bluuebunny/crossref_metadata_2025 huggingface dataset:
On the General Theory of the Approach to Equilibrium. I. Interacting Normal Modes
Authors: Prigogine I. , Henin F.
Published: January 2005

A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.
Zero-Point Energy of an Electron Lattice
Authors: Coldwell-Horsfall Rosemary A. , Maradudin Alexei A.
Published: January 2005

At very low densities an electron gas in a compensating uniform background of positive charge crystallizes into a bcc lattice for which the correlation energy per electron is (−1.792/rs) ry. At higher densities the first correction to this result arises from the zero-point energy of the electrons, which can be expanded in terms of the even moments of the frequency spectrum. We have computed the first five nonvanishing moments and have estimated the contribution to the zero-point energy from the remaining moments using their known asymptotic behavior. This procedure leads to the value (2.638/rs32) ry for the zero-point energy per electron. The low temperature specific heat per electron is found to be 56.21krs9/2· (kT)3 ry. The range of rs values for which these results should be valid is discussed on the basis of Lindemann's melting formula.
Generalization of the Lagrange Expansion with Application to Physical Problems
Authors: Sturrock P. A.
Published: January 2005

Certain problems of fluid dynamics are conveniently discussed in the Lagrangian description by means of the displacement-vector function. The Eulerian variables describing the motion may be obtained from the displacement-vector function by a generalization of the Lagrange expansion which is here established. Two examples of the application of this expansion are given; the derivation of the Fokker-Planck equation, and the description of the surface conditions of a rippling electron stream by the device of equivalent surface charges.
Consistency of the Canonical Reduction of General Relativity
Authors: Arnowitt R. , Deser S. , Misner C. W.
Published: January 2005

The question of consistency of the canonical reduction of general relativity (obtained by eliminating constraints and also imposing coordinate conditions in the action or generator) is examined. It is shown that the equations of motion obtained from this ``reduced'' formalism agree with the original Einstein equations. Agreement is also established for the generators of space-time translations. In order to establish consistency, it is necessary to discard certain well-defined divergence terms in the original Lagrangian. These would otherwise appear as nondivergences in the reduced Lagrangian, incorrectly altering the equations.
Kemmer Wave Equation in Riemann Space
Authors: Da Silveira Adel
Published: January 2005

The Kemmer equation is written in Riemann space. The form of the equation, the supplementary relations and the form of the covariant derivatives of the operators βμ are considered. As a check of the equation it is shown that the equations for the values zero and one of the spin can be obtained from this generalized Kemmer equation with the help of generalized Fujiwara operators. In particular the equation for photons in interaction with the gravitational field is obtained.
Analytical Properties of S Matrix and Uniqueness of the Scattering Potential
Authors: Barut A. O. , Ruei K. H.
Published: January 2005

The Schrödinger equation with the complex momentum k leads to an S matrix with very simple analytical properties. It differs from the conventional S matrix as little as one wishes on the real k axis, but it has, in general, completely different analytical behavior outside the real axis. The present formulation removes some of the unsatisfactory features of the conventional formalism in the sense that no redundant poles can occur and a phase shift determines the scattering potential uniquely. The complete analytical behavior of the S matrix, in particular at infinity, is discussed and the theory is extended to Klein-Gordon and Dirac equations with central potential.
Products of Principal Value Singularities Used in the Formal Theory of Scattering
Authors: Tani S.
Published: January 2005

A mathematically neat derivation is given of the relation between the S matrix and the transformation function for a finite time. It is shown that one can dispense with the adiabatic switching on and off, and yet one reaches the same result as when one employs it. Necessary conditions are discussed for the validity of this statement. Systematic prescriptions are given of handling products of principal value singularities, which is relevant to the scattering theory in momentum space.
Geometry of Light Paths between Two Material Bodies
Authors: Wigner E. P.
Published: January 2005

The pattern of light signals, which was proposed before for the measurement of the curvature, is investigated in a two-dimensional manifold of constant curvature (deSitter space). The pattern consists of light signals between two freely moving bodies, each signal being emitted when the signal from the other body arrives. It is shown that the coordinates of the arrivals (or emissions) of the light signals can be obtained from the coordinates of the emission of the first signal by means of projective transformations [see Eq. (18)] which are iterates of a single such transformation. The same applies to the proper times at which these signals are received.
On Analytic Continuation, Multiplication, and Fourier Transformations of Schwartz Distributions
Authors: Bremermann H. J. , Durand L.
Published: January 2005

It is shown that a class of Schwartz distributions on the real axis can be continued to holomorphic functions in the upper and lower complex half-planes such that the ``jump'' on the real axis represents the distribution. Many operations with distributions can be reduced to operations with the associated holomorphic functions which is of particular interest for the convolution product and for Fourier transforms. By means of the continuations several kinds of multiplications for distributions are being defined, which is of interest for quantum field theory.
Operator Formalism in Statistical Mechanics
Authors: Lieb Elliott
Published: January 2005

