bluuebunny/crossref_metadata_2025 huggingface dataset:
Bound States of a New Kind in Quantum Field Theory
Authors: Zimmerman Robert L.
Published: February 2004

It is the purpose of this article to study the various kinds of bound states and their properties that can occur in quantum field theory. The ability of quantum field theory to create and destroy particles makes the picture of a bound state very complex. However, for this very reason it allows for a larger variety of bound states than can occur in nonrelativistic quantum mechanics. Not only do we consider those bound states that appear as poles in the tau functions, but also as branch points. The bound states that appear as branch points offer many new possibilities in constructing composite theories. An example of this type of bound state is given in which the fermion is a bound state of a boson. The many ambiguities that exist with this kind of bound state are illustrated. In discussing the bound states that appear as poles we have broken them into three subgroups: N-body bound states, mixed bound states, and self-interacting bound states. The N-body bound states are the most analogous to those occurring in nonrelativistic quantum mechanics. The mixed bound states have the peculiar property of not being composed of any unique group of elementary particles. The self-interacting bound states are those that are directly coupled with themselves. The characteristics of these subgroups of bound states are discussed and illustrated.
Partial-Wave Analysis in Terms of SL(2, C)/SU(2) Harmonic Functions
Authors: Maduemezia Awele
Published: February 2004

A new scheme for analyzing the scattering amplitude in the S channel of a two-body elastic collision in terms of the representation functions of the covering group of the homogeneous Lorentz group is presented. The scheme uses representation functions defined on the homogeneous space SL(2, C)/SU(2), which for the equal-mass case considered here is the same as the hyperquadric q2−q02=−m2, where m is the particle mass, q is the relative center-of-mass momentum (in the initial or final state), and 2q0 = S½ is the center-of-mass energy. The corresponding representations are multivalued and belong to the so-called degenerate (but not most-degenerate) series. The scattering amplitude derived has the correct threshold behavior in the S variable.
Analytic Continuation of Appell's Hypergeometric Series F2 to the Vicinity of the Singular Point x = 1, y = 1
Authors: Hahne G. E.
Published: February 2004

The infinite series, absolutely convergent if |x| + |y| < 1, for Appell's F2 (α, β, β′, γ, γ′; x, y) is analytically continued into a linear combination of four infinite series in powers of (x - 1) and (y - 1); each of the latter four series is absolutely convergent if |x − 1| + |y − 1| < 1. The analytic continuation is carried out by manipulation of the Mellin-Barnes integral representations for the hypergeometric functions appearing in the course of the calculation.
Finite Transformations of SU3
Authors: Holland Douglas Francis
Published: February 2004

The properties of SU3 finite transformations are investigated. These transformations on the defining three-dimensional complex space are parameterized in a form employing three factors, two of which are the Euler parameterization of an SU2 subgroup. The irreducible representations of the factored parameterization are found explicitly. The volume element is calculated and the orthogonality relation is verified. Spherical harmonic basis states are derived as a specialization of the transformation matrix. Another result is a definition of triality and a simple proof that it is additive modulus three.
Korteweg-de Vries Equation and Generalizations. III. Derivation of the Korteweg-de Vries Equation and Burgers Equation
Authors: Su C. H. , Gardner C. S.
Published: February 2004

The Korteweg-de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean-invariant systems under the weak-nonlinearity and long-wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.
S-Matrix Singularity Structure in the Physical Region. III. General Discussion of Simple Landau Singularities
Authors: Bloxham M. J. W. , Olive D. I. , Polkinghorne J. C.
Published: February 2004

Drawing on the understanding of unitarity integrals acquired in previous papers [J. Math. Phys. 10, 494, 545 (1969)], we show that unitarity demands that connected amplitudes be singular on the positive-α arcs of all ``simple'' Landau curves in their physical regions. Further, we show that the amplitudes are nonsingular on mixed-α arcs, while on the positive-α arcs their discontinuities are given by the Cutkosky rule. This confirms arguments from perturbation theory and demonstrates how a weak analyticity assumption can generate in an exact way a singularity scheme relating to causality.
Lower Bounds to the Ground-State Energy of Systems Containing Identical Particles
Authors: Calogero F. , Marchioro C.
Published: February 2004

Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
Invariant Imbedding and Case Eigenfunctions
Authors: Pahor S. , Zweifel P. F.
Published: February 2004

