bluuebunny/crossref_metadata_2025 huggingface dataset:
Absorptive Parts and the Bethe-Salpeter Equation for Forward Scattering
Authors: Nussinov S. , Rosner J.
Published: January 2005

We have developed a Laplace transform approach to the Bethe-Salpeter equation for absorptive parts of forward scattering amplitudes. The method appears direct and unsophisticated and is useful for computation. It is essentially identical to decomposition into four-dimensional partial waves, but the inversion formula is more straightforward. We have obtained the high-energy behavior of sums of several types of φ4 graphs in the weak- and strong-coupling limits. These examples illustrate some general results we obtain for Mth order φ4 kernels. Specifically, the absorptive part behaves as sn0(log s)β for high s; for weak coupling λ, n0 ∼ λ½M, while for strong coupling in the ladder graph approximation, n0/λ½ → 1/2π. We have also proven an interesting inequality related to absorptive parts. One of its corollaries is that uncrossed ladder graphs ``majorize'' crossed ones.
Expanding Universe in Conformally Flat Coordinates
Authors: Tauber G. E.
Published: January 2005

Einstein's field equations for a homogeneous isotropic universe are written in terms of a conformally flat coordinate system for both the closed and the open cases. It is shown that these can be readily integrated for equations of state of the form p = aρ in terms of algebraic functions. Explicit solutions and expressions for the radius and Hubble factor are given for a = 1, ⅓, and for p = 0.
Common Interpretation of Phase Transitions in Various Models
Authors: Emch Gérard G.
Published: January 2005

A discussion is presented of the formal analogy which exists between the Bardeen-Cooper-Schrieffer (BCS) model for superconductivity and the several variations of the model suggested by Kac for the exact study of phase transitions. Some of the algebraic techniques, pioneered by Haag for his formulation of axiomatic field theory and his application of this theory to the BCS model, are shown to extend to all these cases. Some arguments are given in the beginning of the paper to point out the necessity of a new interpretation of the existence of various phases in the same physical system. The results obtained so far for equilibrium situations confirm the consistency of the proposed tentative description of the phenomena considered.
Role of the Integral-Operator Method in the Theory of Potential Scattering
Authors: Gilbert R. P. , Howard H. C.
Published: January 2005

A recently developed method in the theory of potential scattering is investigated. The authors supply analytic computations for determining the location and the type of singularities, which occur in several scattering amplitudes that are constructed from physically relevant potentials.
Generalization of the Variational Method of Kahan, Rideau, and Roussopoulos. II. A Variational Principle for Linear Operators and its Application to Neutron-Transport Theory
Authors: Kostin Morton D. , Brooks Harvey
Published: January 2005

Additional applications of a generalized form of the variational method of Kahan, Rideau, and Roussopoulos are presented. Equations used in neutron transport theory, such as the spherical harmonics operator form of the Boltzmann equation, are derived from the generalized variational functional and an interpretation of these operator equations in terms of flux- and source-generating operators is suggested. A relationship between this variational method and the variational method of Lippmann and Schwinger is established, and it is shown that the least-squares variational functional of Becker for linear equations can be derived from a generalized variational functional.
Representations of Inhomogeneous SU(n) Groups with Mixture of Homogeneous and Inhomogeneous Operators Diagonal
Authors: Mirman Ronald
Published: January 2005

Explicit representations of some inhomogeneous compact SU(n) groups and their algebras are constructed with certain mixtures of diagonal homogeneous and inhomogeneous operators.
Two-Meson Solution of the Charged Scalar Static Model with Bound States
Authors: Bronzan J. B.
Published: January 2005

A two-meson solution of the charged scalar static model is constructed for the case in which the coupling is sufficient to produce bound states. This is done by taking the solution given previously for the case in which bound states are absent and increasing the coupling. As usual, bound states appear as poles in scattering amplitudes. However, since one of the mesons in the three-body states can bind to the source, bound-state meson scattering channels also appear. The new amplitudes relating to bound-state meson scattering are obtained, and their unphysical singularities are examined. The enlarged scattering matrix is unitary.
Compatibility Requirements in BBGKY Expansion
Authors: Liboff Richard L. , Perona Giovanni E.
Published: January 2005

The BBGKY equations are written in nondimensionalized form leaving two parameters α and β explicit. The coefficient α is a measure of the ``strength'' of the potential of interaction, while β is a measure of the ``range'' of the interaction potential. The s-particle distribution function is expanded about vanishing correlation with ε entering as a parameter of smallness. Nine cases are considered, depending on the order of magnitude of α and β (viz., ε, 1, ε−1). Lowest order terms, in each of the cases considered, give a kinetic equation, together with subsidiary conditions. In the ``rigid sphere'' limit, the Vlasov equation appears together with a second-order differential constraint. The Vlasov equation alone appears in the Rosenbluth-Rostoker limit. A critical survey is made of the compatibility problem which is inherent to expansions of the BBGKY sequence. For the most part, the problem reduces to the compatibility of ``type-l'' constraint equations. These constraint conditions cause any given order set of equations to be over determined.
Results on Certain Non-Hermitian Hamiltonians
Authors: Wong Jack
Published: January 2005

