bluuebunny/crossref_metadata_2025 huggingface dataset:
Difficulties in the Kinetic Theory of Dense Gases
Authors: Dorfman J. R. , Cohen E. G. D.
Published: January 2005

For the determination of the transport coefficients of a dense gas, the long-time behavior of the pair distribution function F2 for small intermolecular distances is obtained from a density expansion in terms of the first distribution function F1. On the basis of the dynamics of small groups of particles, it is shown that this expansion contains divergences so that it cannot be used for (a) the computation of the long-time behavior of F2 beyond O(n); (b) the demonstration of the decay of the initial state beyond O(n2). Similar divergences are encountered in the computation of the transport coefficients from time-correlation functions. The nature of the divergences suggests (a) there is no kinetic stage in the approach of a dense gas to equilibrium, in the sense of Bogoliubov; (b) a weak logarithmic density dependence of the transport coefficients.
Temperateness of the Absorptive Part of the Scattering Amplitude in the Proof of Dispersion Relations
Authors: Dobson Peter N.
Published: January 2005

A proof is given within the context of the Bremermann, Oehme, and Taylor proof of dispersion relations that the absorptive part of the elastic scattering amplitude remains tempered throughout the relevant region of analytic continuation. The technique of this proof, which avoids the use of the Jost-Lehmann-Dyson representation, may be applicable to some field theories which do not assume local commutativity and for which such a representation cannot, therefore, be constructed.
Thermodynamically Equivalent Hamiltonian Method in Nonequilibrium Statistical Mechanics
Authors: Girardeau M. D.
Published: January 2005

The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.
Relativistic Partial Wave Analysis of Electron Scattering
Authors: Fischer C. R. , Akerib R.
Published: January 2005

The partial wave series for the scattering amplitude for high-energy electron scattering is not uniformly convergent. The singularity responsible for the nonuniform convergence containing terms in (sin ½θ)2iα − 2, (sin ½θ)2iα − 1, and (sin ½θ)2iα is separated from the rest of the series so that an accurate partial wave analysis may be carried out for any scattering angle.
Embedding of a Simple Lie Group into a Simple Lie Group and Branching Rules
Authors: Navon A. , Patera J.
Published: January 2005

A criterion established by Dynkin is used to specify the embedding of a connected simple Lie group G′ into a connected simple Lie group G, and to derive a standard procedure for evaluating branching rules. It is shown that the weight systems of the irreducible parts contained in the representation of G′ induced by a given finite dimensional representation φ of G are obtained by projection of the weight system of φ. The projection mapping is determined directly from the specification of the embedding. The general procedure is supplemented with two constraint equations on the dimensions and indices of the irreducible representations.
Lowest-Order Correction to Many-Fermion Vertex Function
Authors: Layzer Arthur
Published: January 2005

The lowest-order vertex correction, the exchange counterpart of the polarization correction, is evaluated for all values of the energy and momentum transfer variables and for an arbitrary rotationally invariant two-body interaction. The method employs a dispersion theory technique and an angular expansion of the interaction in spherical harmonics. For any partial wave, the result is expressed as a one-dimensional integral over a closed interval that can easily be evaluated numerically once the form of the interaction is specified. Explicit expressions can be obtained for the long wavelength limit, and for an expansion in powers of the momentum transfer variable. In this case the result depends on the value of the interaction and its momentum derivatives when one of the momentum variables lies on the Fermi surface.
Derivations of Hierarchies for N-Particle Systems and Vlasov Systems by Means of the Functional Calculus
Authors: Nakayama Toshio , Dawson John M.
Published: January 2005

The methods of the functional calculus are applied to the Klimontovich equation and to the Vlasov equations. The equation for the generating functional is derived for both cases. By taking moments of these equations, the BBGKY hierarchy is obtained for the case of the Klimontovich equation, and a similar hierarchy is obtained for the case of the Vlasov equation. The two hierarchies are identical, except for some terms involving individual particle interactions. Methods of solving the Vlasov hierarchy are discussed and it is shown how quasi-linear theory can be obtained.
Equations of Motion in Classical Nonlinear Field Theories
Authors: Rosen Gerald
Published: January 2005

A Rayleigh-Ritz procedure is outlined for deriving approximate equations of ``particle'' motion (the over-all motion of a singularity-free particlelike solution) in a superimposed small-amplitude external field. The method is illustrated here for a classical nonlinear model scalar field theory. A relativistic generalization of the approximation procedure is described.
Multiple Scattering of Electromagnetic Waves by Arbitrary Configurations
Authors: Twersky Victor
Published: January 2005

