bluuebunny/crossref_metadata_2025 huggingface dataset:
Multiple Scattering in the Diffusion Approximation
Authors: Salpeter E. E. , Treiman S. B.
Published: January 2005

The passage of classical particles through a grainy scattering medium can be described by a line-arized Boltzmann equation. A discussion is given of the physical conditions which justify the use of the Fokker-Planck diffusion approximation to this equation. Some limiting properties of the solutions of the diffusion equation are first discussed for the initial-value problem in an infinite medium characterized by a diffusion length D. For a total path lengthℓ ≪ D convenient formulas are given for the distribution of scattering angles θ and, for given θ, the first few moments of the final position vector are computed. These results are taken as a basis for approximate treatment of steady-state boundary-value problems. The case of a particle beam incident on a thin plane parallel slab of thickness d ≪ D is considered. Approximate formulas are given for the angular distribution of the transmitted beam and for the (very small) fraction of the beam which emerges from the entrance face. Errors are assessed, and the behavior for grazing angles of incidence or exit is discussed in a conjectural way.
Phase-Space Formulation of the Dynamics of Canonical Variables
Authors: Mehta C. L.
Published: January 2005

Statistical reformulation of quantum mechanics in terms of phase-space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form [u(p, q),υ(p, q)]={f(∂2∂q1 ∂p2−∂2∂q2 ∂p1)u(p1, q1)υ(p2, q2)},evaluated for p1 = p2 = p; q1 = q2 = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase-space distribution functions to describe pure states are also given.
Scattering of Surface Waves on an Infinitely Deep Fluid
Authors: Burke J. E.
Published: January 2005

The reflection and transmission of small amplitude waves incident on a plane barrier submerged in an infinitely deep fluid are investigated; the barrier has a finite width and is parallel to the undisturbed free surface of the fluid. Green's function techniques are used to represent the velocity potential in terms of its discontinuity (pressure difference) across the barrier. The application of the boundary condition at the barrier leads to a one-dimensional integral equation for the pressure difference. This equation is extended by introducing the fluid velocity normal to the plane of the barrier, and then it is analyzed by the complex Fourier transform methods of Wiener and Hopf. The velocity transforms are found to be characterized by a pair of dual inhomogeneous integral equations which allow systematic approximation by iteration. A convolution provides representations for the velocity potential in the different regions of the fluid, and these are the basis for investigating the reflection and transmission of waves. Results for some related problems (e.g., scattering by a semi-infinite barrier, by a finite dock, etc.) are obtained as limiting cases.
Diffraction by a Smooth Transparent Object
Authors: Chen Yung Ming
Published: January 2005

In previous investigations the so-called geometrical theory of diffraction was established. Its application has been limited to problems of diffraction of waves by smooth opaque objects.

In the present paper, the geometrical theory of diffraction is extended and applied to problems of diffraction by a smooth transparent object of any shape. For simplicity only scalar fields and two-dimensional problems are considered. However, this method can be modified easily to apply to vector fields and three-dimensional problems.

In Part I, the fields associated with the geometric rays and the diffracted rays are constructed in detail for the case of a circular cylinder. Afterwards a general method for treating diffraction by a smooth transparent cylinder of arbitrary shape is given. In order to determine the coefficients, the exact solution for the case of a circular cylinder is found and evaluated asymptotically for large ka in Part II. Then it is compared with the solution obtained from geometrical theory in Part I.
Generalizations of the Jost Functions
Authors: Moses H. E.
Published: January 2005

The Jost functions have proved valuable in the study of the analytic properties of the scattering phase for the radial Schrödinger equation. In the present paper we shall present an alternative definition of the Jost functions, prove the equivalence of the new definition to the usual one, and generalize the new definition to the one-dimensional Schrödinger equation (− ∞ <x< ∞), the three-dimensional nonseparated Schrödinger equation, and the three-dimensional nonseparated Dirac equation. It is hoped that these generalizations lead to a better understanding of the analytic properties of the scattering operator for these and related dynamical systems. The generalized Jost functions are shown to be operators in the variables which label the degeneracy of the continuous spectrum of the Hamiltonians which are considered.
Note on Wigner's Theorem on Symmetry Operations
Authors: Bargmann V.
Published: January 2005

Wigner's theorem states that a symmetry operation of a quantum system is induced by a unitary or an anti-unitary transformation. This note presents a detailed proof which closely follows Wigner's original exposition.
Singular Bethe-Salpeter Scattering Amplitudes
Authors: Swift Arthur R. , Lee Benjamin W.
Published: January 2005

