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Volume 13, Issue 7, July 1972

Generalized Lienard‐Wiechert Fields
View Description Hide DescriptionWe show that a moving point multipole source for a spin‐S field gives rise to radiation having (in the rest frame of the source) multipole structure of order S through 2L + S, where L is the highest order moment of the source.

Closed Rotating Cosmologies Containing Matter Described by the Kinetic Theory. Entropy Production in the Collision Time Approximation
View Description Hide DescriptionA collision time approximation for the production of partly collisional particles (``neutrinos'') from a collision dominated fluid (``electrons and photons'') within closed anisotropic Bianchi type IX cosmologies is discussed. Two distinct, constant, times are introduced; the neutrino production rate is distinct from the neutrino destruction rate The conservation of energy and momentum is discussed, and it is shown, for a subclass of rotating closed cosmologies and for Maxwell‐Boltzmannneutrinos, that the local entropy production rate s ^{μ} _{;μ} is positive, aside from ``rest entropy'' terms, i.e., terms representing a fixed amount of entropy permanently attached to each neutrino (which goes out of existence when the neutrino is destroyed and reappears when a neutrino is regenerated). The results here contain as appropriate limits the same conclusions within special relativity and within simpler (Bianchi type I) cosmologies.

Poincaré and TCP Invariance in the Determination of Wave Equations for Particles of Arbitrary Spin
View Description Hide DescriptionA brief outline is given of different types of approaches to the derivation of relativistic wave equations for free particles of arbitrary mass m and spin s. It is pointed out that all such equations can be subsumed in a standard form involving wavefunctions transforming according to the representation of the Lorentz group. Generalizing earlier work employing this standard form, we determine all such equations (in a standard form) which are invariant under the Poincaré group and the combined operation Θ ≡ TCP. It is shown that the resulting equations are automatically invariant under T, C, and P separately.

Bounds for Solution of Nonlinear Wave‐Wave Interacting Systems with Well‐Defined Phase Description
View Description Hide DescriptionIn this paper, upper and lower bounds for the solutions of nonlinear three‐wave interactingsystems with well‐defined phase description are established. The results are used to deduce a sufficient condition for the non‐existence of explosive instabilities.

Some Comments on the Behavior of Acceleration Waves of Arbitrary Shape
View Description Hide DescriptionThis work concerns an application of the general results of Bailey and Chen to the study of acceleration waves of arbitrary shape. A modification of one of their main theorems is also presented; it applies to converging waves. For both converging and diverging waves, general expressions are given in terms of the initial principal curvatures. For diverging waves, a necessary and sufficient condition is given for the initial critical amplitude to be nonzero. In addition, Bailey and Chen's asymptotic formulas are specialized so as to apply to acceleration waves of arbitrary shape.

Harmonic Analysis of Analytic Functions on Hyperspheres
View Description Hide DescriptionThe real analytic functions on the hypersphere S^{n} are shown to be in one‐to‐one correspondence with the family of series of hyperspherical harmonics with exponentially falling coefficients. These functions may be continued onto a larger complex manifold on which they represent holomorphic functions. The convergence of the harmonic expansions for the real analytic functions on S^{n} is governed by the singularity structure of the continued function on this complex manifold.

Structure of the Gravitational Field at Spatial Infinity
View Description Hide DescriptionA formalism is introduced for analyzing the structure of the gravitational field in the asymptotic limit at spatial infinity. Consider a three‐dimensional surface S in the space‐time such that the initial data on S is asymptotically flat in an appropriate sense. Using a conformal completion of S by a single point A ``at spatial infinity,'' the asymptotic behavior of fields on S can be described in terms of local behavior at A. In particular, the asymptotic behavior of the initial data on S defines four scalars which depend on directions at A. Since there is no natural choice of a surface S in a space‐time, the dependence of these scalars on S is essential. The asymptotic symmetry group at spatial infinity, whose elements represent transformations from S to other asymptotically flat surfaces, is introduced. It is found that this group, which emerges initially as an infinite‐dimensional generalization of the Poincaré group, can be reduced to the Lorentz group. A set of evolution equations is obtained: These equations describe the behavior of the four scalars under the action of the asymptotic symmetry group. The four scalars can thus be considered as fields on a three‐dimensional manifold consisting of all points at spatial infinity. The notion of a conserved quantity at spatial infinity is defined, and, as an example, the expression for the energy‐momentum at spatial infinity is obtained.