We present a few results on the spectral properties of a class of physically reasonable non-Hermitian Hamiltonians. These theorems relate the spectral properties of a non-self-adjoint operator (of the aforementioned class) in terms of that of a self-adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrödinger equation has to be solved numerically, a knowledge of the spectral properties of the non-Hermitian Hamiltonian would insure when the eigensolutions exist.
n-Representability Problem for Reduced Density Matrices
Authors: Kummer Hans
Published: January 2005

In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8).

Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.
Phase Transition of a Bethe Lattice Gas of Hard Molecules
Authors: Runnels L. K.
Published: January 2005

A pseudo-lattice, or homogeneous Husimi tree, is simpler statistically than a true lattice in two or three dimensions, since the pseudo-lattice contains only low-order cycles. The prototype is the ``Bethe lattice,'' or Cayley tree, containing no cycles at all. Reported here are the results of studies of the lattice statistics of the Bethe lattice of coordination number three, using the language of the hard molecule lattice gas; the size of an adsorbed molecule prevents simultaneous occupancy of the same site or any of the three nearest-neighbor sites. Using methods related to those used in enumerating graphs, a recursion relation is obtained which must be satisfied by the grand partition function. It is shown that the resulting equation of state is not the quasi-chemical equation expected because of the absence of cycles. There is no phase transition of lower than third order and in all likelihood none at all. The quasi-chemical equation of state is obtained only if ``surface'' effects are eliminated; even then the solution is valid only for activities z < 4, at which point the ``interior'' system undergoes a second-order transition to an ordered state with a finite discontinuity in compressibility.
Complementary Variational Principles and Their Application to Neutron Transport Problems
Authors: Pomraning G. C.
Published: January 2005

Several variational principles are developed which give upper and lower bounds for the linear functional (S, ψ), where ψ is the solution of the inhomogeneous equation Hψ = S with H a self-adjoint, positive-definite, linear operator. Some of the principles bound this functional only with respect to small or local variations, whereas others give bounds for arbitrary variations. Several of our results coincide with those of other authors widely scattered throughout the literature, and we show that these principles have a common origin. Other results given are new. Examples of the use of these principles are taken from the field of neutron transport theory, and we use both the linear Boltzmann or transport equation and the diffusion equation. One interesting result is that certain ``exact'' values of the extrapolated endpoint for the Milne problem which have been reported in the literature fall, due to numerical inaccuracies, outside the bounds computed here.
Internal Holonomy Groups of Yang-Mills Fields
Authors: Loos Hendricus G.
Published: January 2005

Classical Yang-Mills potentials define a Lie group, the internal holonomy group, which is analogous to the ordinary holonomy group defined by the Christoffel symbols in general relativity. The internal holonomy group is an important tool for the dissection and study of the Yang-Mills equations and of local gauge theory in general. A number of theorems on internal holonomy groups is presented, together with some applications.
Timelike Cluster Properties in Nonrelativistic Scattering
Authors: Taylor John R.
Published: January 2005

It is generally assumed that experiments which are sufficiently separated in space or time should be mutually independent. In scattering theory this independence is expressed by the cluster decomposition of the S matrix. It has been proved by Hunziker that for experiments which are separated in space the cluster property does hold in nonrelativistic scattering. The purpose of this paper is to prove the same result for experiments which are separated in time.
Invariant Imbedding Equations for the Dissipation Functions of an Inhomogeneous Finite Slab with Anisotropic Scattering
Authors: Bellman R. , Kagiwada H. , Kalaba R. , Ueno S.
Published: January 2005

Two complete sets of differential-integral equations are derived for the dissipation functions of an inhomogeneous finite slab which scatters anisotropically. The first set is for the case of monodirectional illumination of the upper boundary, and the second is for the monodirectional illumination of the lower boundary. Conservation equations link the reflection, transmission, and dissipation functions.
Group Formulation of Potential Scattering
Authors: Finkelstein R. , Levy D.
Published: January 2005

Potential scattering is formulated in terms of an integral equation on the group space of O3. The potential need not be spherically symmetric except at large distances from the scattering center.
Spin-s Spherical Harmonics and ð
Authors: Goldberg J. N. , Macfarlane A. J. , Newman E. T. , Rohrlich F. , Sudarshan E. C. G.
Published: January 2005

Recent work on the Bondi-Metzner-Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular-momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.
New Form of Characteristic Functional for Cascade Processes
Authors: Scott W. T.
Published: January 2005