A new approach to the solution of transport problems, based on the ideas introduced into transport theory by Ambarzumian, Chandrasekhar, and Case, is discussed. To simplify the discussion, the restriction to plane geometry and one-speed isotropic scattering is made. However, the method can be applied in any geometry with any scattering model, so long as a complete set of infinite-medium eigenfunctions is known. First, the solution for the surface distributions is sought. (In a number of applications this is all that is required.) By using the infinite-medium eigenfunctions, a system of singular integral equations together with the uniqueness conditions is derived for the surface distributions in a simple and straight-forward way. This system is the basis of the theory. It can be reduced to a system of Fredholm integral equations or to a system of nonlinear integral equations, suitable for numerical computations. Once the surface distributions are known, the complete solution can be found by quadrature by using the fullrange completeness and orthogonality properties of the infinite-medium eigenfunctions. The method is compared with the standard methods of invariant imbedding, singular eigenfunctions, and a new procedure recently developed by Case.
Realization of Chiral Symmetry in a Curved Isospin Space
Authors: Meetz Kurt
Published: February 2004

The nonlinear realizations of the chiral group SU(2)⊗SU(2) are studied from a geometric point of view. The three-dimensional nonlinear realization, associated with the pion field, is considered as a group of coordinate transformations in a three-dimensional isospin space of constant curvature, leaving invariant the line element. Spinor realizations in general coordinates are constructed by combined coordinate-spin-space transformations in analogy to Pauli's method for spinors in general relativity. The description of vector mesons and possible chiral-invariant Lagrangians, which yield the various nonlinear models in specific frames of general coordinates, are discussed.
Solution of the Bethe-Salpeter Equation with a Square-Well Potential
Authors: Keam R. F.
Published: February 2004

The ladder approximation Bethe-Salpeter equation for (i) a bound spin-0 boson-boson system of zero total mass and (ii) a bound spin-½ fermion-antifermion system of zero total mass is solved for a four-dimensional square-well potential.
Unitary Representations of the Poincaré Group Contained in a Class of Representations of the Conformal Group
Authors: Klink William H.
Published: February 2004

A unitary irreducible class of representations of the conformal group is constructed and reduced with respect to the Poincaré group to see which unitary irreducible representations of the Poincaré group it contains. In particular, it is shown that this class of representations of the conformal group does not contain the continuous-spin representations of the Poincaré group. It is concluded that the representations of the conformal group cannot be used to eliminate the continuous-spin representations.
Powers of the D Functions
Authors: Gomatam Jagannathan , Rohrlich F.
Published: February 2004

While powers of distributions in general do not exist, [D+]n and [D−]n are the exceptions among all D functions; they do exist and are given explicitly. A ``modified'' power [DΓ]n based on the representation of DΓ as analytic functionals can be defined. It exists for all homogeneous DΓ, i.e., for D±, D, and D1. The relation of [D±]n to [D±]n is given, and explicit expressions are found in x space and in p space. The Källén-Lehmann-Umezawa-Kamefuchi representation of these distributions is derived. The extent to which these considerations are applicable to the ΔΓ(x, m), m > 0, is discussed.
Renormalization of a Finite Matrix Hamiltonian
Authors: Peres Asher
Published: February 2004

We investigate the eigenvalues of a finite matrix Hamiltonian H = H0 + g0V, where H0 is diagonal with eigenvalues 1, 2, …, N, and where all the elements of V are equal to 1. We are interested in the case N → ∞. The radius of convergence of the perturbation series is (ln N)−1, but nevertheless the exact eigenvalues of H tend to well-defined limits when N → ∞. It is shown that if we define g=(g0−1+lnN)−1 and if we let g0 → 0 as N → ∞ in such a way that g is constant, then it is possible to obtain a perturbation series with the ``renormalized'' coupling constant g, provided that suitable counter terms are introduced. We also investigate a different model (where Vmn = mn) and show that no such renormalization is possible there.
Ground-State Many-Boson Problem with Repulsive Potentials
Authors: Eyges Leonard
Published: February 2004

We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.
Functional Methods in Statistical Mechanics. I. Classical Theory
Authors: Brittin W. E. , Chappell W. R.
Published: February 2004

A statistical-mechanical theory of fields is developed. Since a field has an infinite number of degrees of freedom, it is natural and convenient to use functional methods for its description. The most general statistical-mechanical state for a field is represented by a distribution functional which satisfies a functional differential equation analogous to the Liouville equation. The functional Fourier transform (characteristic functional) is introduced and its properties are studied. Multitime functionals and various reduced distribution functions are also discussed. The formalism is applied to the free electromagnetic fields as well as to a system of charged particles (plasma) interacting via the electromagnetic field.
Functional Integrals Representing Distribution Functions in Statistical Mechanics
Authors: Blomberg Clas
Published: February 2004

