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Apr 1977

Volume 18, Issue 4, pp. 537-853

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Reciprocity relations and forward amplitude theorems for elastic waves

V. Varatharajulu

J. Math. Phys. 18, 537 (1977); http://dx.doi.org/10.1063/1.523335 (7 pages) | Cited 21 times

Online Publication Date: 26 August 2008

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Relations are derived between the amplitudes of scattered elastic waves from obstacles of arbitrary shape when the direction of observation and incidence are reversed and interchanged, similar to the reciprocity theorem for the scalar wave equation. For waves which undergo a polarization change on scattering (longitudinal to shear and vice versa), a new type of reciprocity relation is derived which is peculiar to elastic waves. Expressions for the scattering cross section are obtained for shear and longitudinal incident waves, and it is shown to be proportional to the amplitude of the scattered wave in the forward direction alone. Incident waves are restricted to plane, monochromatic waves.
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62.30.+d Mechanical and elastic waves; vibrations

Positive solutions and subcriticality of energy dependent transport systems in slab geometry

C. V. Pao

J. Math. Phys. 18, 544 (1977); http://dx.doi.org/10.1063/1.523336 (7 pages) | Cited 7 times

Online Publication Date: 26 August 2008

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The energy dependent transport system in an anisotropic medium in slab geometry subjecting to internal source and incoming fluxes is investigated. The investigation is based on a corresponding integral equation of the boundary value problem from which a recursion formula for the determination of the solution is obtained. It is shown by using the notion of an upper solution that the convergence or divergence of the sequence of iterations depends solely on the existence or nonexistence of an upper solution. Through the construction of a suitable upper solution one can obtain an explicit estimate for the value of c, which represents the average number of secondary neutrons per collision, in terms of the (optical) slab length 2a so that the system is either critical or subcritical. It is shown in particular that if c<[1−E2(a)]−1, where E2(a) is the exponential integral of order two, then the integral equation has exactly one nonnegative solution for any nonnegative source and incoming fluxes. This result insures the subcriticality of the system as well as a constructive recursion formula for the determination of the solution. Estimates for a more general system and the energy independent system are also obtained.
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51.10.+y Kinetic and transport theory of gases

Partial‐wave analysis for supersymmetric scattering amplitudes

P. D. Jarvis

J. Math. Phys. 18, 551 (1977); http://dx.doi.org/10.1063/1.523337 (13 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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The reduction of the direct product of two unitary irreducible representations of supersymmetry into a direct sum of UIR’s is carried out for the massive case, and the Clebsch–Gordan coefficients are written down. The supersymmetric coupling of the different spin components of a multiplet arises from the use of the ’’superhelicity’’ basis (superhelicity κ=−j0,−j0+1,...,j0−1,j0) in which the spin is not diagonal. Here j0=O,1/2,1,... is the ’’superspin,’’ and the ordinary helicity, λ, is given by λ=κ or κ±1/2. The physical results are retrieved by transforming back to the spin basis after the reduction. The results of the reduction are used to analyze the scattering processes 1→2+3 and 1+2→3+4 for particles belonging to supersymmetric multiplets. The ordinary partial wave helicity amplitudes are given in terms of a small number of reduced partial wave superhelicity amplitudes corresponding to given total superspin. By continuing the latter to complex superspin, it is shown how, in the high‐energy limit in the crossed channel, a singularity in one superhelicity amplitude contributes to the high‐energy behavior of several different spin channels.
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03.65.Db Functional analytical methods
11.80.Cr Kinematical properties (helicity and invariant amplitudes, kinematic singularities, etc.)

