bluuebunny/crossref_metadata_2025 huggingface dataset:
The Lorentz Group Symmetry of the Hydrogen Atom and the Coulomb T Matrix
Authors: Rajagopal A. K. , Shastry C. S.
Published: February 2004

The existence of solutions for the Lippmann-Schwinger (LS) equation for the Coulomb problem is studied by investigating whether its kernel belongs to the Hilbert-Schmidt (HS) class. The kernel for each partial wave is shown to belong to the HS class for complex energies whereas for real energies it becomes unbounded. Unlike the case of short range potentials, e.g., the Yukawa potential, even the suitably symmetrized kernel does not belong to the HS class for real, positive energies. These formal properties strongly indicate that a unique limit for the partial wave off-shell Coulomb T matrix as it approaches the unitarity axis may not exist. It is found that exploiting the O(4) symmetry of the Coulomb Hamiltonian (H) in the subspace of the negative spectrum of H and the O(3, 1) symmetry in the subspace of the positive spectrum of H, one can construct the off-shell Coulomb T matrix in terms of the eigen-solutions (Sturmians) of the kernel of the Lippmann-Schwinger equation. These follow from the work of Perelemov and Popov, and Schwinger on the Coulomb Green's function. On the basis of the generating functions for the Sturmians, various integral, contour integral, and discrete sum representations for the complete off-shell Coulomb T matrix are derived. In this way, we explicitly demonstrate that indeed the T matrix has a nonunique limit as one approaches the unitarity axis. It is also shown that when the asymptotic Coulomb distortion is taken into account, the physical Coulomb amplitude can be deduced from this Coulomb T matrix. These results incidentally rectify some errors in the earlier works. Since for both negative and positive energies the Coulomb T matrix is obtained as the explicit solution of the LS equation, the validity of the generalized unitarity—the Low equation—in a certain sense is guaranteed. This is proved in a general way, by showing that a generalized Low equation follows when the energy is complex only from the LS equations and some defining relations; in the Coulomb case, the generalized unitarity relationship for real energies must be interpreted as the limit when the imaginary part becomes zero.
On the Bounds of the Average Value of a Function
Authors: Yeh Robert H. T.
Published: February 2004

We describe here how one can obtain upper and lower bounds of the average value of a function f(E)¯ in terms of the bounds of the variable E and Ē, E2¯.
The Classical Moment Problem and the Calculation of Thermal Averages
Authors: Lonke A.
Published: February 2004

The physical information contained in the first 2n moments of the single-particle spectral weight function of a fermionic many-body system is investigated. The approach is based on the mathematical theory of the classical moment problem. Under consideration are the thermal as well as dynamical properties of the system. Using this information, approximate n-pole single-particle thermal Green's functions and the corresponding spectral weight functions are constructed. It is shown that these approximations are not unique and depend on a real parameter. This dependence is used for the calculation of the rigorous error bounds of the approximate thermal averages.
A Note on Asymptotically Flat Spaces
Authors: Aronson B. , Lind R. , Messmer J. , Newman E.
Published: February 2004

It is shown that there is a simple alternative to the Bondi type of coordinate condition in asymptotically flat spaces, which leads to a different asymptotic coordinate group whose structure is simpler than that of the BMS group. As a concomitant to this work, there was discovered a scalar function on null infinity. From an analysis of this function one can obtain, starting from the asymptotic shear of one family of hypersurface orthogonal rays, (1) all null asymptotically shear-free congruences, (2) the asymptotic shear of all hypersurface orthogonal rays.
Many-Body Point Transforms. II. An Exact Noncluster Approach to the Hard-Core Many-Body Problem
Authors: Witriol Norman M.
Published: February 2004

The method of many-body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard-core many-body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth-order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.
Some Considerations of Entropy Change
Authors: Okubo S. , Isihara A.
Published: February 2004

An inequality for two positive operators is used to discuss entropy theorems for time smoothing, mixing, and other processes. Especially, it is proved that neglect of nondiagonal matrix elements for the density matrix causes the entropy to increase.
On Clifford Numbers, Dirac and Relativistic Hamilton-Jacobi Equations
Authors: Percus Jerome K. , Petrakopoulos Nikitas L.
Published: February 2004

Conditions are established under which the Dirac equation for an electron in an electromagnetic field has an exact semiclassical solution. In other words, the phase is identified with a solution of the corresponding relativistic Hamilton-Jacobi equation and the spinor amplitude has no explicit dependence upon ℏ. The complete set of admissible fields is determined for one-dimensional and stationary three-dimensional systems, while an extensive class is indicated for the general time-dependent problem.
Path Integrals in Curved Spaces
Authors: McLaughlin David W. , Schulman L. S.
Published: February 2004