We show how to obtain formal solutions of the chain of equations for distribution functions in classical statistical mechanics. These solutions are in the form of complex functional integrals. They are not unique, which fact is a fundamental property of the equations, and the different solutions are recognized by different integration paths in the complex function space. The different manners of integration correspond to different phases, of which some can be identified with the possible physical states. The treatment of the integrals in some cases is also discussed. They are closely related to generalizations of the molecular field approach to the problem. It is also shown that the functional integrals can be written as averages over an external field and that essentially the same form is valid in the quantum-mechanical case.
SU
3 Algebra in the Mixed-Shell Space (0p, 0d, 1s). I
Authors: Ishidzu Takehiko , Kawarada Hideo , Sato Masachiyo
Published: February 2004

The operators of Elliott are extended in such a way that they also describe an SU3 algebra in the mixed-shell space M*, where M* is given as the direct sum of two spaces M(0p) and M(0d,1s) spanned by the single-particle wavefunctions in the (0p) and (0d, 1s) shells of a harmonic-oscillator potential. The representation of SU3 in this space M* is investigated in detail by the aid of the weight diagram in a way analogous to that of Banerjee and Levinson. The basis is expressed in an explicit manner using the one-particle wavefunctions in the usual shell model. The states arising from two- and three-particle systems are classified according to the irreducible representations in this extended space.
Multiple Scattering of Electromagnetic Waves in an Underdense Plasma
Authors: Watson Kenneth M.
Published: February 2004

Scattering of electromagnetic waves by an extended underdense plasma is studied. The analysis begins with expressions for multiple scattering of waves. An explicit account of coherent scatterings leads to modified equations. These modified equations are used to derive a transport equation for the intensity (a tensor expressed in polarization components). It is shown that this transport equation may be applied to the calculation of radar backscatter.
Sufficiency Condition for the Validity of the WKB Approximation
Authors: Northover F. H.
Published: February 2004

In spite of the many applications—in theoretical physics and elsewhere—of this well-known method of approximating to solutions of the differential equation W″ + fW = 0, when f is slowly varying through a region where f ≠ 0, no simple sufficient condition for its validity appears yet to have been given. In the present paper, such a condition is derived. Also, the connection formulas establishing the relation between the constants in the WKB approximation to a given solution in the various regions of the complex plane delineated by the ``Stokes lines,'' which converge on a simple zero of f, are rederived, as existing arguments are open to criticism on at least two major grounds. Finally, a sufficient condition is given for the existence of a common region of validity for the approximation based on the series solutions of the differential equation around a zero of f and the WKB approximation valid sufficiently far from this zero.
First-Order Approximation to a Spherically Symmetric Solution of Einstein's Equations
Authors: Stabell R.
Published: February 2004

In analogy with the metrics of Schwarzschild and de Sitter, a first-order approximation to the spherically symmetric solution of Einstein's field equations is found where the deviation from the flat Minkowski space-time is given by potentials of the gravitational force and the cosmical Λ force. It is shown that in the Newtonian approximation the forces acting per unit mass of the fluid are the assumed gravitational self-attraction and cosmical Λ force.
Random Walks on Lattices. III. Calculation of First-Passage Times with Application to Exciton Trapping on Photosynthetic Units
Authors: Montroll Elliott W.
Published: February 2004

The following statistical problem arises in the theory of exciton trapping in photosynthetic units: Given an infinite periodic lattice of unit cells, each containing N points of which (N − 1) are chlorophyll molecules and one is a trap; if an exciton is produced with equal probability at any nontrapping point, how many steps on the average are required before the exciton reaches a trapping center for the first time? It is shown that, when steps can be taken to near-neighbor lattice points only, as N → ∞, our required number of steps is 〈n〉∼{N2/6,       linear chain,π−1NlogN,  square lattice,1.5164N,     single cubic lattice.The correction terms for medium and relatively small N are obtained for a number of lattices.
Partial Separation of the Schrödinger Equation for Two Charged Particles in a Magnetic Field
Authors: Carter Benjamin P.
Published: February 2004

An effective Hamiltonian is derived which describes the relative motion of two oppositely charged particles in a homogeneous magnetostatic field.
Considerations on the Classical Spinning Electron
Authors: Cohn Jack
Published: February 2004

The present work is concerned with the asymptotic properties of the field radiated by a classical spinning (and nonspinning) point charge in arbitrary motion. It is demonstrated that the expression found for the radiated angular momentum is satisfactory in that it is a 4-tensor which is spacelike surface-independent. The angular-momentum radiation rate is then expressed in terms of the retarded kinematic properties of the particle. In addition, a ``center-of-energy'' theorem is proved for nonspinning charges in arbitrary motion—a result which follows from the demonstrated result that nonspinning charges also radiate angular momentum. Finally, it is demonstrated that the (linear) energy-momentum radiation rate is independent of the spin of the involved charge. The discussion is manifestly covariant throughout and many mathematical details are deferred to appendices.
Criteria for Slater Determinants and Quasiparticle Vacuum States
Authors: Navon A.
Published: February 2004

