bluuebunny/crossref_metadata_2025 huggingface dataset:
Transfer Matrix Solution of Some One- and Two-Medium Transport Problems in Slab Geometry
Authors: Aronson Raphael
Published: February 2004

The transfer-matrix method is used to solve several one- and two-medium problems in plane geometry. The formal solutions are given in terms of operators which make no reference to the interactions or to the nature of the flux approximations. This formulation points up the structural similarities between solutions of various problems.
Approximation of Linear Transport Processes
Authors: Hejtmanek H. J.
Published: February 2004

The semigroup generated by the linear transport operator is approximated by the corresponding discrete linear transport process. It can be shown that, in the case of a 2-dimensional μ space (x, μ), the generating operators converge and the discrete linear-transport process is uniformly quasibounded. A stability condition can be formulated.
Model-Independent Analysis of Nonrelativistic Multiparticle Reactions
Authors: Warren Russell E. , Klink William H.
Published: February 2004

The Clebsch-Gordan coefficients for the reduction of an n-fold tensor product of nonrelativistic free-particle states into one over-all state are calculated. The degeneracy parameters resulting from this decomposition are then cast into Galilean-invariant functions of the particle momenta. Finally, the reduction is applied to the scattering matrix for nonrelativistic reactions involving an arbitrary number of final particles.
Matrix Elements of Relativistic Electrons in a Coulomb Field
Authors: Gargaro W. W. , Onley D. S.
Published: February 2004

Matrix elements for a radiative interaction between states of a Dirac electron in the presence of a Coulomb field are reduced to a closed analytic form in the limit of zero electron mass; corrections for finite electron mass are indicated. The application of these to inelastic electron scattering and radiation problems is discussed.
Nonlinear Stochastic Differential Equations Containing Random Parameters with Small and Large Correlation Time
Authors: Sancho N. G. F.
Published: February 2004

This paper is an extension of a previous one [J. Math. Phys. 11, 762 (1970)], and describes a technique for finding the moment equations of a nonlinear stochastic system containing random parameters with small and large correlation time. The nonlinear term can then be linearized by various methods. Finally, the moment inequalities are preserved using the technique described by Bellman and Richardson.
Coulomb Scattering. I. Single Channel
Authors: Mulherin Denis , Zinnes I. I.
Published: February 2004

With the aid of a Coulomb ``free propagator'' Uc(±)(t) constructed from the asymptotic form of the conventional time-independent solution to the scattering problem in a pure Coulomb field, Mo/ller wave operators are shown to exist for general Coulomb-like (Coulomb + short-range) potentials. It is emphasized that when scattering occurs in such long-range fields, one should distinguish three classes of states: the prepared and detected states, the scattering states, and certain asymptotic states which are peculiar to long-range potentials. The existence of the third class as a class distinct from the first suggests a connection between Uc(±)(t) and the quantum theory of measurements.
The Representations of U(4)⊃U(2)⊗U(2)
Authors: Brunet M. , Resnikoff M.
Published: February 2004

The state vectors associated with an irreducible representation of U(4) restricted to the subgroup U(2)⊗U(2) are explicitly constructed for the general 4-rowed Young diagram, employing Littlewood's rules and a theorem of Cartan. Expressions are given for the degeneracy of a U(2)⊗U(2) state in U(4), and the set of U(2)⊗U(2) states obtained are shown to be independent and complete. Operators which break this degeneracy are constructed, and the behavior of the state vector under the conjugation operations is also discussed. The results are useful for the nuclear quasispin and Wigner supermultiplet models.
Derivation of a Lower Bound on the Free Energy
Authors: Yeh Robert H. T.
Published: February 2004

For a Hamiltonian with a ground-state energy E0 = 0, we show that for any normalized state |ψ〉, 〈ψ| e−βH |ψ〉≤e−β〈ψ|H|ψ〉a, where a=exp [−β 〈ψ| H 2 |ψ〉/〈ψ| H |ψ〉]. From this inequality, a general formula for a lower bound on the free energy is derived.
Localized States on a Hyperplane
Authors: McDonald S. C.
Published: February 2004

The localized states of Newton and Wigner are reconstructed as a superposition of the canonical states of Foldy, and these states are then generalized to an arbitrary hyperplane via a procedure similar to that used in a previous work for the hyperplane generalization of helicity states. The corresponding hyperplane position operator, with mutually commuting components, is constructed and is seen to be equivalent to that local position operator given by Fleming.
Branching of Solutions to Some Nonlinear Eigenvalue Problems
Authors: Dean Edward T. , Chambré Paul L.
Published: February 2004

