bluuebunny/crossref_metadata_2025 huggingface dataset:
Classical Green's Functions in Response Theory
Authors: Herzel John C.
Published: February 2004

The present paper is concerned with the retarded many-time thermodynamic Green's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many-time Poisson-bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many-time thermodynamic Green's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many-time Green's function equations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov-Sadovnikov (B-S) hierarchy from the lowest-order many-time Green's function equation of motion.
TwoGroup Neutron Transport Theory in Spherical Geometry
Authors: Schnatz T. W. , Siewert C. E.
Published: February 2004

A set of normal modes for the two-group steady-state neutron transport equation in spherical geometry is constructed. The singular eigenfunction-expansion technique is then used to develop a rigorous solution to the isotropically emitting spherical shell-source problem in an infinite medium.
On the Solution of a Generalized Wiener-Hopf Equation
Authors: Mittra R. , Lee S. W.
Published: February 2004

This paper deals with the generalized Wiener-Hopf equation G(α)X+(α)+H(α)X+(−α)−Y−(α)+Ψ(i)(α)=0, τ1<τ<τ2,where α (= σ + iτ) is a complete variable, G(α), H(α), and Ψ(i)(α) are known functions, and X+(α) and Y−(α) are unknowns, analytic in upper and lower half-planes, respectively, as indicated by their respective subscripts. This type of equation arises in a class of boundary-value problems in electromagnetic theory, the geometries of which may be described as modified Wiener-Hopf type. The method of approach, which is fundamentally different than those currently available in the literature, is based on a pairing of singularities in the complex α plane. This leads to a functional equation which is exactly solvable in its asymptotic form. The knowledge of this solution permits one to employ one of several rapidly converging numerical procedures available in the literature for a more accurate solution. Two examples illustrating the application of the procedure are included in the paper.
Curvature Collineations in Empty Space-Times
Authors: Collinson C. D.
Published: February 2004

It is shown that the only curvature collineations admitted by an empty space-time, not of Petrov type N, are conformal motions. The curvature collineations admitted by the plane-fronted gravitational waves are found.
General Relativistic Thin-Sandwich Theorem
Authors: Komar Arthur
Published: February 2004

The configuration variables for the gravitational field gmn are assigned arbitrarily on two infinitesimally neighboring spacelike hypersurfaces. We then investigate the extent to which a solution of the vacuum Einstein field equations can be found consistent with the given assignment. A local approach, employing Dirac's Hamiltonian formalism, reveals that solutions can be found locally which are nonunique and highly unstable.
Calculation of Lattice Green's Functions
Authors: Oitmaa J.
Published: February 2004

A proof is given that, in the numerical evaluation of lattice Green's functions, it is possible to restrict the wave-vector summation to an irreducible section of the Brillouin zone. It is shown that, in order to use this simplification, the Green's functions must be appropriately symmetrized.
Relativistically Invariant Classical Hamiltonian Mechanics with Extra Variables
Authors: Pearle Philip Mark
Published: February 2004

It is shown that one can construct a relativistically invariant classical mechanics in Hamiltonian form by utilizing as independent dynamical variables the position of each particle, together with its canonical conjugate variable, and the velocity of each particle, together with its canonical conjugate variable. The ten generators of the Poincaré group, obeying the correct Poisson bracket relations between themselves and the position variables, are constructed in their most general form. By restricting this form, it is possible to construct a Hamiltonian theory where the equations of motion for the particle's positions and velocities depend only upon the positions and velocities (and not upon their conjugate variables): these turn out to be the most general relativistically invariant classical equations of motion of this type. This is not useful as a starting point for constructing a physically interesting quantum theory, since the probability density in the quantized theory obeys the classical Liouville equation. This approach is also applied to the Galilean-invariant classical mechanics.
Exact Finite Method of Lattice Statistics. III. Dimers on the Square Lattice
Authors: Runnels L. K.
Published: February 2004

The 2-dimensional square-lattice model of hard dimeric molecules has been studied over the entire range of dimer densities. The effects of an external field have been included by allowing different activities for the two possible molecular orientations. Thermodynamic properties have been obtained by the transfer matrix technique for lattices of increasing finite circumference and infinite length. The convergence with increasing size has been the most rapid of all systems yet studied by this technique, satisfactory convergence usually being realized with a circumference of only ten sites. The results are in agreement with previous zero-field studies employing other methods. There is no phase transition in the presence or absence of an external field, and the equation of state is remarkably insensitive to the field. The limited information obtained regarding the nature of the approach to the close-packed limit is consistent with recent predictions based on the series method.
Toward a General Theory of Measurement. I
Authors: Horst David R.
Published: February 2004

