bluuebunny/crossref_metadata_2025 huggingface dataset:
Further Exact Solutions of the Dirac Equation
Authors: Stanciu George N.
Published: January 2005

The four exactly solvable problems for the Dirac equation previously found by the author are discussed in detail. The presentation is mainly within the two-component relativistic description of a spin ½ particle. Both bound state and scattering solutions are considered.
Expansion Method for Nonlinear Boundary-Value Problems
Authors: Canosa José
Published: January 2005

A homogeneous nonlinear boundary-value problem, which reduces to the Helmholtz equation when the nonlinearity is removed, is solved by an expansion method using as a basis the eigenfunctions of the linear Helmholtz equation. The nonlinear differential equation is reduced to a nonlinear algebraic system in the expansion coefficients, which can be easily solved with any desired degree of accuracy. It is found that with only three terms in the eigenfunction expansion, a satisfactory agreement with the exact numerical solution of the problem is obtained, even for strongly nonlinear cases. Solutions are presented both in one-dimensional slab and cylindrical geometries. It is also shown that the method can be applied to inhomogeneous problems.
Properties of Velocity-Dependent Potentials
Authors: Ferreira Erasmo M. , Guillén Nilo , Sesma Javier
Published: January 2005

Properties of the solutions of the Schrödinger equation with a velocity-dependent potential are studied. Particular attention is given to the examination of the singularities of the differential equation. In the particular cases of one dimension and of the l = 0 partial wave of a spherically symmetric problem, a simple correspondence is found between the velocity-dependent problem and a static one.
Substitution Group and Mirror Reflection Symmetry in Special Unitary Groups
Authors: Ališauskas S. J. , Jucys A. P.
Published: January 2005

The substitutions leaving the character of the representation of the group SUn invariant are considered. The phases induced by these substitutions on the basis functions are established. The substitution giving the contragrediency transformation has been found. This transformation is interpreted as the reflection of the subspace of commuting operators and the corresponding coordinate systems with respect to the rest subspace. The application of the substitution group to the resolution of multiplicity problem in the case of SU3 is demonstrated.
Functional Derivatives and Vector Meson Fields
Authors: Chen Tuan Wu
Published: January 2005

More general functional derivatives for vector mesons are introduced. As an application, a finite theory is formulated for neutral vector fields based on the ``strong'' Lorentz condition together with the other usual assumptions of asymptotic quantum field theory.
Weak Correspondence Principle
Authors: Klauder John R.
Published: January 2005

The weak correspondence principle (WCP) for a scalar field states that the diagonal matrix elements G(f,g)≡〈f,g|G|f,g〉of a quantum generator G necessarily have the form of the appropriate classical generator G in which f(x) and g(x) are interpreted as the classical momentum and field, respectively. For a field operator φ(x) and its canonically conjugate momentum π(x) the states in question are given by |f,g〉≡exp {i ∫ [φ(x)f(x)−π(x)g(x)] dx} |0〉,where |0〉 denotes the vacuum. The validity of the WCP is established for the six Euclidean generators (plus the Hamiltonian) of a Euclidean-invariant theory, and for the ten Poincaré generators of a Lorentz-invariant theory. Only general properties and certain operator domain conditions are essential to our argument. The WCP holds whether the representation of π and φ is irreducible or reducible; in the latter case, the WCP holds even if the vectors |f, g〉 do not span the Hilbert space, or even if the generator G is not a function solely of π and φ. Thus, the WCP is an exceedingly general and completely representation-independent connection between a classical theory and its quantum generators which is especially useful in the formulation of nontrivial, Euclidean-invariant quantum field theories.
Proof that Successive Derivatives of Boltzmann's H Function for a Discrete Velocity Gas Alternate in Sign
Authors: Harris Stewart
Published: January 2005

It has been conjectured by McKean that the particular property of Boltzmann's H function which singles it out from a wide class of functionals of the Boltzmann solution may be that its successive derivatives alternate in sign. We consider here the proof of this alternating property for a discrete velocity gas. For the linearized-model Boltzmann equation, the proof is trivial. For the full (i.e., nonlinear) model Boltzmann equation, the proof is shown to be equivalent to demonstrating the positivity of a particular polynomial. The proof of this property is then demonstrated. It is also shown that H(n), like H(1), is zero only for the equilibrium distribution.
Time-Dependent Green's Function for Electromagnetic Waves in Moving Conducting Media
Authors: Besieris Ioannis M. , Compton Ralph T.
Published: January 2005