The multiplicity of solutions to nonlinear eigenvalue problems is analyzed by a method originally proposed by Hammerstein, and the results are given in tables which display the relation between the number of solutions for eigenvalues in the neighborhood of a critical eigenvalue and the properties of the nonlinear function and its derivatives. Several general types of nonlinear functions are considered, and a simple method of estimating the critical eigenvalue for each type is presented. Since these functions describe physical phenomena, a stability analysis is given for the cases of multiple solutions in order to determine which solution represents the observed physical state. The results are applied to the following nonlinear eigenvalue problems: (i) nonlinear heat generation and the temperature distribution in conducting solids; (ii) temperature distribution in a heat-conducting gas undergoing chemical reactions, leading to a thermal explosion; (iii) nonlinear effects of temperature-dependent viscosity on the temperature distribution of a fluid flowing in a pipe; (iv) neutron flux distribution in a reactor for temperature-dependent cross sections.
Exact Solution of a Family of Integral Equations of Anisotropic Scattering
Authors: Kagiwada H. , Kalaba R.
Published: February 2004

It is shown that the solution of a certain Cauchy system provides the solution of a family of integral equations occurring in the theory of anisotropic scattering in a finite slab. Numerical experiments show that the Cauchy system is readily solved numerically, even in the case of very strong forward scattering.
Problem of the Disordered Chain
Authors: Lin Ta-Feng
Published: February 2004

The equivalence of the methods of Dyson and Kac is proven in the 1-dimensional problem of the disordered chain, i.e., the determination of the spectrum of the frequencies of normal modes of a system of coupled oscillators, where the masses and/or spring constants are random variables.
Inequalities for Appell Functions
Authors: Minton B. M.
Published: February 2004

The asymptotic expansion of one of Appell's generalizations of the Jacobi function is given for one parameter becoming large while the other is kept fixed. Inequalities are given which may be useful when both parameters become large.
Reduction of the Most Degenerate Representations of SO(p, q) with Respect to SO(p−1,q−1)[multiplication in right half circle]Tp+q−2
Authors: Castell L.
Published: February 2004

The reduction of three principal series of the most degenerate unitary irreducible representations of the identity component of the noncompact rotation groups SO0(p, q) with respect to the identity component of the inhomogeneous rotation group SO0(p−1,q−1)[multiplication in right half circle]Tp+q−2 is carried out, using the results of the reduction with respect to the maximal compact subgroup SO(p) ⊗ SO(q).
Exact Solution of a Family of Matrix Integral Equations for Multiply Scattered Partially Polarized Radiation. II
Authors: Buell June , Casti John , Kalaba Robert , Ueno Sueo
Published: February 2004

In the theory of multiply scattered partially polarized radiation, a key role is played by the integral equation J(t,x,z)=Ie−(x−t)/̄z+ ∫ 0xK(|t−y|)J(y,x,z)dy, 0≤t≤x,where J and K are square matrices, I is the unit matrix, z is a parameter lying in the interval (0, 1), and for the interval length x, 0 ≤ x ≤ x1. It is shown that this family of matrix integral equations can be transformed into a Cauchy problem that is readily solved by modern computing machines.
Inverse Scattering Problem for Dirac Particles. Explicit Expressions for the Values of the Potentials and Their Derivatives at the Origin in Terms of the Scattering and Bound-State Data
Authors: Corbella O. D.
Published: February 2004

The inverse scattering problem for the Dirac equation with two central potentials (one a relativistic scalar, the other the fourth component of a 4-vector) is investigated. Exact expressions are obtained for the values of the two potentials and their derivatives at the origin, as a function of the scattering data and bound-state parameters of the particle and the antiparticle for one angular momentum. The consideration is limited to the case of S and P waves (j = ½) and to potentials satisfying a restrictive condition; explicit expressions for the values of the two potentials and their first three derivatives at the origin are given. In the case with only one potential present (the usual Dirac equation), these relationships provide explicit connections between the scattering and bound-state parameters for the particle and those for the antiparticle. For instance, an explicit and simple expression for the binding energy and for the corresponding normalization constant, in terms of the scattering phase shifts of the particle and of the anti-particle, is given in the case with only one bound state, say, for the particle, present (in S or P wave, with j = ½). It is assumed that the two potentials are central, holomorphic, and satisfy the conditions ∫0∞rn|Vi(r)|dr<∞,i=1,2,n=0,1,2.
Analyticity of the Scattering Amplitude in Field Theories with Exponentially Decreasing Commutators in the Spacelike Region
Authors: Dobson Peter N.
Published: February 2004

