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Random Operator Equations in Mathematical Physics. I
Authors: Adomian G.
Published: February 2004

For stochastic differential equations arising in physical problems, the objectives, limitations, and restrictive assumptions of the various methods are studied and some promising new methods are derived which eliminate various limitations and allow treatment of a wide class of applications in physics. (Among these, will be an adequate treatment of the propagation of an electromagnetic wave in a random continuum or a random d'Alembertian operator without assumptions of ``small randomness'' and other restrictions.)
Crossing Relations for Center-of-Mass- and Breit-System Canonical Amplitudes
Authors: Chakrabarti A.
Published: February 2004

We derive the crossing relations for the canonical amplitudes, starting with the case of four particles of nonzero rest mass. We give the results for both the center-of-mass- and the Breit-frame amplitudes and compare two different points of view. The definition of the ``Breit system'' is adapted to a unified treatment of all the three types of momentum transfer. It is shown that the Breit-frame amplitudes need no preliminary adjustments of constraints or the definition of a ``generalized'' amplitude in the alternative point of view. We also show that the usual convention about the continuation of the covariant spinor amplitudes involve, for the particles to be crossed, a rotation π of the ``physical'' spin about the direction of the continued momentum. The corresponding results for the helicity and the transversity amplitudes are derived as corollaries. Finally, we discuss the case of amplitudes involving zero-mass particles and some remarks are added concerning the kinematic singularities. The definition of the canonical and the transversity amplitudes are compared and some useful Lorentz-transformation formulas are collected together. The transformations connecting the center of mass and the Breit frames are parametrized in terms of the invariants. The well-known invariant amplitudes for πN scattering are used to verify our canonical formula.
O(5) Bases for Nuclear Seniority Model
Authors: Ahmed K. , Sharp R. T.
Published: February 2004

Two complete sets of bases are given for the general representation of O(5), utilizing the SU(2) subgroup which classifies nuclear seniority model states. The first set, which are analogous to the SU(3)⊃O(3) states of Elliott and Harvey, are formed by projecting good SU(2) states out of certain ``intrinsic'' states. The second set are analogous to the SU(3)⊃O(3) states defined by Moshinsky and Bargmann; each state is characterized by a particular simple term which appears in the expansion of that state and no other. Transformation matrices between the two sets of states are given. The utility of the states for the calculation of generator matrix elements and Clebsch-Gordan coefficients is discussed. The seniority model states of Hecht and Parikh are discussed briefly.
Remarks on the Equations of Motion in Odd-Dimensional Spaces and TCP Invariance
Authors: Santhanam T. S.
Published: February 2004

It is shown that the problem of TCP noninvariance of the equations of motion in odd-dimensional spaces is connected with the irreducibility of the representations of the Clifford algebra C2n+1. It is further shown that the above difficulty can be overcome by using representations of C2i+1, i > n, whether the particle is massive or massless.
Global and Infinitesimal Nonlinear Chiral Transformations
Authors: Joseph A. , Solomon A. I.
Published: February 2004

The problem of determining arbitrary nonlinear representations of a given compact Lie group is studied with the object of constructing Lagrangians invariant under the group. To achieve this, expressions for the covariant derivatives are obtained and it is shown how previous treatments based on global and on infinitesimal considerations are related. The noncompact case, the relationship between non-linearity and zero-mass particles, and the possibility of embedding the representation manifold in a higher-dimensional space are all discussed.
Some Remarks on Classical Scalar Radiation from Point Sources
Authors: Cawley Robert G.
Published: February 2004

The radiation from a point source with proper time-dependent coupling strength to a real classical scalar field is worked out. The adiabatic switching limit reproduces the results of the physical case of constant coupling for the radiation. The problem of defining a ``radiated'' field is discussed.
Technique for Finding the Moment Equations of a Nonlinear Stochastic System
Authors: Sancho N. G. F.
Published: February 2004

A technique is described for deriving the moment equations of a nonlinear stochastic system with a random forcing term. The nonlinear term is then linearized by means of minimizing the mean-square error between nonlinear and linear terms. The traditional derivation of the Fokker-Planck equation for the conditional probability density, and hence the moment equations, is bypassed.
Colorings of a Hexagonal Lattice
Authors: Baxter R. J.
Published: February 2004

The number of ways WL of coloring the bonds of a hexagonal lattice of L sites (L large) with three colors so that no adjacent bonds are colored alike is calculated exactly, giving W = 1.20872 …. This is equivalent to counting the number of 4-colorings of the faces of the lattice and can also be regarded as a multiple-dimer problem. If one introduces activities corresponding to certain vertex configurations, then the system is found to have an infinite-order phase transition between two ordered states.
Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field
Authors: Griffiths Robert B. , Hurst C. A. , Sherman S.
Published: February 2004

An inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0. The concavity of magnetization, which holds not only for spin-½ but also for arbitrary-spin Ising ferromagnets, provides a basis for certain thermodynamic inequalities near the ferromagnetic critical point, including one involving the ``high temperature'' indices α and γ.
Generalized Anharmonic Oscillator
Authors: Bender Carl M.
Published: February 2004

The generalized anharmonic oscillator is defined by the Hamiltonian HN, which in the coordinate space representation is given by HN = −d2/dx2 + ¼x2 + g(½x2)N. The analytic properties of the energy levels of HN as functions of complex coupling g are derived and described. Zeroth-order WKB techniques are used in the mathematical analysis. For all N, the results are qualitatively similar to those for the ordinary anharmonic oscillator in which N = 2 and, thus, the results are model independent for this wide class of models. The limiting case N → ∞ is solved exactly without using WKB techniques. The exact solution agrees with the WKB solution to zeroth order. This agreement is most impressive and testifies to the accuracy and utility of WKB methods.
On the Inverse Scattering Problem at Fixed Energy for Potentials Having Nonvanishing First Moments
Authors: Cox Joseph R. , Thompson Kerris W.
Published: February 2004

It has been shown previously by Newton that his solution (and its extension by Sabatier) of the problem of finding a central potential from a knowledge of all phase shifts at fixed energy yields a series whose expansion coefficients converge slowly unless the first moment of the potential vanishes. In particular, any truncation of the series after a finite number of terms necessarily results in potentials which have vanishing first moments. In this paper we propose a new, but formally somewhat similar, series for the potential which, for such truncations, does not suffer from this physically rather severe restriction. The series also furnishes new exact solutions of the Schrödinger equation at fixed energy. A closed-form expression for the scattering amplitude is obtained for a specific example. The problem of constructing the new series from the phase shifts is not discussed.
Note on the Uniqueness of the Solution of an Equation of Interest in the Inverse Scattering Problem
Authors: Cox Joseph R. , Thompson Kerris W.
Published: February 2004

It is proved that the solutions recently obtained by the authors [J. Math. Phys. 11, 805 (1970)] of the Regge-Newton integral equation (of interest in connection with the inverse scattering problem at fixed energy) are, for a given kernel, inhomogeneity, and boundary condition, uniquely determined.
Summability Methods in Perturbation Theory
Authors: Reeken M.
Published: February 2004

The Mittag-Leffler summability method is applied to operator-valued analytic functions and a corresponding procedure for perturbation theory is derived, which has a bigger region of convergence. This region is explicitly described.
Analytical Solution of the Boundary-Value Problem for the Nonlinear Helmholtz Equation
Authors: Wilhelm H. E.
Published: February 2004

A rigorous analytical solution in terms of the elliptic integral function of the first kind is derived for the nonlinear inhomogeneous boundary-value problem δ(d2f/dx2) = αf2 − εf − σ, f(s) = f̃.
Birkhoff Gravitational Field of Milne's Universe
Authors: Boyling J. B.
Published: February 2004

It is shown that Milne's special relativistic cosmology is consistent with Birkhoff's flat-space theory of gravitation and maintains its stability in the presence of gravity. The gravitational correction to the redshift in the light from distant galaxies is calculated.
Degeneracy in the Presence of a Magnetic Monopole
Authors: McIntosh Harold V. , Cisneros Arturo
Published: February 2004

The symmetry of the force field of a magnetic monopole is comparable in its simplicity to that of the hydrogen atom or a harmonic oscillator. Both these latter systems possess a ``hidden'' symmetry which leads to an ``accidental'' degeneracy in the energy spectrum of their Schrödinger equations. Since the monopole field is derived from a vector potential which is not symmetric, but undergoes a gauge transformation under rotation, the concepts of symmetry and constants of the motion must be expressed properly in the presence of velocity-dependent forces. It is found that neither the mechanical nor the canonical angular momentum is conserved in the presence of a monopole field, but rather a total angular momentum which incorporates angular momentum resident in the magnetic field. The total angular momentum defines a cone, on whose surface the motion takes place, whatever central electrostatic potential may exist. Neither the harmonic oscillator nor the hydrogen atom retain their accidental degeneracy when the nucleus possesses a magnetic charge, but if a repulsive centrifugal potential proportional to the square of the pole strength is added, accidentally degenerate systems with a higher symmetry result. The symmetry of such a harmonic oscillator is still somewhat obscure, but the ``charged Coulombic monopole'' has an O(4) symmetry group generated by the total angular momentum together with a Runge vector constructed from the total angular momentum. The irreducible representations of O(4) which occur are not the n2 representations of the hydrogen atom, but the m · n (m − n = 2ε, twice the monopole charge) representations which cannot be realized by four-dimensional spherical harmonics. The magnetic pole strength must be quantized, if admissible solutions of Schrödinger's equation are to exist, and according to Schwinger's quantization (ε = nℏc/e) if the wavefunctions are to be single valued; in any event, the ground state will be degenerate.
Boson Expansions for an Exactly Soluble Model of Interacting Fermions with SU(3) Symmetry
Authors: Li Shou Yung , Klein Abraham , Dreizler R. M.
Published: February 2004