A self-consistent measurement theory is set forth in this paper, which can be applied to the problem of measurement in field theory. Following the formulation of Birkhoff and von Neumann, we construct a system of axioms sufficient to ensure that it is possible to find an implicit mechanism within each theory for the experimental procedure corresponding to each observable proposition of that theory. It is shown that any theory of a type similar to quantum mechanics cannot consistently describe a closed system from the point of view of a single observer.
Search for a Universal Symmetry Group in Two Dimensions
Authors: Cisneros Arturo , McIntosh Harold V.
Published: February 2004

Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well-known and widely accepted solutions of Schrödinger's equation. We examine the reasons for this discrepancy on the quantum-mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non-Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two-dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.
On Heat Conduction and Wave Propagation in Rigid Solids
Authors: Bogy D. B. , Naghdi P. M.
Published: February 2004

This paper is concerned with conduction of heat and the related problem of propagation of thermal waves in stationary rigid solids. Special attention is given to rate-dependent response and the ensuing conditions of propagation in the conducting medium. The case in which the response of the medium is characterized by functions that are nonlinear in temperature and temperature gradient but are linear in the rate of temperature is studied in detail. The conclusions reached are also discussed in the context of the infinitesimal theory.
Nonlinear Electrodynamics: Lagrangians and Equations of Motion
Authors: Boillat Guy
Published: February 2004

After a brief discussion of well-known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born-Infeld theory, and finally, electromagnetic waves are compared with those of Einstein-Schrödinger theory.
Korteweg-deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws
Authors: Kruskal Martin D. , Miura Robert M. , Gardner Clifford S. , Zabusky Norman J.
Published: February 2004

The conservation laws derived in an earlier paper for the Korteweg-deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.
Nonphysical Region Singularities of the S Matrix
Authors: Goddard Peter
Published: February 2004

The work of a previous paper, which examined the relationship between the assumption of Cutkosky-type discontinuity formulas to the positive-α condition for the singularity of anomalous thresholds, is extended using more powerful methods. The method is first described in general terms. It is then justified using homological techniques, and specific expressions are calculated for distortion terms. Finally, some specific cases are considered and more elaboration versions of the cancellation processes previously found are shown to occur.
Trajectory of Charged Particles in Weakly Inhomogeneous Magnetic Fields
Authors: Cook David M.
Published: February 2004

The detailed time dependence of the trajectory, and particularly of the acceleration, of a charged particle moving nonrelativistically in a weakly inhomogeneous, slowly varying electromagnetic field is calculated by a method that involves superimposing a conventional perturbation expansion in the weak field on the expansion of Berkowitz and Gardner and of Kruskal, the latter expansion being asymptotically valid in the limit of small Larmor radius and slowly varying fields. Fourier transforms are introduced to facilitate isolation of rapidly varying terms from the most general solutions for the coefficients in the expansion. The calculations are carried to first order in the weak inhomogeneity and (at least formally) to all orders in Larmor radius. The resulting velocity reproduces the familiar transverse drift velocities of first-order orbit theory, emphasizes that in the presence of fields varying slowly in time the polarization and curvature drifts are to be calculated from the time derivative of the fields as seen from a frame of reference fixed with respect to the guiding center of the particle's orbit, and permits calculation of higher-order contributions to the drift velocities, contributions of second order in Larmor radius being explicitly evaluated; the resulting acceleration prepares the way for a study of the effect of weak inhomogeneities on the radiative output of single particles.
Three-Dimensional Linear Transport Theory
Authors: Case K. M. , Hazeltine R. D.
Published: February 2004

A recent technique for extending the singular eigenfunction method in linear transport theory to problems which are not strictly 1-dimensional is compared to a more naive approach based on the Fourier transform. The latter appears to have advantages with regard to simplicity and directness.
Singular Three-Body Amplitudes in the Theory of the Third Virial Coefficient
Authors: Dashen Roger , Ma Shang-keng
Published: February 2004

We have shown earlier how the virial coefficients can be expressed in terms of traces taken over products of multiparticle (on-energy-shell) scattering matrices. The traces are to be taken in the angular-momentum representation instead of the momentum representation to avoid the infinite forward N-particle scattering amplitude for N > 2 caused by certain singular diagrams. Here, we analyze and sum the contribution of the singular diagrams to the third virial coefficient. It is also shown that one can get the same result if off-shell amplitudes in the momentum representation are utilized.
Spinors in a Weyl Geometry
Authors: Adler Ronald J.
Published: February 2004