We consider the configuration partition function, ZN, of a classical imperfect gas of particles having rigid cores. The rigid cores result in a geometric simplification which makes it possible to find a finite recursion relation between ZN and ZN−1. From this, it is possible to express ZN in terms of the vacuum expectation value of a finite Boson operator raised to the Nth power, N being the number of particles.
Higher-Order Approximations in Multiple Scattering. I. Two-Dimensional Scalar Case
Authors: Zitron Norman , Karp Samuel N.
Published: January 2005

A formula is derived which expresses the perturbed scattering amplitudes of a combination of two arbitrary cylinders as a function of the unperturbed scattering amplitudes of the individual cylinders. The formula is valid when the spacing of the scatterers is large compared to their dimensions. It involves derivatives of the scattering amplitudes with respect to the angles of incidence and of observation. Interaction terms of degrees d−½, d−1, and d−½ are taken into account, where d is the spacing. Verification is obtained in a special case. The result is employed to calculate the total scattering cross section.
Higher-Order Approximations in Multiple Scattering. II. Three-Dimensional Scalar Case
Authors: Zitron Norman , Karp Samuel N.
Published: January 2005

The method of Part I is extended to cover the three-dimensional scalar problem for two bodies of arbitrary shape. All interaction terms of order d−1 and d−2 are given.
Theory of the Dipole Antenna and the Two-Wire Transmission Line
Authors: Wu Tai Tsun
Published: January 2005

The properties of the dipole antenna are studied by an approximate procedure that makes use of the Wiener-Hopf integral equation. In particular, the input admittance and the radiation pattern are found. The present results thus supplement the existing theories, which are concerned mostly with short dipoles. The same procedure is then applied to several related problems. First, the back-scattering cross section of a dipole antenna is found approximately for normal incidence. Secondly, the two-wire transmission line is studied in detail by considering it to be two coupled dipole antennas. The capacitive end correction for an open end is evaluated, and the radiated power and the radiation resistance are found for a resonant section of transmission line with both ends open. Finally, the dielectric-coated antenna is considered briefly.
Boundary-Layer Behavior in the Superconductor Transition Problem
Authors: Cohen Hirsh , Miranker W. L.
Published: January 2005

A superconducting material may be isothermally transformed to a normal conductor by raising the magnetic field to a value greater than a certain critical field. When this is done, the transition takes place along an interface, determined by the critical field value and by a magnetic-flux condition. In the present paper, the effect on the transition rate of magnetic-field penetration into the superconductor is studied. This involves the solution of a free-boundary problem in which the free boundary divides two regions, each governed by a different parabolic differential equation. The problem is solved by using the asymptotic techniques of singular perturbation theory. A boundary layer is shown to exist along the moving interface on the superconducting side. The presence of the boundary layer slows the motion of the free boundary. Also included in the solution is a study of the effect of the magnitude of the initial field on the starting motion of the free boundary. Finally, some numerical results are presented.
Frequency Spectrum of a Disordered One-Dimensional Lattice
Authors: Langer J. S.
Published: January 2005

The frequency spectrum of a disordered one-dimensional lattice is calculated via an investigation of the phonon propagator. The spectrum is evaluated in detail for a low concentration of light impurities inserted at random along a linear chain. It is found that an impurity band occurs near the frequency of the local mode. Higher-order effects resulting from clusters of impurities are calculated and discussed.
Some Cluster Size and Percolation Problems
Authors: Fisher Michael E. , Essam John W.
Published: January 2005

The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed. The cluster size and percolation problems are then solved exactly for Bethe lattices (infinite homogeneous Cayley trees) and for a wide class of pseudolattices derived by replacing the bonds and/or sites of a Bethe lattice by arbitrary finite subgraphs. Explicit expressions are given for the critical probability (density), for the mean cluster size, and for the density of infinite clusters. The nature of the critical anomalies is shown to be the same for all lattices discussed; in particular, the density of infinite clusters vanishes as R(p) ≈ C(p−pc) (p≥pc).
Acnodes and Cusps on Landau Curves
Authors: Eden R. J. , Landshoff P. V. , Polkinghorne J. C. , Taylor J. C.
Published: January 2005

It is shown that the Landau curve for a reduced sixth-order diagram can acquire acnodes and real cusps as the masses are varied. They are associated with complex singularities that under certain conditions are in the physical sheet and cause a breakdown of the Mandelstam representation. The problem of obtaining general criteria for acnodes and cusps is discussed.
New Approach to Einstein's Empty Space Field Equations
Authors: Newman E. T. , Tamburino L. A.
Published: January 2005