An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.
On the Two Majorana Representations
Authors: Pavković Marko I.
Published: February 2004

An interesting property of the Majorana representations, symbolized in the Clebsch-Gordan equations (0,12)⊗(12,32)=(12,0)⊕⋯ and (12,0)⊗(12,32)=(0,12)⊕⋯, is reported. (12, 32) refers to the basic spinorial nonunitary representation of the SL(2, c) group in Gel'fand's notation, while (0, ½) and (½, 0) are two Majorana representations, again in the same notation. The Majorana representations represent the only solution of the equation X⊗(12,32)=∑a⊕Ya, in the sense that, of all the unitary irreducible representations of the SL(2, c) group, only (0, ½), or (½, 0) multiplied tensorially with (12, 32) will yield a unitary result—at least one of the Ys unitary.
Generalized Algebraic Mass Formulas. I
Authors: Dickens E. D. , Levine A. D.
Published: February 2004

It is shown that the SL(2, C)-Poincaré associative algebraic model of Böhm can be extended without essential difficulty to an SL(2, C)-de Sitter model which gives rise to a meson mass formula with both spin and isospin dependence.
Equivalence of Some Generalizations on the Spherical Model
Authors: Bettoney G. V. , Mazo R. M.
Published: February 2004

We define n-spherical models as follows: (1) Divide a lattice of N points into n mutually exclusive subsets with N1, ⋯, Nn points, respectively; ΣNα = N. With each lattice point j, associate a variable εj, and impose the restrictions ∑εj2=Nα,α=1,⋯,n; the sums are over the sites of the αth subset. (2) Adopt the energy expression for the Ising model with the ε's playing the role of the ``scalar spins.'' We then prove that, in the thermodynamic limit (N → ∞, Nα → ∞ for all α), the thermodynamic functions of the n-spherical model are equal to those of the ordinary spherical model.
Integral-Transform Gaussian Functions for Heliumlike Systems
Authors: Bishop David M. , Somorjai R. L.
Published: February 2004

The one-dimensional Laplace transform of the Gaussian exp (−r2x), g(r)=L{G(x)}= ∫ 0∞e−r2xG(x)dx,was used to generate functions which could be useful as basis sets for atomic and molecular calculations. A particular choice of the weighting function G(x) led to functions of the form g(r) = (qr)νKν(qr), where Kν(qr) are modified Bessel functions of the second kind. These functions were used as basis functions for the helium isoelectronic series and accounted for 98.98% (H−), 99.89% (He), 99.96% (Li+), 99.98% (Be++), 99.996% (O6+) of the Hartree-Fock energy.
Evaluation of Certain Radial Coulomb Integrals Using Symmetry Properties of the Coulomb Field
Authors: Swamy N. V. V. J. , Kulkarni R. G. , Biedenharn L. C.
Published: February 2004

The symmetry properties of the nonrelativistic Coulomb field problem allow one to construct an operator calculus for evaluating matrix elements of the multipole operator r−q. By means of this operator calculus an explanation is given for the vanishing of certain radial integrals treated earlier by Pasternack and Sternheimer, as well as for the value of similar integrals occurring in Coulomb excitation.
Effective Permittivity of a Polycrystalline Dielectric
Authors: Molyneux John E.
Published: February 2004

We use statistical variational principles to determine upper and lower bounds for the effective permittivity of a polycrystalline dielectric. We indicate how to derive bounds containing permittivity correlation functions of arbitrary order, and we obtain explicit expressions for bounds depending on one- and two-point correlation functions and for bounds containing one-, two-, and three-point correlation functions. We prove that for two classes of polycrystal, the effective permittivity may be exactly determined, and we use these exact expressions to show that we have obtained the best possible upper and lower bounds.
Reduced Matrix Elements of Tensor Operators
Authors: Briggs John S.
Published: February 2004

Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
O(5) Harmonics and Abnormal Solutions in the Bethe-Salpeter Equation
Authors: Basu Debabrata , Biswas S. N.
Published: February 2004