This paper extends to three-dimensional vector electromagnetic scattering problems our previous development of the scalar problems. We introduce a vector-dyadic formalism that facilitates exploiting the previous results, and derive analogous integral equations which specify the multiple-scattering amplitudes for many objects in terms of the corresponding functions for isolated scatterers. One representation is in terms of the dyadic analog of Beltrami's operator. For arbitrary configurations, the multi-scattered amplitudes are developed as series in inverse powers of the separations of scatterers (with coefficients in terms of isolated scatterer amplitudes and their derivatives); for two scatterers, we derive a corresponding closed form in terms of a differential operator. Another representation is a system of algebraic equations for the many-body multipole coefficients in terms of the isolated scatterer values. Explicit closed forms are derived for two arbitrarily spaced elementary scatterers (electric dipoles, magnetic dipoles, etc.) both by separations of variables, and by working with elementary dyadic fields.
Runge-Lenz Vector and the Coulomb Green's Function
Authors: Hostler Levere
Published: January 2005

The fact that the Runge-Lenz vector is an extra constant of the motion for a charged particle moving in a Coulomb potential is found to account for the especially simple structure of the nonrelativistic coordinate space Coulomb Green's function. Also, the study of the consequences of this extra constant of the motion leads to a separation of variables in the differential equation for the coordinate space Coulomb Green's function, and hence to a new derivation of the closed-form expression for this Green's function which avoids the use of infinite series and the detailed properties of special functions.
Existence of N and D Matrices
Authors: Hartle James B. , Taylor John R.
Published: January 2005

This paper establishes the existence of N and D matrices with the property that the partial-wave T matrix has the form T = ND−1. We consider the case of a finite number of two-body channels and prove that, if T is analytic with right- and left-hand cuts and is suitably bounded, then N and D can be constructed with all the usual properties—namely, N and D have the left- and right-hand cuts, respectively, N is finite at the bound-state poles, and D tends to one as the energy goes to infinity.
Isoparity and Simple Lie Group
Authors: Tanabe Kosai , Shima Kazuhisa
Published: January 2005

The direct generalization of the isoparity (or G-parity), with the defining property that it is commutable with the referring internal symmetry group, is investigated on the basis of the theory of Lie algebra. This is one special problem of the group extension of a simple Lie group by an involution. It is shown that the isoparity of this type can be defined for the simple Lie groups SU(2)(A1 type), SO(2l + 1)(Bl, l ≥ 2), Sp(2l)(Cl, l ≥ 2), SO(2l)(Dl, l ≥ 3), G2, F4, E7, and E8, but not for the SU(l + 1)(Al, l ≥ 2). The relation between the inner automorphism group and the Weyl group of the simple Lie algebra concerned is available to construct the isoparity operator explicitly. Some illustrative examples are presented.
Relation of the O(2, 1) Partial-Wave Expansion to the Regge Representation
Authors: Boyce J. F.
Published: January 2005

The general two-particle scattering amplitude is expanded in terms of partial waves corresponding to the crossed channel little group, O(2, 1). Under the assumption of square integrability over the group manifold, the invariance of the S matrix under the complex Lorentz group, which follows from the Bargmann-Hall-Wightmann theorem, enables this expansion to be identified with the Regge representation in the crossed channel, whenever no dynamical singularities occur to the right of Re j = −½. The identification requires the assumption of the fixed t dispersion relation necessary for the definition of the Regge representation.
Evaluation of Phase-Space Integrals
Authors: Campbell Graham H. , Lepore Joseph V. , Riddell Robert J.
Published: January 2005

The relativistic phase-space integral over the submanifold defined by total momentum zero and fixed total energy is reduced to a single contour integration. The number of particles involved and their masses are arbitrary. It is shown that the contour integration may be readily approximated by the saddle-point technique and yields a result which is easily handled by a computer. In the nonrelativistic and extreme relativistic limits, this method leads to expressions for the phase space which may be obtained from the exact results for these cases by replacing Γ-function factors by the Stirling approximation.
Null Tetrad Approach to Motions in Empty Space-Time
Authors: Collinson C. D. , French D. C.
Published: January 2005

The integrability conditions of conformal motions are written in the null tetrad formalism of Newman and Penrose. The maximum order of the group of conformal motions admitted by nonflat empty space-times of given Petrov type is shown to be at most one greater than the maximum order of the group of Killing motions. The symmetries of those empty space-times which possess hypersurface orthogonal geodesic rays with nonvanishing divergence are determined. Among these space-times is one of type III which admits a group of Killing motions of order three. This provides a counter example to a result of Petrov which states that the maximum order of the group of Killing motions for such space-times is two.
Topology in General Relativity
Authors: Geroch Robert P.
Published: January 2005

A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.
One-Dimensional Three-Body Problem
Authors: Hurley James
Published: January 2005

The three-body problem in one dimension is examined to determine for what class of interactions the Schrödinger equation may be solved by separation of variables.
Equations with Connected Kernels for N-particle T Operators
Authors: Newton Roger G.
Published: January 2005

The Lippmann-Schwinger procedure for single particle scattering is generalized for the calculation of n-particle T matrices describing collisions of a particle with a bound state of (n − 1) other particles. The relevant T operators are expressed in terms of resolvents of compact operators, the equations for which are uncoupled. The two-body interactions appear in the form of their T operators only.
Two Variational Principles for Magnetostatic Calculations
Authors: Schrader David M.
Published: January 2005