Scattering Theory in Fock Space
View Description Hide DescriptionIt is shown that the wave operators which can be introduced in quantum field theories with cutoff interactions can be represented by wave transformers acting on states represented by statistical operators. Integral representations can be found for these wave transformers, which then form the basis of a time‐independent scattering theory for such quantum field theories. It is pointed out that this leads to a time‐independent perturbation theory in Fock space.

On the Equivalence of Dressing Transformations
View Description Hide DescriptionThe equivalence of the representations of the Weyl algebra for in a box induced by the dressing transformation of Glimm and the unitary dressing transformation is studied. Equivalence is shown for the simplified model in which only the most singular portion of the interaction is kept.

A Tachyon Dust Metric in General Relativity
View Description Hide DescriptionSpecial relativity allows the possibility of a class of particles, called tachyons, which travel with speeds greater than the speed of light in vacuum. These particles have spacelike 4‐velocities. Since tachyons have energy and momentum, they will contribute to the gravitational field through the energy‐momentum tensor. One question then is what types of solutions to the Einstein field equations will tachyons yield. We consider a metric which admits a four‐parameter isometry group. When this metric is used in the field equations using a dust energy‐momentum tensor, solutions exist only for spacelike 4‐velocity of the dust. We interpret these as solutions for a tachyon dust. Exact solutions to the field equations are obtained.

Time‐Dependent and Time‐Independent Potential Scattering for Asymptotically Coulomb Potentials
View Description Hide DescriptionWe consider the three‐dimensional quantum mechanical problem of nonrelativistic potential scattering, where the Hamiltonian is H _{2} = H _{1} + V = H _{0} + V_{c} + V;H _{0} is the free particle Hamiltonian, V_{c} is the Coulomb potential, and V(x) is a real‐valued potential function defined for x ∈ R ^{3}. We show the existence of the wave operators W _{±}(H _{2},H_{D} ) = s‐lim e ^{ iH 2 t } e ^{ −iHD (t)} on L ^{2}(R ^{3}) for V = V _{2} + V′, where is the family of unitary operators used by Dollard to show the existence of the wave operators W _{±}(H _{1},H_{D} ) appropriate for the pure Coulomb case. If V is spherically symmetric then W _{±}(H _{2},H_{D} ) are shown to be absolutely continuous complete. If in addition V(r) is continuous in (0,∞), then W _{±}(H _{2},H_{D} ) are continuous complete. In both cases is unitary. The connection between the more physical, time‐dependent wave operator approach and the traditional time‐independent method is made, and phase shift formulas are obtained for the wave and scattering operators.

On Weyl and Lyra Manifolds
View Description Hide DescriptionIt is shown that Weyl's geometry and an apparently similar geometry suggested by Lyra are special cases of manifolds with more general connections. The difference between the two geometries and their relationship with Riemannian geometry are clarified by giving a global formulation of Lyra's geometry. Finally the outline of a field theory based on the latter geometry is given.

Perturbations of Gibbs States. I. Grand Canonical Formalism for Self‐Interacting Fermion Systems
View Description Hide DescriptionThe perturbation series for the statistical operators of quantum statistical mechanics developed earlier is applied to provide a perturbation theory for the grand canonical Gibbs states of a self‐interacting fermion system. Explicit formulas for an interaction Hamiltonian which is a polynomial in creation‐annihilation operators are provided.

The Plasma Inverse Problem
View Description Hide DescriptionA simple and direct development of the theory of the plasma inverse problem is given, and it is shown that the time record of the reflected wave arising from a δ function electric field normally incident on a stratified plasma determines uniquely the plasma density through an integral equation.