Exact solutions of the covariant Bethe-Salpeter equation in the ladder approximation for two scalar particles bound by a massless particle have been obtained for all energies. By using Fock's stereographic projection, the Bethe-Salpeter equation is transformed on to the surface of a 5-dimensional Euclidean sphere and the solutions are then expressed as a series in O(5) harmonics. The normalization condition has been imposed by requiring that the expectation value of appropriate components of the energy-momentum tensor with respect to the bound states is the total energy of the system; it is found that states corresponding to certain values of the quantum numbers do not satisfy the normalization requirement. These are the so-called abnormal asolutions.
Representations of Noncompact Semisimple Lie Groups
Authors: Joseph David W.
Published: February 2004

A method is presented for the construction of unitary representations of semisimple Lie groups (or, more precisely, of the corresponding algebras), proceeding directly from the commutation relations among the canonical generators e±α and hα. In the case of the orthogonal groups, the correspondence between the canonical generators and the more usual tensor generators is written down explicitly. It is shown how this method can be used to construct a certain class of representations (bounded above or below) of the Lie algebras of the noncompact groups SU(p, q) and SO(p, q), which arise in the dynamical group treatment of certain physical systems.
Noncompact Algebras with Additively Labeled Multiplets
Authors: Joseph D. W. , Hieggelke C. J.
Published: February 2004

A large class of noncompact simple Lie algebras is examined to determine those that admit unitary irreducible representations in which the multiplets associated with the compact subalgebra are labeled by the eigenvalues of a generator, with only a finite number of multiplets corresponding to each eigenvalue. It is shown that such representations are precisely those which are bounded above or below, so that the eigenvalues of the labeling generator are bounded above or below. It is found that many of the algebras examined do not have such representations; for example, among the pseudo-orthogonal groups SO(p, q) with p and/or q even, only those of the form SO(p, 2) do admit representations of this kind.
Statistical Average of a Product of Phase Sums Arising in the Study of Disordered Lattice. II
Authors: Sah Priyamvada
Published: February 2004

This recurrence formula, derived in a previous paper [J. Math. Phys. 10, 2263 (1969)] to evaluate 〈S(k1)S(k2) ⋯ S(kn)〉,is extended here, for the case when k1 + k2 + ⋯ + kn = 0 but no other partial sum of the set k1, k2, ⋯, kn is zero. The average is of O(N) as compared to the O(1) when k1 + k2 + ⋯ + kn ≠ 0. Formulas are then proposed for the situations when more than one partial sum of the set vanish. 〈S(k)S(−k)〉 is considered for a parabolic probability distribution s(r) with a cutoff which shows striking similarity with x-ray diffraction pattern of liquids.
Quasiparticle Formalism and Atomic Shell Theory. II. Mixed Configurations
Authors: Cunningham M. J. , Wybourne B. G.
Published: February 2004

The quasiparticle technique and formalism of an earlier paper is extended to describe general mixed configurations. The main features of the equivalent electron case are preserved, viz., a loss of spin quantum number and particle number, compensated by a rich classificatory scheme, and general operator-matrix-element evaluation without the use of coefficients of fractional parentage.
Expected Number of Distinct Sites Visited by a Random Walk with an Infinite Variance
Authors: Gillis Joseph E. , Weiss George H.
Published: February 2004

Consider a random walk of n steps on an infinite, simple cubic lattice. Let p(r) be the (symmetric) probability of a vector jump r, and let Sn be the expected number of distinct lattice points visited in the course of the random walk. In the present paper we calculate asymptotic values for Sn for the particular choice of jump probabilities p(r) = p(−r) = Ar−(1+α), where 2 ≥ α > 0, and p(r1, r2) = Ar−β, where r2=r12 + r22, 2≥β >1, and A denotes the normalizing constant. The results are, in 1D, (1) Sn ∼ An, 1 > α > 0, (2) Sn ∼ Bn/ln n, α = 1, (3) Sn ∼ Cn1/α, 2 > α > 1, (4) Sn ∼ D(n ln n)½, α = 2, where A, B, C, and D are calculable constants, and, in 2D, (1) Sn ∼ An, 2 > β > 1, (2) Sn ∼ Bn/ln ln n, β = 2.
Natural Spin-Orbitals and Generalized Overlap Amplitudes
Authors: Goscinski O. , Lindner P.
Published: February 2004

The generalized overlap amplitudes for ``hole'' states which appear in the one-particle Green's function are, in general, not linearly independent. We show that canonical orthonormalization of them yields the natural spin-orbitals (eigenfunctions of the first-order reduced density matrix).
Strong Coupling Solution to Schrödinger Equation: The Mixing of States
Authors: Garrido L. M. , Navarro L.
Published: February 2004