Tetrad formalism is used to derive a set of 36 scalar field equations which correspond to the ordinary field equations Rμν=0. The scalar equations are obtained by beginning with a given Petrov type of empty-space Riemann tensor and applying the Ricci identity to each of the tetrad vectors. The unknowns or field variables become the 24 Ricci rotation coefficients, the number of which can always be reduced by the Bianchi identities and occasionally by tetrad transformations which leave the form of the Riemann tensor invariant. The use of these scalar field equations is illustrated by their application to a degenerate case of Petrov type I. It is believed that by this method all possible solutions of this particular case have been found.
New Approach to the Einstein and Maxwell-Einstein Field Equations
Authors: Newman Ezra T.
Published: January 2005

The components of a ``vierbein'' system are introduced as field variables in place of the metric tensor in a Riemannian space. The Riemann tensor, which is then written in terms of these new variables, is used to reformulate the Einstein and Einstein-Maxwell equations with or without a cosmological constant. These field equations have as solutions metrics with Riemann tensor of predetermined algebraic properties.
Schrödinger Scattering Amplitude. II
Authors: Grossmann Alex
Published: January 2005

The results of the preceding paper are used to indicate procedures of calculation of the scattering amplitude, obtain several expansions, find bounds on the variation of the amplitude under a change in the potential, and study multiple scattering.
Calculation of the Eigenvalues of a Tridiagonal Hermitian Matrix
Authors: Pierce Louis
Published: January 2005

For real symmetric or Hermitian matrices with tridiagonal form, the secular equation may be written as a continued fraction equation f(λ)=0. f(λ) is a member of a recursively defined sequence R(n)(λ) of n continued fractions if the secular equation is of the nth order. The basis for a new method of computing the eigenvalues of such tridiagonal matrices is given. The method requires the determination of an integervalues function Pn(γ) for a succession of values of γ, where Pn(γ) is a function only of n and the signs of the n terms in R(n)(γ).
Statistical Theory of the Energy Levels of Complex Systems. II
Authors: Dyson Freeman J.
Published: January 2005

The distribution function of spacings S between nearest neighbors, in a long series of energy levels with average spacing D, is studied. The statistical properties of S are defined in terms of an ensemble of systems described in a previous paper. For large values of t = (πS/2D), it is shown that the distribution of S can be deduced from the thermodynamical properties of a certain model. The model, which replaces the eigenvalue distribution by a continuous fluid, can be studied by the methods of classical electrostatics, potential theory, and thermodynamics. In this way the distribution function of spacings S is found to be asymptotically Q(t)=At17/8 exp[−14t2−12t]for large t. The numerical constant A can in principle not be determined from such a continuum model. Reasons are given for considering the remaining factors in the formula for Q(t) to be reliable.
Surface Vibrational Modes in Crystal Lattices with Complex Interatomic Interactions
Authors: Gazis Denos C. , Wallis Richard F.
Published: January 2005

The investigation of surface vibrational modes in crystal lattices is complicated by the necessity for satisfying the free boundary conditions at the surface if the interatomic interactions include next-nearest neighbor interactions, next-next-nearest neighbor interactions, etc. The number of equations specifying the free boundary conditions may then become rather large. A general method is presented in this paper for the investigation of the normal modes of one-, two-, and three-dimensional lattices which are finite or semi-infinite in one dimension and have free boundary surfaces, assuming interatomic interactions of various ranges. The method is illustrated by calculations of the normal modes for the finite and semi-infinite diatomic linear chains with nearest and next-nearest neighbor interactions. Some remarks are made regarding the applications of the mathematical techniques to the evaluation of certain continuant determinants of large order.
Two Folk Lemmas on the Expansion of the S Matrix or the Out Field in Normal Ordered In Fields
Authors: Greenberg O. W.
Published: January 2005

We prove that if the out field or the S matrix is expanded in terms of normal ordered products of the in field, then either the expansion has infinite degree or it is the trivial case Aout=Ain, S=1. From this fact it follows that any field theory model in which the Heisenberg field (local or not) has a terminating normal ordered expansion in terms of a (generalized) free field cannot provide a nontrivial unitary S matrix.
Spin and Statistics with an Electromagnetic Field
Authors: Boulware David G.
Published: January 2005