This paper treats the problem of radiation from sources of arbitrary time dependence in a moving conducting medium. The medium is assumed to be homogeneous and isotropic, with permittivity ε, permeability μ, and conductivity σ, and to move with constant velocity v with respect to a given inertial reference frame. v may have any value up to the speed of light. It is shown how the Maxwell-Minkowski equations for the electromagnetic fields in the moving medium can be integrated by means of a pair of vector and scalar potential functions analogous to those commonly used with stationary media. The wave equation associated with these potential functions is derived, and an associated scalar Green's function is defined. The solution for the Green's function is obtained in closed form, by means of a technique making use of the relation between the fundamental solution of a radiation problem and that of a corresponding Cauchy initial-value problem. The resulting Green's function is found to consist of an oblate spheroidal shell, similar to that which occurs in a lossless medium, plus a residue which persists after the shell. In addition, the Green's function is exponentially damped in both space and time, an effect not present in a lossless medium.
The Half-Space Green's Function for an Atmosphere with a Polarized Radiation Field
Authors: Smith O. J. , Siewert C. E.
Published: January 2005

This paper presents a rigorous solution for the half-space Green's function of the matrix transport equation that describes the flow of polarized radiation in a free-electron atmosphere. The singular normal modes expansion technique is used to construct the solution in such a manner that the expansion coefficients can be determined by applying the proper boundary conditions. The necessary completeness and orthogonality theorems are proved, and thus all expansion coefficients are found by simply taking scalar products. In addition, the albedo problem for a semi-infinite half space is solved explicitly.
Point-Loop Renormalization in Regularized Field Theories. I
Authors: Frank William M.
Published: January 2005

The relation between the augmented and the standard vacuum amplitudes is derived (for a finite-valued amplitude for the point fermion-antifermion loop) by diagrammatic methods. This relation is shown to consist of a mass renormalization, whereby the point fermion-antifermion loop is absorbed into the fermion propagator, plus some additional factors, and the analytical properties of the amplitudes are greatly modified. The effects of the renormalization are the increase of the density of zeros, the increase of the exponential order in the variable z, and the change of the singularities in η from simple poles to essential singularities. The significance of these for massless fermions or bosons is briefly discussed.
Relaxation to Equilibrium of a Dilute Electron Plasma
Authors: McLeod J. B. , Ong R. S. B.
Published: January 2005

The relaxation of the velocity distribution to equilibrium in an electron plasma in which the dominant collisions are described by the Fokker-Planck operator is studied. It is shown mathematically that the linearized collision operator possesses a continuous spectrum of eigenvalues which extends over the entire real interval from zero to infinity. Consequently the decay to equilibrium is not uniformly exponential. For large values of the time the decay to Maxwellian is shown to be of the order of the inverse power of the time variable. Moreover the rate of decay depends also on the initial perturbation.
Relations between Field-Plus Source and Fokker-Type Action Principles
Authors: Anderson James L. , Schiminovich Samuel
Published: January 2005

The problem of constructing a Fokker-type action that will yield equations of motion for the sources of an original field-plus-source system is considered. It is shown that such an action can be constructed for a large class of systems by substituting for the fields appearing in the original field-plus-source action a solution of the field equations in terms of the source variables provided that certain conditions are satisfied. In general, these conditions will be satisfied only by half-advanced, half-retarded-type solutions of the field equations and then only for a field-plus-source Lagrangian that differs from the usual one by a complete divergence. The problem of constructing a Fokker action for a system that possesses a gauge-type covariance is complicated by the existence of differential identities that are satisfied by the left-hand sides of the field equations and that arise as a consequence of the gauge covariance. These difficulties can be overcome by the introduction of gauge conditions on the field variables. The problem of finding solutions to the field equations suitably modified with the help of the gauge conditions that satisfy both the gauge conditions and the conditions for the construction of the Fokker action is discussed. As an application of the method, we have obtained the Fokker action of electrodynamics using a number of different gauge conditions, arriving thereby at several different expressions for this action. We have also considered the construction of a Fokker action when the field equations are solved by an approximation method. In such cases we have shown that to determine the Fokker action up to the nth order of the approximation it is only necessary to solve the field equations up to order ½(n + 2) or ½(n + 1) for n even or odd, respectively. The method is then applied to obtain the Darwin action of electrodynamics.
Microscopic Approach to Kinetic Theory
Authors: Chappell Willard R.
Published: January 2005

A microscopic kinetic theory is developed for a plasma by the use of approximate equations of motion for the microscopic ``exact'' distribution function f̂(r,v,t)= ∑ i=lNδ(r−ri(t)) δ(v−vi(t)). These equations can be solved to obtain asymptotic expressions for f̂(r, v, t) that are then used to calculate correlation functions. These approximate equations are also used to obtain the approximate equations for the correlation functions and the principal results of the test-particle approach. In particular, two sets of equations are constructed. One set describes a homogeneous, slowly varying system and yields the Balescu-Lenard equation. The other set describes a homogeneous, slowly varying system that contains small, inhomogeneous and quickly varying perturbations.
Translational Invariance Properties of a Finite One-Dimensional Hard-Core Fluid
Authors: Leff Harvey S. , Coopersmith Michael H.
Published: January 2005

