bluuebunny/crossref_metadata_2025 huggingface dataset:
Behavior of the Scattering Amplitude for Large Angular Momentum
Authors: Barut A. O. , Dilley J.
Published: January 2005

Langer's theory on the asymptotic behavior of the solutions of differential equations is applied to angular momentum, giving stronger results than were possible hitherto by Born approximation. It is shown that, for potentials V(r) analytic in the right-hand r plane satisfying |r2V(r)| < ∞ at r = 0 and |r| = ∞, the phase shift has the asymptotic form (λ = l + ½) δ→ lim λ→∞−(2λ12)12V(λ/k)2k2−λ−1 ∫ λ/k+λ−12/k∞V(r)r dr(k2r2/λ2−1)12,  Re λ>0,in the λ plane and for all complex k. Consequently, (a) all Regge trajectories are bounded for analytic potentials; there are no poles for |λ| → ∞ in the right-half λ plane, (b) stronger limits can be given for the feasibility of the Watson-Sommerfeld transformation. The pathological behavior of the cut off potentials (e.g., square-well) is attributed to the nonanalyticity of the potential.
On the Representations of the Semisimple Lie Groups. II
Authors: Baird G. E. , Biedenharn L. C.
Published: January 2005

The explicit determination of the matrices of the generators of the unitary groups, SUn, is carried out and discussed in two alternative treatments: (a) by purely algebraic infinitesimal methods, and (b) by Young-pattern techniques employing the Schwinger-Bargmann boson operator methods. The implication of this result for a tableau calculus is discussed and a determination of the [λ] × [1] Wigner coefficient for all SUn is indicated.
Remarks on the Type of Von Neumann Algebras of Local Observables in Quantum Field Theory
Authors: Kadison Richard V.
Published: January 2005

The von Neumann algebras of local observables associated with certain regions of space-time are believed to be factors. We show that these algebras are not of finite type. The commutant of the tensor product of two semifinite von Neumann algebras is analyzed with the aid of this result. The factors in question have the vacuum state as separating and cyclic vector. It is shown that a factor of type I∞ with I∞ commutant, and a subfactor of type I∞ with I∞ relative commutant have a common separating and cyclic vector. This settles negatively some conjectures aimed at proving that these factors are not of type I. An argument of Araki's showing that the factors associated with certain regions are not of type I is presented in simplified form.
Theory of Transport Coefficients in Low-Density Gases
Authors: McLennan James A. , Swenson Robert J.
Published: January 2005

The evaluation of the correlation-function expressions for transport coefficients is discussed. It is shown that for the case of a low-density nondegenerate monatomic gas with short-range interactions, results are obtained which are in agreement with those usually found from the Boltzmann equation. The analysis makes use of a generalized master equation, and of the factorization theorem of Kac. While the main concern is with low-density gases, the methods developed have a wider range of application.
Thermodynamics of Electrical Networks and the Onsager-Casimir Reciprocal Relations
Authors: Meixner J.
Published: January 2005

Electrical networks furnish an excellent example for the application of thermodynamics of irreversible processes. In particular, they provide one of the simplest cases where, besides Onsager coefficients, one has Casimir coefficients in the phenomenological equations. Two thermodynamic formulations of the network equations are given, which closely resemble the Lagrangian and the Hamiltonian formalism, respectively, of classical mechanics. In the first formulation, only Onsager coefficients occur, but the thermodynamic forces are of a peculiar type in that they are Lagrangian derivatives. Incidentally, it is shown that Casimir reciprocal relations can generally be replaced by Onsager reciprocal relations if the independent variables in the linear phenomenological relations are chosen in a proper way. As a generalization of the network equations, Maxwell's equations in continuous matter with dielectric, magnetic, and Joulean heat losses are considered. Matter is assumed to be isothermal, but not necessarily uniform nor isotropic. Under the influence of impressed electric fields, current distributions are produced. The connection of these fields is expressed by a generalized admittance function. A well-known reciprocity theorem for electromagnetic fields is seen to hold even if all types of losses, as mentioned before, are present. This is due to the Onsager-Casimir reciprocal relations for the dielectric tensor, the permeability tensor, and the resistivity tensor. From the reciprocity theorem, a symmetry relation can be derived for the generalized admittance function. A generalized version is given in the presence of a static magnetic field.
On the van der Waals Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions
Authors: Uhlenbeck G. E. , Hemmer P. C. , Kac M.
Published: January 2005