Two variational principles for the estimation of the self-interaction energy of a solenoidal current density are introduced. The two variational principles are direct magnetic analogs of the principles of Thomson and Dirichlet for estimating interactions of charge densities. One of the magnetostatic variational principles provides an upper bound on the self-interaction energy, and the other, a lower bound. It is shown how one may extend the ideas to the estimation of the interaction of dissimilar current densities. Various other refinements and extensions are discussed, and a sample calculation is exhibited.
Theory of Diffraction by a Curved Inhomogeneous Body
Authors: Wait James R.
Published: January 2005

The diffraction of electromagnetic waves by a convex spherical surface is considered. The problem is formulated in terms of the mutual impedance between two radially oriented electric dipoles for a smooth inhomogeneous surface. The general integral equation is simplified when the surface is sectionally homogeneous. In this case, the result is given as a twofold integral which involves the product of pattern functions associated with a homogeneous spherical surface. The reduction of these pattern functions to manageable form is one of the major objectives of the paper. The final result shows explicitly how the classical Fresnel diffraction pattern is modified by the radius of curvature and the properties of the diffracting surface. The special case where the diffracting surface is perfectly conducting except for a small inhomogeneity near the crest reduces to a very simple formula if the grazing angle is zero. It is shown that this result is compatible with the perturbation theory for scattering from an impedance strip on a conducting plane.
Analytic Continuation of Laplace Transforms by Means of Asymptotic Series
Authors: Beyer W. A. , Heller L.
Published: January 2005

Conditions under which a Laplace transform L{F(t)}=f(s) may be analytically continued, by means of an asymptotic expansion of F, outside the half plane of convergence of the Laplace transform integral are investigated. For t > k define RN by F(t)=t−β{∑n=oNant−n+RN(t)} for some fixed β with Re β < 1 and suppose that F is integrable on [0, k) for some k ≥ 0. First, it is shown that if RN(t) = O{N!(σ/t)N+1} uniformly in N and t > k for some σ > 0, then the singular part of f at s = 0 can be determined in terms of ai. If β is an integer, then in some neighborhood of s = 0 it is shown that L{F(t)}=sβ−1 ∑ i=0−βaiΓ(1−i−β)si+(log s)g(s)+h(s),where g and h are analytic at s = 0 and g(s)=∑i=1∞(−1)iai−βsi−1/(i−1)!. If β is not an integer, in some neighborhood of s = 0 it is shown that L{F(t)}=sβ−1g(s)+h(s), where g(s)=∑i=0∞aiΓ(1−i−β)si, with g and h analytic at s = 0. Second, if the estimate on RN(t) holds uniformly in N and in the complex t plane in the region (|t|>k)∩(|arg t|<(12π)+λ) for some λ > 0, then the analytic continuation of f can be determined in terms of the ai. For any k′ > k and for |arg s| < π we have L{F(t)}=∫0k′e−stF(t)dt+∫0∞a(t)Γ(2−β,k′(s+t))(s+t)β−2 dt, where Γ is the incomplete gamma function and a(t) is the analytic continuation of ∑i=0∞aiti/i!. If k = 0 in the hypotheses, then with a slight further restriction on F(t) one has L{F(t)}=Γ(2−β)∫0∞a(t)(s+t)β−2 dt. A generalization and application to a problem in nonrelativistic dispersion theory which includes a Coulomb potential are discussed.
Cluster Expansion of an Inverse Overlap Matrix for Solids
Authors: Arai Tadashi , Gabriel John R.
Published: January 2005

In this paper, a method is developed to compute an inverse overlap matrix based on a linked cluster expansion of a determinant. The inverse is expanded in terms of cluster integrals represented by diagrams and a recurrence relation for generation of all diagrams required is found. Thus the computation of an exact inverse overlap matrix is reduced to solving the recurrence equations self-consistently. An approach for solving the equations is suggested and bounds for errors accompanying this procedure are calculated. The method is applied to the hydrogen lattice.
Bargmann-Wigner Equations
Authors: Critchfield Charles L.
Published: January 2005

The redundancy of the Bargmann-Wigner equation for free particles of spin S greater than one-half is analyzed. Only 2(2S + 1) equations involve an essential time derivative, as is well known. An additional ⅓(S + 3)(4S2 − 1) is required to define all components of the representation. The remaining (S + 1)(4S2 − 1) are time derivatives and divergence conditions; the latter occurring doubly.
Neutron Transport from a Point Source in Two Adjacent, Dissimilar, Semi-Infinite Media
Authors: Erdmann R. C.
Published: January 2005