The Bethe-Salpeter equation for scattering is investigated in configuration space for the class of singular ``potentials'' (i.e., a λφ4 theory in the ladder approximation) which behave as r−4 near the light cone. The discussion relies on the similarity between solutions of the Bethe-Salpeter equation and the Schrödinger equation where the corresponding problem is scattering by a r−2 potential. Through a consideration of the asymptotic properties of the two-particle, free Green's function, the elastic scattering amplitude is shown to be the coefficient of the outgoing wave part of the wave-function, just as it is in the nonrelativistic case. At zero total energy, it is just the coefficient of e−mr/r½. The differential form of the Bethe-Salpeter equation is expanded in four-dimensional spherical harmonics, and the singular part of the potential is incorporated into the differential operator. The resulting equation is formally solved by converting it into what is now a Fredholm integral equation. Care is taken to choose the proper asymptotic behavior for the solutions of the new equation. The discussion of the singular potential is carried out at zero total energy in order to obtain spherical symmetry. The technique for handling the singular potential and extracting the T matrix at zero energy is demonstrated by application to two examples. The exact scattering amplitude is found for exchange of two massless mesons. A first-order solution is obtained for a phenomenological potential that approximate the exchange of two massive mesons. This solution exhibits many of the features expected from a truly physical potential.
On the Canonical Relativistic Kinematics of N-Particle Systems
Authors: Chakrabarti A.
Published: January 2005

The ``canonical'' relativistic kinematics is discussed for n-particle systems. Our aim is to derive the formulas in as simple and symmetrical a way as possible and thus to avoid the extra complications arising for n > 2 in the usual stepwise generalization of the two-particle method. At first we derive the canonical form of the infinitesimal operators (N, M) in a highly symmetrical form. It is then shown that though the restrictions imposed by the condition that our states be also energy eigenstates compel us to sacrifice a part of the symmetry and simplicity of the formulas, we can indeed include the effect of the spins of the component particle in a completely symmetric way. This reduces to a minimum the additional complications introduced when the component particles have nonzero spins. The corresponding, relatively simple, generalized (C-G) coefficients connecting the canonical and the direct-product states are calculated. It is shown that the use of ``spinor'' representation for the individual particles simplifies the deductions considerably. Explicit results are given usually for n = 3 only, since the generalization to n > 3 introduces no essentially new features.
Bergman's Integral Operator Method in Generalized Axially Symmetric Potential Theory
Authors: Gilbert R. P.
Published: January 2005

This paper contains a study of properties of solutions to the equation of generalized axially symmetric potentials. These potentials play an important role in many aspects of mathematical physics, in particular to an understanding of compressible flow in the transonic region. The ideas that have been basic in this investigation are contained in the integral operator method of Bergman. This method allows one to transplant certain properties of analytic functions to the solutions of linear partial differential equations. Results are obtained concerning singularities, residues, bounds, and growth of entire solutions, which are analogous to those found in classical function theory.
On Pais's Charge Correlation Coefficients
Authors: Shpiz J. M.
Published: January 2005

An alternative proof of a theorem of Pais which states that the charge correlation coefficients for an n-π system are independent of the row of the representation of the symmetric group is presented in which no detailed knowledge of the representations is used.
On the Number of Self-Avoiding Walks. II
Authors: Kesten Harry
Published: January 2005

Let χn(d) [resp. γ2n−1(d)] be the number of self-avoiding walks (polygons) of n (2n − 1) steps on the integral points in d dimensions. It is known that βd=limn→∞ [χn(d)]1/n=limn→∞ [γ2n−1(d)]1/2n−1 In this paper βd is compared with βd,2r=limn→∞ [χn,2r(d)]1/n where χn,2r(d) is the number of n-step walks on the integral points in d dimensions with no loops of 2r steps or less. In other words the walks counted in χn,2r(d) may visit the same point more than once as long as there are more than 2r steps between consecutive visits. It turns out that βd,2r−βd=O(d−r)(d→∞) and it follows in particular that 1βd=2d−1−1/2d+O(1/d2)(d→∞) It is also shown that, for suitable constants α6 = α6(d) and α7 = α7(d), χn(d)≤βnexp [α6n2/(d+2)logn] and β2n−1exp [−α7n2/(d+1)logn]≤γ2n−1(d).
Note on the Evaluation of Some Fermi Integrals
Authors: Glasser M. L.
Published: January 2005

A method is illustrated for evaluating Fermi integrals by means of an integral representation for the derivative of the Fermi function. Several examples from free-electron diamagnetism are discussed. A convenient representation for the free energy of a fermi gas is also derived.
Exact Solution of the Schrödinger Equation for Inverse Fourth-Power Potential
Authors: Spector Richard M.
Published: January 2005

The solution of the Schrödinger equation for the potential V(r)=V/(μr)4 and all angular momenta is given in terms of known functions, namely, the solutions to the modified Mathieu equation with complex parameters and complex argument. Scattering solutions for both attractive and repulsive potentials are given and in the latter case the explicit expression for the S matrix is given.
A Proof that the Free Energy of a Spin System is Extensive
Authors: Griffiths Robert B.
Published: January 2005