The connection between spin and statistics is proved in the presence of an electromagnetic field. Two functions, corresponding to the Wightman functions for the two orderings of Lorentz gauge operators, are exhibited as expectation values of radiation gauge operators. These are shown to be Lorentz invariant and have the requisite positive frequency characteristics. The radiation gauge commutation relations on an x0=const surface imply that the corresponding Lorentz gauge functions vanish for spacelike separations. Assuming the ``wrong'' connection between spin and statistics, the anticommutator functions are shown to be zero and continued to the origin. At the origin, the Lorentz gauge functions are simply the radiation gauge operator expectation values. Hence, the wrong commutation relations imply the vanishing of the radiation gauge field anticommutator at the origin, which is inconsistent with a positive definite metric and a non-vanishing field.
Use of the Displacement Vector in Electromagnetic Theory
Authors: Dedrick K. G. , Wilson Robert N.
Published: January 2005

Problems in the electrodynamics of charged fluids often require calculation of the charge and current densities, given prior knowledge of the initial distributions and the displacement vector field. In this paper, solutions are obtained for these new distributions, with the property that if the initial distributions have a sharply defined boundary, terms arise which can be interpreted as distributions of electric charge and current multipoles located on the initial boundary surface. These solutions cannot be considered valid near the initial boundary, but prove to be useful in that many of their properties are given correctly. The electromagnetic potentials due to the charge and current densities in the displaced configuration are calculated, and form the basis for a discussion of the complicated boundary value problem encountered in the description of the electrodynamics near the boundary surface. The results are directly applicable to the theory of high frequency electron tubes, and certain formulas are of use in hydrodynamics and elasticity.
Some Properties of Impedance as a Causal Operator
Authors: Wu Tai Tsun
Published: January 2005

As a model for the various problems connected with domains of analyticity and dispersion relations, the concept of impedance (self-impedance) has been studied in general under the physical requirements of linearity, passivity, reproducibility, and causality. After identifying impedance functions with suitably defined positive functions, any impedance function may be decomposed to the sum of a minimum reactive part and two reactive parts, one of which is of unfamiliar nature. In the time domain, the convolution representation of the current in terms of the voltage and vice versa are obtained rigorously. As an example, a reactance obtained from the Cantor function is studied in detail. Its peculiar delta-function response is computed explicitly, and it is also used to construct a counter example to a representation by van Kampen of the S matrix of a Maxwellian field. The general results are also applied to study the Kronig-Kramers dispersion relations for dielectric constants. It is proved that one of the two relations is true under very general conditions, but the other is false in general.
Ergodicity Conditions in Quantum Mechanics and Van Hove's Hamiltonians
Authors: Prosperi G. M.
Published: January 2005

In this work we start from a formulation of the quantum-ergodic problem developed in a preceding paper. We prove that the ergodicity conditions which we established there in an abstract and very general form are in particular satisfied for the class of Hamiltonians for which Van Hove was able to prove a master equation. Some restrictive assumptions have been introduced to Van Hove's theory which are necessary to identify the unperturbed states of Van Hove with the states ωvi of the ergodic theory.
On the Calculation of Analytic Functions of Cyclic Matrices
Authors: Abraham Phillip B. , Weiss George
Published: January 2005

Recently Löwdin, Pauncz, and de Heer, have discussed the calculation of functions of cyclic matrices. They presented three exact methods for doing this. We show that the last of their methods can be generalized to a method which is convenient for approximate calculations and which in addition can be extended to the calculation of functions of higher order cyclic matrices. A number of examples are presented. It is also shown that by the same techniques it is possible to evaluate functions of a skew circulant matrix.
Approximate Solutions of the Modified Bloch Equations for Low Magnetic Fields
Authors: Runge Richard J.
Published: January 2005

A set of modified Bloch equations proposed by Torrey, Olds, and Codrington for the description of the time dependence of the magnetic moment expectation value in low magnetic fields, is solved using two approximate methods. The equations are solved approximately for the steady state with a constant field and linearly polarized radio-frequency field applied and an error analysis of the approximate solution is given. A low-harmonic solution is developed and alternative numerical integration techniques are given. Results are exhibited for the low-field electron paramagnetic resonance in anthracene negative ion solution and asphaltene. The methods developed are applicable to a wide range of frequencies, applied field strengths, and ratios of transverse to longitudinal relaxation times.
Theory of the Thin Circular Loop Antenna
Authors: Wu Tai Tsun
Published: January 2005

The current distribution on a thin circular loop transmitting antenna driven by a delta-function generator is determined approximately by Fourier series expansion. A difficulty encountered in previous analysis is shown to be due to an inadequate approximation.
Expectation Value Formalism in Quantum Field Theory. II
Authors: Bakshi Pradip M. , Mahanthappa Kalyana T.
Published: January 2005

The expectation value formalism has been developed to treat the problems which require the coupling of quantized Fermi and Bose fields.
Quaternionic Representations of Compact Groups
Authors: Finkelstein D. , Jauch J. M. , Speiser D.
Published: January 2005

