Lorente M. , Gruber B.
Published: February 2004

An explicit classification of the semisimple complex Lie subalgebras of the simple complex Lie algebras is given for algebras up to rank 6. The notion of defining vector, introduced by Dynkin and valid for subalgebras of rank 1, has been extended to the notion of defining matrix, valid for any semisimple subalgebra. All defining matrices have been determined explicitly, which is equivalent to the determination of the embeddings of the generators of the Cartan subalgebra of a semisimple subalgebra in the Cartan subalgebra K of the simple algebra containing this subalgebra. Moreover, the embedding of the root system of the subalgebras in the dual space K* of an algebra is given for all subalgebras. For the S-subalgebras of the simple algebras (up to rank 6), the embedding of the whole subalgebra in an algebra is given explicitly. In addition, the decomposition (branching) of the defining (fundamental) and adjoint representations of an algebra with respect to the restriction to its S-subalgebras has been determined. In the first part of this article a brief review of Dynkin's theory of the classification of the semisimple Lie subalgebras of the simple Lie algebras is given. No proofs are repeated, and at places where concepts have been extended and new results derived, merely an indication for their proof is given. This part of the article will serve as a prescription for a classification of semisimple subalgebras of the simple Lie algebras of rank exceeding 6. Later in the article, explicit expressions are given for the index of an embedding of a simple Lie subalgebra in a simple Lie algebra. These expressions are valid for the classical Lie algebras of arbitrary rank as well as for the exceptional Lie algebras.
Scalar-Tensor Theory of Gravitation in a Lyra Manifold
Authors: Halford W. D.
Published: February 2004

A closed-form exact solution to the field equations of a scalar-tensor theory, formally similar to the Brans-Dicke theory, is obtained. It is shown that the present theory predicts the same effects, within observational limits, as the Einstein theory.
Gauge Invariant Decomposition of Yang-Mills Potentials
Authors: Treat Richard P.
Published: February 2004

The Yang-Mills (YM) potentials are decomposed into an isovector part and a part which transforms nonhomogeneously under local gauge transformations. Two decompositions are shown; one of them is based on a gauge-invariant version of the transversality condition, and the other arises from a gauge-invariant modification of the Lorentz condition. The latter is Lorentz as well as gauge invariant. The gauge invariance of the decompositions is obtained at the expense of locality since the separate parts of the decomposed potential are functionals of the full YM potential. The transverse-longitudinal decomposition is used to throw the YM sourceless field equations into a gauge-invariant Hamiltonian form. Static fields in the Hamiltonian formulation are discussed. The decompositions are used to construct massive, gauge-invariant but nonlocal Lagrangians. A Lorentz and gauge-invariant nonlocal interaction of the YM field with a spinor-isospinor field is formed. The transverse-longitudinal decomposition is used to investigate the geometric structure of a configuration space Ω of YM potentials. The nonexistence of submanifolds of Ω orthogonal to the gauge-invariant manifolds X ∈ Ω is proved in contradistinction to the electromagnetic case. A Green's functional for the Yang-Mills field is represented explicitly by an infinite power series of functionals and is shown to be self-adjoint.
Quantization of a General Dynamical System by Feynman's Path Integration Formulation
Authors: Cheng K.S.
Published: February 2004

The Schrödinger equation is obtained by Feynman's path integration method of quantization for a general dynamical system. The meaning of the results is discussed.
An ``H-Theorem'' for Multiplicative Stochastic Processes
Authors: Fox Ronald Forrest
Published: February 2004

In a recent paper the author showed how multiplicative stochastic processes lead to a potentially comprehensive theory for nonequilibrium phenomena. In this paper an ``H theorem'' is proved from results obtained using multiplicative stochastic processes.
Algebraic Equations for Bethe-Salpeter and Coulomb Green's Functions
Authors: Kyriakopoulos E.
Published: February 2004

The equation for the Bethe-Salpeter Green's function in the case of two scalar quarks interacting via the exchange of a scalar particle of zero mass is transformed to an algebraic equation with the help of the dynamical group SO(5, 2). From this equation a one-parameter integral representation of the Green's function is obtained in the case of maximal binding, and from this representation the Green's function is calculated in terms of a hypergeometric function. The equation for the f-dimensional nonrelativistic Coulomb Green's function is also transformed to an algebraic equation.
Exponential Fourier Transforms for Coupled Harmonic Oscillator Chains
Authors: Lee K. H.
Published: February 2004