The characteristic functional for an energy-dependent cascade process is redefined so as to introduce the arbitrary function without using exponentials. An immediate consequence is that the functional becomes the function at t = 0 when the process is multiplicative with one particle at the initial time. A simple one-component model of a cosmic-ray shower is used to illustrate how this definition leads directly to (a) a simple form of the Chapman-Kolmogoroff equation, (b) forward and backward integro-differential equations for the functional, and (c) the derivation of all probability functions of interest for the process. An example is given of the distributions in number of particles at different depths when the total cross section for multiplication is proportional to the energy. Some general forms of functional are proposed. The extension to electron-photon showers is outlined.
Interpolation Method for the Many-Body Problem
Authors: Kijewski L. , Percus J. K.
Published: January 2005

Variational principles for lower bounds to the energy, or free energy for T > 0°, of many-body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy—or free energy—for the model Hamiltonian HM. Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H − HM approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs-Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.
Matrix Products and the Explicit 3, 6, 9, and 12-j Coefficients of the Regular Representation of SU(n)
Authors: Kaplan L. M. , Resnikoff M.
Published: January 2005

The explicit Wigner coefficients are determined for the direct product of regular representations, (N)⊗(N)=2(N)+…, of SU(n), where N = n2 − 1. Triple products CmCiCm = αFi + βDi, and higher-order products, are calculated, where Ci may be Fi or Di, the N × N Hermitian matrices of the regular representation, and m is summed. The coefficients α, β are shown to be 6-j symbols, and higher-order products yield the explicit 9-j, 12-j, symbols. A theorem concerning (3p)-j coefficients is proved.
Unitary Representations of the Group O(2, 1) in an O(1, 1) Basis
Authors: Mukunda N.
Published: January 2005

We consider the unitary irreducible representations of the group SO(2, 1), belonging to the continuous and the discrete classes. We cast them into a form in which the noncompact generator of an O(1, 1) subgroup is diagonal. We examine some properties of the remaining generators in this basis. We recover the known result that the spectrum of the noncompact generator covers the real line twice for representations of the continuous class, and once for those of the discrete class.
Canonical Realizations of the Rotation Group
Authors: Pauri M. , Prosperi G. M.
Published: January 2005

A general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics, proposed in a preceding paper, is applied to the rotation group. A number of significant physical examples corresponding to nonirreducible realizations is treated in detail: specifically, the mass point, the rotator, and the rigid body with a fixed point. The explicit form of the possible irreducible realizations is worked out. Such realizations do not directly correspond to any realistic physical model but play a relevant role for the introduction of the spin in classical mechanics.
Existence of Solutions of Crossing-SymmetricN/DEquations
Authors: Atkinson David
Published: January 2005

A fixed-point theorem is applied to the N/D equations of ππ scattering, for which crossing is satisfied exactly by the absorptive part of the amplitude, up to a finite, but arbitrary cut off. It is found that ghost-free solutions exist, if the subtraction constants are not too large, but that these solutions are not unique, since the Castillejo-Dalitz-Dyson ambiguity is not resolved by the requirement of crossing symmetry.
Invariant Imbedding as a Generalization of the Resolvent Equation
Authors: Mockel André
Published: January 2005

Invariant imbedding equations for the Green's function of a general linear operator are shown to derive from a generalization of the resolvent equation of the theory of operators. Two applications are given: the neutron, or photon, transport in an arbitrary finite body, and the quantum-mechanical theory of collision. In the last case, a nonlinear differential equation for the amplitude of transition Tβα from a channel α to a channel β with conservation of the total energy is derived from the general equation.
Causality and the S Matrix
Authors: Rañada Antonio F.
Published: January 2005

A formalism is presented in order to impose causality conditions for finite-range interactions which do not require the existence of wave packets with a sharp front. The absence of bound states and particle production is assumed. Two different causality conditions—referred to as strong and weak conditions—are studied. It is shown that in general the convolution kernel that connects the ingoing with the outgoing wave is not zero in the past. Some analytic properties of the S matrix are deduced.
Bethe-Salpeter Equation in Momentum Space
Authors: Saenger Robert M.
Published: January 2005

The Wick transformation in momentum space is modified to include all scattering energies by use of a coordinate surface which possesses limited detours into the complex relative energy plane. This device retains the simple form of the equation for the Bethe-Salpeter amplitude ψ(p, p0). It is shown that the transformation is valid if ψ(p, p0) has the cut structure indicated by field theory, and this structure is shown to be consistent with the Bethe-Salpeter equation provided the interaction V(x) satisfies a simple causality condition. It is further shown that the cut structure of a solution to the transformed equation can be deduced from the causal structure of the interaction alone without reference to field theory. Basic properties of the transformed equation are derived and a numerical treatment for purely elastic scattering is presented.
Energy Requirement for Nonlinear Density Fluctuations in a Vlasov Plasma
Authors: Chen Robert L. W.
Published: January 2005