In this paper we give some ideas that can be useful to solve Schrödinger equations in the case when the Hamiltonian contains a large term. We obtain an expansion of the solution in reciprocal powers of the large coupling constant. The procedure followed consists in considering that the small part of the Hamiltonian engenders a motion adiabatic to the motion generated by the large part of the same.
Relaxation to Quantum Statistical Equilibrium of the Wigner-Weisskopf Atom in a One-Dimensional Radiation Field. II. Finite Systems
Authors: Davidson Russell , Kozak John J.
Published: February 2004

In this paper, a model is studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system. The model chosen for discussion is that of the Wigner-Weisskopf atom in interaction with a massless boson field, here 1-dimensional. The solution takes the form of a constant term plus a time-dependent one, expressed as the sum of residues at a series of poles along the real axis of a Laplace transform variable. Numerical calculations were performed on various aspects of the solution, and although these were too delicate to be quite certain around sensitive values of the time, the general picture is clear: After an initial decay to a value near zero, the series gave rise at fairly regular intervals to rapid and large fluctuations, the size of which never quite attains the initial value, but may nonetheless be large even after very long times. This result seems to be in agreement with the observation made by Zwanzig, namely, that for finite systems the master equation might demonstrate properties associated with the finite size of the system which would become important over certain long-time scales and would be ignored by the use of the customary ``thermodynamic limit'' of statistical mechanics. The relationship between this work and that of Montroll and Mazur and of Rubin is also discussed.
Oblique Reflection of an Electromagnetic Plane Wave from a Striated Impedance Surface
Authors: Wait James R.
Published: February 2004

A method is described for calculating the reflection of a plane wave of arbitrary incidence from a flat surface which has a spatially modulated surface impedance in one particular direction. Choosing a sinusoidal form for the modulation, the reflected wave spectrum is calculated. It is shown that, in general, for any arbitrary incidence, an infinite number of reflected modes are excited. Also, we find that the mode of order n is coupled to modes of order n ± 1.
Physical-Region Discontinuity Equations for Many-Particle Scattering Amplitudes. II
Authors: Coster Joseph , Stapp Henry P.
Published: February 2004

Discontinuity equations are derived for physical-region normal thresholds in all direct and crossed channels. The discontinuity is given as a unitarity-type integral with an integrand that contains, as factors, the two physical scattering functions corresponding to the two vertices of the Landau diagram associated with the normal threshold. There is also a third factor, which for the case of a leading normal threshold with any number of particles is the Hermitian conjugate of the elastic scattering matrix associated with the set of internal lines of the Landau diagram. For nonleading normal thresholds below the lowest 4-particle threshold, the extra factor is defined by an integral equation that resembles unitarity, but has a restricted set of intermediate particles.
Simple Waves in N-Dimensional Propagation
Authors: Boillat Guy
Published: February 2004

A field is assumed to satisfy a quasilinear system of partial differential equations. Solutions are sought which only depend on a single function of the time and space coordinates.
Branching Laws, Inner Multiplicities, and Decomposition of Classical Groups
Authors: Wong M. K. F.
Published: February 2004

A method has been found whereby the inner multiplicity of all classical groups [all irreducible representations of SU(n) and SO(2k + 1) and some simple irreducible representations of SO(2k) and Sp(2k)] can be obtained easily from the branching laws of Weyl (unitary), Boerner (orthogonal), and Hegerfeldt (symplectic). Once the inner multiplicity is known, the same formula can be used again to obtain the decomposition of a classical group into its subgroups without any restriction. Finally, since the inner multiplicity is connected to the outer multiplicity through the Racah-Speiser lemma, this method enables us to obtain the Clebsch-Gordan series for the direct product of all classical groups.
Doppler Shift in Frequency in the Transport of Electromagnetic Waves through an Underdense Plasma
Authors: Peacher Jerry L. , Watson Kenneth M.
Published: February 2004

In an earlier publication, the validity of the radiation transport theory was studied for the calculation of multiple scattering of electromagnetic waves by a turbulent plasma. In the present paper, we extend the transport theory to include a description of the Doppler shift in frequency caused by electron motion.
Nonexistence of Global Solutions to the Nonhomogeneous Wave Equation, Regardless of Boundary Conditions
Authors: Bownds John M.
Published: February 2004

The fact that the ordinary derivative y′ is known to be a Darboux function implies that the right-hand side of the differential equation y′ = f(x, y) must also be such a function. Since it is now known that the mixed hyperbolic derivative uxy is also a Darboux function, this implies that under certain mild conditions the equation uxx − utt = f may not have a classical global solution unless f has the proper Darboux structure. As with the ordinary differential equation, this nonexistence result does not depend on boundary conditions.
Lattice Dynamics of Cubic Lattices with Long-Range Interactions
Authors: Davies John A.
Published: February 2004