A solution to the steady-state, two-dimensional Boltzmann equation is obtained for the flux due to a point source of neutrons located in a system of two adjacent half-spaces having the same mean-free path. Isotropic scattering and monoenergetic neutrons are assumed. The solution for an arbitrary source location is related to that for an interface source which is obtained by the Wiener-Hopf technique. Asymptotic expansions for the flux far from the source, both along and away from the interface, are derived and compared with approximate theory.
Bounds for Certain Thermodynamic Averages
Authors: Harris A. Brooks
Published: January 2005

Upper and lower bounds for thermodynamic averages of the form 〈{A, A†}〉 are presented.
Regularization and Peratization of Singular Potentials
Authors: Gale William A.
Published: January 2005

We prove that the commonly used regularizations for singular potentials are successful. This means that one can investigate the peratization properties with confidence. A general argument for the success of peratization, as an approximation procedure, is presented. The class of failures of peratization is extended to a series of arbitrarily weakly singular perturbations on the inverse fourth potential. The two results present an unresolved contradiction; some resolutions are considered.
Eigenfunction Expansions Associated with the Second-Order Invariant Operator on Hyperboloids and Cones. III
Authors: Limić N. , Niederle J. , Rączka R.
Published: January 2005

The eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones are derived. The global unitary irreducible representations of the SO0(p, q) groups related to hyperboloids and cones are obtained. The decomposition of the quasi-regular representations into the irreducible ones is given and the connection with the Mautner theorem and nuclear spectral theory is discussed.
Bogoliubov-Parasiuk-Hepp Renormalization Theorem and Spacelike Regularization
Authors: Brandt Richard A.
Published: January 2005

Hepp's recent proof that the Bogoliubov-Parasiuk renormalization prescription leads to finite integrals after the employed space-time regularization of the Green's function is removed is extended to a case in which only spacelike regularization is used.
Class of Nonanalytic Perturbations in Quantum Mechanics
Authors: Frank William M.
Published: January 2005

It is shown that the resolvent operator for the Hamiltonian which is the sum of the harmonic oscillator Hamiltonian H0 and a polynomial perturbation gP(x) of degree exceeding two (g is a coupling constant) is not expressible as a convergent power series in g. The source of this nonanalyticity is the failure of the anharmonic perturbation operator to be small in norm compared to H0. The nature of the singularity at g = 0 is conjectured. The result makes clear that the divergence of the Ward-Hurst-Thirring model has nothing to do with the ``difficulties'' of field theory which are related to the infinite number of degrees of freedom of a field.
Double Structure of a Combined Internal and Space-Time Symmetry Group and Mass Splitting
Authors: Fleischman Owen , Nagel John G.
Published: January 2005

Assuming that the group E, obtained by combining the Poincaré group P with a general internal symmetry group S, has a direct-product-like structure as occurs in the McGlinn theorem, the conditions for the existence of an isomorphism of E with a nontrivial semidirect-product-like structure are examined. It is found that a nontrivial double structure can exist if and only if S contains an internal Lorentz group L, P or their covering groups L̄, P̄. Mass splitting then occurs if the Poincaré group of the semidirect-product-like structure is identified as the physical space-time group.
Reduction of Relativistic Wavefunctions to the Irreducible Representations of the Inhomogeneous Lorentz Group. I. Nonzero Mass Components
Authors: Moses H. E.
Published: January 2005

When the space-time coordinates of a relativistic system undergo the transformations of the proper, orthochronous, inhomogeneous Lorentz group, the wavefunction of the system undergoes transformations which may be considered to constitute a representation of the group. We give a simple algorithm for reducing this representation to the irreducible unitary ray representations if we assume that only nonzero mass representations occur. The extension to cases in which zero mass representations occur will be given in a later paper. The form in which the reduction is given is an expansion of the wavefunction as given in configuration space in terms of a basis such that the coefficients transform in accordance with the Foldy-Shirokov realization of the irreducible representations. Any wave equation which the wavefunction satisfies and any auxiliary conditions, such as the Lorentz condition or reality conditions, eliminate or relate in a simple way some of the representations which can appear. As examples, we reduce the scalar wavefunction, the four-vector with and without the Lorentz condition, the Dirac wavefunction, the wavefunction which transforms like the electromagnetic field, and a wavefunction which transforms as a generalization of the Dirac wavefunction. In these examples it is also shown that if one replaces the amplitudes associated with the irreducible representations by annihilation and creation operators in a suitable manner, one obtains the usual canonical formalism for second quantization in configuration space. The reduction technique given herein is a simple application of the results of an earlier paper by the author and J. S. Lomont in which is shown how to reduce any unitary ray representation of the inhomogeneous Lorentz group.
Weyl Coefficients in SU(3)
Authors: Lezuo Klaus J.
Published: January 2005

The Weyl coefficients are by definition the matrix elements of the Weyl operators in SU(3). They are found to be generalized hypergeometric series of the type 4F3, and can be written down in a simple way from the Gel'fand patterns involved.
Dynamics of Pressure-Free Matter in General Relativity
Authors: Ellis G. F. R
Published: January 2005