Exact solutions of an isoperimetric problem are presented which lead to a lower bound on the kinetic energy of the particles in a Vlasov plasma, in terms of the mean-density fluctuation:E≈>32nkT[1+0.4(|Δn|¯n)2],where E is the kinetic energy per unit volume, n the mean number density, k the Boltzmann constant, T the temperature, and Δn the deviation of density. This energy requirement associated with Δn must be taken into account when considering the energetic aspect of the growth of instabilities. The present work, which relies entirely on mathematical analysis, confirms the main results of an earlier nonrigorous calculation.
Single-Particle Green's Function for a One-Dimensional Many-Fermion System
Authors: Theumann Alba
Published: January 2005

In this paper we compute the single-particle Green's function for a system of N-interacting fermions in one dimension. The problem is exactly soluble in the case of a δ-function potential. We consider also the case of a small modification to a δ-function potential and we compute the Green's function up to first order in a perturbation expansion. In both cases we find that the Green's function has a branch cut in the complex energy plane but it has no single-particle poles.
Induced and Spontaneous Emission
Authors: Bergmann Saul M.
Published: January 2005

The problem of induced and spontaneous emission is investigated for an atomic two-level system with incident beams of radiation which are either in a coherent state or in a stationary state (contain a definite number of photons). The treatment is fully quantum-mechanical, and is confined to the case where the frequency spectrum of the incident beam is narrow compared to the natural linewidth of the system. It is shown that, under such conditions, the spontaneous emission for frequencies within the narrow band of the incident radiation is sharply reduced compared to the prediction of the natural lineshape. It is shown that a hole is burned in the natural lineshape within the narrow frequency band, thus effectively quenching the spontaneous emission at some frequency within the band. This effect is shown to occur both for the coherent and stationary beams. Quantities proportional to the induced and spontaneous probability amplitudes and the lifetimes are computed for times comparable to and long compared to the free lifetime of the state. An expression is found for the spectrum of the emergent radiation in terms of these quantities. Its physical meaning is briefly discussed. The density operator of the field for all times is given.
Note on the Robinson-Trautman Solutions
Authors: Foster J. , Newman E. T.
Published: January 2005

A procedure is given which enables one to construct all the type N Robinson-Trautman solutions and an infinite class of type III solutions. Some approximate solutions, consistent with the Bondi-Sachs radiation conditions for bounded source fields, are also given. These approximate solutions, obtained by perturbing the Schwarzschild solution, are Schwarzschild in the asymptotic future. It is also shown that there exists an infinity of exact solutions which are Schwarzschild in the asymptotic future.
Instantaneous Action-at-a-Distance in Classical Relativistic Mechanics
Authors: Hill Robert Nyden
Published: January 2005

The possibility of describing orbits in classical relativistic mechanics in instantaneous action-at-a-distance fashion by second-order differential equations (as in Newton's gravitational theory) is investigated with particular emphasis on the two-body problem of classical relativistic electrodynamics. Differential conditions are stated to guarantee world-line invariance and form-invariance of the equations of motion under Lorentz transformation for such a description of an N-particle system in three dimensions. A pair of integrodifferential equations for the equations of motion are derived to provide an explicit means of passing from a description via direct interaction along light cones to an instantaneous action-at-a-distance description for a two-body problem. These integrodifferential equations are applicable to the two-body problem of classical electrodynamics with either retarded interactions and radiation damping or with half-advanced plus half-retarded interactions.
Functional Approach to Classical Non-Equilibrium Statistical Mechanics
Authors: Hosokawa Iwao
Published: January 2005

A new approach to classical non-equilibrium statistical mechanics is considered. The essential idea is to solve Bogoliubov's functional-differential equation for the generating functional embracing all distribution functions for the particle system with the aid of the technique of functional integration. It becomes possible with a minor modification of his formalism. A general solution is given. The solution for our state functional is reduced to the Hopf characteristic functional for the Vlasov selfconsistent field when the interaction energy of particles is relatively small in each s-body Hamiltonian Hs.
Series Representation Method for Obtaining the Emission Coefficient from the Integrated Intensity Distribution for Asymmetrical Light Sources
Authors: Maldonado C. D.
Published: January 2005

A procedure for inverting the spectroscopic integral equation which relates the emission coefficient to the integrated intensity distribution for an optically thin and asymmetrical light source is presented. In this procedure the emission coefficient is expanded in terms of a complete set of orthogonal polynomials which are ``invariant in form'' to a rotation of axes and use is made of the spectroscopic integral equation to determine the unknown expansion coefficients in terms of known information for the integrated intensity distribution. In order to check its validity, the resultant series representation was summed exactly for a hypothetical example corresponding to an asymmetrical source whose emission coefficient possesses elliptical symmetry and diminishes with position in a Gaussian manner. Also the same hypothetical example was used to test the accuracy of the numerical procedure which was developed for summing the series representation in practical situations where the known information on the integrated intensity distribution is presented in the form of experimental data.
Consistency Conditions on Models for High-Energy Scattering
Authors: Eden R. J.
Published: January 2005