A Lagrangian formulation for noninteracting high‐spin fields

Francisco Antonio Doria

J. Math. Phys. 18, 564 (1977); http://dx.doi.org/10.1063/1.523338 (8 pages) | Cited 5 times

Online Publication Date: 26 August 2008

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We develop here some properties of Clifford algebras and of ’’algebraic’’ or Sauter spinors that lead to a very simple classification scheme for the Lorentz group representations. We apply our results to a Dirac‐like Lagrangian and get a general formulation for noninteracting high‐spin fields; the equations for spin 1/2, 0 and 1, 3/2 and 2 are explicitly calculated; the first three are seen to reproduce some previous results by Teitler.
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03.70.+k Theory of quantized fields

Generalized invariants for the time‐dependent harmonic oscillator

N. J. Günther and P. G. L. Leach

J. Math. Phys. 18, 572 (1977); http://dx.doi.org/10.1063/1.523339 (5 pages) | Cited 51 times

Online Publication Date: 26 August 2008

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A generalized class of invariants, I (t), for the three‐dimensional, time‐dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time‐independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time‐dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.
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31.15.-p Calculations and mathematical techniques in atomic and molecular physics
33.20.-t Molecular spectra
03.65.Db Functional analytical methods

Lattices of effectively nonintegral dimensionality

Deepak Dhar

J. Math. Phys. 18, 577 (1977); http://dx.doi.org/10.1063/1.523316 (9 pages) | Cited 110 times

Online Publication Date: 26 August 2008

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We construct a class of lattice systems that have effectively nonintegral dimensionality. A reasonable definition of effective dimensionality applicable to lattice systems is proposed and the effective dimensionalities of these lattices are determined. The renormalization procedure is used to determine the critical behavior of the classical XY model and the Fortuin–Kasteleyn cluster model on the truncated tetrahedron lattice which is shown to have the effective dimensionality 2 log3 /log5. It is found that no phase transition occurs at any finite temperature.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Stationary gravitational fields of a dually charged perfect fluid

A. Das and S. Kloster

J. Math. Phys. 18, 586 (1977); http://dx.doi.org/10.1063/1.523340 (4 pages) | Cited 5 times

Online Publication Date: 26 August 2008

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Stationary field equations in the presence of a charged perfect fluid with both electric and monopole currents in isometric motion are studied. It is shown that the eight‐parameter group of transformations which preserve the stationary electrovac equations can also be applied to dually charged sources. In the case of dually charged dust an equilibrium condition ρ= (σ∗σ)1/2 implies a functional relationship between ReΓ and the complex potentials Φ and Φ∗. Furthermore, it is proved that when ρ≠0 and σ≠0, the additional assumption of an arbitrary linear relationship between Γ and Φ leads uniquely to the Israel–Spanos class of solutions.
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04.20.-q Classical general relativity
04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

Renormalization group structure for translationally invariant ferromagnets

George A. Baker and Samuel Krinsky

J. Math. Phys. 18, 590 (1977); http://dx.doi.org/10.1063/1.523341 (18 pages) | Cited 28 times

Online Publication Date: 26 August 2008

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We introduce a correlation function description of the renormalization group approach to critical phenomena. Our work is based on treating the renormalization group operator as a linear mapping on the set of Ursell functions, rather than as a nonlinear mapping on the space of Hamiltonians. We mainly consider the ’’mean‐spin’’ renormalization group, but the closely related ’’decimation’’ transformation is also considered. Using this approach, we demonstrate for a suitable class of system that the spectrum of the renormalization group operator is bounded and countably infinitely degenerate. We give counterexamples to the notion that there must be convergence to a renormalization group fixed point. Our formulation of the renormalization group is sufficiently general so that convergence to a fixed point does not necessarily imply hyperscaling, i.e., the vanishing of the anomalous dimension of the vacuum, ω∗. In the case of convergence to a fixed point we find δ= (d+σ−ω∗)/(d−σ−ω∗), with ω∗?0 necessarily. Above the critical temperature we are able to prove that convergence is obtained to the ’’infinite temperature fixed point,’’ which result generalizes the central limit theorem to ferromagnetically correlated variables.
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05.70.Jk Critical point phenomena

Identification of the velocity operator for an irreducible unitary representation of the Poincaré group

Thomas F. Jordan

J. Math. Phys. 18, 608 (1977); http://dx.doi.org/10.1063/1.523342 (3 pages) | Cited 5 times

Online Publication Date: 26 August 2008

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For a particle described by an irreducible unitary representation of the Poincaré group, for either positive mass or zero mass and discrete helicity, it is shown that the velocity operator can be identified by its transformations under the Poincaré group together with the assumption that it is a Hermitian operator whose different components commute with each other.
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11.30.Cp Lorentz and Poincaré invariance
02.30.Tb Operator theory