In this paper we present a simplification of the path integral solution of the Schrödinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Itô integral.
The Phase Shift. III. As a Continuous Functional
Authors: Frank William M.
Published: February 2004

The continuity of the phase shift and scattering length as a functional of the potential is considered. We find that the convergence of ∫dr|Vn| to zero is a sufficient though not necessary condition for the vanishing of the phase shift, while the convergence of ∫dr r|Vn(r)| or ∫dr|Vn(r)|1/2 to zero would be a sharp condition. A number of theorems of this type are proven. Theorems are also proven demonstrating the continuity of the phase shift and scattering length as a sequence of potentials converging in the proper norms to a nonzero potential. Some implications of the existence of a Banach space of potentials are discussed briefly.
Existence of a Solution of a Stochastic Integral Equation in Turbulence Theory
Authors: Padgett W. J. , Tsokos C. P.
Published: February 2004

In the theory of turbulence, the random position of a tagged point in a continuous fluid in turbulent motion, r(t; ω), is a vector-valued random function of time t ≥ 0, ω ε Ω, where Ω is the supporting set of the underlying probability space (Ω, B, P). If u(r, t; ω) is the Eulerian velocity field, then r(t; ω) satisfies the stochastic integral equation r(t;ω)=∫0tu(r(Υ;ω),Υ;ω)dΥ, t≥0. General conditions under which a random solution of this stochastic integral equation exists are given in the form of a theorem, and the theorem is proved using the concepts of admissibility with respect to an operator on a Banach space and fixed-point methods of functional analysis.
Infinite-Dimensional Representations of the Lorentz Group: Complementary Series of Representations
Authors: Carmeli M. , Malin S.
Published: February 2004

The method used by Carmeli to obtain another form for the principal series of representations of the group SL(2, C) is extended to the complementary series of representations of that group. As a result, a new form for the complementary series of representations of SL(2, C) is obtained which describes the transformation law of an infinite set of numbers under the group translation in a way which is very similar, but as a generalization, to the way spinors appear in the finite-dimensional case.
Lattice Green's Function for the Body-Centered Cubic Lattice
Authors: Katsura Shigetoshi , Horiguchi Tsuyoshi
Published: February 2004

The lattice Green's function for the body-centered cubic (bcc) lattice I(t)=1π3∫ ∫ 0π∫dx dy dzt±iε−cosxcosycoszis considered. With the use of the analytic continuation to complex value of t from Maradudin's result for t > 1, the value of the real and imaginary parts of the integral I(t ± iε) for 0 < t < 1, ε → 0, is obtained. The expressions valid for t → ∞, t≳1, t≲1, and t ∼ 0 are given. They are useful for analyzing the nature of the singularity and for carrying out numerical calculations in all regions of t.
Generalized Master Equations for Inhomogeneous Systems
Authors: Dufty James W.
Published: February 2004

A generalized master equation of the Van Hove type is derived for inhomogeneous systems with arbitrary initial conditions in such a way as to display explicitly the renormalized ``free'' propagation, as well as the ``gain-loss'' terms. The corresponding Wigner function formulation of this equation is given as a representation appropriate for the study of kinetic equations for inhomogeneous systems from the point of view of master equations. The results are formally exact.
Approximate Topologies
Authors: Schulman L. S.
Published: February 2004

Several examples illustrate the utility of mathematical models which differ in their topological properties from the ``real'' systems they describe. The examples are the Aharonov-Bohm effect, the periodic lattice, and scattering off an obstacle. By going to the approximate topology, the Hamiltonian loses its essential self-adjointness, but the relation of the approximate to the exact topology leads to a convenient classification of its extensions.
Application of the Semidirect Product of Groups
Authors: Geroch Robert , Newman E. T.
Published: February 2004

A simple example of the application of the semidirect product of groups is presented.
General Involutional Transformations and the Representation of GL(n)
Authors: Santhanam T. S. , Chandrasekaran P. S. , Menon Nalini B.
Published: February 2004

The properties of general involutional matrices satisfying the equation Am = kI, k = const, are studied. Generating equations for the induced representations of an arbitrary (n × n) matrix are explicitly written down. The special case when n = 3 is discussed in detail. It is shown that the conditions which make an arbitrary (n × n) matrix involutional leave any of its induced representation involutional. It is further shown that an involutional matrix in its self-representation can be expanded in the basis of generalized Clifford algebra with coefficients which are the generalized hyperbolic functions. Eigenvalues of induced matrices, in particular when they are involutional, are calculated.
Labeling States and Constructing Matrix Representations of F4
Authors: Patera J.
Published: February 2004

A method is developed for construction of a complete basis for a class of finite-dimensional irreducible representations of the simple Lie algebra F4 and for calculation of matrix elements of F4-generators relative to that basis. The algebra F4 is embedded into A25, and results of Gel'fand and Tsetlin concerning that algebra are used.
Nonrelativistic Current Algebras as Unitary Representations of Groups
Authors: Goldin Gerald A.
Published: February 2004