For the same one-dimensional fluid model discussed in Part I, we have derived general expressions for the two- and three-particle distribution functions. It is seen that these distribution functions depend on all the eigenvalues and eigenfunctions of the basic Kac integral equation, and the dependence is so transparent that the generalization to s particles is obvious. The fluctuation and virial theorems are discussed and shown to be consequences of our general formula. In the van der Waals limit, the behavior of the two-point distribution function is discussed, both for distances of the order of the hard core and for distances of the order of the range of the attractive force. The long-range behavior is, in first approximation, equivalent to the one-dimensional version of the Ornstein-Zernike theory, but only in the one-phase region and not too near the critical point. In the two-phase region, all distribution functions are linear combinations of the two corresponding distribution functions of the saturated vapor and liquid, with coefficients proportional to the mole fractions of vapor and liquid. This is shown for our model; we also give arguments for our belief that these relations are general, and express the geometrical separation of the two phases. The relation to the Ornstein-Zernike theory is discussed in more detail, especially in connection with a recent formulation of this theory by Lebowitz and Percus. We conclude with some comments on the relevance of our results for the three-dimensional problem.
Asymptotic Form of the Structure Function for Real Systems
Authors: Mazur P. , Van Der Linden J.
Published: January 2005

It is shown that the asymptotic formula for the structure function obtained by Khinchin for the case of a system of n noninteracting components, also holds for a classical system of N interacting particles. It is essential for the derivation that the interactions be such that the system behaves in the limit as N tends to infinity, as a thermodynamic system.
Lattice Statistics-A Review and an Exact Isotherm for a Plane Lattice Gas
Authors: Fisher Michael E.
Published: January 2005

Progress in the theory of lattice statistics is reviewed with emphasis on the use of series expansions to study the critical behavior and on the exact results obtained by transformation theory. Recent work is reported which indicates that the magnetization of the three-dimensional Ising model vanishes as (Tc − T)β with β≃516, while the low-temperature susceptibility diverges as (Tc − T)−γ with γ=54(γ=74 in two dimensions). An Appendix is devoted to a detailed tabulation of the exact numerical values and best estimates for the critical temperatures, energies, specific heats, entropies, magnetizations, and the ferro- and antiferromagnetic susceptibilities of four plane lattices and the simple, body-centered, and face-centered cubic lattices.

A square lattice gas with infinite nearest-neighbor repulsions and weak second-neighbor attractions (across alternate squares) is solved exactly for one particular temperature by transformation. The gas undergoes a transition at a density ρt=142, the corresponding form of the isotherm being p−ptkT=α(ρt−ρ)ln α |ρt−ρ|,where α=π(1+342), so that the isothermal compressibility becomes logarithmically infinite.
Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model
Authors: Montroll Elliott W. , Potts Renfrey B. , Ward John C.
Published: January 2005

In this paper we rederive, by simpler methods, the Onsager-Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two-dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager-Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.
Generalized Momentum Operators in Quantum Mechanics
Authors: Robinson Peter D. , Hirschfelder Joseph O.
Published: January 2005

The usual form P0 for the quantum-mechanical operator P conjugate to a generalized coordinate q1 is, in atomic units, P0=−ig−12 ∂/∂q1(g12),where g is the Jacobian of the transformation from Cartesian coordinates to the generalized coordinates. However, in some cases, this plusible form for P is not Hermitian with respect to physically acceptable bound-state wavefunctions, as it must be if it is to represent a real observable quantity. In this paper, the general form P=−i(A·grad+12 div A)is justified. Here A = h1q̂1, where h1 is the metric scale factor corresponding to q1, and q̂1 is the unit vector in the direction of increasing q1. This general form reduces to P0 if the usual formula for the divergence is applied. In the cases where P0 is not Hermitian, it transpires that q̂1 is ill-defined at one or both of the end points α and β of the range of q1, and the divergence formula is thus invalid at such points. It is shown that, in order to obtain a Hermitian form for P, certain terms involving delta functions similar to Dirac's must be added to the usual formula for div A. These terms can be regarded as implicit in div A. If q̂1 is ill-defined at the lower limit q1 = α, then the resulting new Hermitian form for P, which we propose as the correct one, is P=−i[g−12 ∂/∂q1(g12)+12 δ+(q1−α)]. If q̂1 is, in addition, ill-defined at the upper limit q1 = β, then the extra term +½i δ_(β − q1) must be added. Corresponding new forms are obtained for the Laplacian operator. In addition, the new formulas for P are applied to hypervirial relations. In the Appendix, Charles Goebel obtains a similar expression for the momentum operators by replacing the metric scale function by θg where θ is a step function, unity inside and zero outside the range of definition of the generalized coordinates. The differentiation of θ then produces the delta functions.
Relativistic Coulomb Scattering
Authors: Rosen Bernard
Published: January 2005

Contour integration is employed to evaluate analytically the angular dependence of the relativistic correction terms to the Rutherford cross section for the scattering of electrons by nuclei at high energies. The phase shifts obtained by Mott from the solution of the Dirac equation are expanded in powers of the fine structure constant, and the resulting infinite sums are converted into integrals in the complex angular-momentum plane. In turn, these integrals are evaluated by distorting the path of integration and by the use of integral representations. In this manner, the angular dependence of the cross section is obtained in closed form up to terms of the fifth (fourth) order in the cross section (wavefunction). The form of the correction term corresponding to an arbitrary power of the fine structure constant is found in terms of two-dimensional integrals involving elementary transcendental functions. A related problem, the nonrelativistic scattering for an attractive 1/r2 potential is also discussed.
Some Properties of the Reduced Density Matrix
Authors: Yang C. N.
Published: January 2005