The free energy obtained from the canonical partition function for a finite spin system possesses a certain convexity property, of which theorems by Peierls and Bogoliubov are particular applications. This property is used in proving the following result: Consider a regular lattice of spins in the form of a parallelepiped (in two dimensions a parallelogram, in one dimension a linear chain). The free energy of the system divided by the number of spins approaches a definite limit as the linear dimensions of the system become infinite. The limit is not influenced by certain common types of boundary conditions. A similar result, but with convergence understood in a weaker sense, holds for derivatives of the free energy such as entropy, magnetization, and specific heat. In the proof it is necessary to assume that the Hamiltonian has the translational symmetry of the spin system, and that long-range interactions decrease sufficiently rapidly with the distance r between spins. (For example, as r−3−ε with ε > 0 for interactions between pairs of spins in 3 dimensions.)
A Theorem on the Determinantal Solution of the Fredholm Equation
Authors: Manning Irwin
Published: January 2005

For a Fredholm equation with Green's-function-type kernel, a new proof is given of a known theorem relating the Fredholm determinant with the behavior of the solution at the origin. The utility of this theorem in practical calculations is pointed out, as are some of its implications for potential scattering in quantum mechanics. The scattering phase shift is shown to have the property (∂/∂λ)δ(E,λ)=−π 〈ψE| V |ψE〉.
Comments on Nonlinear Wave Equations as Models for Elementary Particles
Authors: Derrick G. H.
Published: January 2005

It is shown that for a wide class of nonlinear wave equations there exist no stable time-independent solutions of finite energy. The possibility is considered whether elementary particles might be oscillating solutions of some nonlinear wave equation, in which the wavefunction is periodic in the time though the energy remains localized.
Exact Quantization Rules for the One-Dimensional Schrödinger Equation with Turning Points
Authors: Bailey Paul B.
Published: January 2005

It is pointed out that, for a number of problems, exact quantization rules exist which closely resemble that of Bohr-Wilson-Sommerfeld. In some cases it is shown how these rules may be derived mathematically from the Schrödinger equation.
Coupled Magnetomechanical Equations for Magnetically Saturated Insulators
Authors: Tiersten H. F.
Published: January 2005

The differential equations and boundary conditions governing the macroscopic behavior of nonconducting magnetically saturated media undergoing large deformations, are derived by means of a systematic and consistent application of the laws of continuum physics to a model consisting of an electronic spin continuum coupled to a lattice continuum. The macroscopic effect of the quantum mechanical exchange interaction is included as are dissipation and the associated thermodynamics. The resulting nonlinear equations are specialized to the important case of a small dynamic field superposed on a large static biasing field. Only the linear approximation in the small-field variables is obtained. This final system of linear equations permits the solution of a variety of magnetomechanical boundary-value problems.
The Wave Equation and the Green's Dyadic for Bounded Magnetoplasmas
Authors: Seto Y. J. , Dougal Arwin A.
Published: January 2005

In studies of electromagnetic wave propagation and radiation in magnetoplasmas, the wave equation takes the form of a dyadic-vector Helmholtz equation. The investigation here shows that the dyadic-vector Helmholtz equation is solvable by the separation method in four cylindrical coordinate systems. Solutions in the form of complete sets of eigenfunctions are possible when boundary surfaces are present. For problems involving current sources in the plasma, the Green's dyadics for finite or semifinite domains can be constructed from the complete sets of eigenfunctions which are solutions to the homogeneous equation. The Green's dyadic for infinite domain is also shown to be obtained from that for a semifinite domain through a limiting process.
Exact Statistical Thermodynamics of Gravitational Interactions in One and Two Dimensions
Authors: Salzberg Allan M.
Published: January 2005

The exact equilibrium statistical mechanics of one- and two-dimensional gases, in which the particles interact through gravitational forces, is obtained. It is found that these systems are characterized by nonextensive thermodynamics leading to behavior somewhat reminiscent of the formation of a star from interstellar dust.
Sufficient Conditions for an Attractive Potential to Possess Bound States
Authors: Calogero Francesco
Published: January 2005

Simple conditions on the potential are given, which are sufficient to secure the existence of at least one bound state for each angular momentum l ≤ L. One such condition is given by the inequality − ∫ 0Rdr rV(r)[rR]2L+1− ∫ R∞dr rV(r)[Rr]2L+1≥2L+1,where R is an arbitrary radius and V(r) an everywhere attractive potential. Upper bounds on the energy of the lower bound state for each angular momentum are also given.
Spherically Symmetric Gravitational Fields
Authors: Bergmann P. G. , Cahen M. , Komar A. B.
Published: January 2005

It is shown that the Schwarzschild solution is the only spherically symmetric solution of the Einstein vacuum field equations, even when the differentiability of the metric is weakened to the extent of permitting solutions which are C0, piecewise C1. Petrov's purported counterexample is analyzed and shown to be essentially equivalent to Schwarzschild's example.
Hilbert Spaces of Type S
Authors: Grossmann Alexander
Published: January 2005