An analytic study is made of the dispersion relations and frequency spectra of unbounded cubic lattices in which there exist long-range pair potentials of the form r−p, where r is the distance between particles. This paper is an extension of work by Davies and Yedinak in which simple cubic lattices are studied. Cubic lattices having two particles per unit cell are now considered. The analytic behavior of the optical branches near the origin of the first Brillouin zone is determined for the case of 1 ≤ p ≤ 3 and the resulting contribution of the longitudinal optical branch to the frequency spectrum is obtained. A special case of such lattices is Kellermann's model for NaCl. This model is studied in detail.
Diffraction of Electromagnetic Waves by Planar Dielectric Structures. I. Transverse Electric Excitation
Authors: Lehman G. W.
Published: February 2004

A formalism is developed for solving a class of electromagnetic diffraction problems involving heterogeneous planar dielectric structures. Using analytic properties of finite Fourier transforms, we show that the electromagntic field distributions can be obtained by solving a single-variable Fredholm integral equation of the second kind. The special case of TE-wave propagation through a finite metallic guide with a dielectric insert is discussed in detail. Our numerical approach spans the Rayleigh to geometric optics range. Field distributions are presented for typical configurations involving positively and negatively absorbing media. Excitation of bound-mode resonances is discussed.
Neutron Transport in a Nonuniform Slab with Generalized Boundary Conditions
Authors: Belleni-Morante A.
Published: February 2004

We prove the existence and the uniqueness of the solution of the initial-value problem for neutron transport in a nonuniform slab with generalized boundary conditions, which include the vacuum and the perfect reflection boundary conditions as particular cases. Moreover, we show that the position-dependent transport operator has at least one real eigenvalue and we indicate the asymptotic behavior of the neutron density as t → + ∞.
E2
¯
−
Parametrization
of SL(2, C)
Authors: Iverson G. J. , Mack G.
Published: February 2004

We consider the E2¯−parametrization of unimodular 2 × 2 matrices A ∈ SL(2, C), which is of the form A=E1e−12aσ3VE2,  with  V=e12iπσ2  and  E=(e−12iφ0ze12iφ)∈E2¯. E2¯ is a covering of the group of Euclidean motions in the plane. We compute the correspondingly factorized matrix elements of the unitary representations of SL(2, C) in an E2¯−basis the result is given in Eq. (6). As a fringe benefit we obtain an integral transform which amounts to expansion in terms of Meijer G-functions and which generalizes the familiar Hankel transform. The results of this paper are useful, e.g., for computing vertex functions in the theory of massless particles with continuous spin.
Diffraction by a Circular Cavity
Authors: Ludwig Donald
Published: February 2004

The reduced wave equation Δu + k2n2u = 0 is considered, where n = 1 in the exterior of a circular cylinder of radius a and n = N < 1 in the interior. The solution and its first derivatives are required to be continuous for r = a. The behavior of the solution for large ka is described, especially in the vicinity of a ray incident upon the cylinder at the critical angle. The principal novelty of the present results is a description of the solution in certain regions by a combination of geometrical optics (ray contributions) and whispering gallery modes (a kind of diffracted modes). These results are in qualitative agreement with those of Chen, but certain details in Chen's treatment require elaboration. The present results do not agree with those of Nussenzveig for N < 1, especially in the geometrical interpretation. This revision of Nussenzveig's work brings it into agreement with Keller's geometrical theory of diffraction. In fact, part of the present work can be combined with Chen's results to give a fairly complete geometrical theory for diffraction by a convex, transparent object. However, the present treatment does not give a uniform asymptotic expansion of the solution, since the expansion is not given in a transition region in the immediate vicinity of the critically reflected ray.
Noninvariance Groups in the Second-Quantization Picture and Their Applications
Authors: Moshinsky M. , Quesne C.
Published: February 2004

We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.
Lorentz-Invariant Gravitational Perturbations and the Evaluation of Generalized Green's Functions
Authors: Goenner Hubert
Published: February 2004

The Lorentz-invariant perturbation theory of classical nonlinear field theories, notably the fast-motion approximation for a system of gravitationally interacting particles, is used to illustrate the appearance of generalized Green's functions (GGF's) in physics. The (retarded) GGF's are defined as the convolution of the usual (retarded) Green's function with certain other linear functionals. A manifestly Lorentz-invariant technique for the evaluation of the convolution integral is described and applied to two important classes of GGF's. Time-symmetric GGF's are also discussed briefly. The technique exhibited provides the tools for the calculation of the higher-order approximations in a Lorentz-invariant perturbational approach to the classical n-body problem.
Breaking of Euclidean Symmetry with an Application to the Theory of Crystallization
Authors: Emch Gérard G. , Knops Hubert J. F. , Verboven Edward J.
Published: February 2004