The main purpose of this paper is to show the conditions under which a finite dimensional representation of a group, irreducible over the complex field, is reducible over quaternions. The answer is simply stated in terms of the Frobenius-Shur classification of group representations.
Complex Scalar Field in General Relativity
Authors: Das A.
Published: January 2005

This paper introduces complex scalar fields in general relativity to describe charged, gravitating particles with zero spin. A class of exact solutions of the combined Maxwell-Einstein-Klein-Gordon field equations is found that may represent complete models of matter. One of the models discussed is spherically symmetric and the remaining class does not assume any specific symmetry. Such exact solutions are possible only if the mass parameter equals the charge parameter in magnitude, so that physically speaking, matter is in equilibrium under the mutual action of electromagnetic and gravitational forces.
Mathematical Deductions from Some Rules Concerning High-Energy Total Cross Sections
Authors: Yang C. N.
Published: January 2005

Mathematical implications of the Pomeranchuk rule and the Pomeranchuk-Okun rule are discussed.
Invariant Solutions of the Exact Bethe-Salpeter Equation in the General-Mass Case
Authors: Nakanishi Noboru
Published: January 2005

The invariant solutions of the exact Bethe-Salpeter equation are found in the case of nonzero exchanged-meson mass. They are expressed as series of ``truncated vertex functions,'' which are certain combinations of the usual vertex functions. The eigenvalues of the coupling constant are determined by a transcendental equation, which is not solved explicitly.
Complex Singularities in Production Amplitudes
Authors: Bessis D. , Pham F.
Published: January 2005

It is shown for a simple model of production that Watson's scheme for final-state interaction may be good even though the amplitude has complex singularities. The method used consists in continuing analytically partial-wave dispersion relations. Stress is put upon the necessity of correctly choosing the cuts starting at the complex singularities. The safest choice in this approach is to take the cut along the line on which the series of partial waves diverges. A simple geometrical recipe is given to see the evolution of this line, the appearance of an anomalous threshold and a complex singularity.
New Small-Angle X-Ray Scattering Calculation
Authors: Rothwell W. S.
Published: January 2005

A method due to Hardy for evaluating finite integrals is used to obtain the structure factors for several rotationally symmetric shapes. The factor for particles formed by joining paraboloids of revolution is shown to be proportional to a second-order Lommel function of two variables.
A Lattice of Von Neumann Algebras Associated with the Quantum Theory of a Free Bose Field
Authors: Araki Huzihiro
Published: January 2005

Von Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von Neumann algebra on another complex Hilbert space (the Fock space) is defined. This correspondence is proved to be an isomorphism between a certain complemented lattice of subspaces and that of the von Neumann algebras. This result has an application to the duality theorem in the theory of a free scalar field, which is to be discussed in a separate paper.

A necessary and sufficient condition on a subspace, in order that the corresponding von Neumann algebra is of type I, is obtained.
Strict Localization
Authors: Licht A. L.
Published: January 2005

A complete characterization for a general quantum field theory is given of the strictly localized states introduced by J. Knight. It is shown that each such state can be generated from the vacuum by a partially isometric operator. Necessary and sufficient conditions are given for the superposition of such states to be also strictly localized. Finally, it is shown that there is a connection between the von Neumann type of the ring generated by the field operator in a finite region and the possibility of constructing strictly localized states.
The Remainder in Mayer's Fugacity Series
Authors: Penrose Oliver
Published: January 2005

Upper and lower bounds are obtained for the remainder after a finite number of terms of the expansions in powers of fugacity z for the pressure p, the s-particle distribution functions, the density ρ, and the fugacity coefficient z/ρ, for a system of particles with two-body interactions. The interaction potential must be either nonnegative or else have a hard core and decrease faster than r−3 at large distances. The results hold for all positive z, and apply to lattice gases as well as to fluids.

For nonnegative potentials, the results imply that successive partial sums of each of these fugacity expansions provide alternate upper and lower bounds, valid even if the series diverges, on the physical quantity the expansion represents.
Solution of Linear Integral Equations Using Padé Approximants
Authors: Chisholm J. S. R.
Published: January 2005

It is shown that the exact solution of a nonhomogeneous linear integral equation with a kernel K of rank n is given by forming the Padé approximant P(n, n) from the first 2n terms of the perturbation series solution. It follows that for a compact kernel K, the solution is limn→∞ P(n, n); this gives meaning to a large class of perturbation series when the perturbation is large. The possible extension of this result to wider classes of equations is discussed.
External-Mass Singularity
Authors: Nakanishi Noboru
Published: January 2005