It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand-Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the physical information is contained in a cylindrical measure μ on J′, the dual of the nuclear space J. The Fourier transform of μ can be interpreted as an expectation functional with respect to the state of lowest energy. The nonrelativistic Fock representation (including the theory of n particles in a box) is examined in this formalism. Conditions on μ are systematically developed which suffice to recover all of the infinitesimal generators on a common, dense, invariant domain in the Hilbert space. For the Weyl group slightly weaker conditions than for the semidirect product case would suffice. Gaussian measures in J′, as well as the measures defining n-particle representations of J ∧K, satisfy these conditions.
The Reduction O(3,1)⊃O(2,1)⊃O(1,1)
Authors: Macfadyen N. W.
Published: February 2004

We set up a representation of the principal series of the group SL(2, C) in terms of the sequence of noncompact subgroups SL(2,C)⊃SU(1,1)⊃O(1,1). The basis functions are just the cross-basis matrix elements of SU(1, 1) between O(2) and O(1, 1) bases, and we derive much material on these before using them to calculate the representation functions. The latter have two pairs of discrete labels distinguishing between equivalent representations occurring in the reduction, and these are related to two discrete reflection operators that are introduced in order to obtain a maximal Abelian set.
The Einstein Tensor and Its Generalizations
Authors: Lovelock David
Published: February 2004

The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.
Coupling of Waves in Nonuniform Plasma
Authors: Chakraborty B.
Published: February 2004

This paper studies mutual coupling of transverse and longitudinal high-frequency waves of both electron and ion fluid motions of perturbation in slowly varying nonmagnetized equilibrium plasma. Besides, the particular problem of coupling of ion acoustic and electromagnetic waves is considered in detail. The power of electromagnetic radiation generated by an ion acoustic wave and also in reverse from the latter to the former have been evaluated.
Some Remarks on the Theory of Trapping of Excitons in the Photosynthetic Unit
Authors: Sanders J. W. , Ruijgrok Th. W. , Bosch J. J. ten
Published: February 2004

The problem of random motion of an exciton in a square lattice with a sink is formulated in terms of finite difference equations. The case where the exciton suffers a fixed delay time at each lattice site is solved numerically. The more realistic case, also investigated by Knox [J. Theoret. Biol. 21, 244 (1968)], where this delay time is inversely proportional to the number of nearest neighbors, is shown to be equivalent to the case with periodic boundary conditions. Consequently, an exact analytic expression for the total delay time can for that case be derived. For lattices of any size this quantity can now easily be calculated. Giving the exciton the possibility of escaping from the lattice will influence the total delay time. This effect is investigated, as well as the effect of distributing the sink over the lattice sites. As a by-product, an amusingly simple formula k(λ)−1c(λ) ∑ μc(μ)is derived for the average path length upon first return in a random walk over an arbitrary network.
Integral Representation Technique for Expansion of Arbitrary Analytic Functions of the Distance between Two Points
Authors: Rafiqullah A. K.
Published: February 2004

The coefficients of expansion of an arbitrary analytic function of the distance r between two points (r1, θ1, φ1) and (r2, θ2, φ2) in terms of the Legendre polynomials Pl(cos θ12) are double Bessel transformed. Assuming that the transformed coefficients are diagonal, consistent with the differential equations satisfied by the original coefficients, we derive the explicit expressions for the latter coefficients. These formulas are identical to those derived by Sack from the solutions of the differential equations in terms of the hypergeometric functions.
Correlation Inequalities for Coupled Oscillators
Authors: Leff Harvey S.
Published: February 2004

Correlations in a system of N classical, coupled oscillators are studied, with a view toward obtaining a more complete understanding of Griffiths-type inequalities. The potential energy is assumed to be U = ½ ΣiΣjJijxixj − ΣHkkxk, with − ∞ ≤ xi ≤ ∞ and Jij = Jji. The N × N matrix with elements {Jij} is assumed positive definite. Sufficiency conditions for the correlation functions to satisfy Griffiths-type inequalities are found to be: (i) Kij ≥ 0 for all i, j, where K = J−1 and (ii) ΣjKljHj ≥ 0 for all l. The class of systems obeying (i) and (ii) contains those for which Jij ≤ 0, i ≠ j, and Hk ≥ 0 for all k; these are direct analogs of Ising ferromagnets. It is proved that a necessary and sufficient condition for Griffiths-type inequalities to hold for arbitrary {Hk ≥ 0} is simply (i) above. The sufficiency conditions (i) and (ii) are broader than those available to date for Ising models (only the sufficiency condition of ferromagnetic coupling is known). The necessary and sufficient condition (i) has no known Ising counterpart at present.
A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold
Authors: Sen D. K. , Dunn K. A.
Published: February 2004