The implications of weakening the axiom of local commutativity to a condition of exponential decrease are studied. A strong operator limit on the behavior of the commutator for large spacelike separations leads to fields which are almost local in the sense of Haag. A further condition on the spatial behavior of matrix elements of the commutator is introduced, which leads to analyticity of the elastic scattering amplitude at fixed momentum transfer in a striplike region in the complex-energy plane.
Spectrum of Casimir Invariants for the Simple Classical Lie Groups
Authors: Gilmore Robert
Published: February 2004

The spectra of the Casimir invariants for the classical compact simple Lie groups are presented. It is proved that these invariants are irreducible and functionally independent. The highest weight of a representation is determined in terms of its invariant spectrum.
Degeneracy of the SU(3) Direct Product and the Symmetric Representations of SU(6)⊃SU(3)⊗SU(2)
Authors: Resnikoff M.
Published: February 2004

The relation between a state of U(n) associated with an m-rowed Young diagram, m ≤ n, and a state of U(m) associated with an m-rowed Young diagram provides the basis for the symmetric state of U(mn)⊃U(m)⊗U(n). As an application, the state vectors associated with the irreducible representations of SU(6) restricted to the subgroup SU(3)⊗SU(2) are explicitly constructed for the symmetric representation, and generalized to the case of a 2-rowed Young diagram. Expressions are given for the degeneracy of an SU(3)⊗SU(2) state in SU(6), and the completeness of the obtained set of states is discussed. The direct product of symmetric SU(6)⊃SU(3)⊗SU(2) states implies a direct product of SU(3) states; the operator which breaks the degeneracy of the ``2-rowed'' SU(6)⊃SU(3)⊗SU(2) state is shown to be Moshinsky's operator X which breaks the degeneracy of the SU(3) direct product.
Eikonal Approximation for Nonlinear Equations
Authors: Emery V. J.
Published: February 2004

An eikonal approximation for nonlinear equations is derived from an expansion in powers of space and time derivatives. For the special case of one dependent variable, the method is equivalent to an averaging method proposed by Whitham and derived by Luke. A general solution for each order of the expansion is obtained and discontinuous solutions are discussed.
Infinite-Dimensional Representations of the Lorentz Group
Authors: Carmeli Moshe
Published: February 2004

The method used by Naimark to obtain symmetrical spinors and their transformation law from finite-dimensional representations of the group SL(2, C) is extended to infinite-dimensional representations. As an infinite-dimensional representation, we use the principal series of representations realized by means of the special unitary group SU2. As a result another form of the principal series of representations of SL(2, C) is obtained which describes the transformation law of an infinite set of numbers under the group transformation in a way which is very similar, but as a generalization, to the way spinors appear in the finite-dimensional case.
Green's Function for the Nonlocal Wave Equation
Authors: Baraff G. A.
Published: February 2004

The Green's function for the nonlocal wave equation in a semi-infinite medium is calculated using the Wiener-Hopf technique and shown to be given by an integral whose integrand contains only the source-free solution of that same equation.
Scattering of Electromagnetic Waves in Uniformly Moving Media
Authors: Censor Dan
Published: February 2004

Various representations are considered for 2- and 3-dimensional electromagnetic wavefunctions in uniformly moving media. Rather than solving the relevant wave equation for the fields, the present formalism exploits the transformation formulas for plane waves in moving media, and as a starting point a spectral (plane-wave) representation is constructed. The procedure applies to arbitrary orders of β, but for simplicity only first-order velocity effects are considered in detail. Then, similarly to the velocity-independent problem, waves in uniformly moving media are represented in terms of complex integrals, special function series, inverse-distance differential-operator series, and surface integrals. Since familiar forms and functions are used, the present representations are extensions of the corresponding velocity-independent expressions. The latter are available at any given stage by letting the velocity vanish or by replacing the moving medium by free space. Scattering by circular cylinders and spheres is considered, and results are specialized to the case of thin cylinders; the new velocity effects introduce additional multipole terms. The original wavefunctions are transformed into the frame of reference of the medium at rest. Scattering by arbitrary objects moving in free space follows as a special case.
Fluctuation Theory for Space-Charge-Limited Currents
Authors: Rosen Gerald , Eckstut Gene
Published: February 2004

A phenomenological fluctuation theory is presented for 1-dimensional space-charge-limited currents in ideal nonmetallic solids. Notwithstanding the essential nonlinearity of the dynamical equation which governs the potential field, the theory is solved exactly for an initial statistical state that is associated with a probability measure of Gaussian form in an auxiliary field variable.
Differential Forms and Maxwell's Field: An Application of Harmonic Integrals
Authors: Ohkuro Shigeru
Published: February 2004