It was noted by London in 1927 that a mathematical connection exists between the unified field theory of Weyl and the old quantum theory of Bohr. We wish to demonstrate that a Dirac electron in an electromagnetic field described by the Weyl formalism appears to couple to the field via the usual minimal coupling recipe. It is generally acknowledged that the Weyl theory is unsatisfactory for describing macroscopic electromagnetic phenomena. The objections, however, may not be entirely applicable on a quantum-mechanical scale.
Invariant Representation of All Analytic Petrov Type III Solutions to the Einstein Equations
Authors: Brans Carl H.
Published: February 2004

The results of the application of previously developed techniques to the analysis of the Petrov type III solutions to the vacuum Einstein equations are presented. The procedure involves the computer aided analysis of the Einstein-Petrov equations to the extent that the functions uniquely and invariantly generating all local analytic solutions are determined. For the case of type III it is shown that, relative to a given fixed point in the manifold, all local analytic solutions are uniquely and invariantly determined by six arbitrary analytic functions of one variable and six others of two variables. These functions, called generating functions, thus provide a representation of all such solutions and may be used for the study of the structure of the family of Einstein empty space metrics.
Reduction of the Direct Product of Representations of the Poincaré Group
Authors: Moses H. E.
Published: February 2004

We expand the direct product of two representations of the Poincaré group into representations of the Poincaré group in the general case that the factors of the direct product may have any mass, whether real, zero, or imaginary, and the total energy may be indefinite. The representations of the Poincaré group, which appear in the expansion of the direct product, have masses which run through a continuous spectrum of real and imaginary values and are irreducible in terms of the mass and sign of energy (for real mass), but are reducible in terms of the infinitesimal generators of the little groups. To obtain the expansion in terms of irreducible representations, one need only reduce the infinitesimal generators of the little groups. This reduction is carried out for the real mass components and, in principal at least, can be carried out for the infinitesimal generators of the little groups for the imaginary mass components. The factors of the direct product and the representations which appear in the expansion are expressed in terms of a particular momentum representation called ``the standard helicity representation'' which enables us to use a uniform notation for all masses, whether real, zero, or imaginary. The earlier portion of the present paper summarizes the properties of these representations.
Closed-Form Solution of the Differential Equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P=0 by Normal-Ordering Exponential Operators
Authors: Wilcox Ralph
Published: February 2004

A closed-form solution to Lambropoulos' partial differential equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P(x, y, t)=0,subject to the initial condition P(x, y, 0) = Φ(x, y), is presented. The applicability of the normal-ordering method to a class of partial differential equations is briefly discussed.
Analysis of the Newman-Unti Integration Procedure for Asymptotically Flat Space-Times
Authors: Dixon W. G.
Published: February 2004

An analysis is given of the procedure of Newman and Unti for solving the vacuum gravitational field equations for all space-times in which Ψ0 = O(r−5). It was found empirically by Newman and Unti that when the nonradial equations and three of the u-derivative equations have been satisfied to their lowest nontrivial order in r−1, they are then found to be identically satisfied to all orders. A general proof of this result is given which avoids the need for direct verification.
Relativistic Fields Due to a Particle in a Grounded Cylindrical Box
Authors: Liboff Richard L.
Published: February 2004

The exact relativistic fields inside a perfectly conducting, closed, grounded cylindrical box of finite length, which are due to a charged particle moving down the axis in vacuum, are obtained. This permits construction of the solution for a line segment propagating down the axis and the return surface current. The fields are shown to be causal so that no field is present before the light front. The technique of solution employs superposition of Lorentz-transformed fields of the positive and negative images. The initial condition that there be no field in the box at time t = 0 is satisfied by adding a superposition of the cavity modes to the inhomogeneous solution. The fields and energy in the wake are also found.
Some Simple Observations on Griffiths' Theorems for the Classical Heisenberg Model
Authors: Yeh Robert H. T.
Published: February 2004

We show that Griffiths' theorems are valid for any classical ferromagnetic Heisenberg model in the weak interaction limit. They are also valid for certain chain-type and ring-type models, regardless of interaction strength.
The Adiabatic Invariant of the Linear or Nonlinear Oscillator
Authors: Symon K. R.
Published: February 2004