It is shown by a concrete example that the scattering amplitude may have an external-mass singularity, which does not correspond to a threshold for any energy variable. If the external mass exceeds the external-mass singularity, the scattering amplitude (even for the forward scattering) is no longer a boundary value of an analytic function of energy variables in the usual sense. This would mean a deadlock of the S-matrix theory based on the analyticity. In the Appendix an example of a threshold in the non-Euclidean case is given.
On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model
Authors: Kac M. , Uhlenbeck G. E. , Hemmer P. C.
Published: January 2005

For a one-dimensional fluid model where the pair interaction potential between the molecules consists of a hard core and an exponential attraction, Kac has shown that the partition function can be determined exactly in the thermodynamic limit. In Sec. II this calculation is reviewed and further discussed. In Sec. III, we show that in the so-called van der Waals limit when the range of the attractive force goes to infinity while its strength becomes proportionally weaker, a phase transition appears which is described exactly by the van der Waals equation plus the Maxwell equal-area rule. In Sec. IV the approach to the van der Waals limit is discussed by an appropriate perturbation method applied to the basic integral equation. The perturbation parameter is the ratio of the size of the hard core to the range of the attractive force. It is seen that the phase transition persists in any order of the perturbation. The two-phase equilibrium is characterized by the fact that in this range of density, the maximum eigenvalue of the integral equation is doubly degenerate and that the corresponding two eigenfunctions do not overlap. In Sec. V we comment on the relevance of our results for the three-dimensional problem.
Asymptotic Behavior of the Radial Distribution Function
Authors: Lebowitz J. L. , Percus J. K.
Published: January 2005

The pair distribution function in a uniform classical fluid is equivalent to the one-body density when one particle is fixed. An implicit relation for this nonuniform density is found by a functional expansion of the difference of chemical potential and external potential about its value for a system of uniform density. A linearization of this expansion, followed by retention of, at most, second derivatives of the inhomogenity, reproduces the Ornstein-Zernicke relations for the asymptotic pair correlation. Linearization alone calls for the sum of internal potential and direct correlation function to vanish asymptotically. This relation is developed for the case of weak long-range forces, resulting in the Debye-Huckel expression for an electron gas, and reproducing the asymptotic correlations of the Kac-Uhlenbeck-Hemmer one-dimensional model. The relation is also shown to follow from the virial expansion for the direct correlation function.
Scattering, Weak Decays, and Final-State Interactions in Model Theories in a Dressed-Particle Picture
Authors: Haller Kurt
Published: January 2005

Elastic and inelastic collisions in the Lee model are treated in a dressed-particle picture. The procedure by which the renormalization constant Z disappears from the integral equations for the transition matrix is studied in detail. A new procedure is given for obtaining the exact transition matrix for N − θ scattering. A hypothetical fermion B is introduced and the decays B → N + θ, B → V + θ, as well as the decay in the Goldberger-Treiman model are studied in the dressed-particle picture. The iterative expansion of the B → V + θ decay amplitude is obtained and is shown to agree with the renormalized power series for this decay, though such is not the case for B → N + θ decay, or the decay in the Goldberger-Treiman model. It is shown that in the final-state interaction of B → V + θ decay, there are cancellation effects between one- and two-meson contributions, which are crucial in avoiding divergences.
Electrostatic Interactions in Complex Electron Configurations
Authors: Wybourne B. G.
Published: January 2005

Simplified expressions for the matrix elements of electrostatic interactions both within and between several types of complex electron configurations have been obtained by the application of angular-momenta recoupling techniques. The use of these recoupling techniques avoids the usual extensive calculation of the sums of products of the matrix elements of tensors of the types Vκk and Uk. The derived expressions involve the sums of products of coefficients of fractional parentage and n − j symbols, and as such, are amenable to machine computation.
Potential Scattering
Authors: Belinfante Johan G. , Unal Burhan C.
Published: January 2005

The potential-scattering model is discussed from the point of view of analyticity properties of the scattering amplitude. Dynamical schemes based on the Mandelstam representation and the use of Regge poles are reviewed. Consequences for strong interaction physics are also briefly reviewed.
Binary Kernel Formulation of a Heisenberg Model of Ferromagnetism
Authors: Greenberg Newton I.
Published: January 2005

An ideal Heisenberg model of a ferromagnet for spin 12 is studied by considering the model in terms of a spin-deviation lattice gas. Utilizing the general methods of Yang and Lee, a binary kernel function is obtained in terms of which the thermodynamic properties of the lattice gas can be completely expressed. As an example, Dyson's results are rigorously obtained.
Equivalence and Antiequivalence of Irreducible Sets of Operators. I. Finite Dimensional Spaces
Authors: Lomont J. S. , Moses H. E. , Yu Kwang
Published: January 2005