Maxwell's equations are derived under the assumption of 4-dimensionality of Euclidean space from a somewhat different definition of time from that considered by Einstein, using differential forms and de Rham's theorem in the theory of harmonic integrals. It is shown that the continuity equation of the current density is an elementary consequence of the Jordan-Brouwer theorem of topology under the requirement of integrability of the field. Matter appears as a singular point of the field, and introduces various kinds of ``currents,'' in the sense of de Rham and Kodaira, according to the topological character of the domain of integration of the field. These ``currents'' describe characters of the material, and are represented not by ordinary functions, but by generalized functions in the Schwartz sense. Examples of these ``currents,'' such as electric and magnetic polarizations and the supercurrent, are given, and the origin of the fundamental difficulties with dimensions in the usual theory of electromagnetism is attributed to this fact.
Solution of a Three-Body Scattering Problem in One Dimension
Authors: Marchioro C.
Published: February 2004

The scattering problem of three equal particles interacting via repulsive inverse-cube forces is solved in one dimension both in the classical and the quantal cases. The quantal S matrix is similar to that produced by repulsive δ-function interaction of infinite strength.
Commuting-Operator Approach to Group Representation Theory
Authors: Killingbeck J.
Published: February 2004

A study is given of the class-sum-operator approach to the representation theory of finite groups, the group D3h being specifically studied. The class sum approach is shown to simplify the decomposition of Kronecker products. By using the class-sum-operator approach, it is shown that the ``indirect'' group-projection operators of Löwdin can be used in finite group theory, where they lead to useful factorizations of the finite group-projection operators. It is also shown that tensor operators of certain symmetry types can be constructed within the group itself, and may be used analogously to the usual operator equivalents of crystal field theory.
Theory of Particles with Variable Mass. I. Formalism
Authors: Greenberger Daniel M.
Published: February 2004

The equivalence principle (through the mechanism of the gravitational red shift) allows one to set up a classical formalism with the proper time as an extra degree of freedom, independent of the coordinate time, and with an immediate physical interpretation. Then proper time and mass occur as conjugate variables in a canonical formalism, leading to a gravitational theory of particles with variable mass. The nonrelativistic theory and a relativistic vector theory of gravity are described as models. The theory is capable of providing a dynamical framework for cosmological models with the creation of matter. Some simple examples are discussed, including the steady-state universe with continuous creation, where the correct relation between the density of matter and the Hubble constant appears automatically, with no free parameters.
Moments and Correlation Functions of Solutions of a Stochastic Differential Equation
Authors: McKenna J. , Morrison J. A.
Published: February 2004

This paper shows how to obtain exact, closed-form expressions for various moments and correlation functions of the solutions of the stochastic, ordinary differential equation d2udz2+β02[1+ηT(z)]u=0,where T(z) is the so-called ``random telegraph'' wave and β02 and η are positive real constants. These moments and correlation functions are calculated by two different methods, one a phase space method and the other a matrix method familiar from optics. It is found that the moments are sums of exponentials. The first-order moments decay exponentially but the second-order moments grow exponentially. The correlation functions are also sums of exponentials and show that the solutions do not form a stationary process. An important application of these results is obtained in the problem of a plane electromagnetic wave normally incident on a randomly stratified dielectric plate. It is shown that, if 𝔖 is the amplitude transmission coefficient of the plate, then 〈1/SS*〉 can be expressed in terms of the second-order moments of the solutions and derivatives of solutions of the stochastic differential equation.
Quantum Statistical Mechanical Derivation of Generalized Hydrodynamic Equations
Authors: Robertson Baldwin
Published: February 2004

Differential conservation equations are derived for the mass-, momentum-, and energy-density operators for a 1-component simple fluid of Bose or Fermi particles with arbitrary pairwise interactions. These equations are used in a statistical mechanical derivation of exact equations of motion for the expectations of these operators. The equations of motion are coupled to equations relating these expectations to the local temperature, chemical potential, and fluid velocity. The coupled equations are closed in the sense that the expectations and their thermodynamic conjugates listed above are the only unknowns, although some of the dependence in the equations on the conjugates is expressed only implicitly. The equations of motion are memory-retaining nonlocal generalizations of the classical hydrodynamic equations and apply to a normal fluid arbitrarily far from equilibrium. The formalism is not carried as far as has the corresponding classical formalism because the local equilibrium expectation of the momentum density here does not equal the fluid velocity times the expectation of the mass density as is true in classical statistical mechanics.
Elastic Radiation in a Half-Space
Authors: Case K. M. , Hazeltine R. D.
Published: February 2004