Certain Hilbert spaces of generalized functions are examined. They contain the space of tempered distributions, are invariant with respect to the Fourier transformation and contain functions that increase rapidly at infinity.
The Structure of Space and the Formalism of Relativistic Quantum Theory. II
Authors: Ahmavaara Yrjö
Published: January 2005

A purely group theoretical treatment of interactions of relativistic particles is shown to be possible in the connection of a finite geometry of space-time. The S operators are constructed by means of the reductions of the irreducible unitary representations of the relativity group of space-time. The dependence of the S-matrix elements on the momenta is still nontrivial, since there is a nontrivial distribution of observable momenta in a finite geometry. The Feynman graphs are easily evaluated by a simple computational technique involving five rules. The coupling schemes corresponding to nuclear electromagnetic, and weak forces in finite geometry are considered.
Reduced Density Matrices of Quantum Gases. I. Limit of Infinite Volume
Authors: Ginibre Jean
Published: January 2005

The reduced density matrices for quantum gases are studied by Banach space techniques. For suitably restricted interactions, they are shown to be analytic functions of the activity. As the volume of the system becomes infinite, they tend in some sense to well-defined limits for which the same analyticity properties hold. As a consequence, the virial expansion is shown to be convergent in a neighborhood of the origin.
Two-Photon Ionization of Atomic Hydrogen. II
Authors: Zernik W. , Klopfenstein R. W.
Published: January 2005

This paper contains a generalization of results described in previous work. A general formula is given for the total cross section for two-photon ionization of a hydrogenic state. An implicit method for evaluating the second-order radial matrix elements that occur in this expression is described in detail. Numerical results were obtained for the states with principal quantum numbers n = 1 through 5. It is concluded that, when both one-photon and two-photon ionization are energetically possible, the effect of the latter may be expressed as an intensity-dependent correction to the Gaunt factor which may therefore be written as G = g1 + ζIλ3g2. Here g1 is the usual Gaunt factor, I is the intensity of the light in W cm−2, λ is the wavelength in cm, g2 is a dimensionless factor of order unity, and ζ is a constant given by ζ = 0.1504 W−1 cm−1. Graphs of g2 are given as a function of electron energy.

These results do not include the effect of three-photon processes, which can also contribute to the first-order intensity-dependent correction to the Gaunt factor as a result of interference between the first- and third-order amplitudes.
Asymptotic Expansion of the Intensity of the Small-Angle X-Ray Scattering from Cylinders of Arbitrary Cross Section
Authors: Schmidt Paul W.
Published: January 2005

From a previously developed expression for the intensity of the small-angle x-ray scattering from randomly oriented right cylinders of arbitrary cross section and with uniform electron density, an asymptotic expansion is developed which can be used to approximate the scattered intensity at relatively large angles of the small-angle region. The asymptotic expansion is useful for numerical calculations at these large angles, where calculations by the usual techniques of numerical integration and series expansion are most difficult. The terms in the asymptotic expansion are found to be determined by the behavior of the characteristic function β(r) of the cross section at certain values of r. The general asymptotic expansion is used to calculate asymptotic expressions for the scattering from right circular cylinders and rectangular parallelepipeds.
Isoperimetric Problem with Application to the Figure of Cells
Authors: McKean H. P. , Schreiber M. , Weiss G. H.
Published: January 2005

Biological applications suggest the following geometrical problem. Consider n three-dimensional cells, touching or not, and assume that the free energy of their figure is the sum H = A + αB of the area A of the cell walls adjacent to the ambient fluid plus an adjustable constant 0 ≤ α ≤ 2 times the area B of the walls separating two cells. Given the partial volumes of the cells, the problem is to describe the shape of the (optimal) figure that renders H as small as possible; the analogous problem for two-dimensional cells is the subject of this paper. Geometrical proofs of the following features of optimal two-dimensional figures are presented below: (a) the edges bounding the cells are circular arcs; (b) at an inside corner, three edges meet at angles 2π/3; (c) at an outside corner, three edges meet with outside angle 2 cos−1 α/2; (d) pressures can be ascribed to the cells so that the pressure drop across an edge is proportional to its curvature; (e) bubbles appear at each inside corner as α passes 3½. All these facts have three-dimensional analogues with similar proofs.
Algebraic Construction of the Basis for the Irreducible Representations of Rotation Groups and for the Homogeneous Lorentz Group
Authors: Alcarás J. A. Castilho , Ferreira P. Leal
Published: January 2005

The basic functions for a class of the irreducible representations of the rotation groups in n dimensions (Rn) are explicitly constructed by an algebraic method in which the basic functions are taken to be homogeneous polynomials in the variables of En. The solutions correspond to the hyperspherical harmonics of the mathematical literature and are of interest for problems exhibiting invariance under a certain Rn. The method is also applied to derive a basis for the infinite-dimensional irreducible representations of the homogeneous Lorentz group if we then look for homogeneous functions in the variables of the corresponding pseudo-Euclidean space.
Generalized Boltzmann Equation for a Quantum Gas Obeying Classical Statistics
Authors: Fujita S.
Published: January 2005

