Volume 18, Issue 4, April 1977
Index of content:

Reciprocity relations and forward amplitude theorems for elastic waves
View Description Hide DescriptionRelations are derived between the amplitudes of scatteredelastic 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 scatteredwave in the forward direction alone. Incident waves are restricted to plane, monochromatic waves.

Positive solutions and subcriticality of energy dependent transport systems in slab geometry
View Description Hide DescriptionThe 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−E _{2}(a)]^{−1}, where E _{2}(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.

Partial‐wave analysis for supersymmetric scattering amplitudes
View Description Hide DescriptionThe 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=−j _{0},−j _{0}+1,...,j _{0}−1,j _{0}) in which the spin is not diagonal. Here j _{0}=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.

A Lagrangian formulation for noninteracting high‐spin fields
View Description Hide DescriptionWe 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.

Generalized invariants for the time‐dependent harmonic oscillator
View Description Hide DescriptionA 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,I _{ m n }(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.

Lattices of effectively nonintegral dimensionality
View Description Hide DescriptionWe 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 X Y 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.

Stationary gravitational fields of a dually charged perfect fluid
View Description Hide DescriptionStationary 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.

Renormalization group structure for translationally invariant ferromagnets
View Description Hide DescriptionWe 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 l i n e a r m a p p i n g 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.

Identification of the velocity operator for an irreducible unitary representation of the Poincaré group
View Description Hide DescriptionFor 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.

Ashkin–Teller model as a vertex problem
View Description Hide DescriptionIt 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.

Periods on manifolds, quantization, and gauge
View Description Hide DescriptionIt 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.

Superfields as an extension of the spin representation of the orthogonal group
View Description Hide DescriptionWe 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.

A C*‐algebra formulation of the quantization of the electromagnetic field
View Description Hide DescriptionA 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.

Gauge equivalence of the electrodynamics of charged bosons
View Description Hide DescriptionThe 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.

A simple derivation of a closed formula for Bogoliubov boson transformations
View Description Hide DescriptionConsidering 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.

Moment‐theory approximations for nonnegative spectral densities
View Description Hide DescriptionMoment‐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.

Some examples of transparent potentials in the classical approximation
View Description Hide DescriptionWe 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.

Free electromagnetic fields on a compact Lie group manifold
View Description Hide DescriptionMaxwell’sequations 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.

A class of solutions of the Dirac equation in the Kerr–Newman space
View Description Hide DescriptionIn a region of spacetime that may be described by the Kerr–Schild metric, the gravitational fieldequations define a field of O(3) matrices. By examining the spin representations of these rotations it is first shown how the gravitational fieldequations 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.

Characterization of the Szekeres inhomogeneous cosmologies as algebraically special spacetimes
View Description Hide DescriptionThe 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.