A Green's function for the elastic wave equation, which satisfies certain boundary conditions on the surface of a homogeneous half-space, is derived by means of the Fourier transformation. This half-space Green's function is then applied to the computation of radiative effects due to the earth's surface when a radiating source is located on or within that surface. The results obtained are to be taken as an extension of a previous and similar formulation for the infinite medium due to Case and Colwell.
SL(2, C) Symmetry of the Gravitational Field Dynamical Variables
Authors: Carmeli Moshe
Published: February 2004

We represent the spin coefficients and the Riemann tensor in the form of linear combinations of the infinitesimal generators of the group SL(2, C). This representation is similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices. The spin coefficients take the role of the Yang-Mills-like potentials, whereas the Riemann tensor takes the role of the fields.
Nonlinear Light Propagation in a Resonant Medium and Causality
Authors: Da Costa R. C. T.
Published: February 2004

The propagation of an optical pulse through a system of two-level atoms embedded in a host medium of constant refractive index n is considered. It is shown that the values of the solutions of the nonlinear system of coupled Maxwell and Schrödinger equations at time t at a point z, in the McCall-Hahn approximation, depend only on data contained in the interval [z - c/nt, z] at t = 0. Thus, information cannot be transmitted through this medium with velocity greater than c/n. Apparent violations of this property in the case of self-induced transparency are explained.
Thermodynamic Limit of Time-Dependent Correlation Functions for One-Dimensional Systems
Authors: Gallavotti G. , Lanford O. E. , Lebowitz Joel L.
Published: February 2004

We investigate the time evolution of the correlation functions of a nonequilibrium system when the size of the system becomes very large. At the initial time t = 0, the system is represented by an equilibrium grand canonical ensemble with a Hamiltonian consisting of a kinetic energy part, a pairwise interaction potential energy between the particles, and an external potential. At time t = 0 the external field is turned off and the system is permitted to evolve under its internal Hamiltonian alone. Using the ``time-evolution theorem'' for a 1-dimensional system with bounded finite-range pair forces, we prove the existence of infinite-volume time-dependent correlation functions for such systems, limρΛ(t;q1,p1;⋯;qn,pn), as Λ→∞, where Λ is the size of the finite system. We also show that these infinite-volume correlation functions satisfy the infinite BBGKY hierarchy in the sense of distributions.
Simple Procedure for Determining the Number of Components of an Irreducible Tensor
Authors: Pietenpol J. L. , Speiser D.
Published: February 2004

We show that the number of linearly independent components of a tensor in n dimensions with specified symmetry properties is given by a polynomial in n. This polynomial can be determined in a simple way from the Young diagram associated with the tensor.
Distributions and Their Hermite Expansions
Authors: Simon Barry
Published: February 2004

We present a self-contained treatment of the technical parts of distribution theory needed in quantum field theory. The treatment is particularly suited for physicists since an absolute minimum of abstract functional analysis is used: In fact, only the Baire category theorem is needed. The simple nature of some proofs depends on extensive use of the expansion of a distribution as a sum of harmonic oscillator wave-functions. While this Hermite expansion is not new, the fact that it provides elementary proofs of several theorems does appear to be new.
Path Integrals and Product Integrals
Authors: Hamilton John F. , Schulman L. S.
Published: February 2004

The product integral is shown to be the appropriate mathematical tool for implementing the path integral for spin in its simplest form.
Completely Orthogonalized Plane Waves
Authors: Girardeau M. D.
Published: February 2004

Orthogonalized plane waves (OPW's) are orthogonal to the core states by construction, but they are not mutually orthogonal. Furthermore, the set of all core states plus all OPW's is overcomplete. In this paper we construct a set of ``completely orthogonalized plane waves'' (COPW's) which share with OPW's the properties of being orthogonal to the core states and simply related to plane waves, but are superior to OPW's in that the set of all core states plus all COPW's is both orthonormal and complete (not overcomplete).
Equivalence of Basis-Dependent and Basis-Independent Approach to Canonical Field Operators
Authors: Hegerfeldt Gerhard C.
Published: February 2004