By means of the connected-diagram expansion a generalized Boltzmann equation is obtained for the Wigner distribution function f describing distinguishable quantum particles subject to constant electric and magnetic fields. This equation is exact and meaningful in the thermodynamic limit provided that the main interaction processes which cause the change in f involve finite numbers of particles. If the interaction processes are assumed to be localized in time, and long-time solutions are looked for, the equation could be simplified so that its collision term can be described in terms of f only just like the conventional Boltzmann equation for a dilute gas.
Unitarity and On-Mass-Shell Analyticity as a Basis for S-Matrix Theories. I
Authors: Gunson J.
Published: January 2005

Many-particle unitarity relations are analyzed and related to maximal analyticity conjectures with particular reference to continuations into unphysical sheets. Poles and cuts associated with unstable particles are treated in detail and generalized unitarity relations derived.
Long-Time Behavior of the Electric Potential and Stability in the Linearized Vlasov Theory
Authors: Sáenz Albert W.
Published: January 2005

In this paper we study in a mathematically rigorous manner how the electric potential, produced by small electronic charge density oscillations of definite wavenumber vector k in a plasma, behaves in the long-time limit and the connection between this behavior and the stability of a given steady, spatially uniform, distribution of the plasma electrons. Our work is based on the linearized Vlasov equation and on the associated Poisson equation. We formulate a very general initial-value problem concerning this system of equations, writing the above electric potential at a given position vector r and time t as φ(t)eik·r multiplied by a suitable constant, where φ(t) is independent of r. We establish the existence and uniqueness of solution of this problem by exploiting the fact that, in the linear theory, φ(t) obeys an inhomogeneous Volterra integral equation of convolution type, which is rigorously derived here. A detailed study of the asymptotic properties of the solutions of this equation for t → ∞ is made, including the establishment of necessary and sufficient conditions on the initial perturbations (perturbations of the steady electron distribution function at t = 0) for φ(t) to be of negative exponential order as t → ∞. As a byproduct of this asymptotic investigation, we give a precise discussion of the Landau damping of long wavelength plasma oscillations in an initially Maxwellian plasma, concluding that in this case φ(t) exhibits such damping for a broad range of initial perturbations and that the damping decrement is essentially that first computed by Landau. We introduce criteria of stability and instability based on the boundedness and unboundedness, respectively, in the limit t → ∞ of certain nonnegative quantities Wp(t), which are defined as suitable norms of the perturbed electron distribution function. New sufficient conditions for stability and instability are proved for extensive classes of initial distributions and initial perturbations. These results are compared with conclusions on stability and instability reached by Backus.
Spatially Inhomogeneous States of Many-Body Systems
Authors: Eger M. , Gross E. P.
Published: January 2005

To treat many-body systems in the presence of a static potential or problems of highly collective spatially inhomogeneous motions such as vortex lines, wavefunctions of a type Ψ= ∏ i=1N g(xi)Φ(x1,…,xN)have been proposed. Here Φ is the exact ground state of the homogeneous system and g(x) is a one-particle state introduced to describe the effect of the spatial inhomogeneity. However, to determine g(x) by the variation principle, one needs to know the spatial correlation functions of all orders for the homogeneous many-body system. It is shown that the method of point transformations allows one to work with qualitatively similar but different states. The description of the system in the presence of a static impurity or of a state representing a vortex line in liquid helium requires a knowledge of only the average kinetic energy and x-ray scattering factor for homogeneous liquid helium. Both of these are available from experiments. The treatment of a recoiling impurity atom, strictly speaking, requires a knowledge of the current correlation tensor for the ground state of the homogeneous many-body system. This term vanishes in the Hartree limit of the theory for bosons.
Mathematical Study of the Cutoff Procedure for Divergent Integrals in Bound-State Problems
Authors: Mandelbrojt J. , Mandelbrojt S.
Published: January 2005

We consider the class of conveniently normalized functions having a given rate of decrease at infinity on the real axis. The lower bound of the length of the supports of the Fourier transforms of such a given class of functions is evaluated. This could be considered as a refined form of the uncertainty relation. The condition, which is necessary and sufficient for this lower bound to be finite is indicated. In the physically important case when this lower bound is infinite, a length, which we call the ``natural cutoff of this class of functions'' is nevertheless defined, it characterizes the interval outside of which the value of the Fourier transform of a function of the given class may be negligible. A construction which gives the cutoff, or the natural cutoff, is given; it is valid for practically any given rate of decrease. We carry out the calculation explicitly in the case of a decrease of the functions characterized by |F(x)| ≤ A e−μ|x| (in this case the support of the Fourier transform is necessarily infinite). A canonical function is built such that the abscissa for which it begins to become ``small'' should be reasonably considered to be the natural cutoff of the Fourier transform of all the functions satisfying this inequality. The same method can be used for the class of functions satisfying an inequality of the form |F(x)| ≤ A exp [−C(x)], with ∫1∞ C(u) u−3 du < ∞.
Conservation Laws for Free Fields
Authors: Kibble T. W. B.
Published: January 2005