A new scalar-tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well-known Brans-Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.
Magnetic Corrections to the Boltzmann Distribution Function
Authors: Thomas R. B.
Published: February 2004

By means of an expansion in powers of the magnetic field, where classical statistics are assumed, an expression is found for the gauge-independent equilibrium distribution properly corresponding to the ordinary Boltzmann transport equation. The validity of the new distribution is demonstrated by determining the current density induced in a simple system of free electrons simultaneously accelerated by crossed electric and magnetic fields and comparing the current with that obtained using the gauge-dependent density matrix formalism. To terms of second order in the magnetic field, the results are shown to be in exact agreement.
Continuity of Automorphisms of Quasilocal Algebra
Authors: Chen Etang
Published: February 2004

We show that automorphisms of quasilocal algebra A, a C*-algebra with identity on Hilbert space H, is extendable to the weak operator closure of A under a certain physical assumption.
Theorem on Invariant Amplitudes
Authors: Guertin Ralph F.
Published: February 2004

It is shown that any basis of covariant polynomials for a two-particle scattering process yields invariant amplitudes free of kinematical singularities, provided (a) the total number of basis polynomials equals the number of spin space components of the scattering amplitude and (b) the polynomials of each of the two parity signatures are separately linearly independent at all points where three of the particle 4-momenta are linearly independent. This result allows one to directly identify good basis sets without going through the very tedious algebra involved in comparing them to the sets of Hepp and Williams. The latter are not useful for practical applications because the spinor indices belonging to different particles are coupled and these sets do not transform into themselves under the relevant discrete symmetry operations.
Electromagnetic Diffraction by a Thin Conducting Annular Disk
Authors: Jain D. L. , Kanwal R. P.
Published: February 2004

The problem of diffraction of time harmonic electromagnetic waves by a perfectly conducting thin annular disk is solved when the incident wave is a plane wave traveling in a direction perpendicular to the annulus. The electromagnetic problem is reduced to two scalar problems with the help of the Hertz vector representation formulas. Each of these scalar problems is formulated in terms of a Fredholm integral equation of the first kind, which is subsequently reduced to a system of four simultaneous Fredholm integral equations of the second kind. This system is then solved by the straightforward iteration scheme. Low-frequency approximations are presented for various physical quantities such as the components of the induced surface current density at the disk and the scattering cross section.
Asymptotic Theory of Wave Propagation in Spatial and Temporal Dispersive Inhomogeneous Media
Authors: Chen Kenneth C.
Published: February 2004

The recently developed asymptotic theory of wave propagation is extended to slightly inhomogeneous and slowly varying anisotropic media which exhibit both spatial and temporal dispersion. A particular form of the constitutive relation is first introduced. Asymptotic solutions are then obtained by assuming a series solution ``ansatz'' into Maxwell's equations and the constitutive relation. The eikonal equation and the transport equation are obtained, by a procedure similar to that of Lewis, for lossless media, in which a Hermitian operator is involved. The modified transport equations obtained from other forms of the constitutive relation are given. They are interpreted as generalized Poynting theorems for appropriate physical situations. Finally, as a by-product of this work the space-time conductivity tensor of anisotropic plasmas σ(r, t), which is the 4-dimensional Fourier-Laplace transform of σ̃(k,ω), is found to be very simple.
Two-Dimensional Quarter Space Problems in One-Speed Transport Theory
Authors: Leonard A.
Published: February 2004

Methods derived from the theory of several complex variables are used as a means of analyzing a class of two-dimensional transport problems in a scattering and absorbing quarter space (0 ≤ x1, 0 ≤ x2, − ∞ ≤ x3 ≤ ∞) described by a linear, one-speed Boltzmann equation. Using Fourier transformation and the Bochner decomposition, the multivariable analog of the Wiener-Hopf factorization, we find the Green's function in transform space, which solves all source problems having a solution bounded at infinity. The transform of the density asymptotically far from the corner (x1 = x2 = 0) is determined explicitly, while the remainder is given in terms of the solution to a pair of Fredholm equations.
Properties of the Residues in the Veneziano Model
Authors: Dobson P. N. , Simmons W. A.
Published: February 2004

Explicit formulas for the expansion coefficients of the Veneziano amplitude are derived. We show that the problem of positivity can be reduced to a study of the zeros of certain polynomials.
Low-Frequency Scattering by Perfectly Conducting Obstacles
Authors: Asvestas J. S. , Kleinman R. E.
Published: February 2004