A formalism is developed for expressing the n-particle distribution functions Dn(x1 ≤ x2 ≤ … ≤ xn) explicitly in terms of the configurational partition function for one-dimensional fluids with hard-core repulsive and nearest-neighbor attractive forces. The translational invariance properties of the Dn functions are investigated for the case of no attractive forces when the system is finite. When the number density is less than half the close packing density, there exists a central region in which D1(x) is constant and all the Dn functions, n ≥ 2, are functions of the (n−1) nearest-neighbor separation distances. Several relevant theorems are proved and limiting cases are investigated.
Non-Usual Topologies on Space-Time and High-Energy Scattering
Authors: Atkinson D. , Halpern M. B.
Published: January 2005

Motivated by recently observed deviations from quantum electrodynamical theory, we study the possibility that our notions of space-time may need revision at small distances. In this work, we wish to call attention to certain techniques which are available for studying different space-time structures within the framework of topology. Our main effort is in the consideration of a non-usual topology on space-time in which is embedded an elementary length. By working separately in each n-particle subspace, the embedding is done in an inhomogeneous Lorentz invariant way, and we avoid any lattice structure in space-time. Particles in this topology are in general extended structures, and we find the surprising feature that, at high energies, the topology enhances backward and large-angle scattering. From these preliminary investigations, we are not as yet able to make more than qualitative comparison with experiment. Along the way, we have the opportunity to remark on ways of embedding an intrinsic breakdown of certain invariances (e.g., parity) in the topology of space-time.
New Characterization of the Ray Representations of the Galilei Group
Authors: Bernstein Herbert J.
Published: January 2005

An elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy-Leblond, U(b′,a′,v′,R′)U(b,a,v,R)=±exp [i(12m)(a′·R′v−v′·R′a+bv′·R′v)]×U(b′+b,a′+R′a+bv′,v′+R′v,R′R).
Time-Dependent Green's Function for Electromagnetic Radiation in a Conducting Moving Medium: Nonrelativistic Approximation
Authors: Besieris Ioannis M.
Published: January 2005

The electromagnetic radiation emanating from a source immersed in a linear, homogeneous, conducting medium moving with a uniform velocity v with respect to the rest frame of the source distribution is investigated. It is shown that in the nonrelativistic limit, that is for v/cv and v/cm ≪ 1, where cv is the velocity of light in free space and cm denotes the phase velocity of a wave in the medium considered at rest, the electromagnetic field intensities can be expressed in terms of a pair of scalar and vector potentials by specifying a modified Lorentz condition. The time-dependent Green's function associated with the hyperbolic partial differential equations satisfied by these potentials is determined explicitly.
Stability of Matter. I
Authors: Dyson Freeman J. , Lenard A.
Published: January 2005

The stability problem of a system of charged point particles is discussed, and a number of relevant theorems are proven. The total energy of a system of N particles has a negative lower bound proportional to N53 when no assumption is made on the statistics of the particles. When all particles belong to a fixed number of fermion species, a lower bound exists proportional to N.
Translational Invariance Properties of a One-Dimensional Fluid with Forces of Finite Extent
Authors: Coopersmith Michael H. , Leff Harvey S.
Published: January 2005

The translational invariance properties of a one-dimensional fluid with finite range forces are investigated. For N particles in the interval [0, L], with a two-body interaction potential w(x) = 0 for x ≥ R, we find the following: (a) If w(x) has a hard core of diameter d and R ≤ 2d, each n-particle distribution function Dn(x1, …, xn) is translationally invariant if and only if L > 2(N − n)R and x1, …, xn lie in [(N − n)R, L − (N − n)R]. (b) For arbitrary finite values of R, with or without a hard core, the above conditions are sufficient for translational invariance of the Dn. These conditions hold for all temperatures.
Symmetry Group of the Hydrogen Atom
Authors: Finkelstein R. J.
Published: January 2005

It is shown how the complete dynamics of the hydrogen atom is related to the three-dimensional rotation group.
Density Fluctuations of a Fluid
Authors: Jones Gerald L. , Kennedy Robert E.
Published: January 2005

A proof is given, in the canonical and the grand canonical formalism, to show that the density fluctuations in a macroscopic region of a fluid are large in the two-phase states and small in the one-phase states. The definition of large and small density fluctuations is one used previously by Dobrushin in connection with the Ising model. The density fluctuations at the critical point are, using this definition, small if the critical isotherm has no flat portion.
New Derivation of the Ising Model Cluster Expansion
Authors: Jasnow David , Wortis Michael
Published: January 2005

A new and simple derivation of the cluster expansion for the free energy of an Ising model with arbitrary range of interaction is presented. The proof explicitly isolates the dependence of the free energy on the strength of individual interaction bonds. Several points of principle are discussed. An illustrative appendix gives simple applications.
Weber's Mixed Boundary-Value Problem in Electrodynamics
Authors: Laporte O. , Fowler R. G.
Published: January 2005