An orthonormal tetrad system and associated coordinate system is obtained, which may be used to locally describe any dust-filled space-time. This is used to study dust-filled spaces in which there exist multiply transitive groups of motions; all such spaces are classified in detail. Spaces containing shear-free dust are also considered; it is shown that σ = 0 ⇒ ωΘ = 0. Three classes of solution with σ = 0, ω ≠ 0 are studied. Several new solutions of the field equations are contained in these results.
Clebsch-Gordan Series for Symmetrized Tensor Products
Authors: Andersen C. M.
Published: January 2005

An explicit method for obtaining the Clebsch-Gordan series for the symmetrized n-fold direct products of a finite-dimensional representation of a semisimple Lie group is discussed and illustrated. The method is based on the use of weight diagrams.
Deformation and Contraction of Lie Algebras
Authors: Levy-Nahas Monique
Published: January 2005

This paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it is shown that the only groups which can be contracted in the Poincaré group are SO(4, 1) and SO(3, 2).
Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs
Authors: Pang Sing Chin , Hecht K. T.
Published: January 2005

Normalized lowering and raising operators are constructed for the orthogonal group in the canonical group chain O(n) ⊃ O(n − 1) ⊃ … ⊃ O(2) with the aid of graphs which simplify their construction. By successive application of such lowering operators for O(n), O(n − 1), … on the highest weight states for each step of the chain, an explicit construction is given for the normalized basis vectors. To illustrate the usefulness of the construction, a derivation is given of the Gel'fand-Zetlin matrix elements of the infinitesimal generators of O(n).
Renormalized Coupling Constants and the Inversion Problem
Authors: Fonda Luciano , Ghirardi Gian Carlo
Published: January 2005

It is shown how, starting from the (experimental) knowledge of scattering phase shift, energies of bound states and renormalized coupling constants, one is able to determine completely the parameters of a quantum field theoretic model previously considered by the authors, the so-called Dyson model. The very same conclusion holds for the case of potential scattering, which is also briefly considered.
Basic Algebra of Antilinear Operators and Some Applications. I
Authors: Herbut Fedor , Vujičić Milan
Published: January 2005

The primary aim of this work is to find the canonical (i.e., simplest) form of antilinear operators which is the analog of the diagonal form of linear ones, as well as to obtain that class of antilinear operators which corresponds to the class of normal, i.e., diagonalizable, linear ones. To achieve this aim two basic tools are used: polar factorization of an arbitrary antilinear operator into a linear, Hermitian, positive, semidefinite, and anti-unitary operator Âa = Ĥ1Ûa = ÛaĤ2, and representation of antilinear operators by antilinear matrices, which are products of a matrix factor transforming by unitary congruence transformations and the operation of conjugation which is the same for all antilinear operators and all bases. The canonical form is defined as the simplest form of the matrix factor. The criteria of simplicity are: a quasi-diagonal form with smallest possible submatrices, a maximal number of zeros in them, and as many positive numbers as possible among the nonzero elements. It is found that the analog of the diagonal form of linear operators is the second-order canonical form consisting of a diagonal part with nonnegative elements, which is as large as possible, and of two-by-two submatrices with zeros on the diagonal and with at least one positive element. The operators having this form are those whose polar factors can be simultaneously canonical, taking for the anti-unitary factor, essentially the Wigner canonical form. These antilinear operators are called normal ones, and they can also be defined by the following relations between the polar factors: [Ĥ1, Ĥ2]− = 0 and [Ĥ1, Ûa2]−=0 or by the single commutator, [Âa, (Âa†)2]−=0. A simple procedure to obtain the canonical form of a given normal antilinear operator is developed. A few applications of the results obtained are outlined. They belong to different fields such as electric network theory, quantum mechanics, and self-consistent Hartree-Bogoliubov theory.
Time Behavior of a Reactor and Ergodic Theory of Semigroups
Authors: Hejtmanek Hans
Published: January 2005

The solution of the time-dependent neutron transport equation is a semigroup of linear transformations acting on a Banach space. There are some ergodic theorems that can be used to describe the asymptotic behavior of the solution under very general conditions on the semigroup. The results are compared with Wing's famous approach.
New Approach to the Ising ModeI. II
Authors: Gibberd R. W. , Hurst C. A.
Published: January 2005

Onsager's results for the partition and correlation functions for the Ising model on a two-dimensional rectangular lattice are rederived using a Green's function technique. The definition of the Green's function is based on a recently published casting of the Ising model into a many-body fermion problem. The relation between this approach and other methods used for solving the Ising problem are indicated.
A Unifying Principle in Statistical Mechanics
Authors: Lewis Robert M.
Published: January 2005