New criteria characterizing Slater determinants and quasiparticle vacuum states are obtained. These criteria are expressed as quadratic homogeneous equations in the coefficients of the development of the trial wavefunction in a basis of Slater determinants. A redefinition of quasiparticle vacuum states permits the introduction of quasiparticle transformations which are more general than the generalized Bogoliubov transformations.
Lie-like Approach to the Theory of Representations of Finite Groups
Authors: Gamba A.
Published: February 2004

Using characters, one can set up the theory of representations for both finite and continuous groups. For continuous groups, another approach—the Lie theory—is also possible. It is shown that a similar theory, based on commutators, can be developed also in the case of finite groups. Essentially, the group algebra of a finite group is converted into a Lie algebra by replacing the usual associative product by the product x∘y=xy−yx. Then the resulting Lie algebra is a direct sum of special unitary Lie algebras.
Analytical Solutions of the Neutron Transport Equation in Arbitrary Convex Geometry
Authors: Gibbs Alan G.
Published: February 2004

The integral equation describing the transport of monoenergetic, isotropically scattered neutrons in a one-, two-, or three-dimensional body of arbitrary convex shape, containing distributed sources, is considered. An exact representation of the neutron density ρ(r) is obtained, involving a superposition of functions belonging to the null space of a simple differential operator. In general, when a countable basis is chosen to span the null space, the coefficients in the expansion of ρ(r) satisfy a coupled system of singular integral equations which is reducible to a system of Fredholm equations. If no sources are present, an exact criticality condition is also obtained. Some techniques for evaluating the expansion coefficients are given and several examples are considered.
Expansion of Lie Groups and Representations of SL(3, C)
Authors: Mukunda N.
Published: February 2004

We show that in any unitary representation of SL(3, C) belonging to the principal series, the generators can be expressed as functions of the generators of a suitably chosen unitary representation of the non-semisimple group SU(3) × T8. Both the nondegenerate and the degenerate series are considered. Since the representations of SU(3) × T8 in an SU(3) basis are known, this provides a solution to the multiplicity problem in setting up the SL(3, C) representations in an SU(3) basis.
Analytic Representations of Two-Point Functions with Noncanonical Light-Cone Singularities. I
Authors: Lukierski Jerzy
Published: February 2004

We introduce the classes of analytic functions describing by means of its boundary values the sets of commutator functions with singularities δ(k)(x2) and (x2)+−k(k=1,2,…) on the light cone. Their relation with the Källen-Lehmann representation having nonintegrable spectral functions is given. The generalized wave-renormalization constants, measuring noncanonical singularities, are introduced. The formulas exhibit in a clear way the light-cone behavior and provide a proper scheme for studying the equal-time limits.
Determination of Invariant Amplitudes from Experimental Observables
Authors: Moravcsik Michael J. , Yu Wing-Yin
Published: February 2004

Necessary and sufficient conditions are formulated for deciding whether a set of 2n − 1 bilinear products of n complex amplitudes determines these amplitudes with no continuum of ambiguity or not. The conditions can be translated into very simple geometrical prescriptions which in most cases provide quick and easy practical tests for such decisions.
Hamilton-Dirac Theory of Hamilton's Equations
Authors: Cawley Robert G.
Published: February 2004

It is shown that in a very general way two distinct canonical formalisms can be used to describe a classical system. No corresponding nonuniqueness is introduced into the canonical quantization procedure if the Dirac bracket correspondence to the quantum-mechanical commutators is employed.
Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. II
Authors: Gutzwiller Martin C.
Published: February 2004

The phase-integral approximation of the Green's function in momentum space is investigated for a particle of negative energy (bound state) which moves in a spherically symmetric potential. If this potential has a Coulomb-like singularity at the origin, it is shown that any two momenta can be connected by an infinity of classical trajectories with a fixed energy. The summation of the usual phase and amplitude factors over these trajectories is the approximate Green's function. If there is orbital precession, there are not only poles along the negative energy axis, but also weaker singularities which are not examined in detail. The poles are found at the energies which are given by the semiclassical quantum conditions: angular momentum = (l + ½)ℏ and action integral for the radial motion = (n + ½)2πℏ, where l and n are integers ≥ 0. The residues at these poles give the approximate bound-state wavefunctions as a product of the asymptotic formula for Legendre polynomials with the asymptotic solution of the radial Schrödinger equation. It is conjectured that the occurrence of poles in the approximate Green's function is a direct consequence of the periodic character of the classical motion.
Representations of the Orthogonal Group. II. Polynomial Bases for the Irreducible Representations of the Orthogonal Group
Authors: Wong M. K. F.
Published: February 2004