The basis-independent approach to canonical commutation relations (CCR's), which allows arbitrary test-function spaces for smearing the field operators, is a generalization of the basis-dependent approach, in which the fields are smeared with an orthonormal system (or finite linear combinations thereof) to obtain an infinite set of qk and pk. Using recent results on continuity properties of representations of the CCR's, we show that every representation of the basis-independent type in a separable Hilbert space can be obtained by continuous extension of a suitable representation of the basis-dependent type where properties like irreducibility, cyclicity, etc., remain unaffected. In this sense, both approaches are equivalent, and the classification problem for CCR's is reduced from the simultaneous consideration of all representations for all possible test-function spaces to those for a single one (up to isomorphism).
Probabilistic Interpretation of the Classical Scattering Cross Section
Authors: Babbitt Donald G.
Published: February 2004

The probabilistic interpretation of the classical scattering cross section is discussed in a mathematically rigorous framework. In particular, extensive use is made of the notion of a Poisson process based on a plane and on a sphere. The case of Rutherford scattering is given as a detailed illustration.
Nearly Symmetric Advection
Authors: Soward A. M.
Published: February 2004

A class of exact solutions of the heat conduction equation and the magnetic induction equation are constructed for the special case where there is no diffusivity or time dependence. It is shown how these elementary solutions may facilitate the solution of a more general class of problems in which diffusivity and time dependence may also be present. Moreover, the existence of effective variables, first introduced by Braginskii for the kinematic dynamo problem in the lowest-order approximation, is a direct consequence of these elementary solutions.
Two-Time Spin-Pair Correlation Function of the Heisenberg Magnet at Infinite Temperature
Authors: Morita Tohru
Published: February 2004

The expansion coefficients in powers of time of the two-time spin-pair correlation function are obtained up to terms of order t8 for the Heisenberg magnet of spin ½ at infinite temperature for the square and cubic lattices and up to terms of order t10 for the linear chain. The result is applicable to the isotropic as well as anisotropic Heisenberg magnet where the exchange integrals in the z direction and in the orthogonal plane are assumed to be different. Analysis of the result is left to future work.
Vector Operators and a Polynomial Identity for SO(n)
Authors: Bracken A. J. , Green H. S.
Published: February 2004

It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n-vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.
The Canonization of Nice Variables
Authors: Stern David P.
Published: February 2004

In a perturbed periodic classical motion, the angle variable may be eliminated either by Kruskal's transformation to ``nice variables'' or, if the system is canonical, by the Poincaré-Von Zeipel method. For systems that possess a Hamiltonian, the present work (i) shows that Kruskal's transformation may be made canonical order by order, (ii) derives a practical formula for achieving this result, and (iii) shows that the two methods are equivalent and may be matched order by order.
Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters
Authors: Wegner Franz J.
Published: February 2004

It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 < n < d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.
Conservation of Energy and Momentum in Relativistic Electromagnetic Scattering
Authors: Twersky Victor
Published: February 2004

In a previous paper we transformed the plane wave excitation Φ(r,t) for an obstacle moving with constant velocity in free space (Σ) to obtain Φ′(r′,t′) in the scatterer's system (Σ′), and considered the scattered wave U′ in Σ′ and its transform U in Σ. The wave Φ has period T in t, and Φ′ and U′ have period T′ in t′; the wave U is not periodic in r,t, but in retarded (R, t) and present coordinates it has the period Ts equal to the dilation of T′. Now we transform t′-averages (over T′) of quadratic functions of Φ′ + U′ (the time-averaged energy density, the energy flux vector, and the momentum flux tensor) to obtain r, t-dependent forms in Σ (which in R, t may be interpreted as t-averages over Ts). We consider the average power absorbed by the scatterer and the force that acts on it in Σ′, and the theorems that relate these to the scattered and interference terms. Then we show how the known Σ′ conservation theorems are exhibited in Σ, and determine in Σ the corresponding (r, t-independent) power imparted to the scatterer and the force that acts on it, and the reradiated and interference terms. Since Ts is, in general, small enough to regard the present position as practically fixed in Σ, we also consider the corresponding differential reradiated cross section, etc.
A Technique for Solving Recurrence Relations Approximately and Its Application to the 3-J and 6-J Symbols
Authors: Neville Donald
Published: February 2004

It is shown that not only some second-order differential equations, but also some second-order difference equations (three-term recurrence relations) possess solutions of WKBJ type. WKBJ-like solutions are derived and used to study the ``classical'' (large J) limits of both the 6-J symbol and (in an appendix) the 3-J symbol.
One-Particle Operators and Local Internal Symmetries
Authors: Herbst Ira W.
Published: February 2004

The structure of local operators linear in the creation and destruction operators of a finite number of particles is investigated. It is shown that these operators determine a set of local and relatively local free fields relative to which the original operators are local. This result is used to study local internal symmetries in a theory of interactions and to show that these symmetries commute with the Lorentz group. The assumptions of Haag and Ruelle which lead to a complete particle interpretation are also discussed. Asymptotic many-particle states are constructed under less restrictive assumptions.
Electromagnetic Radiation in Curved Spaces
Authors: Sigal Richard
Published: February 2004