Electrodynamics problems with mixed boundary values promise to assume increasing practical importance in fields such as plasma physics. A new method of attacking such problems in three dimensions is presented and discussed.
Relationship of the Internal and External Multiplicity Structure of Compact Simple Lie Groups
Authors: Macfarlane A. J. , O'Raifeartaigh L. , Rao P. S.
Published: January 2005

As a preliminary step in a program aimed at developing the Racah algebra of an arbitrary compact simple Lie group L, this paper gives a unified review with various extensions of the work of Biedenharn, Speiser, and others on the relationship between the internal and external multiplicity structures of L, the former being that of the weights of the representations of L, the latter being that of the terms of the Clebsch-Gordan series of L.
General Theory of Resistive Beam Instabilities
Authors: Weinberg Steven
Published: January 2005

This paper considers all resistive instabilities of a self-pinched cylindrically symmetric beam of charged particles in a finite or an infinite Ohmic plasma channel. The problem is reduced to an ordinary second-order linear differential equation for the longitudinal component of the perturbed electric field. The equation can be solved for a uniform beam shape, yielding an implicit transcendental equation whose roots define the various modes. We find that for each azimuthal ``quantum number'' m there are two infinite sequences of modes and two exceptional modes, except that some of these modes are missing for m = 0, 1, and 2. In all modes we find stable oscillation at very low and very high frequencies, and instability at intermediate frequencies, the growth rates generally reaching maxima somewhat less than the betatron frequency ωβ. The largest maximum growth rate is in the ``hose'' mode (the only exceptional mode for m = 1), where it is approximately 0.29 ωβ. For a general smooth beam shape, the catalog of modes is similar to that for a uniform beam, except that there also appears a continuous spectrum. It is also proved for general beam shape that at low frequencies the ``hose'' dispersion relation becomes the same as that derived earlier under the assumption of rigid beam displacement; this is not the case at higher frequencies.
Induced Representations of Strong Coupling Groups
Authors: Cook T. , Sakita B.
Published: January 2005

Mackey's method of induced representations is applied to the strong coupling group G = K·T, where K is compact and T is Abelian, to obtain the general irreducible representations. The form of the meson-isobar-isobar couplings is obtained by reducing these, with the use of the Peter-Weyl theorem, to irreducible representations of the compact subgroup K. The results are applied to the cases: pseudoscalar octet mesons, K = SU2 ⊗ SU3, T = T24, and SU6 35-plet mesons, K = SU6, T = T35. Explicit representations are obtained which are consistent with mass formulas M∝f2(μ)−58j(j+1) and M∝G2 in the respective cases.
Symmetry Properties of the 3j Symbols for SU(3)
Authors: Derome Jean-Robert
Published: January 2005

For an arbitrary compact group it is in general not possible to choose the 3j symbol (j1j2j3)r, m1m2m3 such that its absolute value is invariant under every permutation of the j's and of the corresponding m's. Still, it is commonly assumed that for SU(3) such a choice is possible. In this paper it is shown that this assumption is indeed justified.
Determination of Weight Factors in Linked-Cluster Expansions for Lattice Systems
Authors: Essam John W.
Published: January 2005

In a previous paper, it was shown that any function φ(G), defined for a general linear graph G and having the extensive property, can be expanded in terms of the lattice constants of connected subgraphs of G. In this paper, a graphical interpretation of the weight factors occurring in this expansion is given. The usefulness of the expansion in deriving series expansions for properties associated with crystal lattices is discussed with particular reference to percolation problems, dilute ferromagnets, and lattice gases. A result in the theory of linear graphs, recently proved by Rushbrooke in a paper concerned with dilute ferromagnets, is rederived.
CTP Invariance of the S-Matrix in a Theory of Local Observables
Authors: Epstein Henri
Published: January 2005

In a theory of local observables as proposed by Haag and Araki, the assumptions which make possible a collision theory also guarantee the CTP invariance of the S-matrix.
Lie Algebra Extensions of the Poincaré Algebra
Authors: Galindo A.
Published: January 2005

The ``linear'' counterpart of the problem of analytic group extensions of the Poincaré group is presented in terms of the considerably simpler (but less general) analysis of Lie algebra extensions of the Poincaré algebra P. After easily proving with this technique that every C kernel (P, θ) has an extension and that every such extension is inessential, the problem of analyzing the central extensions of P is carried out with the well-expected result that every such extension is trivial. But contrary to some claims, we exhibit an example which explicitly shows an essential noncentral extension of P.
Expansion Theorem for the Linearized Fokker-Planck Equation
Authors: Lewis Jordan D.
Published: January 2005

The linearized Fokker-Planck kinetic equation for each component of a homogeneous, nondegenerate, fully ionized plasma is separated by means of a spherical harmonic expansion into an infinite set of singular intergo-differential equations. Each equation is shown to generate a continuous set of eigen-functions, for which asymptotic high-speed forms are found. By extending the theory of singular differential equations an expansion formula is developed, which is shown to be complete with respect to functions square integrable in velocity space.
A Set of Harmonic Functions for the Group SU(3) as Specialized Matrix Elements of a General Finite Transformation
Authors: Nelson T. J.
Published: January 2005