Polynomial bases for the irreducible representations of the orthogonal group, which are characterized by the Gel'fand pattern, have been obtained. The method used is very similar to Moshinsky's and is a generalization from the unitary group to the orthogonal group. The Wigner coefficients of O(3), commonly called the Clebsch-Gordan coefficients of R(3), are rederived by means of the polynomial bases obtained in this paper.
Group Representation in a Continuous Basis: An Example
Authors: Itzykson C.
Published: February 2004

Given an irreducible unitary representation of a noncompact group, what happens if one tries to diagonalize one of the noncompact generators? We study some aspects of this question on an example, chosen to be a representation of the discrete series with j = −½ of the special real linear group in two dimensions.
Gel'fand-Kirillov Conjecture on the Lie Field of an Algebraic Lie Algebra
Authors: Chow Yutze
Published: February 2004

This is an article written to review with sufficient detail the so-called Gel'fand-Kirillov conjecture concerning the isomorphisms between the quotient fields of the algebras generated by canonical variables [pi, qj] = δij1 and the quotient fields of the universal enveloping algebras of algebraic Lie algebras. This conjecture sheds new light on the relation between the universal enveloping algebra of an algebraic Lie algebra and the Lie algebras of dynamic groups in quantum mechanics.
Equations of the de Broglie Wavefield in the Case of Spherical Symmetry
Authors: Kulhanek J.
Published: February 2004

Two methods of integrating the equations of the de Broglie wavefield are introduced. The first integrating method does not seem to be as suitable for the domain of the microphysics as the second one. According to the second integrating method, the equations of the de Broglie wavefield are solved for the metric with spherical symmetry. In this case, the equations of the de Broglie wavefield admit of a solution which includes both Schrödinger and Hamilton-Jacobi methods of description as particular cases.
Wave Equations, Multipole Solutions, and Characteristic Hypersurfaces for Massless Particles
Authors: Hajj Jack P. , Weaver David L.
Published: February 2004

In this paper, the canonical wave equation and auxiliary conditions for a massless particle with arbitrary integral or half-integral spin s are derived from a symmetric spinor formulation. The wave equation is solved for the multipole solutions using the appropriate parity operator for each spin. Finally, the characteristic hypersurfaces for the wave equation are derived and found to agree with that of the photon. The special cases of spins one, three-halves, and two are detailed in conclusion.
SU(6) Clebsch-Gordan Coefficients for the Product 35⊗70
Authors: Carter J. C. , Coyne J. J.
Published: February 2004

A method is presented which makes explicit use of Young diagrams to calculate SU(3)⊗SU(2) multiplet-coupling coefficients in SU(6). The multiplet-coupling coefficients for 35⊗70 are given.
Covariant Electromagnetic Potentials and Fields in Friedmann Universes
Authors: Peters P. C.
Published: February 2004

Electromagnetic potentials and fields are found for arbitrary four-current densities in Friedmann universes. A choice of gauge is made so that the potentials are similar to the flat-space potentials. A formalism is developed which allows the construction of potentials and fields which are covariant with respect to spatial transformations. It is shown explicitly how these potentials are related to the flat-space potentials through conformal and gauge transformations. Some features of the solutions in the finite models are discussed with reference to problems of interpretation raised recently by Katz.
Involutional Matrices Based on the Representation Theory of GL(2)
Authors: Kim Shoon Kyung
Published: February 2004

Involutional matrices M(a, b, c) with three arbitrary parameters are introduced, based on a matrix representation M(R) of the linear homogeneous transformation R ∈ GL(2). Symmetry properties, eigenvalues, and recursion formulas for the representation M(R) are obtained and specialized to the involutional matrices M(a, b, c). A set of special involutional matrices A(ξ), B(ξ), C(ξ), and E(ξ) with one arbitrary parameter ξ are introduced as special cases of M(a, b, c). Their relations are discussed.
Partial-Wave Expansion in the Crossed Channel for Scattering Amplitudes Invariant under the Galilei Group
Authors: Gervais J. L. , Velo G.
Published: February 2004

A partial-wave analysis in the crossed channel is performed for a Galilean-invariant scattering matrix using the irreducible unitary representations of the Euclidean group in two dimensions. As in the relativistic case, a formula is obtained which is useful in determining the high-energy behavior of the scattering amplitude for fixed-momentum transfer. In particular, the Born term is shown to correspond to a δ function in this representation. Moreover, this parametrization is related by a group contraction to the corresponding background term of the relativistic case.
Diffraction of Electromagnetic Waves by a Right-Angle Dielectric Wedge
Authors: Kraut Edgar A. , Lehman Guy W.
Published: February 2004