The recently discovered conservation laws for the ``zilch'' of the electromagnetic field (of which a simple derivation is presented) are examined in the context of a general discussion of bilinear conserved quantities in free-field theories. It is shown that there is always an infinite set of these quantities, and a method of finding them all is presented and illustrated by applications to simple theories. The existence of these conserved quantities is shown to be a consequence of the fact that the momentum-space density is constant in time.
Bosons and Fermions
Authors: Penney R.
Published: January 2005

It is proven that one cannot construct boson creation and annihilation operators from a finite number of fermion operators. The proof follows from the isomorphism of the fermion algebra and the algebra of Dirac matrices.
Exact Eigenstates of the Pairing-Force Hamiltonian. II
Authors: Richardson R. W.
Published: January 2005

The restrictions on a previously reported class of exact eigenstates of the pairing-force Hamiltonians are removed and it is indicated that all the eigenstates of this Hamiltonian can be included in this class. Explicit expressions are given for the expectation values of one- and two-body operators in the exact, seniority-zero eigenstates of this Hamiltonian. In particular, a simple expression for the occupation probabilities of the levels of the single-particle potential is given. This expression may be easily evaluated for realistic nuclear systems.
Quantum Mechanical Time
Authors: Rankin Bayard
Published: January 2005

Some suggestions on the probabilistic structure of quantization are first obtained by studying local properties of random variables such as continuity and differentiability with respect to a topology in the sample space. This analysis is extended to analytic functions on the complex plane and conditions are formulated under which probability densities are expressed as sums of absolute squares of complex numbers. Physical restrictions are introduced through the Schrödinger equation for a single bound-particle system. As a result, observations on a physical system become identified with the random selection of points in a topological measure space and the physical observables such as energy, position, and momentum, as well as time, are identified with measurable functions on appropriate spaces. Time, considered as a function, appears to be multiple-valued with spacings between multiple values and a detailed functional structure that characterizes and is characterized by the physical system under observation and the physical observables being measured. Its detailed functional structure is related to the physically measurable probabilities of quantum theory and it is seen to serve in the capacity of a conditioning random variable in the computation of quantum mechanical expectations.
Functional Expansion in Peratization Theory
Authors: Fried H. M.
Published: January 2005

A functional peratization expansion is defined in the charged vector meson theory of weak interactions. The simplest approximation is carried through and shown to yield an allowed lepton-lepton scattering amplitude, properly damped on the light cone. A basic deficiency in this procedure is the absence of any criteria for the estimation of the character and size of successive corrections.
Schrödinger Basis for Spinor Representations of the Three-Dimensional Rotation Group
Authors: Pandres Dave
Published: January 2005

It is shown that the double-valued spherical harmonics provide a basis for the irreducible spinor representations of the three-dimensional rotation group. Pauli's assertion to the contrary is shown to be false. Both infinitesimal and finite rotations are discussed in some detail. It is also shown that there remains a twofold degeneracy in the spherical harmonic Yim when j and m are specified.
Asymptotic Reduced-Width Amplitude Distributions
Authors: Ullah Nazakat
Published: January 2005

The method of moments is used to derive the reduced-width amplitude distributions. The explicit dependence of the distribution function on the dimension N of the random orthogonal matrix for large values of N is obtained. It is shown that in the limit N → ∞, the distribution is the same as the one obtained using the explicit assumption of level independence.
Scattering by Singular Logarithmic Potential
Authors: Aly H. H. , Riazuddin  , Zimerman A. H.
Published: January 2005

Potential of the form (gr−4 ln2 r − g½r−3)θ(r − r0) is considered in connection with the applicability of peratization technique. The advantage of this potential is the fact that while it is dominated by a logarithmic part near the origin, the exact solution of the zero-energy and s-wave Schrödinger equation is obtained in a closed form. We show that the peratization technique gives the correct answers.
Effect of n-Spin-Wave Interaction on the Low-Temperature Spontaneous Magnetization
Authors: Morita Tohru , Tanaka Tomoyasu
Published: January 2005

For a Heisenberg model of a ferromagnet, it is known that the commutator of the Hamiltonian ℋ and the operator S0− = ΣjSj− is proportional to S0−. From this fact, a conjecture is made on the lowlying energy levels of two-, three-, …, n-spin-wave problems. With the aid of the conjecture and the assumption that there is no low-lying n-spin-wave bound state, it is concluded that the contributions to the low-temperature expansion of the spontaneous magnetization due to two-, three-, …, n-spin-wave problems are of O(T4), O(T13/2), …, O(T5n/2−1), respectively.
Solution of the Klein-Gordon and Dirac Equations for a Particle with a Plane Electromagnetic Wave and a Parallel Magnetic Field
Authors: Redmond Peter J.
Published: January 2005