A fundamental problem of statistical mechanics is to obtain simplified descriptions of complex systems. A general principle is presented for obtaining equations of motion for such descriptions. The principle involves maximizing an appropriate entropy functional. It also involves the particle dynamics through the Liouville equation. Various special cases are presented in which the principle yields the Vlasov equation, the Boltzmann equation, Euler's hydrodynamic equations, a generalization of Grad's ten-moment approximation, the Gibbs distribution (i.e., equilibrium statistical mechanics), and Onsager's equations of irreversible thermodynamics. The principle also yields, trivially, the Liouville equation and Hamilton's equations of classical mechanics. Some of these results have been derived elsewhere by very similar procedures, but apparently the generality of the principle has been unrecognized. In terms of the general principle, the origin of irreversibility in the various equations of motion is easily seen, and the relation between the numerous definitions of entropy is clarified. No a priori justification of the principle itself is given.
Integral Representations of Invariant States on B* Algebras
Authors: Lanford O. , Ruelle D.
Published: January 2005

Let U be a B* algebra with a group G of automorphisms and K be the set of G-invariant states on U. We discuss conditions under which a G-invariant state has a unique integral representation in terms of extremal points of K, i.e., extremal invariant states.
Null Electromagnetic Field in the Form of Spherical Radiation
Authors: Bartrum Peter C.
Published: January 2005

An example is given of a differential from which leads, through Einstein's gravitational equations, to an energy tensor representing a null electromagnetic field in the form of spherical radiation.
Decomposition of Tensors of the Classical Groups
Authors: Tompkins Donald R.
Published: January 2005

A tensor symmetrization procedure obtained in a recent publication [Phys. Rev. Letters 16, 1058 (1966)] is shown to support rather than disprove Weyl's tensor symmetrization theorem. This ``extended'' symmetrization procedure differs from Weyl's approach in that to construct a subspace irreducible under GL(n, c) one starts with a set of formal states (symmetrized tensors with formal index values) spanning an irreducible representation of the permutation group rather than starting with a single formal state. Extended symmetrization is often more useful than Weyl's approach because the states obtained are highly organized and because it also yields an efficient independent state selection method for the symmetrization procedures using modified Young symmetrizers and Wigner projection operators. The state organization obtained makes it possible to show that the nonorthogonality which is present for bases obtained with Young symmetrizers can be easily removed. The state organization also makes it possible to simplify the task of recoupling symmetrized tensor representations to gain a simply-coupled form. This form enlarges the class of Clebsch-Gordan and recoupling coefficients which can be evaluated by tensor methods. Group matrices and Lie group generator matrix elements are also obtained by tensor methods. Extended symmetrization using unitary representation Wigner projection operators based on unitary representations is shown to result in orthogonal states although usually not the orthogonal states desired. The usual Young symmetrizers are shown to often be more useful than modified Young symmetrizers or Wigner projection operators.
Ground-State Energy of a Finite System of Charged Particles
Authors: Dyson Freeman J.
Published: January 2005

A trial wavefunction of superconducting type is postulated for the ground state of a system of N positive and N negative charges with Coulomb interactions in the absence of any exclusion principle. The ground-state binding energy is rigorously proved to be greater than AN75 Ry, where A is an absolute constant. Results of earlier perturbation-theoretic calculations for an infinite system are confirmed. The author, with A. Lenard, has previously proved that the exclusion principle, holding for particles with one sign of charge only, is a sufficient condition for the stability of matter; the present paper shows that the exclusion principle is also necessary for stability.
Lower Bound on the Mean Kinetic Energy of a System of Particles
Authors: Mazo R. M.
Published: January 2005

It is shown that K ≥ ⅗μ0 for a system of fermions. K is 〈P2/2m〉, and μ0 is the Fermi energy of an ideal gas of fermions of the same mass, and at the same density, as the system under consideration.
Drukarev Transformation of Dirac Equation
Authors: Brysk Henry
Published: January 2005

Drukarev has transformed the Fredholm equation that is the Green's function formulation of the solution of the Schrödinger equation into a Volterra equation. The present paper exhibits the corresponding result for the Dirac equation. The advantages of this technique in the numerical evaluation of phase shifts (and wavefunctions) are discussed.
Some Theorems Concerning the Phase Problem of Coherence Theory
Authors: Dialetis D.
Published: January 2005

The spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self-coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm-Liouville type on a semi-infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self-coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.
Application of the Theory of Padé Approximants to the Solution of the N/D Equations
Authors: Common A. K.
Published: January 2005

Approximate methods for the solution of the N/D equations are discussed. Two methods in particular are considered, one being the replacement of the unphysical singularities of the scattering amplitude by a series of poles and the other being a pole approximation to the spectral integral of the kinematical factor over the physical region. It is shown that Padé approximants may be used to define a sequence of the above approximate solutions, which converges to the exact solution of the N/D equations in those cases when the Fredholm method also gives a solution.
Expansion of the T Matrix for Resonance Collisions
Authors: Micha David A.
Published: January 2005