A fundamental problem which arises in determining whether two quantum mechanical systems are essentially identical is whether a unitary or antiunitary transformation exists which maps one set of dynamical variables into another. Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, we are led to investigate the following problem: Given two irreducible sets of operators with a one-to-one correspondence between them, find the algebraic properties of the two sets which make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilbert spaces. The rigorous statements of the theorems and the proofs will be given in a second paper. Finally, in the Appendix there is given a definition of invariance of elementary quantum-mechanical systems based on the above theorems, giving the same results as Wigner's definition in terms of transition probabilities.
Axiomatic Perturbation Theory. II. Partial Sum Formalism
Authors: Fried H. M.
Published: January 2005

A generalization of axiomatic perturbation theory is described in which propagator structure is introduced into the basic formalism, effectively resulting in a systematic rearrangement of the perturbation expressions representing a given process. The exact retarded functional of a simple Boson field theory is decomposed into a sum of approximate functionals, and all the n-point functions of the first few approximate functionals are determined from modified unitarity equations and a knowledge of the corresponding perturbation functions. The resulting amplitudes differ from the perturbation amplitudes by the replacement of the perturbative spectral functions with related spectral functions depending upon the latter and on the propagator structure. Sets of integral equations for the higher functionals are proposed and solved.
Group Properties of Free-Field Equations
Authors: Amar V. , Lucchini M. , Sona P. G.
Published: January 2005

We show that a set of n free-field equations and suitable commutation relations relevant to spin 0 or ½ fields, is invariant with respect to a continuous group of transformations isomorphic to the n-dimensional real orthogonal group. We then find that such a set can always be derived starting from a given particle multiplet. The continuous group found in this way, includes as subgroups the isotopic rotations and the gauge groups related to baryon number and strangeness. Hence this method enables us to reobtain the results reached by other authors in some particular cases, and to extend them to all particle multiplets.
N-Body Bose System with a Finite Number of States. I. Irreducible Representations
Authors: Aviles Joseph B.
Published: January 2005

For the purpose of investigating the N-Body Bose System with a finite number of states, we analyze Xij = Ai†Aj, i, j ≤ K. (A and A† are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite-dimensional) irreducible representations of the Xij. A set of fundamental equations is defined for the irreducible representations Xij(N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the Xij(N, K) yields a simple, systematic method for listing all possible ways in which n1+ … ni+ … + nK = N may be satisfied for ni positive integral or zero. This leads to a particularly simple method for constructing the Xij(N, K), i, j ≤ K.
Eigenfunction Form of the Nonrelativistic Coulomb Green's Function
Authors: Mano Koichi
Published: January 2005

By treating the contour integrals in the momentum plane, the conventional eigenfunction form is deduced from the characteristic form for the nonrelativistic Coulomb Green's function. A straight-forward extension of the argument employed here leads to the most general form of the eigenfunction form.
Exact Statistical Mechanics of a One-Dimensional System with Coulomb Forces. III. Statistics of the Electric Field
Authors: Lenard Andrew
Published: January 2005

The statistics of the electric field in a one-dimensional ``sheet'' plasma is studied. It is shown that the electric field as a function of the space coordinate is a Markov process, whose basic probabilities are related to the Fourier coefficients of certain functions that play an important role in the previously found development of the grand partition function. Some implications regarding the statistics of particle locations are explored. The one-point probability distribution of the electric field is found to be asymptotically Gaussian in the plasma limit. It is pointed out that the one-dimensional analog of the Holtsmark calculation leads to an incorrect conclusion because electrostatic shielding is not properly taken into account.
Structure of the Two-Point Function
Authors: Steinmann O.
Published: January 2005

The two-point functions of relativistic quantum field theory are discussed in the framework of distribution theory. A derivation in this framework of the Källén-Lehmann representation is given. The singularities in x space are studied for the example of the DP function. It is in general not possible to extract a singularity with support on the light cone. It is shown however that DP is at best a measure in any neighborhood of the light cone. Thus Lehmann's statement that DP is at least as singular as the corresponding free-field function is confirmed.
Representation of Annihilation Operators in the Space of Periodic Functions
Authors: Derrick G. H.
Published: January 2005

A representation is given of fermion annihilation operators ak which satisfy the usual anticommutation relations {ak, ak′+} = δkk′. In this representation the ak operate on periodic function f(ζ) of a real variable ζ. The Hilbert space is the set of all square integrable functions f(ζ) of period 2π, with the scalar product defined by (f, g)= ∫  −ππf*g dζ.
Empty-Space Generalization of the Schwarzschild Metric
Authors: Newman E. , Tamburino L. , Unti T.
Published: January 2005