Two conjectures made in a previous paper are proved. A further remark is made concerning the largest eigenvalue of the reduced density matrices.
On Strain Energy Functions for Isotropic Elastic Materials
Authors: Bragg Lincoln E. , Coleman Bernard D.
Published: January 2005

This article deals with isotropic elastic materials which possess a strain energy function. For such materials the strain energy of a material point is given, of course, by a symmetric function σ̄ of the principal stretches v1, v2, v3 at that point. It is known that a necessary condition to have an isotropic material compatible with the axioms for thermostatics proposed by Coleman and Noll is that the function σ̄ be jointly and strictly convex in its three variables v1, v2, v3. Here we show that this condition is not sufficient for compatibility with the thermostatic axioms.
Note on the Riccati Equation
Authors: Calogero Francesco
Published: January 2005

A simple modification of a method introduced by R. Bellman is proposed, which under certain circumstances produces both upper and lower bounds for the solution of the Riccati equation. An application to scattering theory is suggested.
On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions
Authors: Biedenharn L. C.
Published: January 2005

It is intended in the present series of papers to discuss explicit constructive determinations of the representations of the semisimple Lie groups SUn by an extension of the Racah-Wigner techniques developed for the two-dimensional unimodular unitary group (SU2). The present paper defines, and explicitly determines, a symmetric vector-coupling coefficient for the group SUn. These coefficients are utilized to construct a series of canonical invariants for SUn, of which the first I2 is the familiar Casimir invariant, and it is proved (by construction) that these invariants form a complete system of independent invariants suitable for uniquely labeling the irreducible inequivalent representations of SUn.
Kinematics of the Relativistic Two-Particle System
Authors: Macfarlane Alan J.
Published: January 2005

Knowing the (canonical) forms that the generators of the infinitesimal transformations of P (the restricted Poincaré Group) take when operating within the Hilbert space of states of an arbitrary irreducible representation of P, such as that which affords a kinematic description of a single relativistic particle, we develop the forms in which they appear when operating within the Hilbert space of states of the (reducible) direct product representation of P which describes a system of two non-interacting relativistic particles. On introducing the Clebsch-Gordan (C-G) series of P which expresses the resolution of this latter Hilbert space into a direct integral over the Hilbert spaces wherein operate the irreducible constituents of the direct product representation, we prove that the generators operate in canonical form within each of these subspaces. If we adopt the alternative viewpoint that the C-G series of P must be written exactly so as to achieve this, our analysis may be regarded as providing a fully explicit and mathematically complete derivation of the formula for the C-G coefficients of P that appear in the C-G series. This formula has previously been suggested only on the basis of heuristic physical argument, and its proof is the principal accomplishment of the present work. One important fact which receives further clarification from the explicit nature of our analysis is the following: in order to see how the intrinsic angular momentum of the two-particle system, in a state of given linear momentum, is compounded from the relative angular momentum and intrinsic spins of the particles, one must view the state from a frame of reference wherein it appears to have zero momentum.
High-Energy Behavior in Perturbation Theory
Authors: Polkinghorne J. C.
Published: January 2005

The dominant-high energy behavior of a wide class of Feynman diagrams is investigated. When the leading contributions are summed they are shown to give a behavior consistent with the Reggepole hypothesis. Series expansions for the trajectory and residue of the dominant Regge pole are obtained in this approximation.
Some Analytic Properties of Green Functions and Self-Energy Parts for a System of Interacting Bosons
Authors: Parry W. E. , Turner R. E.
Published: January 2005

Some analytic properties of the single-particle Green functions and proper self-energy parts for a system of interacting bosons at nonzero temperatures are derived. In particular, it is shown how to obtain the analytical continuations of simple functions of the proper self-energy parts, which can be obtained from perturbation theory at an infinite set of points along the imaginary energy axis, to the whole of the complex energy plane. This is useful in determining the poles of the Green functions.
Generalized Master Equation and t-Matrix Expansion
Authors: Swenson Robert J.
Published: January 2005

Some mathematical aspects of the generalized master equation are discussed. The resolvent operator is expanded in terms of a two-body scattering matrix and this result is used to express the quantities in the generalized master equation in terms of expansions in the scattering matrix. The expansions are utilized in studying the density dependence of these quantities.
Quantum Field-Theory Model whose Truncated Vacuum Expectation Values Vanished beyond Some Order
Authors: Greenberg O. W. , Licht A. L.
Published: January 2005