We present a systematic study of the processes by which the original Euclidean invariance of a quantum statistical theory can be broken to produce pure phases with a lower symmetry. Our results provide a rigorous basis for Landau's argument on the nonexistence of critical point in the liquid-solid phase transition. A classification of the possible residual symmetries is obtained, and its connection with spectral and cluster properties is established. Our tools are those of the algebraic approach to statistical mechanics; in particular, we make an extensive use of the KMS condition. None of our proofs involves the separability of the algebra of quasilocal observables.
Some Mathematical Properties of Oscillator Phase Operators
Authors: Lerner E. C. , Huang H. W. , Walters G. E.
Published: February 2004

A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation-type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.
Electromagnetic Scattering from an Expanding Dielectric Slab
Authors: Pogorzelski Ronald J.
Published: February 2004

The situation where a normally incident plane monochromatic electromagnetic wave is reflected by an expanding dielectric slab is analyzed by means of the invariant-imbedding concept. An approximate expression for the spectrum of the reflected radiation is obtained.
Radiation from a ``Classical Spinning Electron''
Authors: Kolsrud Marius
Published: February 2004

The electromagnetic field given by Cohn, according to which the radiation from a classical spinning point charge is independent of the particle's spin, is shown not to satisfy Maxwell's equations. References are given to earlier expressions, which also correspond with quantum theory.
Conservation Theorem and the Method of Moments in Quantum Scattering
Authors: Galindo V. , Amitay N. , Lee Y. C.
Published: February 2004

The unitary condition, or the constraint imposed by the conservation of probability, is examined for certain approximate solutions to physical scattering problems. The unitary condition is shown to be preserved for an approximate nonrelativistic S matrix derived by a rather generalized moments method. The unitary condition, or conservation theorem, is shown to be satisfied independent of the accuracy of the approximate solution under certain specifically defined conditions.
Orthogonalization Methods
Authors: Schweinler H. C. , Wigner E. P.
Published: February 2004

An essentially unique method of forming an orthonormalized set of vectors from an independent set of n vectors is presented. The resulting set is proven to maximize a certain quartic form in the original vectors.
Scalar-Tensor Theory in Canonical Form. I. The Question of Mach's Principle
Authors: Toton Edward T.
Published: February 2004

By first casting the scalar-tensor theory into canonical form, the constraint equations are obtained from which it is possible to discuss the self-energy problem of the neutral particle and the electron. With the assumptions of Minkowski space and a constant scalar field as boundary conditions at infinity, it is shown by variational arguments that the inertia of these objects is unchanged from the corresponding general relativity values. Accordingly, if the boundary value of the scalar is altered by a change of matter at infinity, the mass of the electron is unchanged. From this point of view, the scalar-tensor theory is no more compatible with Mach's principle than is general relativity.
Surface Layer in General Relativity and Integral Form of Field Law
Authors: Gupta Amritava
Published: February 2004

Assuming that the metric tensor gij depends on a continuous parameter ε, we write Einstein's field equations of general relativity in the form of a divergence equation. The theory of surface layer in general relativity is then worked out. Finally, an integral flux law is proposed from which both the field equations and the boundary conditions for a surface layer can be deduced.
Symmetric Dirac Bracket in Classical Mechanics
Authors: Franke Wilhelm H. , Kálnay Andrés J.
Published: February 2004

It is known that, in classical systems that have second-class constraints which relate the canonical coordinates and momentum, the ordinary skew-symmetric Poisson bracket must be replaced by the skew-symmetric Dirac bracket. It is also known that in the process of quantization of such systems, the Dirac bracket replaces the Poisson bracket in its correspondence with the quantum commutators. In this paper we obtain the symmetric partner of the Dirac bracket, which is of interest not only to classical mechanics but also in regard to the quantization procedure; i.e., the quantization rules for systems which are restricted by second-class constraints such that the commutation rules involve anticommutators (instead of commutators, as in certain fields with Fermi-Dirac statistics) can be given in terms of this new symmetric bracket. This symmetric bracket is related to the Poisson-Droz-Vincent symmetric bracket.
Analytic Solutions to Two Albedo Problems
Authors: Noerdlinger Peter D.
Published: February 2004

A random walk approach leads to rigorous solution of certain albedo problems usually solved by Monte Carlo techniques or other numerical methods.
Unitary Representations of the Group SL(3, C) in an SU(3) Basis
Authors: Mukunda N.
Published: February 2004