The exponential Fourier transform is used to study the dynamics of semi-infinite and infinite chains of interacting harmonic oscillators. In addition to the harmonic coupling between nearest neighbors, each oscillator is subjected to frictional and other external time-dependent forces. In contrast with previous studies on such systems, the initial conditions (at t = 0) are not specified, and the motion of all the oscillators is expressed in terms of the given applied forces only. The analytic structure of the transforms as well as some properties of the propagators are studied for all possible values of physical constants including the limiting values for uncoupled oscillators. The inverse transforms not readily available from tables are obtained by carrying out the integrations explicitly.
Relation between the Statistical Representations of Real and Associated Complex Fields in Optical Coherence Theory
Authors: Agarwal G. S. , Wolf E.
Published: February 2004

In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.
Existence and Uniqueness Questions for the Unitarity Equation
Authors: Tortorella Michael
Published: February 2004

In early articles on this subject, it was asserted that the transformation which arises from the generalized optical theorem for inverse scattering of scalar waves is completely continuous in certain spaces. In this article we show this is not correct: This transformation is not compact in these spaces. We also obtain an improved uniqueness result for the case where there is no spherical symmetry. The article concludes with a discussion of the proper setting of the unitarity equation (generalized optical theorem) in the larger context of the inverse scattering problem.
A Study of Relaxing Waves
Authors: Hsieh D. Y.
Published: February 2004

The equation ut+(1+u)ux−∫0t(1+u)uxf(t−τ)dτ=0 is studied when an initial finite pulse u(x, 0) is given. For the linearized equation, general solutions in terms of Laplace transforms are obtained and more explicit expression for exponential kernels is given. An iteration expansion scheme is established for general kernels. For positive kernels, it is found that the stability condition for the solution is ∫0∞f(t)dt<1. Then the large time solution as well as the solution representing the main disturbance is obtained. For the nonlinear equation, the condition for the shock formation is obtained for the special case f(t) = μe−μt, or when the nonlinearity is weak.
Identities Related to Variational Principles
Authors: Gerjuoy Edward , Rau A. R. P. , Spruch Larry
Published: February 2004

The existence of a well-known identity associated with variational principles for any scattering parameter Q, and often serving as the starting point for the development of a variational bound on Q, strongly suggests that it might be useful to construct identities associated with variational principles for quantities other than scattering parameters. An identity associated with the variational principle for the determination of inner products of the linear form g†φ, a generalization of the aforementioned identity, is presented. Here, g is a known function, and φ is an unknown function satisfying Mφ = ω and specified boundary conditions, where M is a known linear operator and ω is a known function. The generalized identity is obtained from a variational principle for g†φ, this variational principle being itself a generalization of the usual Kohn variational principle for scattering amplitudes and phase shifts. An identity associated with a variational principle for the quadratic form φ†Wφ, with φ as above and W a known linear operator, is also obtained. Finally, we obtain an identity for φ(∞), where φ is defined as the solution of a nonlinear differential equation. The generalized identity related to g†φ is verified for a simple exactly solvable problem.
On the Correlation of Field Quantities in a Random Linearly Elastic Solid
Authors: McCoy John J.
Published: February 2004

The response of a random linearly elastic solid is considered. General formulisms are derived that govern the one- and two-point moments defined on the stress, strain, and displacement fields. The weakly inhomogeneous solid is discussed within the framework of the derived formulisms. Also, we consider the simplifications that are introduced for problems that involve two length scales, one defined by the randomly varying material parameters and one defined by the characteristic dimensions of the over-all geometry of the solid and of all forcing mechanisms.
Remarkable Connection between the Multi-Veneziano Integrand and the Character of Special Unitary Groups
Authors: Santhanam T. S.
Published: February 2004

The remarkable connection that exists between the multi-Veneziano integrand at a fixed point of the external momenta and the character of special unitary groups is exhibited explicitly. Some remarks are made on such a connection for arbitrary external momenta.
Nonnegativity of the Yukawa Hamiltonian
Authors: Piepenbrink John
Published: February 2004

In a recent paper [J. Math. Phys. 8, 423 (1967)], Dyson and Lenard gave a lower bound of the N-particle Hamiltonian, Coulomb interaction, which was quadratic in N. An essential step in their proof is the fact that the one-particle Hamiltonian with Yukawa potential −r−1e−μr is nonnegative if μ is not less than some value. The aim of this paper is to improve this positivity result and thereby improve the lower bound of Dyson and Lenard.
Converging bounds for the free energy in certain statistical mechanical problems
Authors: Baker George A.
Published: February 2004