H. R. Lewis's treatment of the time-dependent harmonic oscillator is extended to the general time-dependent nonlinear oscillator. A prescription is given for obtaining an adiabatic invariant as a power series in the amplitude and to any order in the time variation of the coefficients. An exact invariant of the same algebraic form as the adiabatic invariant is also derived. The exact invariant can be so chosen that, at any particular time, it is equal to the adiabatic invariant. Comparison of the adiabatic and exact invariants permits a calculation of the nonadiabatic changes in the adiabatic invariant in the course of time. Applications to a linear and a nonlinear example are presented and the statistical distribution of the long time changes in the adiabatic invariants are determined in the two cases.
High Symmetry Fields and the Homogeneous Field in General Relativity
Authors: Aichelburg P. C.
Published: February 2004

The usual definition of a homogeneous field in general relativity implies a space with Riklm = 0, thus admitting a group of motions isomorphic to the Poincaré group. After discussing the symmetry group of the homogeneous field in Newtonian space, we point out that there exists no space with Rik = 0, which is a ``true'' field, i.e., Riklm ≠ 0, and which admits an analogous relativistic group. We then study fields, solutions of Rik = 0, which define spaces that admit a 4-parameter group of motions locally isomorphic to the groups T1 ⊗ [T2⊗sO(2)] and T1 ⊗ [T2⊗sO(1,1)]. We compare the motion of a test particle in these fields with the motion in the usual homogeneous field.
Stability of Time-Dependent Particlelike Solutions in Nonlinear Field Theories. I
Authors: Anderson D. L. T. , Derrick G. H.
Published: February 2004

The stability of time-dependent particlelike solutions of the form ψ=φ(r)e−iωt is examined for the nonlinear field ∇2ψ−c−2∂2ψ/∂t2=κ2ψ−μ2ψψ*ψ. It is found that such solutions are unstable for all ω.
Exact State and Fugacity Equations for the Ideal Quantum Gases
Authors: Nieto Michael Martin
Published: February 2004

The fully relativistic expressions for the density, pressure, and energy density of the ideal quantum gases are solved exactly for the fugacity. This allows the exact, fully relativistic equations of state to be obtained for the ideal Fermi and Bose gases. All these expressions are studied in detail, including a discussion of physical implications and various limiting cases.
Lorentz-Covariant Matrices for Elementary-Particle Theories as Polynomials in the Spin Matrices
Authors: Nelson T. J. , Good R. H.
Published: February 2004

The problem of expressing the (2s + 1) × (2s + 1) covariantly defined matrices Sμνρ… (used in theories of particles with spin) in terms of the angular-momentum matrices s is shown to be equivalent to the problem of finding a certain type of polynomial. Explicit expressions for the polynomials, recursion formulas, and differentiation properties are given.
SO
+(n, 1)-Invariant Finite-Component Wave Equations and Reflection Invariance
Authors: Mathews P. M.
Published: February 2004

First-order finite-dimensional equations of the generalized Bhabha type in a space with n spacelike and one timelike dimensions are considered. It is observed that invariance under the connected part SO+(n, 1) of the pseudo-orthogonal group in this space requires the use of a wavefunction transforming according to a representation of the larger group SO+(n + 1, 1). It is shown that though the wavefunction [and each part of it obtained by reduction with respect to SO+(n, 1)] transforms into itself under the strong reflection R also, nevertheless the wave equation in an odd-dimensional space (n + 1 = 2k + 1) may or may not be invariant under R, when the wavefunction employed transforms irreducibly under the connected group SO+(n + 1, 1). Thus, contrary to the suggestion contained in the recent literature, where specific equations are considered, the mere fact of oddness of the dimension of the space does not force the use of representations reducible with respect to the latter group (the matter is representation dependent). It is shown, however, that, irrespective of the representation used and of the dimension of the space, the equation is invariant under a transformation Θ, defined as a combination of an improper transformation which reverses the sign of time and another improper transformation of reflection in any hyperplane in the spacelike variables. It is suggested that TCP in an arbitrary space be identified with Θ rather than with R, which simply reverses the sign of all coordinates.
Asymptotic Evaluation of Multidimensional Integrals
Authors: Sirovich Lawrence
Published: February 2004

The asymptotic evaluation of a wide class of multidimensional integrals occurring in mathematical physics is considered. In this class are included integrals of the form 1(2π)N ∫ −∞∞ρ(k)exp[ik·x−σ(k)t]dk. A semiconstructive method is proven and certain classes of integrals are asymptotically evaluated. Examples involving problems in partial differential equations and a transport equation are given.
Generalized Hypergeometric Function of Unit Argument
Authors: Minton Barry M.
Published: February 2004