Lattice Wind‐Tree Models. I. Absence of Diffusion
View Description Hide DescriptionSome lattice versions of the wind‐tree model of Ehrenfest are introduced and studied. They include two versions in which overlapping of tree particles is forbidden, a third in which it is allowed, and a fourth in which it is allowed but results in a finite repulsive force. In every case it is found that the mean‐square displacement Δ(t) of the wind particles at time t is bounded independently of t at sufficiently high density of the trees. This is in sharp contrast with the Einstein relation Δ(t) = O(t) as t → ∞, which might be expected to hold at low densities. Randomization of the initial velocity of a wind particle is also shown to occur in a certain sense, and upper bounds on a recurrence time are obtained at high density. On the other hand, it is shown that thermalization does not occur even at low densities.

General Theory of Spherically Symmetric Boundary‐Value Problems of the Linear Transport Theory
View Description Hide DescriptionA general theory of spherically symmetric boundary‐value problems of the one‐speed neutron transport theory is presented. The formulation is also applicable to the ``gray'' problems of radiative transfer. The Green's function for the purely absorbing medium is utilized in obtaining the normal mode expansion of the angular densities for both interior and exterior problems. As the integral equations for unknown coefficients are regular, a general class of reduction operators is introduced to reduce such regular integral equations to singular ones with a Cauchy‐type kernel. Such operators then permit one to solve the singular integral equations by the standard techniques due to Muskhelishvili. We discuss several spherically symmetric problems. However, the treatment is kept sufficiently general to deal with problems lacking azimuthal symmetry. In particular the procedure seems to work for regions whose boundary coincides with one of the coordinate surfaces for which the Helmholtz equation is separable.

Analytic Renormalization of the Exponential Interaction: The Three‐Point Time‐Ordered Product with Minimum Light Cone Singularity
View Description Hide DescriptionA method of analytic renormalization is developed to define the three‐point time‐ordered product of massless fields of exponential type as a strictly localizable distribution. The uniqueness property, known for the two‐point T‐product, is verified for the three‐point product, for a special choice of fine renormalization. It is characterized by minimum singularity on the light cone: There are no delta function type singularities concentrated on the surface x _{1} = x _{2} = x _{3}.

Thermodynamics and Hydrodynamics for a Modeled Fluid
View Description Hide DescriptionThis article presents a general study of a two‐dimensional fluid model with microscopic discrete velocities. The rather unusual properties of this model lead to precise thermodynamical laws. The Navier‐Stokes hydrodynamical equations are obtained, which contain a transport coefficient given by a Green‐Kubo integral, and it is shown that this integral does not converge for a reason common to all two‐dimensional fluids.

Algebraic Solution for the Källén‐Pauli State Vectors in the Vθ Sector by a Congruence Transformation
View Description Hide DescriptionFrom the representation of Bolsterli of the Vθ sector physical states in a basis having singular integral equations of the separable type, a congruence transformation carries the representation into the Källén‐Pauli Heisenberg field components. This solves the Källén‐Pauli singular integral equation algebraically. The resulting state vectors are proven explicitly to be both complete and orthonormal and furnish a new Möller wave matrix.

Incoherent Exciton Quenching on Lattices
View Description Hide DescriptionThe time‐dependent excitation function φ(t) for exciton quenching in lattices of any dimension is derived from a general master equation in terms of perfect lattice Green's functions. Attention is largely restricted to nearest‐neighbor interactions for which the Green's functions have simple forms. The quenchers are characterized by three dimensionless rate parameters: λ, for nearest neighbor host‐quencher energy transfer; μ, for back transfer from quencher to host; and Q, for irreversible degradation of excitation on the quencher. Simple expressions for the Laplace transform of φ(t) under two different initial conditions are given for low concentrations of periodically placed quenchers that have identical but arbitrary λ, μ, and Q. Randomly placed quenchers are treated by adapting the coherent potential approximation; in three dimensions, this method gives a φ(t) that is identical to that with periodic quenchers of the same low concentration. Lattices with two types of defects‐one with μ = 0 and one with μ > 0, a case of particular interest in organic crystals‐are treated in some detail. Finally, it is shown that energy transferanisotropies have little effect on φ(t), unless the smallest relative transfer rate is at least as small as the quencher concentration.