We present sequences of converging upper and lower bounds to the partition function per spin specifically for a ferromagnetic Ising model which are valid in the entire finite, magnetic-field, inverse-temperature plane. They are based on the exact high-temperature expansions for a finite system, properties of generalized Padé approximants, and the Villani limit theorem. The results depend only on the general structure of the partition function and certain monotonicity with system size properties which hold fairly generally.
A spectral representation for Coulomb wavefunctions
Authors: Townsend J. , Feldman G. , Fulton T.
Published: February 2004

We obtain a simple spectral representation for the momentum space wavefunctions of the Schrödinger equation with a Coulomb potential in the form of a contour integral. Both the bound state and the scattering solutions are evaluated as residues at the poles enclosed by this contour.
The nonstandard λ:φ24(x): model. I. The technique of nonstandard analysis in theoretical physics
Authors: Kelemen Peter J. , Robinson Abraham
Published: February 2004

The methods of nonstandard analysis are demonstrated as a preliminary step for the construction of the nonstandard λ:φ24: model. Elementary quantum mechanical problems are solved and the renormalization of the scalar field (Yukawa interaction) is investigated.
The nonstandard λ:φ24(x): model. II. The standard model from a nonstandard point of view
Authors: Kelemen Peter J. , Robinson Abraham
Published: February 2004

As a second step in the construction of the nonstandard λ:φ24: model we analyze Glimm and Jaffe's work from the nonstandard point of view.
Applications of the intertwining operators for representations of the restricted Lorentz group
Authors: Lind Robert W. , Messmer James , Newman Ezra T.
Published: February 2004

Some properties of the intertwining operations, which carry over certain infinite-dimensional, reducible (but not completely reducible) representations of the restricted Lorentz group into finite-dimensional (irreducible) representations are studied.
Equations of motion for the sources of asymptotically flat spaces
Authors: Lind Robert W. , Messmer James , Newman Ezra T.
Published: February 2004

We present an exact calculation that leads to the equations of motion (which naturally contain gravitational radiation reaction terms) of a system subject to no external forces. The novelty of our approach lies in the fact that the system is to be considered as the source of an asymptotically flat space and that all the revelant physical quantities such as the velocity νμ, 4-momentum pμ, angular momentum-center of mass tensor Sμν (as well as higher moments) are then defined in terms of surface integrals taken at infinity. A subset of the Einstein equations (equivalent to Bondi's supplemantary conditions) then yields the time-evolution equations for these variables.
Irreducible tensor operators for finite groups
Authors: Buch T.
Published: February 2004

Normalized tensor operators for a finite group G are defined by means of coefficients U which formalize the descent in symmetry from R3 to G. The properties of these coefficients are demonstrated and tables given. Some examples of application show their use and utility.
Solutions of a nonlinear integral equation for high energy scattering. III. Analyticity of solutions in a parameter, explored numerically
Authors: Bart G. R. , Warnock R. L.
Published: February 2004

Solutions of the Ball-Zachariasen equation, discussed in two previous papers, depend analytically on a parameter c which measures the strength of particle production. Numerical experiments, designed to elucidate the structure of the Riemann surface, are reported. The results are consistent with a very pretty hypothesis which describes the Riemann surface completely.
Scattering of a scalar wave from a random rough surface: A diagrammatic approach
Authors: Zipfel G. G. , DeSanto John A.
Published: February 2004

The solution to the problem of a scalar wave scattered from a rough surface is given under the conditions that the normal derivative of the field vanish at the surface and that the surface height be a single-valued function with Gaussian statistics. The solution is in terms of a series with a diagram representation. Partial summation of the series in terms of linear integral equations is briefly discussed.
Noumenon: Elementary entity of a new mechanics
Authors: Rocher E. Y.
Published: February 2004

If the postulate of symmetry on which special relativity is built is rejected, a generalization of the relativistic notions of event and space-time can be proposed. The generalization leads to the notion of noumenon. The noumenon possesses a handedness; it is a seven-parameter entity obtained by associating with an event the angular momentum corresponding to a particular evolution of that event. The noumenon is defined in the complex extension C4 of space-time R4. The main purpose of the paper is to prove the acceptability of the new concepts when confronted with experimental results which are generally regarded as supporting the classical theory of relativity. The potential fruitfulness of the new concepts is shown by a short review of similar ideas developed independently by several authors in different fields of physics: A Maxwellian theory of gravitation is developed; interactions between gravitation and electromagnetism appear, which have common characteristics with weak interactions; and it is suggested that the extra degrees of freedom of the noumenon are related to the quantum numbers of elementary particles.
On the possibility of observing first-order corrections to geometrical optics in a curved space-time
Authors: Dwivedi I. , Kantowski R.
Published: February 2004

Gravity's effect on the polarization of test electromagnetic fields is presented. It is shown that under ordinary circumstances the effect is not measurable.