From unitarity, analyticity in t, and analyticity in E, a variety of conditions can be derived for the high-energy behavior of a crossing symmetric two-body scattering amplitude F(E,t). These conditions are studied as consistency conditions for a class of models for high-energy scattering based on smoothly varying functions. For this class of models they lead to: (1) the Froissart bound without making explicit use of Martin's enlargement of the Lehmann ellipse, (2) conditions on the phase of the amplitude and its derivative with respect to t in the forward direction. As an example of the use of the phase conditions, it is shown how the Lehmann ellipse can be enlarged to give analyticity in the circle |t| < R, where R is fixed. Although the method is different, this result is closely related to the general results of Martin, but with our smoothness assumptions a little more detail can be stated about the behavior of F(E,t).
Homogeneous Lichnerowicz Universes
Authors: Ozsváth István
Published: January 2005

Lichnerowicz has developed a general-relativistic theory of an electrically charged fluid with infinite conductivity. No exact solutions of the corresponding Einstein-Lichnerowicz equations have previously been given. The present paper establishes homogeneous solutions of these equations.
Singular Potentials and the Z = 0 Condition
Authors: Aly H. H. , Müller H. J. W.
Published: January 2005

We consider an interaction Lagrangian consisting of renormalizable and nonrenormalizable terms. After a brief discussion of the vertex function and the boundary conditions determining the renormalization constant corresponding to a pseudoscalar bound state, we derive an explicit expression for the nonrelativistic S-matrix for scattering by the potential 1/r4. These results are then used to evaluate Z. The explicit expressions obtained contain a strong cut-off dependence which cannot be factorized out. However, a specific value of the cut-off reduces the equation to a well-known eigenvalue problem, so that in this case a discrete spectrum of bound states may be obtained.
Modification of the Ehrenfest Model in Statistical Mechanics
Authors: Banta Edwin D.
Published: January 2005

The Ehrenfest model has been used to explain the ``irreversibility'' of thermodynamics and statistical mechanics. The modification described in this paper allows transitions to occur in both directions between the two ``boxes'' at each step of the model procedure. The equilibrium probability distribution is given in the form of a finite product, or in an iterated form particularly suitable for machine calculation. The analysis is illustrated by a simple model of an ionization-recombination process.
Linear Representation of Spinors by Tensors
Authors: Cercignani Carlo
Published: January 2005

A linear representation of spinors in n-dimensional space by tensors is proposed. In particular, in three-dimensional space a set composed by a scalar and a vector is associated to any two-component spinor, while in four-dimensional space the set corresponding to a four-component spinor is composed by a scalar, a pseudoscalar, a vector, a pseudovector, and an antisymmetrical tensor of second order. The resulting formalism is then applied to Schrödinger's and Dirac's equations. In three-dimensional space it turns out that the proposed procedure automatically assigns an intrinsic magnetic moment to an electron in a magnetic field without introducing any relativistic ideas or ad hoc assumptions. In four-dimensional space we can write the Dirac equation in a generally covariant fashion, without introducing new concepts with respect to the usual tensor analysis. The zero-mass Dirac equation splits into two sets of equations, describing respectively the neutrino and the photon. The possible bearing of the proposed approach upon the theories of elementary particles is briefly discussed.
Mixed Irreducible Representations for U3
Authors: Flores Jorge
Published: January 2005

In this paper we study some properties of irreducible representations of the unitary group in three dimensions U3 with positive and negative indices. These representations are useful for the group theoretical classification of particle-hole states in nuclear shell theory, as well as in elementary particle physics. We show how the rules for reducing the direct product of two given representations should be modified when we include positive and negative indices and we use these rules to obtain an algebraic expression for the irreducible representations contained in the direct product.
Strong Coupling Limit in Potential Theory. I
Authors: Frank William M.
Published: January 2005

Analytic properties of the Jost function in g, the coupling constant, are studied for potentials which are L½, i.e., ∫0∞dr |V(r)|12<∞, and have a negative power or exponential tail for large distances. For the bound state and scattering problems, it is found that the Jost function has exponential order ½ for large g, which implies that the scattering phase shift and the number of bound states in an attractive potential grow like |g|½ for large g. The latter result is the best possible and is a considerable improvement over earlier estimates.
Variational Principles and Weighted Averages
Authors: Goldstein Rubin
Published: January 2005