A new class of empty-space metrics is obtained, one member of this class being a natural generalization of the Schwarzschild metric. This latter metric contains one arbitrary parameter in addition to the mass. The entire class is the set of metrics which are algebraically specialized (contain multiple-principle null vectors) such that the propagation vector is not proportional to a gradient. These metrics belong to the Petrov class type I degenerate.
Generalized ``Cross-Correlation'' Field Quantities as Solutions of Field Equations
Authors: Karr Philip R.
Published: January 2005

It has been shown by Wolf that, if a field amplitude obeys the wave equation, then a derived field quantity (the cross correlation or mutual coherence function), also obeys the wave equation. This result helps clarify the subject of partially coherent light; for example, a Huygens' principle for the propagation of intensity is a consequence. A generalization is presented: For that class of linear partial differential equations (p.d.e.'s) in which at least one independent variable, say t, does not appear in the coefficients, one can construct from a solution f(P, t), (P representing collectively the other independent variables) a generalized ``cross-correlation'' function F(P, t; P1, t1, P2, t2 ⋯) satisfying the same p.d.e. as f(P, t); F is an integral over s of f(P, t + s) g (s, P1, t1, P2, t2 ⋯), where (P1, t1, P2, t2 ⋯) are values of (P, t); for particular choices of the highly arbitrary g, F can be, for example, (a) the usual cross-correlation function, used by Wolf, (b) a higher-order correlation function, (c) the Hilbert transform of f(P, t), and (d) derivatives of f(P, t). For linear p.d.e.'s in which two or more independent variables have the above property, higher-dimensional or vector versions of F are obtainable. The existence of this (two-fold) generalization of Wolf's result suggests the possibility of other physical applications.
Formulation of the Many-Body Problem for Composite Particles
Authors: Girardeau M.
Published: January 2005

The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
Asymptotic Wave Vector and Nonrelativistic Perturbation Theory
Authors: Teitler S.
Published: January 2005

The dual nature of perturbations in causing both transitions and persistent effects is investigated. A wave vector initially represented in terms of a set of unperturbed eigenvectors is subjected to a nondissipative perturbation. After a long time, it is assumed that this unperturbed wave vector evolves into an asymptotic wave vector in which both persistent and transition effects are present. These two effects are considered separately and the asymptotic wave vector is formally expressed as an expansion in terms of asymptotically stationary states. A time-operator form of nonrelativistic perturbation theory which formally is very similar to the resolvent formalism is presented. In a manner similar to the resolvent formalism, diagonal and nondiagonal contributions to the development operator are considered separately. A further classification of the development operator into asymptotic and nonasymptotic parts is made. This latter form is used to obtain an explicit form for the asymptotic wave vector, and the asymptotically stationary states are identified.
An Exactly Soluble Model of a Many-Fermion System
Authors: Luttinger J. M.
Published: January 2005

An exactly soluble model of a one-dimensional many-fermion system is discussed. The model has a fairly realistic interaction between pairs of fermions. An exact calculation of the momentum distribution in the ground state is given. It is shown that there is no discontinuity in the momentum distribution in this model at the Fermi surface, but that the momentum distribution has infinite slope there. Comparison with the results of perturbation theory for the same model is also presented, and it is shown that, for this case at least, the perturbation and exact answers behave qualitatively alike. Finally, the response of the system to external fields is also discussed.
Motion of Test Bodies in General Relativity
Authors: Taub A. H.
Published: January 2005

Two distributions in the sense of Schwartz are introduced to describe a test body. Papapetrou's equations for a pole-dipole test body are derived in a coordinate-independent manner. The supplementary condition SσμUμ = 0 arises in a natural fashion in the course of the derivation. The method for treating test bodies with higher-order poles is indicated.
Approach to a Phase Transition in a One-Dimensional System
Authors: Helfand Eugene
Published: January 2005

The manner in which a one-dimensional system approaches a phase transition as the range of its attractive pairwise interactions becomes large is analyzed, as a continuation of studies on such systems by Kac, Uhlenbeck, Hemmer, and the author. Explicitly, consideration is given to an Ising ferromagnetic model and the corresponding lattice gas with interaction −αγ exp (−γ |i − j|) as γ → 0. One finds that the fluid isotherms are actually analytic, but that in a certain region they approach the zero slope characteristic of a phase transition in the essentially singular fashion exp (−const/γ).
Applicability of the Pfaffian Method to Combinatorial Problems on a Lattice
Authors: Hurst C. A.
Published: January 2005

For the solution of certain problems in statistical mechanics, a formulation in terms of the combinatorial problem of counting open or closed polygons drawn on a lattice has been found useful. The solution of such combinatorial problems has been given as a generating function which can be expressed as a Pfaffian. Necessary and sufficient conditions are given here to determine when such a Pfaffian representation is possible. If certain consistency conditions are satisfied, then the solution to the combinatorial problem can be written down immediately. The method makes some use of the ideas of emission and absorption operators for fermions.