A straightforward method of converting the commutation rules of the generators of a group into a set of differential equations is developed. The detailed development is applied to SU(3). Differential operators having the correct commutators are constructed and then combined into the quadratic and cubic combinations of the two Casimir operators. The solution of the resulting set of differential equations is the matrix element of a general finite transformation with the right-hand state specialized to the isotopic spin singlet. The set of such matrix elements is equivalent to the set of SU(3) harmonic functions derived by Beg and Ruegg. The isotopic spin-hypercharge content of the irreducible representations of SU(3) is deduced from the form of the matrix element, though prior knowledge of the fact that each irreducible representation of SU(3) possesses an isotopic spin singlet is assumed.
Null Fields in Einstein-Maxwell Field Theory. II
Authors: Trollope J. R.
Published: January 2005

If the propagation vector of an electromagnetic null field is proportional to a gradient, then the metric tensor can be expressed in terms of only four unknown functions. In the present analysis, it is assumed that the divergence of this vector does not vanish. It is shown that the field equations reduce to a single partial differential equation. Particular solutions which represent electromagnetic and purely gravitational waves are given.
Properties of a Harmonic Crystal in a Stationary Nonequilibrium State
Authors: Rieder Z. , Lebowitz J. L. , Lieb E.
Published: January 2005

The stationary nonequilibrium Gibbsian ensemble representing a harmonic crystal in contact with several idealized heat reservoirs at different temperatures is shown to have a Gaussian Γ space distribution for the case where the stochastic interaction between the system and heat reservoirs may be represented by Fokker-Planck-type operators. The covariance matrix of this Gaussian is found explicitly for a linear chain with nearest-neighbor forces in contact at its ends with heat reservoirs at temperatures T1 and TN, N being the number of oscillators. We also find explicitly the covariance matrix, but not the distribution, for the case where the interaction between the system and the reservoirs is represented by very ``hard'' collisions. This matrix differs from that for the previous case only by a trivial factor. The heat flux in the stationary state is found, as expected, to be proportional to the temperature difference (T1 − TN) rather than to the temperature gradient (T1 − TN)/N. The kinetic temperature of the jth oscillator T(j) behaves, however, in an unexpected fashion. T(j) is essentially constant in the interior of the chain decreasing exponentially in the direction of the hotter reservoir rising only at the end oscillator in contact with that reservoir (with corresponding behavior at the other end of the chain). No explanation is offered for this paradoxical result.
Symmetries of the Bethe-Salpeter Equation for Relativistic Bound-State Problem
Authors: Biswas S. N.
Published: January 2005

Possible dynamical symmetries of the Bethe-Salpeter equation in the ladder approximation for the bound state of two neutral scalar mesons interacting via a massless scalar boson have been investigated. It is shown that the Bethe-Salpeter equation exhibits an O(5) or O(4) symmetry according as E, the total energy of the system, is zero or nonvanishing and the corresponding no-invariance group being O(5, 1) when E = 0 and the O(5, 1) spectrum splits into Σ⊕ O(4, 1) when E ≠ 0 as pointed out by Salam et al. The method employed here differs from that of Cutkosky and of Salam et al.
Leading Landau Curves for a Larger Class of Feynman Diagrams
Authors: Pattnaik Mamata
Published: January 2005

The leading parts of Landau curves for a larger class of Feynman diagrams are found to give no singularities on the physical sheet provided certain conditions are imposed upon the external and internal masses. The number of such masses involved is found to be fixed, no matter how complicated a member of this class is considered.
Complete Excitation Spectra
Authors: Anderson J. T.
Published: January 2005

A type of nonhomogeneous integral equation frequently encountered in interaction problems is solved by means of the Heaviside expansion theorem. The energy spectrum is obtained in the usual way from the solutions of the dispersion equation for the excitations, D(k, s) = 0. The momentum spectrum is obtained as the solutions of ∂D/∂k = 0, which is shown to be also a form of the Cauchy-Riemann equations. It is proved that the existence of a solution of ∂D/∂k = 0 is a necessary condition for the existence of an analytic region on the momentum plane. It is also proved that the existence of a solution of dD/dk = 0 and ∂D/∂s = 0 is a necessary and sufficient condition for the existence of analytic regions in both variables, provided the group velocity is finite and continuous. As an example, two linearized self-consistent equations with arbitrary coupling are solved.
Branching Theorem for the Symplectic Groups
Authors: Hegerfeldt G. C.
Published: January 2005

A theorem analogous to the Weyl branching law for the unimodular groups is derived for Sp(2n).
Decomposition of the Unitary Irreducible Representations of the Group SL(2C) Restricted to the Subgroup SU(1, 1)
Authors: Sciarrino A. , Toller M.
Published: January 2005