An examination of two functionals, which are in common use for making variational estimates of weighted averages, reveals that one may be preferred over the other in certain cases. In particular, for a positive-definite self-adjoint operator, the normalization-independent functional always yields the better approximation to the stationary value.
Correlations in Ising Ferromagnets. I
Authors: Griffiths Robert B.
Published: January 2005

The following results are proved for a system of Ising spins σi = ±1 in zero magnetic field coupled by a purely ferromagnetic interaction of the form −Σi<j Jijσiσj with Jij ≥ 0, for arbitrary crystal lattice and range of interaction: (1) The binary correlation functions 〈σkσl〉 are always nonnegative (〈 〉 denotes a thermal average). (2) For arbitrary i, j, k, and l, 〈σiσjσk σl〉 ≥ 〈σiσj〉 〈σkσl〉. Consequences of these results, in particular the second, are: (i) 〈σkσl〉 never decreases if any Jij is increased. (ii) If an Ising model with ferromagnetic interactions exhibits a long-range order, this long-range order increases if additional ferromagnetic interactions are added. This last fact may be used to prove the existence of long-range order in a large class of two- and three-dimensional Ising lattices with purely ferromagnetic interactions of bounded or unbounded range.
Correlations in Ising Ferromagnets. II. External Magnetic Fields
Authors: Griffiths Robert B.
Published: January 2005

Results of a previous paper showing that 〈σkσl〉 and [〈σkσlσm σn〉 − 〈σkσl〉 〈σmσn〉] are always positive for a system of Ising spins σi = ±1 coupled by a purely ferromagnetic interaction (〈 〉 denotes a thermal average) are extended to the cases where (i) certain spins are constrained to have the value +1 or (ii) the system is placed in an external (``parallel'') magnetic field H. The theorems thus obtained provide a simple proof of the existence of ``bulk'' values for 〈σkσl〉 and for 〈σk〉; the latter is identical with the usual bulk magnetization per spin. The correlation functions 〈σkσl〉 are monotone nondecreasing in |H| for fixed temperature T. Both 〈σkσl〉 and 〈σk〉 (and thus the bulk magnetization) are monotone nonincreasing in T for fixed H ≥ 0.
Two-Body Scattering with Wave Packets
Authors: Hammer C. L. , Weber T. A.
Published: January 2005

Two-body scattering is analyzed using a time-dependent formalism and wave packets of arbitrary shape as the initial wavefunctions. In particular, the case of single-channel resonant scattering is discussed in some detail. In addition to the usual amplitudes resulting from the potential term and the resonant term, a third amplitude is found which interferes with the other two. Another interesting result is that in this formalism the final state is not necessarily represented by spherical outgoing waves, but can be interpreted instead as two separate outgoing wave packets whose shape is different from the initial packets. The Wigner time delay is obtained as a natural consequence of the analysis, and the resonant cross sections are expressed as density matrices constructed from the wave packets.
Upper and Lower Bounds on Knudsen Flow Rates
Authors: Strieder William , Prager Stephen
Published: January 2005

Reciprocal variational principles are used to formulate upper and lower bounds on the low-pressure (Knudsen) flow rate of a gas through a channel of arbitrary geometry. The upper bound is equivalent to one obtained earlier by DeMarcus, but we believe the lower bound to be new. Explicit calculations are given for a short parallel-plate channel. The variational principles discussed here may be applied to a wide range of problems involving linear inhomogeneous integral equations.
Phase Problem in Coherence Theory
Authors: Nussenzveig H. M.
Published: January 2005

For a class of quasi-monochromatic spectra, including Lorentzian and Gaussian line shapes, it is shown that knowledge of the modulus of the complex degree of temporal coherence γ(τ) does not suffice to reconstruct the spectrum. This is due to the existence of zeros of γ(τ) in the complex τ plane, giving rise to a significant contribution to the phase of γ(τ). The position of the zeros and their physical interpretation are investigated. The case of band-limited spectra is also treated, and some general properties of the distribution of zeros in this case are given.
Random Walk with an Excluded Origin
Authors: Rubin Robert J.
Published: January 2005

The mean square end-to-end distance RN2 is calculated for the subset of all random walk configurations on a D-dimensional simple cubic lattice which do not return to the starting point. Explicit results are obtained in the limit N ≫ 1 for the one-, two-, and three-dimensional lattices. The values of the first two terms in the asymptotic series for RN2 are, respectively, N + N, N + N/log N, and N + 0.435/N−½. An unexpected relation is obtained between RN2 and SN, the average number of different lattice sites visited in an N-step random walk on a perfect lattice. It is RN2=SN(SN+1−SN)−1.
Time-Dependent Green's Function for a Moving Isotropic Nondispersive Medium
Authors: Tai C. T.
Published: January 2005

The time-dependent Green's function for a moving isotropic nondispersive medium is hereby obtained by taking the ω-integration of the time-harmonic solution which was previously obtained by means of an operational method and by making use of the known result of the two-dimensional Klein-Gordon equation.
Total Number of Particles and Fock Representation
Authors: Dell'Antonio G. F. , Doplicher S.
Published: January 2005