The eigenfunctions of the kernel of the Lippmann-Schwinger collision equation, corresponding to outgoing waves in all channels, are used to obtain an expansion of the T matrix valid for multichannel collisions, including rearrangements. From this expansion, the transition amplitude in the general case of overlapping resonances is obtained and a characterization of bound states and resonance states is given.
A Class of Eigenvalues of the Fine-Structure Constant and Internal Energy Obtained from a Class of Exact Solutions of the Combined Klein-Gordon-Maxwell-Einstein Field Equations
Authors: Das A. , Coffman C. V.
Published: January 2005

This paper deals with the combined Klein-Gordon-Maxwell-Einstein field equations, which govern completely and self-consistently the spinless, charged, gravitating matter distribution. One of the theorems that have been proved here states that, from a static, purely gravitational universe, a class of electrogravitational universes containing a stationary matter field can be constructed, provided a single differential equation is satisfied. The construction of the electrogravitational universe from the Schwarzchild solution hinges on the solubility of the ordinary differential equation U″+α2(x csch2 x)2U3=0, where the prime denotes differentiation and α2 stands for the fine-structure constant. Next, the following nonlinear eigenvalue problem related to this differential equation has been posed. Are there some positive values of α corresponding to which solutions U(x) exist such that (i) Uis analytic and positive in x ∈ (0, ∞] (⇒ the volume element has one sign), (ii) U(0) = 0 (this condition is physically unpleasant but forced by the differential equation itself), (iii) U′(∞) = 0 (⇒ no force at the center of spherically symmetric mass and charge distributions), (iv) U′(0) = α (⇒ the total charge of the material distribution is α)? The answer is ``yes'' and it has been rigorously proved that there exists a unique solution of the problem. The corresponding value of α comes out to be 1.4343(ℏc)½, which, unfortunately, does not agree with the experiment (the discrepancy may be attributed to the neglect of the second quantization). If the restriction in U(x) to be positive is withdrawn, then a countable number of solutions exist with the corresponding eigenvalues for the electronic charge, internal energy, and mass. These solutions give rise to universes which are topologically inequivalent to Euclidean space and contain a finite number of shells. It should be mentioned that the present eigenvalue problem appears as a consequence of the ``Weyl-Majumdar'' condition on the electrogravitational universe. There may well exist other eigenvalue problems for the fine-structure constant within the framework of the Klein-Gordon-Maxwell-Einstein field equations without the ``Weyl-Majumdar'' condition. ``… es kann dann in jedem Punkte das Krümmungsmass in drei Richtungen einen beliebigen Werth haben, wenn nur die ganze Krümmung jedes messbaren Raumtheils nicht merklich von Null verschieden ist …''—Riemann.
High-Energy Phase Shifts Produced by Repulsive Singular Potentials
Authors: Paliov A. , Rosendorff S.
Published: January 2005

An expansion in inverse powers of the energy (usually fractional powers) of the WKB phase shifts δl produced by repulsive potentials, singular at the origin, has been derived. In most cases this expansion is valid for angular momenta l < lmax, where lmax increases with energy. For large l a power series expansion in the ``coupling constant'' g is developed. The two regions of validity complement each other and sometimes even overlap. The potentials considered have an r−p, p ≥ 1, singularity and a km, m < 2, energy dependence. Define q = p + m − 2; then for q > 0, the case of strong interaction, we get δl ∼ −(gkq)1/p, when k → ∞. The constant of proportionality is independent of l. However, the next term in the expansion depends on l. For q = 0, the case of intermediate interaction, δl becomes independent of energy when k → ∞, but it depends in a complicated way on both g and l. Finally, for q < 0, the case of weak interaction, δl ∼ −gkq(l + ½)1−p, p > 1.
Coordinate and Momentum Observables in Axiomatic Quantum Mechanics
Authors: Gudder Stanley
Published: January 2005

An axiomatic model for quantum mechanics is formulated using physically significant axioms. The model contains a slight strengthening of Mackey's first six axioms, together with two axioms which ensure the existence of coordinate and momentum observables. The symmetries or rigid motions are an essential part of the structure, and a link is constructed between these and the quantum proposition system. Coordinate and momentum observables are defined in terms of abstract coordinate systems and one-parameter groups of motions. It is then shown that as far as the statistical properties of these observables in certain canonical states are concerned, the abstract model may be represented by the usual Hilbert space formulation. Spectral properties of σ-homomorphisms are also investigated. Also included are two appendices of a technical nature: the first considers one-parameter groups of derivables, and the second absolutely continuous σ-homomorphisms.
Representations of the Orthogonal Group. I. Lowering and Raising Operators of the Orthogonal Group and Matrix Elements of the Generators
Authors: Wong M. K. F.
Published: January 2005

A parallel method to that of Pang and Hecht for the construction of normalized lowering and raising operators for the orthogonal group O(n)⊃O(n−1)⊃…⊃O(2) is presented. The generators are defined in a slightly different way from those of Pang and Hecht, and the lowering and raising operators are constructed without using graphs. The Gel'fand-Zetlin matrix elements of the infinitesimal generators of O(n) have also been obtained.
Infra-Red Spectra of Hydrocarbons I. Some Investigations of the Temperature Dependence of Absorption Bands
Authors: Avery W. H. , Ellis C. F.
Published: January 2005