Coupled Fredholm integral equations of the second kind are derived for the electric and magnetic fields scattered when a smooth, bounded, perfectly conducting three-dimensional obstacle is illuminated by a time harmonic, monochromatic, otherwise arbitrary incident field. The kernels of the equations are dyadics constructed from potential functions associated with the scattering surface, i.e., solutions of Laplace's equation satisfying particular boundary conditions. If the frequency of the incident field is sufficiently low, the integral equations may be solved in a standard Neumann series. This is demonstrated in an example, scattering of a plane wave by a sphere.
Density of States
Authors: Deliyannis Platon C.
Published: February 2004

A characterization of the weak limits of states is given, leading to an extension of a theorem of Gudder and the verification of a conjecture by Mackey
Continuous Representations and the Kernel Function
Authors: Gomatam Jagannathan , Gardiner C. W.
Published: February 2004

New examples of continuous representations are obtained utilizing the theory of kernel functions for a finite domain D in the complex z-plane. It is shown how, by starting with the continuous representation for a circular domain, the Riemann mapping theorem makes it possible to obtain the continuous representations with respect to any finite, simply connected schlicht domain with at least two boundary points. A method of constructing orthonormal polynomials with respect to a class of weight functions for both finite domains and infinite plane is given. The standard coherent states employed in quantum optics and quantum field theory emerge as a particular case of the present investigation.
Infinite-Spin Limit of the Quantum Heisenberg Model
Authors: Millard Kenneth , Leff Harvey S.
Published: February 2004

The canonical partition function Q(H, N, s) for a spin-s anisotropic Heisenberg model with N sites, in external magnetic field H, is examined in the limit s → ∞. The coupling coefficients and the magnetic moment per spin are taken to be inversely proportional to s2 and s, respectively. It is proved that lim lim s→∞s−NQ(H,N,s)=(2π)−NQ(H,N),where Q(H, N) is the partition function for a corresponding classical Heisenberg model with spins of unit magnitude. This theorem makes precise the widely believed, but heretofore unproved equivalence between the classical Heisenberg model and the infinite-spin limit of the quantum Heisenberg model. No appeal to the thermodynamic limit is necessary.
Gravitation as a Consequence of the Self-Interaction of the Γ Fields
Authors: Novello Mario
Published: February 2004

We investigate an algebraic-geometric approach to field theory. We try to prove that it is possible to understand Einstein's gravitational equation as a consequence of the self-interaction of the fundamental Γ fields of the assumed Clifford algebra. Furthermore, the antisymmetric object of the algebra obeys a set of equations that has the same structure as Maxwell's equations.
Remarks on a Class of Representations of Current Density Algebras
Authors: Chakrabarti A.
Published: February 2004

A class of representations of the current density algebras is considered by introducing a factorized form involving algebras containing the one satisfied by the integrated charges as a subalgebra.
Relaxation to Quantum-Statistical Equilibrium of the Wigner-Weisskopf Atom in a One-Dimensional Radiation Field. III. The Quantum-Mechanical Solution
Authors: Davidson Russell , Kozak John J.
Published: February 2004

A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak-coupling solution of the generalized Prigogine-Résibois master equation for the model of the Wigner-Weisskopf atom in a one-dimensional radiation field. The Schrödinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak-coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincaré recurrence times for finite systems are made. It is found that considerable differences exist between the statistical-mechanical and quantum-mechanical results, but that both manifest nonanalyticity in the coupling parameter as it tends to zero. Suggestions are given for further work toward the resolution of these discrepancies and a better understanding of the weak-coupling limit.
A Method for Generating Solutions of Einstein's Equations
Authors: Geroch Robert
Published: February 2004

A method is described for constructing, from any source-free solution of Einstein's equations which possesses a Killing vector, a one-parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space-times having a Killing vector.
Scattering by a Nonspherical Potential
Authors: Li Ming Chiang
Published: February 2004

The scattering of a simple nonspherical potential with spheroidal symmetry is discussed. The spheroidal partial wave analysis method is used. It is found that the spheroidal phase shifts have the similar physical meaning as the conventional spherical phase shifts. The integral equation for the spheroidal phase shift is given. The scattering by a discontinuous spheroidal potential is also presented. The WKB approximation is used to evaluate the spheroidal phase shift at high energy. For small angle scattering our formulation reduces to a simple form as in the eikonal description.
Analytic Properties of the Phase Matrix
Authors: Dragt Alex J.
Published: February 2004

We write the S matrix in the form S = exp (iη) and study the analytic properties of the 2-body matrix elements of η as a function of two complex variables. We find that the 2-body matrix elements of η have analytic properties similar to those of the K matrix and the second sheet scattering matrix. In particular, they do not possess Mandelstam analyticity.
Separable Approximations to Matrices and Functions of Two Variables
Authors: Bolsterli M. , Norton J. L.
Published: February 2004

Considerable computational time can be saved by the use of separable approximations to matrices and functions of two variables. Such approximations are considered in general.
Density and Current Density as Coordinates for a System of Interacting Bosons at Absolute Zero
Authors: Chang B. Tsu-Shen
Published: February 2004