The classes of trajectories of charged particles, for which the radiation reaction force vanishes, in curved spaces are examined. The trajectories in the Robertson-Walker cosmologies are given in detail and it is shown that the only reaction-free geodesic trajectories are those at rest relative to the local matter density.
Combined Neutrino-Gravitational Fields in General Relativity
Authors: Trim D. , Wainwright J.
Published: February 2004

The combined gravitational-neutrino field equations are solved, subject only to the restriction that the energy-flow vector of the neutrino field be timelike or null. The principal null congruence (pnc) of the neutrino field is necessarily geodesic and shear-free, and coincides with a repeated pnc of the gravitational field, which is thus algebraically special. The twist of the neutrino pnc plays an important role, and is zero if and only if the neutrino energy-flow vector is null, in which case the neutrino field represents pure radiation.
Quasigroups of Coordinate Transformations in Curved Space-Time
Authors: Halpern M. , Malin S.
Published: February 2004

The Poincaré group is well known to imply a deep connection between the structure of space-time and particle properties. Searching for the analogous mathematical structure for curved space-time, we investigate the role of 10-parametric classes of coordinate systems in curved space-times. It is shown that all coordinate systems can be divided into 10-parametric classes according to the values of gμν and certain nontensorial combinations of its derivatives to an arbitrarily high order at a given point. It is also shown that 10-parametric classes arise naturally in Riemannian space-time when the maximum freedom in assigning numerical values to such nontensorial quantities is used. The set of transformations between members of a given class does not form a group, in general, but a novel mathematical structure (a ``quasigroup''). New results concerning the transformations between normal coordinates and the analytic characterization of geodesic Fermi coordinates are derived.
Crossing Symmetric Expansions of Physical Scattering Amplitudes; The O(2, 1) Group and Lamé Functions
Authors: Macfadyen N. W. , Winternitz P.
Published: February 2004

An explicitly symmetric 2-variable expansion, convergent in the physical regions of both the s and t channels, is obtained for 2-body relativistic scattering amplitudes. Use is made of a symmetric mapping of the variables s and t onto a 2-dimensional hyperboloid, leading to an expansion in terms of Lamé functions, which are the basis functions for an irreducible representation of the group O(2, 1) in our parametrization.
Motion in Electric and Magnetic Fields. I. Klein-Gordon Particles
Authors: Lam Lui
Published: February 2004

Exact solutions of the Klein-Gordon equation are found for a new solvable configuration of external electromagnetic fields, namely, a static uniform electric field inclined at an arbitrary angle to a static uniform magnetic field of arbitrary strength. Applications are discussed at the end.
Conserved Particle Structures in General Relativity
Authors: Williams J. G.
Published: February 2004

When general relativity is examined from the point of view of algebraic topology, it is found that the theory exhibits conserved particlelike structures. The number of such particles associated with a given metric is determined by the homotopy class of the metric. This paper is concerned with showing that if the theory is quantized, then the quantum mechanical states corresponding to 1-particle systems in the unquantized theory may have properties similar to fermions. In particular, it is proved that it is possible for wavefunctionals to exist which are double-valued under 2π rotation.
Unitary Irreducible Representations of SU(2, 2). III. Reduction with Respect to an Iso-Poincaré Subgroup
Authors: Yao Tsu
Published: February 2004

The unitary irreducible representations of SU(2, 2), the covering group of the conformal group, are reduced with respect to an iso-Poincaré subgroup E(3, 1). Explicit representations of the 15 generators of SU(2, 2) in terms of differential operators in the function space of |η,ξ,ε;s,t,m〉, |ω,ξ,ε;r,t,m〉, and |ξ,ε;t,m〉, the basis vectors for ``timelike,'' ``spacelike,'' and ``lightlike'' UIR's of E(3, 1), respectively, are given. The matrix elements between a basis vector in the maximal compact subgroup SU(2) × SU(2) × U(1) and a basis vector in E(3, 1) are calculated for all 14 different classes of UIR's of SU(2, 2). We find that two classes are reducible with respect to ``lightlike'' representations, two classes are reducible with respect to ``timelike'' representations, eight classes are reducible with respect to ``spacelike'' representations, and two classes contain both ``timelike'' and ``spacelike'' representations of E(3, 1).
Periodic Orbits and Classical Quantization Conditions
Authors: Gutzwiller Martin C.
Published: February 2004