The problem of diffraction of a plane-polarized electromagnetic wave incident on a right-angle dielectric wedge is formulated as a singular integral equation in k space. A solution of the singular integral equation is constructed as a power series in the index of refraction. This series converges when the index of refraction is near unity. Using this solution, the electric-field amplitude at the tip of the wedge is examined. We also prove as incorrect a closed-form analytic expression claimed in the literature to be a global solution of the problem considered here.
Motion of a Charged Particle in a Spatially Periodic Magnetic Field
Authors: Coffey Timothy P.
Published: February 2004

Degenerate perturbation theory is employed to discuss the motion of a charged particle in a constant magnetic field on which is superimposed a weak, transverse, spatially periodic magnetic field. A first-order solution of the equations of motion is presented. It is shown that the secular motion is periodic in time. The significance of this result with respect to the stability of protons in the inner Van Allen belt is discussed.
Critical Behavior of Several Lattice Models with Long-Range Interaction
Authors: Kac Mark , Thompson Colin J.
Published: February 2004

We consider a one-dimensional model with infinite-range interaction, a two-dimensional model, and a three-dimensional model, whose free energies can be expressed in terms of the largest eigenvalue of an integral equation. High- and low-temperature expansions in powers of the reciprocal of the range of the exponential part of the interaction, with the classical Curie-Weiss theory as leading term, are developed and studied in the critical region. We find that to leading order in the critical region the resummed high- and low-temperature expansions are analytic at the classical critical point but are nonanalytic at a displaced critical point. The modified singularities, which are no longer of Curie-Weiss type, give critical exponents which are identical with those obtained by Brout and others, and are almost surely not the true exponents. The technique, however, suggests a possible general method of successive approximation to true critical behavior.
On Next-Nearest-Neighbor Interaction in Linear Chain. I
Authors: Majumdar Chanchal K. , Ghosh Dipan K.
Published: February 2004

Ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2(σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground-state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest- as well as the next-nearest-neighbor interaction.
On Next-Nearest-Neighbor Interaction in Linear Chain. II
Authors: Majumdar Chanchal K. , Ghosh Dipan K.
Published: February 2004

Continuing our work on the ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2, −1≤α≤1,we have completed the study of 10 spins. The results of short-chain calculations provide better upper and lower bounds of the ground-state energy per particle as N → ∞, but no simple formula can be fitted to the data to get this limit for all α. For J > 0 and α = ½, however, this is exactly found to be −¾J. Some upper and lower bounds for the free energy are also derived.
Inverse Functions of the Products of Two Bessel Functions
Authors: Sollfrey William
Published: February 2004

Special cases of the inverse function of the product of two spherical Bessel functions have been found recently by other writers as 1F2 hypergeometric functions. We give the general expression as the derivative of a product of spherical Bessel functions. These results have also been found in the classical literature.
Mth Power of an N × N Matrix and Its Connection with the Generalized Lucas Polynomials
Authors: Barakat Richard , Baumann Elizabeth
Published: February 2004

The Mth power of an N × N matrix is expressed via the Cayley-Hamilton theorem as a linear combination of the lower powers of the matrix. The polynomial coefficients of the lower powers of the matrix are expressed in terms of polynomials in N variables, termed the generalized Lucas polynomials. The independent variables in the generalized Lucas polynomials are the traces of the lower powers of the matrix.
Bounds Associated with Integral Equations of Kirkwood-Riseman Type
Authors: Anderson N. , Arthurs A. M. , Robinson P. D.
Published: February 2004

Variational principles are used to find approximate solutions of integral equations arising in the Kirkwood-Riseman theory of the properties of flexible macromolecules. Upper and lower bounds are obtained for the translational diffusion constant.
Initial Conditions in General Relativity: Lapse and Shift Formulation
Authors: Belasco Elliot P. , Ohanian Hans C.
Published: February 2004

We examine the system of coupled differential equations to which the constraints on the Cauchy data reduce if expressed in terms of the ``shift'' vector Nk and ``lapse'' N0. If (3)gij and ∂ (3)gij/∂t are given and Dirichlet boundary conditions are imposed, the solution Nk is found to be unique if 2 × (energy density) − (three-curvature) > 0, but need not be unique when this inequality is not satisfied. No general existence theorem is known, but we list some conditions which make solutions impossible.
Single-Particle Condensate and Pair-Correlation Theory of Inhomogeneous Boson Systems
Authors: Kobe Donald H.
Published: February 2004

A theory of inhomogeneous boson systems is developed which takes into account both the single-particle condensate and pair correlations. A set of three coupled nonlinear integrodifferential equations is derived for the condensate wavefunction and the two Bogoliubov quasiparticle wavefunctions. Half-integral h/m circulation can exist only when the single-particle condensate is completely depleted. General considerations of off-diagonal long-range order and circulation quantization also lead to this conclusion.
Equal-Time Pseudolimits of Current Commutators and Schwinger Terms
Authors: Otterson Paul
Published: February 2004