The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.
Invariants of πN → πBs
Authors: Khan M. N. , Arif-Uz-Zaman
Published: February 2004

The kinematic singularities and threshold relations of helicity amplitudes for inelastic pion-nucleon scattering involving higher baryons in the final state are discussed. A recipe for writing the s- and t- channel helicity amplitudes is obtained using Wigner-Bargmann formalism. Explicit expressions for helicity amplitudes are worked out for the processes where the final state baryon has spin 32 or 52.
Static Gravitational Field of Multipoles in the Second Approximation
Authors: Pechlaner E. , Das A.
Published: February 2004

A method of obtaining the explicit form of the gravitational field of multipoles in the second approximation is furnished here. The generalization of this method to any superposition of multipoles is straight-forward. We obtain as an example the second-order metric for the superposition of a monopole and a quadrupole.
Particles with Spin in a Gravitational Field
Authors: Mashhoon Bahram
Published: February 2004

Papapetrou's covariant equations of motion for a spinning particle in a gravitational field are discussed. The equations of motion for the spin of a particle at rest outside a rotating mass are derived using the Kerr metric. It is shown that Schiff's formula for the mass-current effect follows from these equations in the lowest approximation.
Equal-Time Commutation Relations between Components of a Current and the Energy-Momentum Tensor
Authors: Bergström K. , Yngström S.
Published: February 2004

We consider an arbitrary physical system possessing a set of currents and an energy-momentum tensor. Without the use of canonical commutation rules we derive in a unified way equal-time commutation relations between the components of the currents and the energy-momentum tensor. The currents and energy-momentum tensor are defined by the response of the physical system to the variations of external fields. The formalism of infinite-dimensional continuous transformation groups is utilized.
Invariant Approach to a Space-Time Symmetry
Authors: Debney George C.
Published: February 2004

A study is made of Killing vector fields in vacuum Einstein spaces with a restriction primarily to those fields whose associated bivector is nonnull. However, a well-known theorem of Robinson is modified slightly to show that if such a space admits a null bivector associated with a Killing vector, the space must be algebraically special. Consequently, all algebraically general spaces admit a nonnull Killing bivector (KBV) if they admit a symmetry at all. Furthermore, it is shown that if a vacuum Einstein space admits a spacelike or timelike Killing vector field whose associated KBV is nonnull, the Killing trajectories are not geodesics. Computation of invariants from the curvature tensor and the KBV allows an approach which gives a general classification to such spaces which admit at least one hypersurface orthogonal Killing vector field. A few geometrical properties involving the principal null directions of the KBV are also derived for the hypersurface orthogonal cases. In addition, a topological result follows immediately from the behavior of the invariant J.
Eigenvalue Problem for Lagrangian Systems. IV
Authors: Barston E. M.
Published: February 2004

The constants of the motion of the linear Lagrangian system ξ¨+Aξ̇+Hξ(t)=0 are shown to generate a useful class of orthogonality relations. For systems with complete sets of eigenvectors, these are used to derive the expansion coefficients for arbitrary initial data. We show that every stable system possesses a complete set of eigenvectors and that this set is the union of two basis sets.
Some General Inequalities in Quantum Statistical Mechanics
Authors: Okubo Susumu
Published: February 2004

Let H = H0 + λH1 be the Hamiltonian of a system where λ is a coupling parameter for the interaction H1. The partition function Z = Tr exp (−βH) is then a function of both λ and β. We investigate various general inequalities involving derivatives of log Z with respect to β and λ.
Effect of Electromagnetic Radiation on the Secular Stability of a Homogeneous, Charged, Rotating Drop
Authors: Stettner R.
Published: February 2004

It is shown that electromagnetic radiation induces a secular instability in a homogeneous rotating charged drop, held together by surface tension, at the point of bifurcation at which a triaxial sequence of equilibrium figures branches from the axially symmetric figures.
T Matrix for the Exponential Potential
Authors: Fuda Michael G.
Published: February 2004

An analytic expression for the s-wave part of the exponential potential T matrix is obtained. The separability of the residue of the T matrix at bound state or virtual state poles is demonstrated explicitly.
Exact Stationary States in |φ|2N Field Theories
Authors: Rosen Gerald
Published: February 2004

Exact physical solutions to the Schrödinger stationary state equation are reported for local, relativistic, real scalar theories featuring an interaction energy density of the form g |φ|2N, where the coupling constant g is positive and the exponent N is a constant parameter (not necessarily an integer) greater than unity. No interaction or scattering between the field quanta is manifest in the multiparticle state solutions, implying that the real scalar theories are effectively linear in the absence of mass or coupling constant renormalization.
Self-Adjoint Operators in Indefinite Metric Spaces
Authors: Dobson Peter N.
Published: February 2004