The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
Scattering of Waves in Many Dimensions
Authors: Adawi I.
Published: February 2004

Scattering by a spherical potential is discussed in all dimensions by one formulation using the partial-wave expansion method. The optical theorem relating the total scattering cross section σ to the forward scattering amplitude f(0) is derived.
Some Aspects of the Relationship between Mathematical Logic and Physics. II
Authors: Benioff Paul A.
Published: February 2004

In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials
Authors: Calogero F.
Published: February 2004

The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).
Structure of the Combinatorial Generalization of Hypergeometric Functions for SU(n) States
Authors: Wu Alfred C. T.
Published: February 2004

The combinatorics of the boson operator formalism in the construction of the SU(n) states provides a natural scheme for the appearance of certain generalized hypergeometric functions. It is shown that, while special cases exist where the functions thus generated belong to the class of generalized hypergeometric functions defined by Gel'fand et al. as being the Radon transforms of products of linear forms, the general cases apparently do not. This is already so at the SU(4) level.
Dual Transformations in Many-Component Ising Models
Authors: Mittag L. , Stephen M. J.
Published: February 2004

Dual transformations in many-component Ising models in two dimensions on a square lattice are studied. The models considered include those of Ashkin and Teller and of Potts. In certain cases the dual transformation is a relation between the partition function of a lattice at high and low temperatures and can be used to determine a unique critical temperature if one exists. Dual transformations are considered both from a topological and an algebraic point of view. The topological arguments are a natural extension of those used by Onsager for the 2-component Ising model. The transfer matrices for these models are constructed, and it is shown how the dual transformation arises in this formulation of the problem. The algebras generated by these models are investigated and provide a generalization of the spinor algebra introduced by Kaufman in the 2-component Ising model.
Effect of Molecular Collision Frequency on Solutions of the Linearized Boltzmann Equation
Authors: Richardson Thomas G. , Sirovich Lawrence
Published: February 2004

The linearized Boltzmann equation is considered for steady-state oscillations. Denoting molecular collision frequency by ν(ξ) and writing ν = O(ξα) for ξ large, we show that solutions for x → ∞ behave like exp (−x2/(3−α)). This shows that the continuous spectra dominates hydrodynamics for all except the rigid sphere or radial cutoff case (α = 1).
Eigenvalues of the Hill Equation to Any Order in the Adiabatic Limit
Authors: Guyard Jacques , Nadeau André , Baumann Germain , Feix Marc R.
Published: February 2004

To study the time-dependent linear oscillator, Lewis has recently introduced an auxiliary function w. One of the advantages of this function is that, in the adiabatic limit, a formal expansion of w in ε is possible (ε characterizing the slowness of the time variation). We show that, in this adiabatic limit, the eigenvalues of the Hill equation can be very easily deduced from w. Moreover, the computation of the ε2n-order solution is much simpler if we use the Chandrasekhar method of higher invariants, which is shown to be equivalent to the Lewis expansion. Compact formulas, easy to handle on a computer, are obtained, and the method, which must be considered as a generalization to higher order of the WKB solution, is finally tested on the Mathieu equation.
Diffraction Radiation by a Line Charge Moving past a Comb: A Model of Radiation Losses in an Electron Ring Accelerator
Authors: Hazeltine R. D. , Rosenbluth M. N. , Sessler A. M.
Published: February 2004

A calculation is given of the radiated-energy loss from a charged rod which moves at constant speed past an infinite set of parallel semi-infinite conducting plates of infinitesimal thickness, with the rod taken parallel to and at a fixed distance from the plate edges. The problem is analyzed using the Wiener-Hopf technique, and the resulting formulas are evaluated analytically in the limits of high rod speed and low rod speed, and compared with numerical evaluation over the full range of speeds.
Perturbation Treatment of the Bethe-Salpeter Equation for Tightly Bound Fermion-Antifermion Systems
Authors: Keam R. F.
Published: February 2004

A perturbation theory is developed for the ladder approximation Bethe-Salpeter equation that describes bound spin-½ fermion-antifermion systems of small total mass. A new exact analytical solution for a zero-mass system is presented and the perturbation theory is applied to it. Some time-reversal properties of the equation are also examined.
Eigenvalue Moments for Ensembles of Random Matrices
Authors: McDonald J. F.
Published: February 2004

A straightforward method is given for calculating the moments of the joint eigenvalue distribution corresponding to an ensemble of random Hermitian matrices. The method enables one to calculate the averages using the matrix element distribution. Thus an explicit expression for the joint eigenvalue distribution is not required.