Unitary irreducible representations of the group SL(3, C) are obtained in an SU(3) basis. The principal nondegenerate and principal degenerate series of representations are considered, and the matrix elements of the generators are worked out by the use of expansion formulas expressing them as functions of the generators of the group SU(3) × T8. The forms of the Casimir operators of SL(3, C) and their values in these representations are also given.
Expansion of the Inhomogeneous Pseudo-unitary Symplectic Lie Algebras IUSp(2v1, 2v2) to USp(2v1 + 2, 2v2) or to USp(2v1, 2v2 + 2)
Authors: Nagel John G.
Published: February 2004

We show that the inhomogeneous pseudo-unitary symplectic Lie algebra IUSp(2v1, 2v2) can be expanded to USp(2v1 + 2, 2v2) or to USp(2v1, 2v2 + 2). We also identify a (2v1 + 2v2 + 2)-dimensional linear Lie group G(2ν1+2ν2+2,R)≅JUSp(2ν1,2ν2), which under this expansion is being globally expanded to a (2v1 + 2v2 + 2)-dimensional linear group E(2ν1+2,2ν2;R)≅USp(2ν1+2,2ν2) or to E(2ν1,2ν2+2;R)≅USp(2ν1,2ν2+2).
Recurrence Relations for the Multiplicities in the Classical Groups
Authors: Gruber B.
Published: February 2004

Recurrence relations for the ``outer multiplicity'' (multiplicity of irreducible representations occurring in the reduction of the direct product of two irreducible representations) as well as the ``branching multiplicity'' (multiplicity of irreducible representations into which an irreducible representation of a group decomposes, if this group is restricted to a subgroup) are obtained. These recurrence relations are shown to be closely related to each other. Racah has obtained a recurrence relation for the ``inner multiplicity'' (multiplicity of weights). It turns out that the recurrence relation for the inner multiplicity is a special case of the recurrence relation for the outer multiplicity which, in turn, can be looked upon as a special case of recurrence relation for the branching multiplicity. Moreover, Kostant's recurrence relation for the partition function as well as Racah's recurrence relation for the inner multiplicity have been generalized. This generalization introduces a parameter which can be used to simplify actual calculations. In an appendix, finally, a formula is given which allows one to calculate the branching multiplicity from known inner multiplicities of the subgroup to which it is restricted.
Energy Spectrum According to Classical Mechanics
Authors: Gutzwiller Martin C.
Published: February 2004

The phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ-function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz-Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spectrum of Balmer terms with appropriately reduced degeneracy. However, the orbits which penetrate the screening charge give discrete levels obeying the Bohr-Sommerfeld conditions with the correct degeneracy. The continuous background in the approximate density of states can be discussed on the basis of the formulas derived in this paper. This is necessary as an introduction to the problem of a particle in a potential where the motion is not multiply periodic.
Further Generalization of the Generalized Master Equation
Authors: Leaf Boris
Published: February 2004

For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.
Generalized Group-Theoretical Analysis of Spontaneous Symmetry Breaking
Authors: Chan Choi-Lai
Published: February 2004

A group-theoretical analysis of spontaneous symmetry breaking is carried out in an extension of Glashow's work. A general theorem is proved which is then used to give several interesting results. Apart from a rederivation of Glashow's result, we are able to show, among other things, the following: If a hadronic triplet exists in nature which is nontrivially coupled to the rest of the hadrons, then octet enhancement immediately follows independently of any dynamical detail; in models containing the vector nonet, φ - ω mixing can, in principle, occur as a particular form of spontaneous symmetry breaking in the octet pattern; and if octet enhancement holds, the very accurate mass formula mφ2+mω2=2mk2* is established without recourse to any dynamical detail. Under the same general assumption, a further relation is found relating the mass of the nonet with the mixing angle, which is 4mk2*−3(cos2θmφ2+sin2θmω2)−mρ2=0.
Current Algebras, the Sugawara Model, and Differential Geometry
Authors: Hermann Robert
Published: February 2004

The Lie algebra defined by the currents in the Sugawara model is defined in a way that is natural from the point of view of Lie transformation theory and differential geometry. Previous remarks that the Sugawara model is associated with a field-theoretical dynamical system on a Lie group manifold are made more precise and presented in a differential geometric setting.
Numerical Analysis of an Integral Equation with a δ Function in the Kernel
Authors: Hetherington J. H.
Published: February 2004

An integral equation which has a two-part kernel, where one part contains a δ function, is analyzed with respect to its analytic structure in the λ plane and with respect to numerical approximations to it. The analytic structure of the approximate solution is also investigated. It is found that there is no difficulty in approximating the Dirac δ function by a Kronecker δ; even though the approximate kernel does not approach the true kernel, the solution corresponding to the approximate kernel does approach the true solution. In keeping with the fact that the kernel does not meet the Fredholm conditions, we find that the solution has branch cuts in the λ plane. A form for the solution analogous to Fredholm's solution which emphasizes the analytic structure (i.e., a branch cut) is obtained.