Some properties of self-adjoint operators in indefinite metric spaces are explored, with emphasis on the problem of the completeness of the set of eigenvectors. For operators in spaces of finite dimension, some simple criteria are deduced regarding the existence of such a complete set. Implications of completeness of eigenvectors for operators in infinite-dimensional spaces are discussed, and some partial extensions of the results for finite dimensions given.
Propagation of Random Electromagnetic Fields
Authors: Torrieri Don J. , Taylor Leonard S.
Published: February 2004

The propagation of a random electromagnetic field in a uniform medium is investigated. It is assumed that the two-point mutual coherence γ is stationary over an initial plane. It is shown that the mean intensity will be conserved if and only if γ(kx, ky), the spectrum of γ, is zero for kx2+ky2>k2, where k is the wavenumber in the medium, meaning that evanescent waves are not considered; and it is proved that, under this condition, the transverse coherence is unchanged from plane to plane.
Eigenfunctions of the Integral Equation for the Potential of the Charged Disk
Authors: Wolfe Peter
Published: February 2004

The integral equationf(r,θ)= ∫ 02π ∫ 01u(r′,θ′)[r2+r′2−2rr′ cos (θ−θ′)]12·r′(1−r′2)12dr′dθ′arises in connection with the problem of the electrostatic potential due to a charged disk. We solve the equation by computing a complete set of eigenfunctions and eigenvalues for the integral operator. The eigenfunctions have the form Fm,n±(r,θ)=Pnm[(1−r2)12]e±imθ, 0 ≤ m ≤ n, m + n even. Here Pnm(x) is the associated Legendre function of the first kind.
Geometrization of a Massive Scalar Field
Authors: Ludwig Garry
Published: February 2004

This paper presents the necessary and sufficient conditions that a Riemannian geometry have as its source a real or complex massive scalar field.
Myriotic Fields in Quantum Statistical Mechanics
Authors: Alexanian Moorad
Published: February 2004

A Hilbert space formulation of quantum statistical mechanics is developed by using the notion of myriotic fields. The nonseparable nature of the Hilbert space is investigated, and it is shown that in the classical limit, ℏ → 0, the Hilbert space approaches, in the absence of point eigenvalues, that Hilbert space appropriate to classical statistical mechanics.
Dispersion Relations, Froissart-Gribov Transforms, and Distributions
Authors: Auberson G. , Mahoux G.
Published: February 2004

We construct a family of distribution spaces K′λ. Each K′λ completely characterizes the (tempered) distributions (i) which are the discontinuity of functions holomorphic in a cut plane with given boundedness properties and (ii) the Froissart-Gribov transforms of which are holomorphic in a half-plane with given asymptotic behavior. The main results are stated in two reciprocal theorems. Such a reciprocity shows the adequacy of our spaces K′λ. Some mathematical properties of these ones are given after introducing the proper test function spaces Kλ. In particular, we prove a representation theorem for the distributions in K′λ.
Periodic Solutions of a Nonlinear Differential Equation
Authors: Bradbury Ted Clay , Brintzenhoff Alton
Published: February 2004

Periodic solutions of the nonlinear differential equation ẍ + f(x) = 0 are investigated. The qualitative characteristics of the solution are explored and several theorems about the basic nature of the solution are presented. The solution is then obtained in terms of a Fourier series and numerical calculations are carried out to fifth-order accuracy. The dependence of the frequency of the solution on the amplitude is studied. As an example, the solution is applied to the problem of obtaining the orbit of a charged particle moving in a radial electric field.
A Method of Quantization for Relativistic Fields
Authors: Hammer C. L. , Tucker R. H.
Published: February 2004

A procedure for quantization is given which does not require the canonical formalism. Only the equation of motion for the field and the resulting conserved current are needed to derive all the necessary commutators or anticommutators, the operators which represent the physical observables, the reduction formulas, and vacuum expectation values for the particle fields. The formalism is sufficiently general so that it holds for fields, or a superposition of fields, of arbitrary mass and spin.
Phase-Plane Analysis of Nonlinear, Second-Order, Ordinary Differential Equations
Authors: Dresner Lawrence
Published: February 2004

Group invariance is used to analyze the solutions of several well-known differential equations.
Propagators of Unrenormalizable Fields
Authors: Pfaffelhuber E.
Published: February 2004

A general definition of propagators of causal unrenormalizable fields corresponding to rapidly increasing spectral functions is given in terms of indicatrices. The order of growth of the propagators is determined in terms of that of the spectral functions. The high energy bound is rigorously proved and the indeterminacy of the propagators is shown to be concentrated at the origin of the light cone. Theorems are proved concerning the approximation of unrenormalizable propagators by renormalizable ones.
Remarks on a Paper of Srivastava
Authors: Gould H. W.
Published: February 2004