Solutions of the Klein-Gordon and Dirac equations are obtained for a particle moving in an arbitrary plane electromagnetic wave in vacuo plus a uniform static magnetic field parallel to the direction of propagation of the electromagnetic wave.
Branching Rules for Simple Lie Groups
Authors: Whippman M. L.
Published: January 2005

If Γ is an irreducible representation of a group 𝒢, and ℋ is a subgroup of 𝒢, then Γ furnishes a representation of ℋ which is, in general, reducible, and the branching rules specify which irreducible representations of ℋ occur in the decomposition of this representation. Branching rules are derived for various choices of 𝒢 and ℋ, including most possibilities that have been discussed as higher symmetry groups.
Formalization of the Lagrangian, the Hamiltonian, and Related Concepts
Authors: Pease M. C.
Published: January 2005

We consider linear system of finite dimensionality in which the system matrix is a well-behaved function of the independent variable—z or t. The system is assumed to obey a conservation law with zero signature, but is otherwise unrestricted. Using what we call the ``vectorial derivative,'' it is shown that we can obtain Lagrangian and Hamiltonian functions in terms of which the system behavior is described in the usual way. Thus, the methods of classical physics are made available for the study of such systems, without the physical connotations usually carried by such functions.

In carrying out this formalization, we define a functional—e.g., the Hamiltonian—and, at the same time, define the operator that generates the function. It is shown that the interrelation of two functionals involves the appropriate commutator of the operators involved. For example, the Poisson bracket of a functional, F, and the Hamiltonian, H, is the functional obtained from the operator Γ[ΓH, ΓF], where Γ=(O I−I O)and F and H are the operators involved in F and H. The analogy to quantum mechanics is striking.

Finally, it is shown that the contact transformations that preserve the Hamiltonian operator form a rotational system in the Lie algebra, L, in which ΓH is embedded. That is, they obey equations of the form M = [ΓH, M]. It follows, then, that all H-preserving transformations can be split into components in the intersection of the Γ-unitary group and either L or a subspace derived from L.
Electromagnetic Field inside a Current-Carrying Region
Authors: Fikioris John G.
Published: January 2005

The improper integrals, appearing in the course of evaluating the vector potential A and the electric field E inside a current-carrying region, are carefully examined. It is found that the integrals exist and have a well-defined meaning only when the current density function satisfies a Hölder condition. A definite and precise way of evaluating them is derived and A, E are shown to satisfy the usual inhomogeneous equations.
Threshold Regge Poles for Coupled Channels
Authors: Carnahan W. H.
Published: January 2005

The threshold Regge poles are investigated in a variety of many-channel problems. The cases considered are a simple j-independent interaction between two spin-zero channels, the tensor force problem for two spin-½ particles and a truncated interaction between a particle and a bound system of two particles.
Group Theory of Harmonic Oscillators in n-Dimensional Space
Authors: Louck James D.
Published: January 2005

It is shown how the states of N particles which move in a common n-dimensional harmonic oscillator potential can be classified according to the irreducible representations of the unitary groups UN or Un. The complete set of Nn independent integrals of the motion is obtained, their simultaneous eigenvectors being generalized Gel'fand basis vectors which can be enumerated either as bases for irreducible representations of UN or Un. Explicit formulas are given for the linear transformations induced on the basis vectors by the infinitesimal operators of UN and Un. The relation of the present work for n = 3 to that of Bargmann and Moshinsky is noted.
New Variational Scheme for Finding the Excited States of One-Dimensional Systems
Authors: More Richard M.
Published: January 2005

A new relaxation-variation scheme for solving one-dimensional Schrödinger equations is described. The method approximates the nth eigenvalue and eigenfunction directly, without requiring approximations to the lower eigenfunctions. It is applicable to any one-dimensional Sturm-Liouville system.
Inequality with Applications in Statistical Mechanics
Authors: Thompson Colin J.
Published: January 2005

We prove for Hermitian matrices (or more generally for completely continuous self-adjoint linear operators in Hilbert space) A and B that Tr (eA+B) ≤ Tr (eAeB). The inequality is shown to be sharper than the convexity property (0 ≤ α ≤ 1) Tr (eαA+(1−α)B) ≤ [Tr (eA)]α[Tr (eB)]1−α, and its possible use for obtaining upper bounds for the partition function is discussed briefly.
Expansion of a Function of Noncommuting Operators
Authors: Kumar Kailash
Published: January 2005

The Taylor series expansion of a function f(A + B) of noncommuting operators A and B is written in several ways. Recursion relations, generating functions, and some explicit formulas for operator coefficients of this expansion are given.
Isotopic Space, Complex Conjugation, and U3 Symmetry for Particles
Authors: Teitler S.
Published: January 2005