A local quantum field which has a complete asymptotic field of mass m, and whose truncated vacuum expectation values vanish beyond some order, can lead to no scattering or reactions.
High-Energy Behavior of a Certain Class of Scattering Amplitudes in Perturbation Theory
Authors: Fuchs N. H.
Published: January 2005

The high-energy behavior for a certain class of Feynman diagrams is determined. The results obtained indicate that Regge-type behavior of scattering amplitude may be associated with the existence of M-particle bound states for arbitrary M.
Three-Particle Scattering. I. Planar Case
Authors: Whitten R. C.
Published: January 2005

A formal theory of three-particle scattering in a plane is developed using integral equation methods. Expressions for the scattering amplitude and cross section of elastic and inelastic collisions are derived. The effects of indistinguishability of the colliding particles are discussed.
New Method in the Theory of Imperfect Gases and Liquids
Authors: Lieb Elliott
Published: January 2005

We derive a new expansion method for the thermodynamic properties and correlation functions of imperfect gases. When the interparticle potential is repulsive, we obtain an alternating series of upper and lower bounds for the activity, free energy, pressure, entropy, internal energy, and for the correlation functions. These bounds are valid even if there is a transition. If the potential has an attractive part, we can obtain upper bounds for the activity, free energy, internal energy, and the correlation functions.
Asymptotic Form of the Coefficients of Some Ising-Model Series
Authors: Ninham B. W.
Published: January 2005

The exact asymptotic form of the coefficients of some two-dimensional Ising-model series is derived. A preliminary comparison with some three-dimensional series suggests that the asymptotic nature of their coefficients is not inconsistent with the same analytic behavior.
Nonlinear Coupled Oscillators. II. Comparison of Theory with Computer Solutions
Authors: Jackson E. Atlee
Published: January 2005

A study is made of a particular system of coupled oscillators to determine how the amount of energy exchange and the recurrence time depend on the parameters of the system, and the accuracy with which these are predicted by the perturbation theory developed in part I of this series. The system consists of N + 1 particles, connected by springs which have a quadratic force term, the strength of which, λi, can vary between different particles (``defects''). The equations of motion for the perfect chain (λi = λ) were solved on a computer for the cases N = 4, λ = ¼, ½, ¾; and N = 9, λ = ½; and supplemented by the previous calculations of Fermi, Pasta, and Ulam, in which N = 32, λ = ¼; 1. The ``energy'' in the linear modes,Ek(t)=12(ȧk2+ωk2ak2) are determined from these computations and compared with the perturbation theory in the first paper of this series. As was found by Fermi, Pasta, and Ulam, when only E1 is initially excited, then only the first few modes acquire an appreciable amount of energy, after which nearly all the energy returns to the first mode in a time τλ. The second-order perturbation theory is found to give an accurate estimate (within 15%) of both τλ and the amount of energy exchange for all the above cases, except N = 32, λ = 1, which requires higher-order analysis. It is shown that the nonergodic behavior of this system does not result simply from the incommensurability of the uncoupled frequencies {ωk}, but also from the particular form of mode interaction and the initial conditions used in all calculations, both of which affect the coupled frequency spectrum {Ωk}. To alter the mode interaction a preliminary study is made of ``imperfect'' chains (variable λi), which directly couples low and high modes. It is found that either a large, or else systematic, variation of λi is necessary to appreciably alter the nonergodic behavior of this system. A theoretical examination of the dependence of the coupled frequencies {Ωk} on the initial conditions shows that the ergodic behavior will, in general, be strongly dependent upon the initial conditions.
Singularities of the Four-Point Function in Perturbation Theory with Variable Masses
Authors: Halpern Francis R.
Published: January 2005

The Landau surface for the four-point function with one or more variable external masses is parameterized. The real singularities of the four-point function with one external mass are explicitly determined. The location of the complex singularities of the function depend on the choice of the cuts for the three-point function. Several suggestions are made as to how these could be taken.
On the Number of Self-Avoiding Walks
Authors: Kesten Harry
Published: January 2005

Let χn be the number of self-avoiding walks on the integral points in Euclidean d space and γn the number of n-stepped self-avoiding polygons. It is shown that χn+2/χn−β2 and γ2n+3/γ2n+1−β2 tend to zero as n → ∞ where β=limn→∞χn1/n. Asymptotic estimates for these differences are given. β is also characterized as the unique positive root of Σ1∞λkx−k = 1 where the λk are the number of certain k-stepped self-avoiding walks.
Relativistic Potential Scattering
Authors: Prosser Reese T.
Published: January 2005

The scattering properties of the relativistic two-body problem, governed by the Dirac equation, are investigated. It is shown rigorously that the associated Hamiltonians are self-adjoint, that the associated wave operators exist, and that the scattering operator exists and is unitary, all under suitable conditions on the potential. These conditions on the potential are analogues of those required for the nonrelativistic two-body problem governed by the Schrödinger equation.
Continuous-Representation Theory. II. Generalized Relation between Quantum and Classical Dynamics
Authors: Klauder John R.
Published: January 2005