It is pointed out in this note that two series studied by Rosenbaum [J. Math. Phys. 8, 1977 (1967)] and later given rapid, elementary proofs by the present writer [J. Math. Phys. 10, 49 (1969)] are virtually the same series because of properties of the factorial ratio involved. This shows that the very complicated generalized series introduced by Srivastava [J. Math. Phys. 11, 2225 (1970)] is not needed, besides which his general series seems not possible to sum in general and serves no purpose other than to subsume the two original series in a single series. Since the series are, in fact, almost identical, this is simpler than Srivastava's formula makes out. Finally, the proofs in the writer's previous note are again discussed for their value as elementary techniques avoiding complicated infinite series and unnecessary generalizations.
On the Construction of Orthogonal Matrix Basis Elements of Finite Group Algebras Symmetry Adapted to a Subgroup
Authors: Sullivan John J.
Published: February 2004

The matrix basis elements of a finite group g(Γ, mn) are put in a form |Γm〉 〈Γn| suggestive of a dyadic in operator space. It is shown that if a subgroup H exists for which Γ ↓ H = Σγi, then these operators may be symmetry adapted to the subgroup via |Γγjm〉 Qj†ba. Necessary and sufficient conditions for constructing the elements Qj†, b and a are given. Particular application of the procedure to the bipartition representations of the symmetric group is made.
Microscopic Theory of a Multicomponent System of Charged and Neutral Particles. I. General Quantum Statistical Formulation
Authors: Mohling Franz , Smith Charles Ray
Published: February 2004

Recent contributions to the Lee-Yang-Mohling theory of single-component quantum fluids have enabled us to develop a new theory of the quantum statistics for a multicomponent nonrelativistic system of charged and neutral particles in thermal equilibrium. With the emphasis as much as possible on the physical content of the theory, this paper presents the new formulation of quantum statistics with explicit rules for calculating the grand potential and particle and photon momentum distributions. The present formalism not only simplifies and corrects an earlier version, but also it has made possible clear and systematic procedures for resolving some divergence difficulties that occur in the many-body theory of fully ionized gases.
Microscopic Theory of a Multicomponent System of Charged and Neutral Particles. II. Investigation of Properties of Photons
Authors: Bhowmik Anup , Smith Charles Ray
Published: February 2004

On the basis of the master-graph formulation of quantum statistics in the preceding paper, the properties of photons in a nonrelativistic, multicomponent, fully ionized gas in thermal equilibrium are investigated. The photon self-energy is calculated by summation, to all orders, of selected diagrams, and it is proposed that the solution is formally exact. Next, the photon momentum distribution is calculated, in the high-temperature, low-density limit, to second order in the fine-structure constant. Several nonperturbative results are obtained which are significant even in lowest order. The lowest-order results have an interesting relation to the pair-Hamiltonian approximation and the Bogoliubov transformation, and this relation is discussed. Finally, the counterterm technique of the master-graph theory is employed to accomplish mass renormalization through second order in the fine-structure constant. The investigation is valid for particle densities ρ ≪ 1024 particles/cm3 and for absolute temperatures T ≤ 106°K (but large enough for a high degree of ionization).
Conformal Group in a Poincaré Basis. I. Principal Continuous Series
Authors: Macfadyen N. W.
Published: February 2004

In the first of a series of papers on the global representation theory of SU(2, 2), with particular emphasis on the Poincaré subgroup, we study the two principal continuous series of unitary irreducible representations. These are defined by operators on Hilbert spaces of functions over a six-dimensional manifold, and after an automorphism of the group by a real orthogonal transformation [in precise analogy to that used in the mapping SU(1, 1) → SL(2, R)] we display these in such a form that the Poincaré subgroup SL(2, C) × T4 appears in a simple fashion. The generators of translations, dilations, Lorentz and special conformal transformations are given as differential operators, and by using these we find explicit expressions for the eigenvalues of the three Casimir operators of the group. Finally, we perform the reduction of the two series when restricted to the Poincaré group. It is found that all those principal series representations of P enter which allow a certain value of helicity.
S Matrix and Low Energy Theorem in the Theory of Correlation Functions
Authors: Dashen Roger , Ma Shang-keng
Published: February 2004

We have examined the current correlation functions is statistical mechanics from a scattering theory viewpoint. In the light of low energy theorems, we study the connection between the long wavelength, low-frequency correlation functions of conserved currents and the on-shell scattering matrix elements which describe the collision processes taking place in the thermodynamical system of interest. Explicit general formulas are derived for the leading correction terms to the ideal gas correlation functions. These correction terms include the effect of two-body scatterings and double scatterings in three-body events. They are expressed in terms of two-body cross section, time delay, and the quantum analog of the distance of closest approach. Their connection to the second virial coefficient is established through sum rules. Relationship between our results and the Boltzmann equation is discussed.