The unitary irreducible representations of the group SL(2C) belonging to the principal series restricted to the subgroup SU(1, 1) are decomposed into a direct integral of unitary irreducible representations of SU(1, 1). The matrix elements of the unitary operator which performs the decomposition are given explicitly and used to obtain a relation between the matrix elements of the unitary irreducible representations of the groups SL(2C) and SU(1, 1). Similar identities between the matrix elements of nonunitary representations of these groups are obtained by means of analytic continuation. The relevance of these results to the theory of complex angular momentum and of high energy nearly forwardscattering is pointed out.
Asymptotic Behavior of Stieltjes Transforms. I
Authors: Woolcock W. S.
Published: January 2005

Several theorems are proved concerning the asymptotic behavior of Stieltjes transforms as |z| approaches infinity, in a sector of the complex z plane which does not include the cut in the transform. The asymptotic behavior of the transform is related to the asymptotic behavior, for large values of the argument, of the function whose transform is taken.
Remarks on Relativistic Statistical Mechanics. I
Authors: Hakim Rémi
Published: January 2005

The modifications introduced by the specific forms of relativistic dynamics of many-particle systems are shown to give rise to a different (with respect to the nonrelativistic case) manner to set the problems involved in a tentative construction of relativistic statistical mechanics. Although the difficult problems of relativistic dynamics are not solved, it is possible to define relativistic generalizations of phase space, distribution functions, Gibbs ensembles, and average values. In particular, phase space is chosen for convenience and is no longer related (as is usually the case) to the ``initial data,'' whose nature is yet unknown. As a consequence, only those observables which depend on the variables characterizing phase space give rise to easily computed average values. However, it is possible to enlarge at will the basic phase space and to define subsequent densities from which average values may be calculated. [Example: The calculation of average values of observables A(…xiμ,uiμ…) needs only densities of the form N(…xiμ,uiμ…) and observables involving acceleration variables need the enlarging of phase space so as to include the latter. On this enlarged phase space, densities of the form N(…xiμ,uiμ,γiμ…) may be defined and are used to compute average values, etc.] The notion of equilibrium is discussed and suggestions for reaching the solution of this unsolved problem are made.
Remarks on Relativistic Statistical Mechanics. II. Hierarchies for the Reduced Densities
Authors: Hakim Rémi
Published: January 2005

In Paper I the basis of relativistic statistical mechanics was discussed and notions such as phase space, Gibbs ensemble, distribution functions were defined. Paper II deals with hierarchies of equations for the reduced densities. Extensive use is made of the beautiful methods of Klimontovich. In this way ``classical mesic interactions'' are dealt with neglecting ``radiation'' effects (i.e., ``classical'' emission of mesons). It is shown how the ``renormalization of mass'' affects the hierarchies obtained. Electromagnetic interactions are dealt with: (a) neglecting radiation effects, (b) including radiation effects. The latter case is treated on the basis of the Lorentz-Dirac equations and with the help of suitable modifications of the formalism. In this way a new approach to radiation phenomena (for instance, in plasmas) is obtained. Finally, as a matter of illustration, several well-known relativistic kinetic equations are rederived in a slightly improved manner (i.e., Vlasov and Landau equations).
Stress-Tensor Commutators and Schwinger Terms
Authors: Boulware David G. , Deser S.
Published: January 2005

We investigate, in local field theory, general properties of commutators involving Poincaré generators or stress-tensor components, particularly those of local commutators among the latter. The spectral representation of the vacuum stress commutator is given, and shown to require the existence of singular ``Schwinger terms'' at equal times, similar to those present in current commutators. These terms are analyzed and related to the metric dependence of the stress tensor in the presence of a prescribed of a prescribed gravitational field and some general results concerning this dependence presented. The resolution of the Schwinger paradox for the Tμν commutators is discussed together with some of its implications, such as ``nonclassical'' metric dependence of Tμν. A further paradox concerning the vacuum self-stress—whether the stress tensor or its vacuum-subtracted value should enter in the commutators—is related to the covariance of the theory, and partially resolved within this framework.
Note on the Kerr Metric and Rotating Masses
Authors: Cohen Jeffrey M.
Published: January 2005

Kerr's metric is often said to describe the geometry exterior to a body whose mass and rotation are measured by Kerr's parameters m and a, respectively, even though no interior solution is known. In this paper we give an interior solution valid in the limit when the rotation parameter a is sufficiently small so that terms of higher power than the first are negligible, but the mass parameter m is allowed to be large. This is accomplished by bringing Kerr's exterior metric into the form of the metric for a slowly rotating mass shell. Also, the connection is found between Kerr's parameters and the physical parameters characterizing the rotating body.
Quantum Correlation Function for the Noninteracting Particle System
Authors: Jaen Jong K.
Published: January 2005