Ashkin–Teller model as a vertex problem

F. Y. Wu

J. Math. Phys. 18, 611 (1977); http://dx.doi.org/10.1063/1.523343 (3 pages) | Cited 15 times

Online Publication Date: 26 August 2008

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It is shown that the Ashkin–Teller model on any planar lattice is equivalent to an eight‐vertex model on a related lattice. The exact equivalence is given for finite lattices with a boundary. We show, in particular, that the AT model on the triangular or honeycomb lattice is related to an eight‐vertex model on a Kagomé lattice. The occurrence of two phase transitions in the AT model in general is also discussed.
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05.70.Fh Phase transitions: general studies
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Periods on manifolds, quantization, and gauge

R. M. Kiehn

J. Math. Phys. 18, 614 (1977); http://dx.doi.org/10.1063/1.523317 (11 pages) | Cited 13 times

Online Publication Date: 26 August 2008

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It is suggested that the quantization of flux, charge, and angular momentum be interpreted as a set of independent natural concepts which physically exhibit certain topological properties of the fields on a space–time manifold. These quantum, or topological, properties may be described in terms of one‐, two‐, and three‐dimensional periods, respectively. In terms of this viewpoint, topological constraints between the one‐, two‐, and three‐dimensional periods can be put into correspondence with various gauge theories. If a dynamical system is to be nondissipative, in the sense that its one‐, two‐, and three‐dimensional topological periods are reversible invariants of the motion, then it is proved herein that the dynamical field V must be a Hamiltonian vector field, the field currents must be proportional to V, and the Lagrangian difference between the elastic and inertial energy density must be twice the interaction energy density, respectively.
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03.65.Fd Algebraic methods

Superfields as an extension of the spin representation of the orthogonal group

J. O. Winnberg

J. Math. Phys. 18, 625 (1977); http://dx.doi.org/10.1063/1.523344 (4 pages) | Cited 5 times

Online Publication Date: 26 August 2008

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We show that a superfield can be interpreted as a spinor belonging to the spin representation of a Clifford algebra. A subset of this algebra is connected by a linear automorphism to the orthogonal group. With this interpretation it should be possible to give a physical meaning to Grassmann variables.
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11.30.Pb Supersymmetry
03.70.+k Theory of quantized fields
02.20.Qs General properties, structure, and representation of Lie groups

A C∗‐algebra formulation of the quantization of the electromagnetic field

A. L. Carey, J. M. Gaffney, and C. A. Hurst

J. Math. Phys. 18, 629 (1977); http://dx.doi.org/10.1063/1.523318 (12 pages) | Cited 10 times

Online Publication Date: 26 August 2008

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See Also: Erratum

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A presentation of the Fermi, Gupta–Bleuler, and radiation gauge methods for quantizing the free electromagnetic field is given in the Weyl algebra formalism for quantum field theory first introduced by Segal. The abstract Weyl algebra of the vector potential is defined using the formalism of Manuceau. Then the Fermi and Gupta–Bleuler methods are given as schemes for constructing representations of the algebra. The algebra of the physical photons is shown to be a factor algebra of a certain subalgebra of the original algebra of the vector potential. In this formalism, the application of the supplementary condition in the Fermi method, and the supplementary condition and indefinite metric in the Gupta–Bleuler method, can be interpreted as the means by which a representation of this factor algebra is obtained. The Weyl algebra of the physical photons is the Weyl algebra associated with the radiation gauge method. It is also shown that in the Fock representation of the Weyl algebra given by the Fermi method, automorphisms of the algebra corresponding to Lorentz transformations cannot always be implemented by unitary transformations. This leads us to construct a new representation of the Weyl algebra which provides a covariant representation for the vector potential.
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03.70.+k Theory of quantized fields