It is proved that an average number of particles can be defined only in those representations of the canonical commutation (or anticommutation) relations which are multiples of the Fock-Cook representation.
Energy Tensor of the Null Electromagnetic Field
Authors: Bartrum P. C.
Published: January 2005

The differential conditions are obtained which must be satisfied by the energy tensor of a null, source-free electromagnetic field. At the same time a method is indicated of determining the electromagnetic field when the energy tensor satisfying the necessary conditions is given.
Bootstrap Prediction of Symmetry for a Soluble Static Model
Authors: Cunningham Andrew A.
Published: January 2005

The Huang-Low bootstrap criterion of self-consistency, in the form of Levinson's theorem, is imposed on the exact solutions of the two-channel Low equation with an arbitrary crossing matrix. It is found that this condition, together with a number of dynamical conditions are sufficient to restrict the continuous crossing-matrix parameter to discrete values corresponding to an SU2 symmetry.
Group Embedding of Space-Time and Internal Symmetry Groups
Authors: Doebner H. D. , Hegerfeldt G. C.
Published: January 2005

An embedding of SU(2) and an internal symmetry group G into a larger group G̃ containing SU(2) and G as subgroups is constructed for all G possessing a generalized spin-½ quark model. The starting point is a set of three mathematical conditions for the embedding group G̃ which are derived from physically plausible assumptions. By group theoretical techniques due to Dynkin and Malcev, it is shown that the embedding group is already uniquely determined by the proposed conditions, with only one set of groups G for which two solutions are obtained. The results for G̃ are given explicitly. Identifying SU(2) as covering of the rotation group the spin extension G̃ is enlarged to an embedding G̃h of the homogeneous Lorentz group Lh and G. It is shown that Ḡh can also be obtained without using the spin extension. The minimal translation group which can be attached to Ḡh is calculated. The results are also taken over to the Budini-Fronsdal identification of SU(2).
Peculiarities of the Eight-Dimensional Space
Authors: Gamba A.
Published: January 2005

A mathematical review of the peculiar properties of the space of eight dimensions is presented with the view of possible applications in the study of symmetries of elementary particles. This paper, written for physicists, is self-contained in that it does not require any previous knowledge of the subject nor any advanced mathematics.
Real Spinor Fields
Authors: Hestenes David
Published: January 2005

The Dirac equation is expressed entirely in terms of geometrical quantities by providing a geometrical interpretation for the (−1)½ which appears explicitly in the Dirac equation. In the modification of the Dirac electron theory which ensues, the (−1)½ appears as the generator of rotations in the spacelike plane orthogonal to the plane containing the electron current and spin vectors. This amounts to a further ``relativistic'' constraint on the spinor theory and so may be expected to have physical consequences. It does not, however, conflict with well-substantiated features of the Dirac theory.
Spin and Isospin
Authors: Hestenes David
Published: January 2005

Spinor fields can be classified according to the invariants of their derivatives. It is suggested that different invariants describe different interactions of elementary particles. Then the classification of spinor fields becomes a classification of elementary particles and their interactions. A geometric interpretation of isospin is suggested and is used in a model of nucleon interactions. This theory entails an intimate connection between spin and isospin which appears to have some experimental support. Among other things, it connects the pseudoscalar and isospin properties of the pion and accounts for the polarization of the third axis in isospace by the electromagnetic interactions.
Quasi-Spin Methods and Pairs of One-Particle CFP in the Seniority Scheme
Authors: Innes Frederick R.
Published: January 2005

The use of creation and destruction operators to form irreducible tensors in quasi-spin (triple tensors) is advanced to give the important points of the seniority scheme for linear and bilinear forms (CFP, matrix elements, and products). A number of such forms as studied by Racah and others are taken up with emphasis on cases diagonal in the seniority number, v. Values of CFP for seniority v + 1 can then be determined from those of seniority v − 1. The possibilities in quasi-spin are explored to give a number of additional formulas for products of two CFP. Also, Redmond's formula and some specializations of it are found in the seniority scheme and their properties worked out. The calculation of CFP in the seniority scheme by these formulas is described and comparison made with other methods.
Theory of Diffraction and Matched Asymptotic Expansions
Authors: Kanwal R. P.
Published: January 2005

We discuss the theory of diffraction as a singular perturbation problem. Due to the presence of two different lengths—the wavelength and the characteristic geometric length of the scatterer—there exists no uniformly valid series expansion of the scattered field in the entire region. As such, two different expansions are needed; one of them is valid in the near region and the other in the far region. These two series are then matched in an appropriate fashion. It emerges from this analysis that first two terms of the required asymptotic expansion can be obtained by solving two simple problems of the potential theory.
Closure Relations for the Eigenfunctions of the One-Speed Transport Equation
Authors: Kuščer I. , Shure F.
Published: January 2005