A method for determing the quantum correlation functions of the noninteracting particle system in thermal equilibrium is developed. It is designed to reduce the labor involved in treating the large number of permutation operators of the symmetric group that occurs. An alternate form of the n-particle correlation function is obtained in order to simplify computation. The London-Placzek formula is derived as a check. The error in Kirkwood's superposition approximation for this system is investigated, and the exact relationship between the two- and three-particle correlation functions is found. Finally, a method determining the pair correlation function in the Hartree-Fock approximation of a pair-interacting particle system at a finite temperature is presented.
Dispersion of Electromagnetic Waves in Molecular Crystals
Authors: Mavroyannis C.
Published: January 2005

The dispersion of electromagnetic waves in molecular crystals has been studied using the second quantization formalism. The excitation spectrum, the Green's functions for the optical exciton and photon field, as well as the corresponding distribution functions, are calculated and discussed. The ground state energy of the crystal is derived in a closed form with the polarization of the medium taken into account explicitly. In the low-density limit the expression for the ground state energy corresponds to the summation of an infinite sequence of terms in a perturbation-theory approach.
Superconductivity in One and Two Dimensions. II. Charged Systems
Authors: Rice T. M.
Published: January 2005

In an earlier paper the Ginzburg-Landau free energy functional was used to calculate the effect of thermodynamic fluctuations on the off-diagonal correlation function and we found no off-diagonal long-range order in one- and two-dimensional systems. It has been pointed out that, for a charged system, the use of the Ginzburg-Landau free energy functional is in error for arbitrary nonequilibrium values of the order parameter since the electrostatic energy of the charge fluctuations associated with an arbitrary order parameter is not included in the free energy functional. We have not succeeded in a direct generalization of the free energy functional so we are forced to proceed by inference from the generalized random phase approximation (RPA). We find that, for uncharged systems, the RPA gives a linearization of the results obtained earlier using the Ginzburg-Landau theory. For charged systems we find in the RPA results similar to those obtained for uncharged systems. From this we conclude that it is very likely that, as in uncharged systems, there will be no ODLRO in charged infinite one- and two-dimensional systems.
Relativistic and Causal Theories with Four-Fermion Interaction Hamiltonians
Authors: Blood F. A.
Published: January 2005

Possible ways of constructing field-theoretic operators satisfying the commutation relations of the inhomogeneous Lorentz group are investigated along lines laid down by Dirac. They can be satisfied in both the instant form, in which operators representing rotations and translations in space remain unchanged, and the point form, in which operators representing the homogeneous Lorentz group G41 remain unchanged, provided one can find a causal Hamiltonian density such that [H(t, x), H(t, y)] is proportional to δ(x − y) and which transforms as a scalar under G41. Less restrictive sufficient conditions in the instant form are found, similar to those of Dirac. The commutator can be proportional to derivatives of δ(x − y) if the coefficients on the derivatives satisfy a certain condition. The only way found to satisfy these conditions for an interaction Hamiltonian constructed from fields for identical spin ½ particle (in the interaction picture) is to have the commutator proportional to δ(x − y), which implies local coupling with no derivations. The possibility of having relativistic theories in the instant form without causality is also investigated for the case of a four-fermion interaction Hamiltonian constructed from creation and annihilation operators for a spin ½ particle, but no definite conclusion is arrived at.
Existence and Uniqueness in the Large for Boundary Value Problems in Kinetic Theory
Authors: Cercignani Carlo
Published: January 2005

The boundary-value problem for the linearized Boltzmann equation is shown to have a unique solution for a bounded domain (two walls separated by an arbitrary distance). The proof applies to a general class of models with finite collision frequency and appears to be easily extendible to similar problems in two and three dimensions. It differs essentially from previously known proofs because no limitations are put on the distance between the walls.
States of Classical Statistical Mechanics
Authors: Ruelle D.
Published: January 2005

A state of an infinite system in classical statistical mechanics is usually described by its correlation functions. We discuss here other descriptions in particular: as (1) a state on a B* algebra; (2) a collection of density distributions; (3) a field theory; (4) a measure on a ``space of configurations of infinitely many particles.'' We consider the relations between these various descriptions and prove, under very general conditions, an integral representation of a state as superposition of ``extremal invariant states'' corresponding to pure thermodynamical phases.
Near-Steady Transition Waves of a Collisionless Plasma
Authors: Meyer Richard E.
Published: January 2005

Plane waves of small amplitude ε in a cold plasma which propagate into an equilibrium state and the head of which approaches steadiness are studied on the basis of a one-fluid model with transverse magnetic field. Their asymptotic behavior is shown to depend on whether the limits are taken in the order ε → 0, t → ∞ or t → ∞, ε → 0. The latter, nonclassical limit is the physically relevant one, and an approach is developed which yields uniqueness results for it. It is shown that the wave must ultimately become nonlinear, and a steady solution can be approached only in a conditional sense. If the wave lowers the magnetic pressure, it must do so monotonely. If not, then it must begin with a near-periodic wave train approaching steady solutions locally, but different ones in different places.
Uniqueness of the Haag-Ruelle Scattering States
Authors: Streater R. F.
Published: January 2005