Two summation theorems are given for the terminating generalized hypergeometric function pFp−1, for arbitrary p, with certain restrictions on the parameters.
Spherical Symmetry and Mass-Energy in General Relativity. I. General Theory
Authors: Cahill Michael E. , McVittie G. C.
Published: February 2004

The mass-energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energy-momentum tensor and that mass-energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass-function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative-mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point-mass solution.
Spherical Symmetry and Mass-Energy in General Relativity. II. Particular Cases
Authors: Cahill Michael E. , McVittie G. C.
Published: February 2004

A solution of Einstein's field equations for the motion of a spherically symmetric distribution of perfect fluid is investigated in an isotropic comoving coordinate system. It is shown that this solution includes all solutions discussed by McVittie in which the density depends on both the radial and time coordinates. Necessary and sufficient conditions for the solution to be singularity-free and for the density and pressure to be nonnegative and monotonically decreasing from the center of the material outwards to an outer boundary are found. The material is surrounded by empty space. Examples of both oscillating and ``bouncing'' solutions are produced. It is shown that the outer boundary of the material never penetrates the Schwarzschild radius in all singularity-free solutions.
Off-Shell T-Matrix Elements in Potential Scattering
Authors: Underhill J.
Published: February 2004

We give derivations of integral equations satisfied by the off-shell extension of the scattering amplitude (in nonrelativistic potential scattering) defined by the Lippmann-Schwinger equation. The input information in these equations consists of phase shifts. The derivation is based on the work of R. G. Newton on the inverse scattering problem in potential scattering.
Limitable Dynamical Groups in Quantum Mechanics. II. A Model Including Arbitrary Spin
Authors: Doebner H. D. , Melsheimer O.
Published: February 2004

In the framework of limitable dynamical groups, a group Dst=Dt⊗SO(N)I,  Dt=GE0(N) [multiplication in left half circle]Sp(2N, R),is discussed. It is shown that the physical representation U(Dst) describes interacting nonrelativistic particles of arbitrary spin in N dimensions, with linear combinations of second-order polynomials in momentum and position operators Pi, Qi with i = 1, ⋯, N, and spin-flip operators as Hamiltonians. GE0(N) is the central extension of the inhomogeneous, pure Galilei group, and SO(N)I a rotation group in N dimensions. Sp(2N, R) is the noncompact form of the symplectic group. It is proven that there exists no irreducible unitary representation of Dt describing particles with spin s > 0. Hence it was necessary to enlarge Dt to Dst. The physical free-particle group D0⊂Dst possesses projection parts in Sp(2N, R) as well as in SO(N)I. To calculate them, we decompose the physical representation U(D0) by group-theoretical methods such that the angular momentum splits into a spin and an orbital part. D0 is isomorphic to the central extension of the inhomogeneous Galilei group. The physical representation U(Dst) is calculated by the postulate that U(Dst) and U(D0) can be limited into each other, and that mass and spin conservation hold. The limitation is realized as a group contraction leading to a nonfaithful representation of the contracted group. We prove that U(Dst) is uniquely determined by mass and spin of the free particle which is obtained if the interaction in Dst is turned off. Furthermore, the generators are identified as functions of Pi, Qi, and spin-flip operators. Hence the limitation postulate leads to the physical representation of the abstract group Dst and to the identification of the described physical system.
Branching Rules for U(n)⊃SO(n)
Authors: Brunet M. , Resnikoff M.
Published: February 2004

Using Littlewood's theorem, we derive branching rules for the restriction of U(n) to the special orthogonal group SO(n), for the case of a 3-row Young diagram. The special cases n = 4, 5 are treated by use of Murnaghan's modification rules.
Expansion of the Inhomogeneous Symplectic Lie Algebras T(2n)+ lim ←Sp(n) to Sp(n+2)
Authors: Nagel John G. , Shah K. Tahir
Published: February 2004

We show that the semidirect sum G(n+N)≅T(Nn)+ lim ←Sp(n), with T(Nn) the ideal, can be expanded to Sp(n + N) iff N = 2. We derive a general formula for invariants of T(2n)+ lim ←Sp(n) and, besides, two general formulas for invariants of ISO(n1, n2).