Orthogonality relations for the eigenfunctions of the one-speed transport equation are used to derive the corresponding closure relations. These express in a concise form the completeness properties previously proved by Case.
Einstein Equations and Electromagnetism
Authors: Lanczos C.
Published: January 2005

Einstein's equations for infinitesimal gravitational fields are investigated from the standpoint of replacing Einstein's flat Minkowskian background space by a strongly agitated Riemannian lattice space, with the added demand that the perturbation field be of very low frequency compared with the lattice frequency. It is found that the strongly nonlinear lattice transmits low-frequency perturbations only if the perturbation field satisfies a set of field equations which become practically identical with Maxwell's equations. While this result demonstrates the close relation between Einstein equations and electromagnetism, it also opens a new door toward the deeper understanding of the quantum phenomena.
Phenomenological Theory of Magnetic Hysteresis
Authors: Pipkin A. C. , Rivlin R. S.
Published: January 2005

Constitutive equations specifying magnetization as a functional of the magnetic field strength history are discussed. Limitations on the form of the response functional arising from material symmetries are described. It is pointed out that even in initially isotropic materials, an applied magnetic field lying always in one plane may produce magnetization normal to that plane. Various integral and differential approximations to the response functionals are considered.
Exponential Stability of Non-uniform Guiding Center Plasma
Authors: Kadish Abraham
Published: January 2005

Sufficient conditions are derived for the exponential stability of non-uniform collisionless guiding center plasma when the magnetic field in equilibrium is unidirectional. These conditions are local analogs of the criteria obtained for uniform plasma equilibria. Growth rates slower than exponential ones are considered.
Plane Wave Propagation in Kinetic Theory
Authors: Sirovich Lawrence , Thurber James K.
Published: January 2005

A number of results for plane wave propagation in kinetic theory are obtained. Among these are the following. It is shown that the Boltzmann equation for bounded collision operator has a cutoff frequency beyond which plane waves cease to exist. The analytic continuation of the dispersion relation is discussed and asymptotic results beyond the critical frequency are obtained. Using a kinetic model it is shown that the low-frequency formal power series for complex wavenumber in terms of frequency (and vice versa) are divergent. The last is contrary to the recent result obtained for the rigid sphere Boltzmann equation.
Construction of the Irreducible Representations of SU4
Authors: Sen R. N.
Published: January 2005

The irreducible representations of SU4 are explicitly constructed from those of SU3 in the basis which is relevant for elementary particle physics. The technique seems to be applicable for the construction of the representations of SUn from those of SUn−1 for any n.
General Method for the Inverse Scattering Problem at Fixed Energy
Authors: Sabatier Pierre C.
Published: January 2005

A method for studying the inverse scattering problem at fixed energy is given; it enables one to get at a fairly large class of potentials C. It is shown that the problem has in C an infinity of solutions, depending on an infinity of parameters. If the study is restricted to the potentials of C which can be continued as even analytic functions in a circle centered at the origin, the problem has only one solution. A linear approximation is given and is shown to yield inverse formulas of the Born approximation. For even potentials of C, the relevance of this approximation is studied as follows: Comparison is made between a given static potential and the potential obtained from the scattering amplitude through the inverse formula. The standard deviation between these two potentials is shown to go to zero as the energy goes to ∞. Miscellaneous properties of potentials, wavefunctions, and Jost functions are given in the framework of this method.
Herglotz-Noether Theorem in Conformal Space-Time
Authors: Wahlquist Hugo D. , Estabrook Frank B.
Published: January 2005

The Herglotz-Noether theorem is proved for rigid motions in conformal space-time.
Evaluation of Expressions of the Debye-Waller Form
Authors: Terwiel R. H.
Published: January 2005

A very simple evaluation of expressions of the Debye-Waller form is given.
Clebsch-Gordan Series in the Exceptional Groups
Authors: Zaccaria F.
Published: January 2005

Conditions are given in order that the number of terms and multiplicities in the Clebsch-Gordan series for the exceptional groups are dependent on only one of the irreducible representations whose direct product is decomposed.
Permutational Symmetry of Reduced Density Matrices
Authors: Poshusta R. D.
Published: January 2005

The usual restriction on the permutational symmetry of density matrices is removed. An Nth-order density matrix of general permutational symmetry is reduced and the symmetry of the resulting pth-order density matrix is found. This gives rise to necessary conditions for the N-representability of spin-free density matrices. We indicate the importance of these results for symmetry under the groups GL(n), U(n), and SU(n). We generalize the question of N-representability to the tensor algebra, where contraction of tensors plays the role of reduction of density matrices. Symmetry properties of the tensors are studied and density matrices are considered as special cases.