The equal-time limits of some current commutators are studied in perturbation theory. Although the orthodox equal-time limit does not exist as a distribution, a Schwinger term may be defined as a pseudolimit—a continuous linear functional on a proper subspace of S(R3). The results are compared with those obtained from alternative approaches. Simple examples are given to explain why these methods give incorrect results for the cases we consider.
Conserved Quantities in the Einstein-Maxwell Theory
Authors: Exton Albert R. , Newman Ezra T. , Penrose Roger
Published: February 2004

It is shown that the 10 gravitationally-conserved quantities defined in asymptotically flat, empty, space-times are, when suitably modified, also conserved in asymptotically flat Einstein-Maxwell space-times. Furthermore, the implied selection rules for transitions between stationary Einstein-Maxwell states are the same as those in the pure gravitational case.
On the Wigner Supermultiplet Scheme
Authors: Hecht K. T. , Pang Sing Chin
Published: February 2004

Calculation of Wigner and Racah coefficients for the group SU(4)⊃[SU(2)×SU(2)] make it possible to perform the spin—isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one- and two-body operators in the Wigner supermultiplet scheme. The SU(4) coefficients needed to evaluate one- and two-particle cfp's, the matrix elements of one-body operators, and the diagonal matrix elements of two-body operators are calculated in general algebraic form for many-particle states characterized by the SU(4) irreducible representations [yy0], [y y − 1 0], [yy1], [y11], [y y − 1 y − 1], [y10], [yy y − 1], [y00], and [yyy], whose states are specified completely by the spin and isospin quantum numbers (y = arbitrary integer). Applications are made to the calculation of the matrix elements of the complete space-scalar part of the Coulomb interaction and the space-scalar part of the particle-hole interaction for nucleons in different major oscillator shells.
New Formulation of Stochastic Theory and Quantum Mechanics
Authors: de la Peña-Auerbach L.
Published: February 2004

The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
Properties of ``Quadratic'' Canonical Commutation Relation Representations
Authors: Klauder John R. , Streit L.
Published: February 2004

A class of representations of the canonical commutation relations is studied, each of which is characterized by an expectation functional that is the exponential of a Euclidean-invariant quadratic form of the test functions. The underlying field operators are realized as the direct product of two Fock representations and the consequences of this realization are analyzed. Compatible Hamiltonians are constructed and an extensive study of the most general quadratic Hamiltonians is presented. In order to include thermodynamic examples, the analysis includes indefinite Hamiltonian spectra as well as the usual definite spectra. Finally, conditions are given for a theory to be local in the sense that all time derivatives of the field operator commute with one another at equal times but unequal spatial arguments.
Conserved Quantities of Newman and Penrose
Authors: Robinson David C.
Published: February 2004

Green's theorem is used to obtain the Newman-Penrose constants in flat and asymptotically flat space-time and to investigate the invariant transformations which these quantities generate. The zero-rest-mass free fields and the coupled Einstein-Maxwell fields are considered and the relationship of this approach to Noether's theorems is discussed.
Degenerate Representations of the Symplectic Groups II. The Noncompact Group Sp(p, q)
Authors: Pajas P.
Published: February 2004

Using a method based on geometrical properties of homogeneous spaces of rank one homeomorphic to coset spaces of Lie groups, a series of degenerate unitary irreducible representations of the noncompact symplectic group Sp(p, q) is investigated. The representation spaces for a discrete series determined by two integer numbers and a continuous series determined by one real and one integer parameter are given, the corresponding basis functions being formed by the linear combinations of eigenfunctions of the Laplace-Beltrami operator of the considered space. Explicit formulas for the action of generators of Sp(p, q) in these representations are obtained. The results provide a deeper insight into the structure of the two-parameter ``not most degenerate'' unitary representations.
New Methods for Reduction of Group Representations. III
Authors: Gabriel John R.
Published: February 2004

The reduction of a space S into its irreducible components S = Σ S(α, p) with respect to (w.r.t.) a group G is discussed in the case where G is the union of two subgroups G1 and G2, and S has already been reduced w.r.t. G1 and G2 separately.
Wiener-Hopf Method and Generalized Integral Transforms
Authors: Bassetto A. , Paccanoni F.
Published: February 2004

We consider the class of singular integral equations that can be solved by means of the Wiener-Hopf technique. The resolvent is given as a dispersion relation in the plane of the parameter λ and some physical applications are discussed.
Generalized Addition Theorem for Spherical Harmonics
Authors: Clapp Roger E.
Published: February 2004