As a conclusion we find that an expansion E(G) of an arbitrary Lie algebra G, in general, is not isomorphic to a deformation D(G) of G, but that there exists a Lie algebra G′ ⊇ G and a deformation D(G′) of G′ such that E(G) ≅ D(G′).
Representations of the Lorentz Group: New Integral Relations between Legendre Functions
Authors: Pajas P. , Winternitz P.
Published: February 2004

New integral relations between Legendre functions of the first and second kind are derived. These functions figure as the basis functions of the irreducible representations of the homogeneous Lorentz group, so that the derived formulas have direct applications to invariant expansions of relativistic amplitudes.
Kronecker Products and Symmetrized Squares of Irreducible Representations of Space Groups
Authors: Bradley C. J. , Davies B. L.
Published: February 2004

Application is made to space groups of a theorem by Mackey on the symmetrized and antisymmetrized squares of induced representations. A new and constructive proof is given of Mackey's theorem which enables the representations appearing in the decomposition of the symmetrized squares to be easily identified. The method is used to extend Birman's tables for Td2, the zinc-blende structure, to include products involving its double-valued representations
Groups of Curvature Collineations in Riemannian Space-Times Which Admit Fields of Parallel Vectors
Authors: Katzin Gerald H. , Levine Jack , Davis William R.
Published: February 2004

By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CC's. It follows that every space-time with an expansion-free, shear-free, rotation-free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.
Irreducible Cartesian Tensors. III. Clebsch-Gordan Reduction
Authors: Coope J. A. R.
Published: February 2004

The reduction of products of irreducible Cartesian tensors is formulated generally by means of 3-j tensors. These are special cases of the invariant mappings discussed in Part II [J. A. R. Coope and R. F. Snider, J. Math. Phys. 11, 993 (1970)]. The 3-j formalism is first developed for a general group. Then, the 3-j tensors and spinors for the rotation group are discussed in detail, general formulas in terms of elementary invariant tensors being given. The 6-j and higher n-j symbols coincide with the familiar ones. Interrelations between Cartesian and spherical tensor methods are emphasized throughout.
Geometrical Derivation of the Conservation Laws
Authors: Alderman Jerry L. , Bergmann Otto
Published: February 2004

Conservation laws which are customarily obtained by invoking the invariance of a Lagrangian density under the transformations of some intrinsic symmetry group may be given a completely geometrical treatment within the context of the theory by Yang and Mills. The formalism is extended from unitary transformations to general linear transformations and the concepts of parallel transfer, covariant differentiation, and intrinsic curvature tensor are discussed. Conservation laws follow from the assumed invariance of the Lagrangian under parallel transfer defined with a general affine connection which is a direct sum of intrinsic and space-time affinities. Conservation of generalized charge is a consequence of the arbitrariness of that part of the affinities which operates in the intrinsic space, and conservation of energy and momentum is related to the arbitrariness of the part of the affinities which operates on the space-time indices. Some comments on Palatini's derivation of Einstein's equation of general relativity are made.
Explosive Instabilities in the Well-Defined Phase Description
Authors: Wilhelmsson H. , Stenflo L. , Engelmann F.
Published: February 2004

The paper presents a general theoretical investigation of 3-wave interaction in the well-defined phase description with special regard to nonlinear explosive instabilities in the presence of linear damping or growth.
Application of the Method of Matched Asymptotic Expansions to a Problem in Linear Transport Theory
Authors: Hendry W. L.
Published: February 2004

The method of matched asymptotic expansions is used to reduce the time-dependent, one-velocity neutron transport equation to a set of more tractable equations. This reduction is accomplished subject to general initial conditions on both the neutron flux and delayed neutron precursors.
Quantum Corrections to the Second Virial Coefficient for a Square-Well Potential
Authors: Theumann Alba
Published: February 2004

The first two quantum corrections to the classical second virial coefficient are evaluated in the case that the intermolecular potential is V(r)=∞,0<r≤a,=−U<0,ab.

The first terms of the expansion of B(T) in powers of λ = (h2/2πmkT)½ are B(T)=23πNa3eUβ[1+(√98)λ/a+(π)−1(λ/a)2]+23πNb3(1−eUβ+(√98)(λ/b)[1+eUβ−2e12UβI0(12Uβ)]+(π)−1(λ/b)2{12(eUβ−1)−38;(πUβ)12e12Uβ[3I12(12Uβ)+I52(12Uβ)]+[1+12(Uβ)−1]3eUβ(Uβ)−12 Erf [(Uβ)12]−[1−12(Uβ)−1]3(Uβ)−12 Erfi [(Uβ)12]−32(Uβ)−1×(1+eUβ)})+O(λ3).The contribution of the bound states to the first quantum correction is evaluated and the behavior of B1(b)/B1 vs T is plotted.
Spin Representations of the Orthogonal Groups
Authors: Gilmore Robert
Published: February 2004