Theory indicates that the separation of the maxima of the P and R branches of an infra-red band should be proportional to the square root of the absolute temperature. This relation has been tested by experiments with ethane, propylene and 2, 2-dimethyl butane in which the temperature was varied between − 195°C and room temperature. It has been found that the relation is approximately valid, the band widths being somewhat less than theory predicts in the liquid and considerably less in the solid. The positions of the bands are independent of the temperature in 2, 2-dimethyl butane and only slightly affected in the other compounds. Some bands of propylene display a pronounced variation in relative intensity with temperature. The usefulness of spectra obtained at low temperatures in the theoretical assignment of fundamentals, as a tool for the study of the liquid state and for analytical purposes, is pointed out.
Raman Spectra of Dialkyl Ketones
Authors: Cleveland Forrest F. , Murray M. J. , Coley J. R. , Komarewsky V. I.
Published: January 2005

Displacements, estimated intensities and depolarization factors have been measured for acetone, 2,6-dimethyl-4-heptanone, 5-nonanone, 7-tridecanone, 8-pentadecanone and 5-tetradecanone. The results for acetone are compared with the previous data: Eight lines previously reported by one or more investigators are believed to be non-existent. A new weak line was observed at 1676 cm−1. The depolarization factors obtained agree well with those reported by Simons, except that the 1356 line was found to be considerably less polarized and data for five additional lines were obtained.
Infra-Red Spectra of Axially Symmetric XY3Z Molecules I. Vibration-Rotation Energies
Authors: Shaffer Wave Henry
Published: January 2005

The vibration-rotation energies of the axially symmetric XY3Z molecular model are investigated to second order of approximation, including cubic and quartic anharmonic terms in the potential energy, all Coriolis interactions between rotation and vibration, centrifugal stretching and dependence of the rotational constants on the vibrational state. The energies are expressed in term value form, E=hc(GV+FR—FC), where GV, FR and FC are, respectively, the vibrational, rotational, and Coriolis terms which are given explicitly as functions of the molecular constants appearing in the Hamiltonian.
Thermodynamics of High Polymer Solutions
Authors: Flory Paul J.
Published: January 2005

A statistical mechanical treatment of high polymer solutions has been carried out on the basis of an idealized model, originally proposed by Meyer, which is analogous to the one ordinarily assumed in the derivation of the ``ideal'' solution laws for molecules of equal size. There is obtained for the entropy of mixing of n solvent and N linear polymer molecules (originally disoriented), ΔS=−k[(n/β) ln v1+N ln v2] where v1 and v2 are volume fractions and β is the number of solvent molecules replaceable by a freely orienting segment of the polymer chain. This expression is similar in form to the classical expression for equal-sized molecules, mole fractions having been replaced by volume fractions. When the disparity between the sizes of the two components is great, this expression gives entropies differing widely from the classical values, which accounts for the large deviations of high polymer solutions from ``ideal'' behavior. The entropy of disorientation of a perfectly arranged linear polymer is found to be of the order of R cal. per chain segment. After introducing a suitable heat of mixing term, partial molal free energies are computed, and the calculations are compared with experimental data for all concentrations. Phase equilibria have been calculated in the region of partial miscibility. The theory predicts, in agreement with experiment, that the critical composition for partial miscibility lies at a low concentration of polymer. Low intrinsic viscosities of polymers dissolved in poor solvents are attributed to the tendency for the molecule to assume a more compact configuration in such an environment.
The Elimination of Errors due to Electrode Polarization in Measurements of the Dielectric Constants of Electrolytes
Authors: Shaw Thomas M.
Published: January 2005

An investigation was made of the errors resulting from electrode polarization in measurements of the dielectric constants of solutions containing electrolytes. The applicability of two methods, one used by Fricke and Curtis and the other by Oncley, to eliminate these errors, was considered. There is introduced a modification of Oncley's method, which was found applicable at higher electrolyte concentrations than Oncley's method. The relative effectiveness of the modified method and that of Fricke and Curtis was tested by employing the two methods to correct experimental data obtained for aqueous solutions of glycine containing various amounts of electrolytes. The measurements were made at frequencies ranging from 10 to 5000 kc, and at conductivities ranging from 0.29×10−4 to 1.74×10−4 ohm−1 cm−1. The corrected values of the dielectric constant were compared with known values in the literature and found to be in good agreement. The limitations of the methods are considered and the usefulness of the methods in cases where the dielectric constant varies with frequency is discussed.
A Systematic Method of Obtaining the Relations Between Thermodynamic Derivatives
Authors: Tobolsky Arthur
Published: January 2005

An elementary systematic method for obtaining the relations between thermodynamic coefficients is presented. This method is applicable for any desired set of independent variables, and is also easily adapted to obtain the derivatives of any functions of the thermodynamic variables.