In analogy with the familiar addition formula for Legendre polynomials, a generalized addition theorem is proved. A general spherical harmonic, depending on the two angles θ1 and φ1, is expressed as an expansion involving spherical-harmonic functions of (θ, φ) and of (θ′, φ′). The six angles are related to each other through the equationscosθ1=cosθcosθ′−sinθsinθ′cos(φ′−φcosθ),sinθ1cos(φ1−φ)=sinθcosθ′+cosθsinθ′cos(φ′−φcosθ),sinθ1sin(φ1−φ)=sinθ′sin(φ′−φcosθ).The expansion is then used in the proof of an integral theorem for spherical harmonics.
Relativistic Continuum Theory for the Interaction of Electromagnetic Fields with Deformable Bodies
Authors: Grot R. A.
Published: February 2004

A variational principle is formulated which yields the balance laws and constitutive equations of a nonconducting, charge-free elastic solid interacting with electromagnetic fields. It is found that the form of the total energy-momentum tensor and the constitutive equations that follow from a Lagrangian action which depends arbitrarily on the inverse deformation gradients and the electromagnetic field tensor are identical to those obtained by formulating a constitutive theory of a nondissipative material based on the basic mechanical, thermodynamical, and electromagnetic balance laws of a continuum.
New Method for Finding Eigenvalues
Authors: Anderson J. T.
Published: February 2004

The eigenvalue equation of a function of a single complex variable is shown to be equivalent to the Cauchy-Riemann equations so that the eigenvalue problem is reducible to the problem of finding the analytic regions of the function. The eigenvalues of a function of several complex variables are also found from the analytic regions on each plane. A self-consistent treatment of two relativistic fields is developed and applied to the interaction of spinor and scalar fields.
Degeneracy of the Dirac Equation with Electric and Magnetic Coulomb Potentials
Authors: Berrondo Manuel , McIntosh Harold V.
Published: February 2004

Investigation is made of the symmetry and degeneracy of the Dirac equation for a Coulomb potential with a fixed center bearing both electric and magnetic charge. Seen from the viewpoint of classical mechanics, relativistic precession removes the accidental degeneracy of the nonrelativistic potential, and may be so severe as to lead to spiral rather than precessing elliptic orbits. The degeneracy may be restored by the introduction of a vector potential which combats the precession and leads to closed relativistic orbits. An angular momentum and a Runge vector are found for the ``symmetric'' potential for arbitrary values of electric and magnetic nuclear charges. A related symmetric Hamiltonian and constants of the motion may be constructed for the Dirac equation, which reduce to those of Biedenharn and Swamy in the absence of magnetic charge. Magnetic charge must be quantized—a requirement seen from the angular part of the wavefunction exactly as in the nonrelativistic problem. The Dirac Hamiltonian is singular for the lowest admissible angular momentum state, corresponding to the spiral orbits, when the magnetic charge is nonzero. The remaining states show an accidental doubling of degeneracy, whose presence may be deduced from an operator which reduces to that of Johnson and Lippman, or the algebra of Malkin and Manko, without the magnetic charge.
Magnetic Field Generated by a Transient Current Distribution
Authors: Clapp Roger E. , Huang Luc
Published: February 2004

The preceding paper [J. Math. Phys. 11, 9 (1970)] gave the retarded Hertz vector in terms of a general transient source-current distribution. Here, this Hertz vector is differentiated to give the vector potential and the magnetic field vector. All of these vector quantities have been expanded in vector spherical harmonics, and in each case it is the expansion coefficients, which are functions of the radial distance r and the time t, which are expressed in the (r, t) plane as two-dimensional integrals over the source-current expansion coefficients having the corresponding values of n and m, where these are the mode parameters characterizing the vector spherical harmonics Pnm(θ,φ),Bnm(θ,φ), and Cnm(θ,φ).
On the Asymptotic Stability of Reactors with Arbitrary Feedback
Authors: Akcasu A. Z. , Akhtar P.
Published: February 2004

A theorem for the boundedness and asymptotic stability of a point reactor with an arbitrary feedback is stated and proved. The criteria obtained are shown to be essentially the same as those given by Akcasu and Dolfes. The theorem is applied to a reactor with an arbitrary linear feedback and to a xenon-controlled reactor with a flux reactivity coefficient whose feedback mechanism involves quadratic non-linearity. It is also compared to a criterion obtained by Corduneanu in the case when delayed neutrons are ignored and the feedback mechanism is linear.
Electric Field Generated by a Transient Current Distribution
Authors: Clapp Roger E. , Huang Luc , Li H. Tung
Published: February 2004

The retarded Hertz vector, obtained earlier [J. Math. Phys. 11, 9 (1970)], is differentiated to give the scalar potential and the electric-field vector associated with a general transient source-current distribution. The scalar and vector quantities are expanded in terms of scalar and vector spherical harmonics. For each mode, characterized by specific values for n and m in the expansions, the scalar potential and electric-field vector are expressed in the (r, t) plane as two-dimensional integrals over the causally accessible portion of the source-current distribution.