This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.
Evaluation of Certain Sums Arising in Chemical Physics
Authors: Abraham Philip B. , Weiss George H.
Published: January 2005

In this paper we show that the Poisson transformation of finite sums of the form ∑ i=1NF (cos 2πj/N) cos 2πjn/Ngenerally leads to a quickly convergent sum for problems of physical interest. Examples related to molecular orbital theory and lattice dynamics are discussed.
The Number of Distinct Sites Visited in a Random Walk on a Lattice
Authors: Vineyard George H.
Published: January 2005

A general formalism is developed from which the average number of distinct sites visited in n steps by a random walker on a lattice can be calculated. The asymptotic value of this number for large n is shown to be (8n/π)12 for a one-dimensional lattice and cn for lattices of three or more dimensions. The constant c is evaluated exactly, with the help of Watson's integrals, for the simple cubic, body-centered cubic, and face-centered cubic lattices. An analogy is drawn with an electrical network in which unit resistors replace all near-neighbor bonds in a lattice, and the resistance of such a network on each of the three cubic lattices is evaluated.
Integral Representations for Vertex Functions
Authors: Fronsdal C. , Norton R. E.
Published: January 2005

The third-order Feynman graph is studied as a function of the three external masses squared for arbitrary real values of the internal masses. Single and double ``dispersion'' relations are derived which, for arbitrary real values of the undispersed variable(s), involve integrations only over real contours. Tables list the spectral functions for both the single and double integral formulas. In several cases, a non-Landau singularity (on the forward scattering curve) appears on the ``physical sheet'', but not as a singularity of the physical boundary value.
Position and Polarization Operators in Relativistic and Nonrelativistic Mechanics
Authors: Bacry H.
Published: January 2005

The author shows how the position operator and the polarization operator are obtained in nonrelativistic quantum mechanics from their equivalents in relativistic quantum mechanics. This problem is investigated with the aid of group methods; it is furthermore shown that the two operators cannot be studied separately.
Tensor Artificial Viscosity for Numerical Hydrodynamics
Authors: Schulz William D.
Published: January 2005

Arguments are presented to the effect that a tensor artificial viscosity is a more suitable quantity to use than the usual scalar viscosity. A specific tensor viscosity is constructed for the case of a two-dimensional Lagrangian system.
Motion of a Relativistic Damped Oscillator
Authors: Struble R. A. , Harris T. C.
Published: January 2005

Quantitative effects of damping in a nonlinear system are illustrated for the case of a relativistic oscillator. The elementary techniques employed furnish details of the decay phenomena which are more extensive than those normally furnished by stability theorems.
Von Neumann Algebras of Local Observables for Free Scalar Field
Authors: Araki Huzihiro
Published: January 2005

Basic properties of von Neumann algebras of local observables for the free scalar field are investigated. In particular, the duality theorem for regions at a fixed time is proved. It is also shown that von Neumann algebras for regions at a fixed time are factors of type II∞ or III∞. An argument which excludes the possibility of a factor of type I for a general class of theory and for a certain class of regions is given.
Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation
Authors: De Dominicis Cyrano , Martin Paul C.
Published: January 2005

Quantum statistical mechanics is renormalized, that is, the thermodynamical functions and the ``bare'' ν-body potentials vν occurring in the Hamiltonian are expressed as functionals of the ν-body distribution functions Gν (the expectation values of 2ν field operators). The field operators are considered as having two components (creation and annihilation), and Gν thus has (2)2ν components. The renormalization is carried out for superfluid Bose systems for which it is necessary to consider the expectation values of 2ν = 1, 2, 3, 4 operators. Superfluid Fermi systems and normal systems appear as particular cases, described by expectation values of 2ν = 2, 4 operators.

This problem, dealt with by algebraic methods in part I, is attacked here by diagrammatic methods. These methods are more dependent on convergence properties but the resulting functionals are obtained explicitly as power series of the Gν′s, the general term being represented by a class of diagrams characterized by their topological structure.

In the final result, the entropy is exhibited as an explicit functional of the distribution functions Gν, (2ν = 1, 2, 3, 4), or more precisely, of functions directly related to them. This functional no longer involves the potentials (nor the equilibrium parameters). It is stationary under independent variations of the Gν′s, subject to the constraints of constant energy and particle number. The four equations of stationarity exhibit each function vν as a functional of the Gν′s, self-consistently defining these distribution functions.
On the van der Waals Theory of the Vapor-Liquid Equilibrium. III. Discussion of the Critical Region
Authors: Hemmer P. C. , Kac M. , Uhlenbeck G. E.
Published: January 2005