It is remarked that no complete proof exists in the literature that the asymptotic states of quantum field theory are independent of the spacelike surfaces chosen to define them. In this paper we present a proof which is valid for any plane surface; this follows the Haag-Ruelle method, supplemented by lemmas showing that the usual bounds on truncated Wightman functions and smooth solutions of the Klein-Gordon equation are uniform in certain spacelike regions. The same lemmas immediately show that asymptotic states may be defined using any spacelike surface possessing a normal, which lies inside a closed timelike cone at all points. The proof of the convergence of these states has hitherto been incomplete. It is then important to show that these states are independent of the surface; we sketch a proof of this.
Reduction of Some Representations of the Generalized Bondi-Metzner Group
Authors: Cantoni Vittorio
Published: January 2005

The infinitesimal operators of a class of representations of the GBM group described in a previous paper are given in explicit form. All unitary representations of this type with positive ``rest mass'' are shown to be completely reducible; each irreducible component is characterized by the eigenvalue m2 of the rest-mass operator, by the lowest eigenvalue s(s + 1) of the ``spin'' operator and by the sign of an eigenvalue of a suitably defined operator Rs=.
Linear Closure Approximation Method for Classical Statistical Mechanics
Authors: Richardson John M. , Levitt Leo C.
Published: January 2005

A description is given of a general closure principle involving the minimization of the mean square error. The procedure based upon this principle can be applied to the truncation of the BBGKY hierarchy at various stages and to the approximation of unwanted terms arising in the equation of motion method by linear combinations of the observables to be retained. On a general level the significance of the closure principle is described in terms of the geometry of function space, and several useful general properties of the principle are derived. A discussion is devoted to the relation between the closure error (i.e., the least mean square error) and the error in the end result (e.g., the free energy, the radial distribution function, etc.); however, the results, while providing some insight, are not sufficiently refined to provide upper bounds to errors in all problems of statistical mechanics where the method is applicable. On the level of specific application it is shown that the principle yields results identical to the random phase approximation and to the linearized version of the Kirkwood superposition approximation in two special cases. Later sections of the paper describe in greater than usual generality, the formalism connecting thermodynamic properties and other equilibrium properties with the microscopic equations of motion in which closure approximations have been introduced. Two illustrative examples of the application of the over-all method were made to the case of a classical system of electrons in a uniform background of compensating charge, one leading to the well-known results of Debye and the other to a more accurate and elaborate theory developed in quantitative detail elsewhere.
Generalized WKB Method with Applications to Problems of Propagation in Nonhomogeneous Media
Authors: Bahar E.
Published: January 2005

A generalized WKB method is derived for the solution of the general second-order differential equation. The problem is reduced to the solution of two coupled first-order differential equations. By an appropriate choice of auxiliary functions, the coupling coefficients may be made sufficiently small to facilitate the solution of the coupled equations. It is shown that these solutions can be used in a range of problems in which the regular WKB solutions fail. These generalized solutions may also be used to derive asymptotic expansions of known functions. Applications of the method to higher-order differential equations are indicated, and solutions to the nonlinear Riccati equation are considered.
Recursion Relations for Coulomb Matrix Elements
Authors: Epstein Saul T. , Epstein Jean H. , Kennedy Barbara
Published: January 2005

It is pointed out that, by using the identity (ψ, (HW − WH′)ψ′) = (E − E′)(ψ, Wψ′), it is possible to derive useful relations among physically interesting matrix elements.
Analytic Functionals in Quantum Field Theory
Authors: Gorgé V. , Rohrlich F.
Published: January 2005

A class of Lorentz invariant generalized functions can be defined as analytic functionals, i.e., as continuous linear functionals which are contour integrals over a suitable space of test functions. These generalized functions include in particular the invariant functions of quantum field theory, but also include ``propagators with higher order poles.'' The analysis shows exactly which of these are well defined, especially in the important special case of zero mass. Various applications to quantum field theory are indicated.
Relation between Creeping Waves and Lateral Waves on a Curved Interface
Authors: Rulf Benjamin
Published: January 2005

Diffraction effects at a gently curved interface between two media are investigated. Particular attention is paid to the behavior of the field on the diffracted rays which propagate along the interface into the shadows. It is found that far from the launching point of such a ray the field comprises of a series of modes which decay exponentially, due to the continuous leakage of energy away from the interface. At moderate distances, in the penumbra region, this series is poorly convergent. It can be converted into an integral, which can be evaluated asymptotically there, yielding a field with an algebraic decay. The field is like that diffracted along a plane interface, the so-called lateral wave, and reduces to it when the radius of curvature becomes infinite. The regions of transition from one representation to the other are determined, and uniform asymptotic expressions, valid across those regions, are given. All our results apply to a two-dimensional scalar problem, but results for three dimensions and for vector problems can be derived in a similar way.