Gauge equivalence of the electrodynamics of charged bosons

R. Sohn and K. Haller

J. Math. Phys. 18, 641 (1977); http://dx.doi.org/10.1063/1.523319 (7 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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The quantum electrodynamics of charged scalar and vector bosons is formulated in the Lorentz gauge, and the effect of the charged particle–photon interaction on the subsidiary condition is explicitly taken into account. The results are extensions of earlier work on spinor quantum electrodynamics, but the presence of seagull vertices and anomalous current commutators in the case of the charged bosons make the extensions nontrivial. An operator gauge transformation that encompasses equations of motion as well as the commutator algebra of the field operators is developed; it is used to transform the theory from the Lorentz gauge to the Coulomb gauge.
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12.20.-m Quantum electrodynamics

A simple derivation of a closed formula for Bogoliubov boson transformations

A. A. Raduta, M. Badea, and E. Badralexe

J. Math. Phys. 18, 648 (1977); http://dx.doi.org/10.1063/1.523320 (3 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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Considering the states with an arbitrary number of bosons and their transformed states under Bogoliubov transformations as wavefunctions of an oscillator type, a very simple derivation for the matrix elements of the Bogoliubov transformations is given.
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11.10.-z Field theory

Moment‐theory approximations for nonnegative spectral densities

C. T. Corcoran and P. W. Langhoff

J. Math. Phys. 18, 651 (1977); http://dx.doi.org/10.1063/1.523321 (7 pages) | Cited 61 times

Online Publication Date: 26 August 2008

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Moment‐theory approximations constructed from finite numbers of spectral power moments are described for continuous, nonnegative spectral densities and associated Stieltjes integrals. Derivatives of the mean (Stieltjes) values of the nth‐order Tchebycheff bounds on nondecreasing distributions provide the appropriate approximations to the associated spectral densities. The nth‐order Tchebycheff density so defined is shown to be real, nonnegative, and continuous on the real axis, to have 2n−4 continuous derivatives there, and to support 2n‐2 positive‐integer power moments. Related approximations to the associated Stieltjes integral are obtained from corresponding principal‐value quadratures. The Tchebycheff densities are convergent in the limit of large numbers of spectral moments for determined moment problems, but they are not solutions of reduced moment problems of appropriate finite order. An illustrative application in the case of normal‐mode lattice vibrations in a diatomic chain indicates that the Tchebycheff densities are suitably convergent, and provide faithful images of the forbidden band gap and Van Hove singularities present.
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02.30.Sa Functional analysis
71.10.-w Theories and models of many-electron systems

Some examples of transparent potentials in the classical approximation

Michel Cuer

J. Math. Phys. 18, 658 (1977); http://dx.doi.org/10.1063/1.523322 (4 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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We give classes and examples of potentials which give a classical deflection function equal to zero (mod 2nπ) for any value of the impact parameter at a fixed energy. Thus these potentials are completely transparent for a classical collision at this energy.
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45.05.+x General theory of classical mechanics of discrete systems

Free electromagnetic fields on a compact Lie group manifold

Leighton L. Henry

J. Math. Phys. 18, 662 (1977); http://dx.doi.org/10.1063/1.523323 (2 pages) | Cited 1 time

Online Publication Date: 26 August 2008

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Maxwell’s equations for free fields are studied on the underlying C∞ manifold of a compact Lie group. The formulation is in terms of exterior differential forms as given by Wheeler. It is found that on a compact connected Lie group there are no free electromagnetic fields. The results obtained are essentially a physical interpretation of the well‐known theorem that the second Betti number of a compact semisimple Lie group is zero.
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04.50.-h Higher-dimensional gravity and other theories of gravity
02.20.Qs General properties, structure, and representation of Lie groups

A class of solutions of the Dirac equation in the Kerr–Newman space

S. Einstein and R. Finkelstein

J. Math. Phys. 18, 664 (1977); http://dx.doi.org/10.1063/1.523324 (8 pages) | Cited 7 times

Online Publication Date: 26 August 2008

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In a region of spacetime that may be described by the Kerr–Schild metric, the gravitational field equations define a field of O(3) matrices. By examining the spin representations of these rotations it is first shown how the gravitational field equations define a spinor field, and it is then shown how this spinor field is related to special solutions of the massless Dirac equation in the Kerr–Newman space. These special solutions have arbitrary angular momentum about the axis of rotation and in the classical limit correspond to orbits that coincide with the principal null congruences.
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04.20.Cv Fundamental problems and general formalism
03.65.-w Quantum mechanics