The inherent U4 symmetry over spinor spaces of the regular decomposition indicated previously is discussed from the viewpoint of a single spinor space. It is noted that this symmetry does not yet include isotopic spin. Isotopic space is introduced by considering that the space-time vectors and its vector Clifford algebra are imbedded in the algebra of a six-dimensional Euclidean space. Three of the latter's dimensions are identified with ordinary space, and three are identified with isotopic space. The ``time vector'' corresponds to the pseudoscalar element in isospace, and the imaginary unit is expressed as a linear combination of the bivectors in isospace. The relation among complex conjugation, time inversion, and space inversion is thereby clarified. Some comments on the application of these ideas to particle symmetry discussions are made. In particular one obtains as the first extension of the SU2 symmetry of isotopic spin, a U3 symmetry for particles.
Spin-Matrix Polynomials and the Rotation Operator for Arbitrary Spin
Authors: Weber T. A. , Williams S. A.
Published: January 2005

A technique for the expansion of an arbitrary analytic function of a spin matrix is developed in terms of a complete set of polynomials based on the characteristic equations of the spin matrices. The expansion coefficients are determined by use of the ascending difference operator from the calculus of finite differences. The expansions are valid not only for functions of a spin matrix but also for functions of a complex variable. In the latter case the eigenvalues of the spin matrices are the zeros of the polynomials. In either case the expansion coefficients are the same although of course in the case of a spin matrix the series terminates. As an example the rotation operator for arbitrary spin is developed via its functional analog in terms of these polynomials, and the region of convergence of the series to the function is investigated.
Null Fields in Einstein-Maxwell Field Theory
Authors: Wyman M. , Trollope R.
Published: January 2005

The role of the electromagnetic null field, within the framework of classical relativistic electrodynamics, has never fully established. The object of this paper is to present a large class of exact solutions to the Einstein-Maxwell null field equations with the hope that the physical implications of these solutions may lead to a more thorough understanding of these fields.

In the present analysis, it appears that the nature of the solutions depends on the form of the propagation vector. In the cases for which the divergence of this vector vanishes, an extensive class of solutions is obtained and particular examples are considered. The analysis of the case when the divergence is nonzero is far from complete and only a particular solution is given.
Transformation from a Linear Momentum to an Angular Momentum Basis for Relativistic Particles of Nonzero Mass and Any Spin
Authors: Moses H. E.
Published: January 2005

The infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear-momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In a previous paper we showed how the two bases were related for representations corresponding to zero mass and any finite spin. In the present paper we show how the two bases are related for representations corresponding to nonzero mass and any spin. Thus this paper and the preceding one enable us to expand relativistic plane waves into relativistic spherical waves and vice-versa for particles of any spin and any mass.

In the previous paper we used the relation between the linear- and angular-momentum bases to integrate the infinitesimal generators in the angular-momentum basis and thereby obtain closed expressions which show how the angular momentum of particles of zero mass and any finite spin transform under changes of frame of reference. Similar use could be made of the transformation of the present paper. Results of such use will be given in a later paper.
All Homogeneous Solutions of Einstein's Field Equations with Incoherent Matter and Electromagnetic Radiation
Authors: Ozsváth István
Published: January 2005

In this paper I prove the theorem that the Einstein Field equations with dust and electromagnetic null field have only two homogeneous solutions. I have given a discussion of these two solutions in in another paper.
On Projective Representations of Finite Groups
Authors: Rudra P.
Published: January 2005

The algebra of projective representation, belonging to a factor system, has been presented. Exact expressions for the projection operators, Kronecker (inner) direct-product representation, have been obtained. Formulas for obtaining the characters of all the inequivalent irreducible projective representations and the Clebsch-Gordan coefficients have been derived.
Born Reciprocity Principle and Unitary Symmetry
Authors: Shin Ernest E. H.
Published: January 2005

In view of the past attempts by Born et al. to explain elementary particles using the Born reciprocity principle as a postulate and of the recent success of unitary symmetry schemes, it is sought to establish a contact between the reciprocity principle and unitary symmetry in connection with the problem of elementary particles.
General Operator Potential for the Two-Nucleon System
Authors: Tharrats J. , Cerceau O. , Rojo O.
Published: January 2005

The interaction between two nucleons is assumed to be described by a general Hermitian operator having certain symmetry and invariance properties.

It is shown that the usual phenomenological potentials employed in dealing with the two-nucleon system, i.e., nonlocal, velocity-dependent and hard-core, are related by means of unitary transformations of the interaction operator. In addition it is shown that another such transformation of the total Hamiltonian operator leads to the effective mass formalism. Some numerical results on phase shifts are compared with most accepted values.
Diffraction by a Smooth Transparent Object. II. Diffraction by a Cylindrical Cavity
Authors: Chen Yung Ming
Published: January 2005

The exact solution corresponding to the field produced by a time-harmonic line source in the presence of a circular cavity of radius a(N < 1) is found and evaluated asymptotically for large ka. Upon comparing it with the geometric rays and the diffracted rays constructed in Chen, the coefficients, especially the diffraction coefficients, are thus determined.