The discussion of the properties of the Kac one-dimensional fluid model presented in Parts I and II of this series of papers breaks down near the critical point. In Sec. II of the present paper we develop a new successive-approximation method for the eigenvalues and eigenfunctions of the Kac integral equation which is valid in the critical region and which connects smoothly with the developments in the one- and two-phase regions given in Part I. The perturbation parameter is (γδ)½ where γδ is the ratio of the ranges of the repulsive and attractive forces. The main physical consequence is that in the critical region the long-range behavior of the two-point distribution function is represented by an infinite series of decreasing exponentials with ranges all of order 1/γ(γδ)½, and with amplitudes of order (γδ)⅔. This leads to deviations from the Ornstein-Zernike theory and to a specific heat anomaly which are discussed in Sec. V. We conclude with some comments on the possible relevance of our results for the three-dimensional problem.
On the van der Waals Theory of the Vapor-Liquid Equilibrium. IV. The Pair Correlation Function and Equation of State for Long-Range Forces
Authors: Hemmer P. C.
Published: January 2005

A fluid where the pair interaction potential between the particles consists of a hard core and a weak attraction of long range is considered. The pair correlation function and the equation of state in the one-phase region are calculated by a perturbation method with the ratio of the volume of the hard core to the interaction volume as perturbation parameter. The technique used is to expand the well-known density expansion in this perturbation parameter, and finally resum to all orders in the density. To lowest order, the equation of state is of the van der Waals type, and the asymptotic behavior of the pair correlation function near the critical point is that described by the Ornstein-Zernike theory. Higher approximations to these results are given.
Generalized Master Equations for the Density Matrix
Authors: Peterson Robert L. , Quay Paul M.
Published: January 2005

Generalized master equations determining the time dependence of both diagonal and off-diagonal density matrix elements are derived. The equations are valid for systems of arbitrary size and initial conditions, and have all the generality of the von Neumann equation for the density matrix. They are general enough for use in determining all higher-order contributions to transport coefficients. Their advantage over the von Neumann equation is that they are in forms suited to reduction by time-independent many-body approximation techniques.
The Concept of Nonlocalizable Fields and its Connection with Nonrenormalizable Field Theories
Authors: Schroer B.
Published: January 2005

From an investigation of the two-point function in nonrenormalizable field theories, it is shown that at least in certain approximations, a nonrenormalizable field is nonlocalizable. This is intimately connected with the occurrence of essential singularities on the light cone in the Wightman functions of the field. Green's functions cannot be defined and the observables, in particular the scattering matrix elements, have to be expressed directly in terms of the unordered expectations values.
Remarks on the Polynomial Boundedness in the Mandelstam Representation
Authors: Jin Y. S. , Martin A.
Published: January 2005

The Mandelstam representation is a statement about the region of analyticity and asymptotic behavior (polynomial boundedness) of a scattering amplitude. In virtue of the unitarity condition, however, these two are not completely independent. Some physical consequences, e.g., uniqueness, polynomial boundedness of the total cross section, etc., which have been already derived from the Mandelstam representation, are shown to be preserved, even if the polynomial boundedness is replaced by a somewhat weaker assumption. By making use of unitarity, analyticity, and crossing symmetry, the following type of scattering amplitude F = E + M, where E is an entire function in both variables s and t, while M denotes a Mandelstam-type function with finite number of subtraction, is shown to be ruled out. Similarly, F = EM is also ruled out, if one imposes the additional restriction that E should increase less fast than an exponential in one variable while the other is finite.
Multiple Scattering of Waves. II. ``Hole Corrections'' in the Scalar Case
Authors: Fikioris J. G. , Waterman P. C.
Published: January 2005

Scalar multiple scattering effects due to a random distribution of spheres are considered in detail. Transformation from a volume to a surface integral allows one to take account of the ``hole corrections'' involved in the equation of multiple scattering, and yields a secular equation for the propagation constant K of the composite medium. In the low-frequency limit a result is given which appears to be exact over the entire range 0 ≤ δ ≤ 1, where δ is the fractional volume occupied by scatterers. Also in this limit, the boundary conditions appropriate to the boundary of the composite medium are established from examination of the total transmitted and reflected fields.
Application of Operational Methods to the Analysis of the Motion of Rigid Bodies
Authors: Silven Saul
Published: January 2005

Various types of gyroscopic precessional motion are discussed by the application of operational methods to the vector Euler equation describing the motion of a rigid body about a fixed point. Discussed herein are free precession and forced precessions due to external torques both fixed in the body and fixed in space.
An Iterative Solution of the N/D Equation
Authors: Bander Myron
Published: January 2005

An iterative scheme is presented for solving the N/D equations in the case where the left-hand cut consists entirely as a sum of poles. At no step is recourse to a matrix inversion of an algebraic system required. A scheme for approximating arbitrary cuts by sequences of poles is also presented.
Fundamental Properties of Perturbation-Theoretical Integral Representations. III
Authors: Nakanishi Noboru
Published: January 2005