Characterization of the Szekeres inhomogeneous cosmologies as algebraically special spacetimes

J. Wainwright

J. Math. Phys. 18, 672 (1977); http://dx.doi.org/10.1063/1.523325 (4 pages) | Cited 21 times

Online Publication Date: 26 August 2008

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The Szekeres inhomogeneous cosmological models are invariantly characterized as a subclass of the algebraically special type {22} solutions of the Einstein field equations for irrotational dust, and their relationship to the locally rotationally symmetric dust solutions is clarified.
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98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

Acoustic emission and the plate Green’s function

W. J. Pardee

J. Math. Phys. 18, 676 (1977); http://dx.doi.org/10.1063/1.523326 (11 pages) | Cited 1 time

Online Publication Date: 26 August 2008

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See Also: Erratum

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The stress at an arbitrary point in an elastic medium due to an acoustic emission is written in a stress Green’s function formalism. This formalism separates the mathematical description of acoustic radiation into separate factors describing the source and the medium response, which, it is hoped, will facilitate identification of source characteristics from remote observations. The description of the medium response is shown to be equivalent to a well posed elliptic boundary value problem for a 36 component Green’s function. This (frequency analyzed) Green’s function is calculated for the infinite isotropic elastic medium and for a similar plate. The latter example includes a complete description of dispersion, mode conversion, and the normal modes of the slab, although the latter need not be explicitly identified.
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43.20.+g General linear acoustics
46.25.Cc Theoretical studies

Ergodic state with least mean recurrence time in Markovian kinetics

Koichiro Matsuno

J. Math. Phys. 18, 687 (1977); http://dx.doi.org/10.1063/1.523327 (3 pages)

Online Publication Date: 26 August 2008

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We present a proof of the theorem that the ergodic state with the least mean recurrence time in an irreducible Markov process having countable states is identical, with probability unity, to the state with the least irreversible decay rate if neither microscopic reversibility nor the doubly stochastic property holds.
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02.50.-r Probability theory, stochastic processes, and statistics
05.20.Dd Kinetic theory

Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique

S. ‐C. Chiu and J. F. Ladik

J. Math. Phys. 18, 690 (1977); http://dx.doi.org/10.1063/1.523328 (11 pages) | Cited 18 times

Online Publication Date: 26 August 2008

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By means of generalized Wronskian relations an operator formulation is developed allowing one to generate a class of discrete evolution equations which are soluble by inverse scattering. This class includes nonlinear difference–difference equations as well as nonlinear differential–difference equations. The development of the method exhibits the significance of the dispersion relation of the associated linearized equation. The generalized Wronskian technique also permits one to obtain generalized Bäcklund transformations.
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02.30.Hq Ordinary differential equations
02.30.Ks Delay and functional equations

Solutions of nonlinear network equations by the inverse scattering method

John F. Ladik and Shiu‐Chu Chiu

J. Math. Phys. 18, 701 (1977); http://dx.doi.org/10.1063/1.523329 (4 pages)

Online Publication Date: 26 August 2008

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Two coupled equations for a nonlinear network are solved by the inverse‐scattering method. N‐soliton solutions are obtained. The effect of the continuum spectrum is investigated.
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02.30.Hq Ordinary differential equations
02.30.Ks Delay and functional equations

General relativistic magnetofluids

F. Paul Esposito and E. N. Glass

J. Math. Phys. 18, 705 (1977); http://dx.doi.org/10.1063/1.523330 (3 pages) | Cited 4 times

Online Publication Date: 26 August 2008

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General relativistic hydromagnetic systems are studied in the infinite conductivity limit. It is shown that when the frozen‐in magnetic field lies in surfaces of constant pressure, the fluid acceleration and magnetic field are orthogonal. The theorem is proved that a shear‐free infinite conductivity magnetofluid is irrotational if and only if the Weyl tensor is pure electric type. A restricted steady state is formulated and discussed, and within that state it is shown that an infinite conductivity magnetofluid cannot be electrically neutral.
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95.30.Qd Magnetohydrodynamics and plasmas
95.30.Lz Hydrodynamics
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