The spin representations of the algebras Bn and Dn are constructed from the spin representations of the canonical subalgebras Bn−1 and Dn−1, respectively.
Partial-Tensor-Product Representations of the Canonical Commutation Relations: Classification
Authors: Reents G.
Published: February 2004

Partial-tensor-product representations of the canonical commutation relations are a generalization of direct- (or tensor-) product representations as discussed by Klauder, McKenna, and Woods. A classification is given of those representations up to unitary equivalence. The equivalence or inequivalence, respectively, of two representations is expressed in terms of the corresponding reference vectors.
Relativistic Theory of Microelectromagnetism
Authors: Eringen A. Cemal , Kafadar Charles B.
Published: February 2004

A theory of relativistic (special theory) microelectromagnetism is proposed for the treatment of the class of physical phenomena arising from a high degree of polarization, magnetization, and rapid local field fluctuation in material bodies. The master electromagnetic balance laws are given for the resultant generalized EM fields. The local field equations and jump conditions are obtained for the moments of the electromagnetic fields up to any order p. The equations of the zeroth-order moments turn out to be Maxwell's equations and associated jump conditions. The Euclidean form of the first-order theory is presented in full.
Nonuniqueness of the Energy Correction in Application of the WKB Approximation to Radial Problems
Authors: Engelke Ray , Beckel Charles L.
Published: February 2004

An infinite set of transformations is given that convert a radial energy-eigenvalue problem into a 1-dimensional problem. Langer's transformations are included in this set. The transformations introduce no spurious, real-axis singularities into the effective potential, and yield wavefunctions that meet 1-dimensional boundary conditions and are asymptotically satisfactory for virtually all power law potentials. The first-order radial WKB quantization condition is transformation dependent through two parameters. It may be distinct from the Langer-Kemble condition obtained by substituting (l + ½)2 for l(l + 1) in the effective potential. Here, l is the rotational quantum number. For the vibrating rotator, a rough test of the usefulness of the (l + ½)2 substitution is described; flexibility advantages provided by the new transformations are pointed out.
De Sitter Symmetric Field Theory. II. Tensorial and Spinorial Realizations
Authors: Bakri M. M. , Ghaleb A. F. , Hessein M. I.
Published: February 2004

Plane-wave solutions with definite helicity are obtained for the field equations (Sμνpν + mγμ)ψ = pμψ. These equations are then rewritten in spinorial form. The relations between spinors, 4- and 5-dimensional tensors and spin-tensors, and the components of ψ are derived.
Peratization of Logarithmically Singular Potentials
Authors: Frank William M. , Land David J.
Published: February 2004

First and second peratizations are performed for a wide class of repulsive potentials with singular behavior near r = 0 of the form L(r)r−m (m > 3), where L(r) is ``logarithmically singular'' at the origin. More precisely, when L(r) is of the class of ``slowly varying functions,'' i.e., functions such that rL′(r)/L(r) → 0 as r → 0, first peratization gives a (generally infinite) constant multiple of the result for the r−m potential. Second peratization for a wide class of slowly varying function offers no substantive improvement.
One-Dimensional Phase Transitions in the Spherical Lattice Gas from Inverse Power Law Potentials
Authors: Carpenter Kenneth H. , Strecker Joseph L.
Published: February 2004

The spherical model of the lattice gas of Gersch and Berlin is studied by making an integral approximation to the Fourier coefficients of the interaction potential. A phase transition is found in one dimension for potentials −gr−α with 1 < α < 2. A necessary condition which a potential must satisfy to allow a 1-dimensional phase transition is given. General expressions are derived for specific heat Cv and isothermal compressibility KT. Discontinuities are found in the thermodynamic properties at the transition for α less than 32 times the number of dimensions. Asymptotic expansions are given for Cv and KT near the critical point.
Maxwell's Equations Having a Gradient as Source
Authors: Muraskin M.
Published: February 2004

We study Maxwell's equations having a source given by a gradient. The source function f obeys the empty-space wave equation as a consequence of current conservation. We find that this system can be written as a mass-zero Dirac equation. Our relationship between tensor and spinor equations is an improvement over other published results, so far as simplicity is concerned.
Self-Interaction of Gravitational Radiation
Authors: Couch W. E. , Torrence R. J.
Published: February 2004

A nonsingular, sourceless, first-order pulse of gravitational radiation imploding from infinity to a focus and then exploding back out to infinity is examined to second order. It is found that, contrary to what might be expected, the nonsingular second-order field contains no radiation. In space-time regions outside of the pulse, the second-order field is nonvanishing only in the region with retarded times earlier than the passing of the exploding pulse and advanced times later than the passing of the imploding pulse. In this region, the second-order field is that of a Schwarzschild mass, plus a nonradiative quadrupole, plus a nonradiative sixteen pole.
Some Simple Lattice-Spin Systems
Authors: Lieh Jong-Won
Published: February 2004