Asymptotic behavior and subtraction problem of the perturbation-theoretical integral representation (PTIR) are investigated in detail. Six theorems are rigorously proved in this connection. It is shown that a function represented by an unsubtracted PTIR may asymptotically increase in particular directions. The relation between the asymptotic behavior and the subtraction number is clarified for the subtracted PTIR. As a by-product one obtains a consistent definition of a finite part of the integral involving x−1θ(x).
Inequalities Relating the Nearest-Neighbor Spin Correlation and the Magnetization for the Heisenberg Hamiltonian
Authors: Falk Harold
Published: January 2005

Convexity properties of the free energy are used to obtain inequalities relating the nearest-neighbor spin correlation and the magnetization for the Heisenberg Hamiltonian.
Representation of Fields in a Two-Dimensional Model Theory
Authors: Tarski Jan
Published: January 2005

The two-dimensional model recently suggested by Schroer is examined. The free scalar massless field in two dimensions is discussed in detail. The infrared-myriotic representations of this field which arise in the model are described; it is shown that, up to unitary equivalence, the required representation depends only on the net total charge of the fermions. Discussion is also given of distribution-theoretic aspects of the fields, and in particular, of possible restrictions on test functions.
Addendum: One-Speed Neutron Transport in Two Adjacent Half-Spaces
Authors: Mendelson M. R. , Summerfield G. C.
Published: January 2005

The interface current for the problem of two half-spaces with a constant source in one half-space is obtained in closed form.
Group-Theoretical Models of Local-Field Theories
Authors: Joos Hans
Published: January 2005

Local fields on a finite, discrete, space-time model are introduced as a guide for axiomatic discussion of quantum field theory.
Three-Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics
Authors: Sack R. A.
Published: January 2005

For any vector r = r1 + r2 an expansion is derived for the product of a power rN of its magnitude and a surface spherical harmonic YLM(ϑ, φ) of its polar angles in terms of spherical harmonics of the angles (ϑ1, φ1) and (ϑ2, φ2). The radial factors satisfy simple differential equations; their solutions can be expressed in terms of hypergeometric functions of the variable (r)2, and the leading coefficients by means of Gaunt's coefficients or 3j symbols. A number of linear transformations and three-term recurrence relations between the radial function are derived; but in contrast to the case L = 0, no generally valid expressions symmetric in r1 and r2 could be found. By interpreting the terms operationally, an expansion is derived for the product of YLM(ϑ, φ) and an arbitrary function f(r). The radial factors are expansions in derivatives of f(r>); for spherical waves, they factorize into Bessel functions of r1 and r2 in agreement with the expansion by Friedman and Russek. The 3j symbols are briefly discussed in an unnormalized form; the new coefficients are integers, satisfying a simple recurrence relation through which they can be arranged on a five-dimensional generalization of Pascal's triangle.
Two-Center Expansion for the Powers of the Distance Between Two Points
Authors: Sack R. A.
Published: January 2005

The powers rn of the distance between two points specified by spherical polar coordinates relating to two different origins, or of the modulus of the sum of three vectors, are expanded in spherical harmonics of the angles. The radial factors satisfy simple partial differential equations, and can be expressed in terms of Appell functions F4, and Wigner or Gaunt's coefficients. In the overlap region, first discussed by Buehler and Hirschfelder, the expressions are valid for integer values of n ≥ −1, but in the other regions, for arbitrary n. For high orders of the harmonics, individually large terms in the overlap region may have small resulting sums; as a consequence the two-center expansion is of limited usefulness for the evaluation of molecular integrals. Expansions are also derived for the three-dimensional delta function within the overlap region, and for arbitrary functions f(r), valid outside that region.
Regge Poles and Potentials with Cores
Authors: Durso John W. , Signell Peter
Published: January 2005

The analytic behavior of the S matrix as a function of real momentum and complex angular momentum is examined for the case of nonrelativistic scattering by potentials with hard-core interiors. The presence of a hard core in a potential introduces a type of symmetry which, combined with unitarity, places restrictions upon the movement of the Regge poles and upon the allowable representations of the scattering amplitude. In particular, it is found that the Watson-Sommerfeld transform representation of the scattering amplitude is not valid. An alternate representation of the S function, more practical for such cases, is postulated and examined numerically. Numerical results are presented for the case when the hard core is replaced by a finite barrier added to a repulsive Yukawa potential interior. It is found that additional trajectories appear. The mechanism for producing the threshold behavior of the phase shifts is compared to that of the case where a hard core is not present.
Generalized Adiabatic Invariance
Authors: Garrido L. M.
Published: January 2005

In this paper we find the quantities that are adiabatic invariants of any desired order for a general slowly time-dependent Hamiltonian. In a preceding paper, we chose a quantity that was initially an adiabatic invariant to first order, and sought the conditions to be imposed upon the Hamiltonian so that the quantum mechanical adiabatic theorem would be valid to mth order. [We found that this occurs when the first (m − 1) time derivatives of the Hamiltonian at the initial and final time instants are equal to zero.] Here we look for a quantity that is an adiabatic invariant to mth order for any Hamiltonian that changes slowly in time, and that does not fulfill any special condition (its first time derivatives are not zero initially and finally).
Convergence of Yukawa Theories with a Finite Number of Interacting Boson Modes
Authors: Frank William M.
Published: January 2005

Convergence of the perturbation expansion of the S matrix is demonstrated for a field theory where a quantized fermion field with regularized propagator interacts with a finite number of quantized boson modes. Conventional graphical techniques and combinatorial analysis are used to establish this result, derived earlier by Edwards, but we believe not properly clarified. A zero radius of convergence is demonstrated for the theory where a Yukawa-type interaction takes place between one boson field interacting in only a finite number of modes with another boson field coupled to it bilinearly. The relation of this result to the convergence of infinite coupled mode theories is discussed.
Energy Sharing and Equilibrium for Nonlinear Systems
Authors: Northcote Robert S. , Potts Renfrey B.
Published: January 2005

A study is made of a one-dimensional system of identical particles in which the forces between neighbors are linear. The system is nonlinear because it is assumed that collisions occur between adjacent particles, which each have an effective diameter d. The energies Ei in the linear normal modes are computed numerically to show that energy is freely exchanged between all the modes in the system, as predicted by the theory. Furthermore, the time averages 〈Ei〉 of these energies show a strong tendency towards equipartition of energy among the modes. This is in distinct contrast to the computations of Ulam, Fermi, and Pasta, which showed that some nonlinear systems appear to be nonergodic. An equation of state and an expression for the total energy of the system as a function of thermodynamic coordinates are derived via statistical mechanics. Expected values for the pressure and temperature of the assembly may then be computed. A comparison of these with the numerical values of those variables arising from the computations shows that the nonlinear system approaches equilibrium.
Some Exact Radial Integrals for Dirac-Coulomb Functions
Authors: Reynolds J. T. , Onley D. S. , Biedenharn L. C.
Published: January 2005

The zero energy loss Dirac-Coulomb integrals are evaluated using the technique of contour integration. The expressions obtained have a closed analytic form, showing that these integrals are formally similar to the corresponding classical and nonrelativistic quantum mechanical, zero energy loss integrals which also have exact elementary solutions.

Application of the zero energy loss Dirac-Coulomb integrals occurs in inelastic electron scattering and similar problems. The investigation of the finite energy loss Dirac-Coulomb integrals requires a study of the zero energy loss integrals as a preliminary.
Relativistic Particle Dynamics and the S Matrix
Authors: Fong R. , Sucher J.
Published: January 2005

Direct-interaction theories are examined from the viewpoint of relativistic scattering theory and the associated concept of ``asymptotic covariance.'' It is pointed out that with any two-particle Hamiltonian which has no bound states there can be associated a variety of representations of the Lie algebra of the inhomogeneous Lorentz group (IHLG), although the S matrix is in general not covariant. It is shown that the requirement of asymptotic covariance ensures both the covariance of the S matrix and the existence of a unique representation of the IHLG to be associated with the relativistic two-particle system. The connection between the Lie algebra, the covariant form of the S matrix, and the uniqueness of K, the generator of pure Lorentz transformations, is thereby clarified. The extension of these considerations to include bound states is made. The form of H given by Bakamjian and Thomas is shown to satisfy asymptotic covariance and, moreover, to be the most general form of interest from the viewpoint of relativistic scattering theory, thereby including as a special case a form of H suggested by Sudarshan. It is also proved that relativistic Hamiltonians of this type do not admit the usual notion of a coupling constant.
New Methods for Reduction of Group Representations Using an Extension of Schur's Lemma
Authors: Gabriel John
Published: January 2005

The generators of the group or group algebra are used in an analog of the Lie-Cartan method, which can be applied to finite or infinite groups. This gives a mean for reduction of an arbitrary group representation, using only the matrix representatives of the generators. It is a set of algorithms using pivotal condensation and can easily be coded as a digital computer program. Connections with Lie-Cartan theory are suggested, the reduction of the symmetric group discussed, and methods for the reduction of representations of the n × n unitary, orthogonal, and proper orthogonal groups suggested.
Theory of Vibrational Structure in Optical Spectra of Impurities in Solids. II. Multiplets
Authors: McCumber D. E.
Published: January 2005

The results of a previous paper describing vibrational structure in the spectra of singlet-to-singlet optical transitions of impurities in solids are extended to a general class of multiplet-to-multiplet transitions. The transitions considered are those between degenerate or nondegenerate multiplets which are not interconnected by the interaction coupling the impurities to the lattice phonons. The phonon field couples impurity states within each multiplet but does not couple states in different multiplets. The results apply to systems which display strong Jahn-Teller distortions as well as to those which do not. As a specific application of our results we consider the singlet-to-doublet transitions of simple three-level systems having C3 and C3v symmetry.