A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved ``Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.
Criteria of Accuracy of Approximate Wavefunctions
Authors: Weinhold Frank
Published: February 2004

The accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ | ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.
Finite Integral Equations for Green's Functions for :φ4: Coupling. I
Authors: Johnson Rodney W.
Published: February 2004

A fully renormalized set of integral equations is derived for the Green's functions of a theory with :φ4: coupling.
Local Gauge Fields in General Relativity
Authors: Treat Richard P.
Published: February 2004

The carrier space for internal degrees of freedom is assumed to be the 6-dimensional space spanned by generators Liκλ of local Lorentz transformations defined at each event of a Riemann space. Gauge potentials are introduced in this internal space which are like the Yang-Mills potentials except that they are related to parameters of the event connection Γμκλ by a map which involves the Liκλ as connecting quantities. The map is similar to the one which relates the Dirac spin space and spin connection to the Riemannian geometry of event space. An algebraic uniqueness condition is shown which is necessary and sufficient for the map to be one-to-one in a neighborhood of a solution Γμκλ of the mapping equations: The metric field is undetermined by a uniform scale transformation. If the gauge potentials have an internal holonomy group which is a subgroup of the Lorentz group and satisfy the free Yang-Mills equations, and if the uniqueness condition is satisfied, then the Γμκλ committed by the inverse map automatically satisfy the free Einstein field equations, indicating that only the gravitational field is included in this mathematical framework. In an attempt to include nongravitational properties of matter, we have then considered internal holonomy groups larger than the Lorentz group. The part of the gauge potentials causing the enlargement is shown to form a current which may be interpreted as a source of the gravitational field.
Expansions in Spherical Harmonics. V. Solution of Inverse Legendre Transforms
Authors: Chambers Ll. G. , Sack R. A.
Published: February 2004

Explicit formulas are derived for inverse Legendre transforms, i.e., for the solutions of the integral equation f(r)= ∫ b1GL(ur)PL(u)dufor arbitrary integer L and general functions f(r). It is shown that the functions f(r), defined by 2-sided transforms (b = −1), follow uniquely from the 1-sided ones (b = 0), but an arbitrary function of parity opposite to L (supplementary function) may be added to GL in the 2-sided case. For L ≥ 2, an arbitrary polynomial of the same parity as L, but of degree lower than L (complementary function), may be added to GL. The complementary and supplementary functions do not affect the values of the integrals for the radial dependence in the expansion of a function of a vector sum in spherical harmonics. A power term in f(r) leads to a power term in GL, except for those powers which occur in the complementary function for which GL involves logarithms. Inverse transforms are also obtained for a restricted number of negative powers and some recurrence relations are derived.
Separable Schrödinger Equations for Two Interacting Particles in External Fields
Authors: Seiler R. , Pratt R. H.
Published: February 2004

We derive the most general form of the Schrödinger equation for two interacting particles in an external field which can be separated by a linear transformation of the coordinates. In particular, the transformation to center-of-mass and relative coordinates is shown to separate only those Hamiltonians with external potentials of the 3-dimensional harmonic oscillator type. The most general allowed form contains in addition specified cubic terms in the potentials.
Spatially Homogeneous Anisotropic Cosmological Models Containing Relativistic Fluid and Magnetic Field
Authors: Vajk J. Peter , Eltgroth Peter G.
Published: February 2004

We review spatially homogeneous, axially symmetric universes containing either an ideal fluid (with a γ-law equation of state) or a uniform magnetic field parallel to the symmetry axis or both. In many cases, the field equations may be solved by the technique (described in detail) of replacing the cosmic time by a suitably chosen timelike parameter. We systematically derive all known exact solutions for such universes.
Isotropy of Space
Authors: Gorini Vittorio , Zecca Antonio
Published: February 2004

It is well known that it is possible to deduce the Lorentz transformations solely on the basis of the principles of relativity, space-time homogeneity, and isotropy of space, without making any use of the principles of reciprocity and of the invariance of the velocity of light. Here we put forward a rigorous and simple formulation of the isotropy principle, which appears to be more general than those usually presented and which, together with the results of a preceding paper, allows us to carry out such a deduction in the physical case of a 4-dimensional space-time in a straightforward and considerably simplified way.