Volume 19, Issue 4, April 1978
Index of content:

Generating functions for polynomial irreducible tensors
View Description Hide DescriptionA general method for constructing a generating function for all irreducible polynomialtensors, with respect to a given compact semisimple group, out of a given set of tensors is derived. The method is applied to the construction of polynomial bases for the IR’s of a group reduced according to a subgroup and to the finding of subgroup scalars in the enveloping algebra of a group. A number of examples are worked out.

Gel’fand lattice polynomials and irreducible representations of U(n)
View Description Hide DescriptionA finite difference equation defines the exponential of a square tableau, extension of the usual Gel’fand pattern. These exponentials or ’’K powers’’ are homogeneous polynomials useful in the theory of group representations. The theory of these polynomials is developed, and some important addition and multiplication theorems are deduced. The application to the group U(n) gives explicitly the Gel’fand states for n=4, and it is conjectured that the given relation is true in general for any dimension. The matrix elements with respect to this basis are calculated for n=3 and the Clebsch–Gordan decomposition of the n product of U(2) is also given.

A conformal invariant model of localized spinning test particles
View Description Hide DescriptionA purely classical model of massless test particle with spin s is introduced as the dynamical system defined by the ten‐dimensional O(4,2) co‐adjoint orbit with Casimir numbers (s ^{2},0,0). The Mathisson, Papapetrou e t a l. equations of motion in a gravitational field are recovered, and the particle appears to travel on null geodesics. Several implications are discussed.

The moments of the multiple distribution functions
View Description Hide DescriptionThe equations of moments equivalent to BBGKY and to the master equations are established with the introduction of an appropriate formalism. We end with the deduction of equations generalizing hydrodynamics. This article is limited to the monoparticle populations.

Scalar–vector instantons in n dimensions: Surface terms
View Description Hide DescriptionBy an analysis that respects surface terms, it is shown that the equations of motion for a non‐Abelian gauge fieldA ^{ a } _{μ} coupled to a scalar field Φ^{ i } possess no regular, finite‐action solutions in n‐dimensional Euclidean space except for n=4, with Φ trivial, and for n<4.

Killing vectors in plane HH spaces
View Description Hide DescriptionEmploying the spinorial approach to the structure of hyperheavens, we integrate completely the Killing vector equations for plane (case I) hyperheavens, reducing them to one master equation of an extremely plausible form. (In this process, optimally simple gauges are demonstrated for each Petrov type.) The mechanism of generating Ernst potentials by Killing vectors is then investigated, and explicit forms are given. Also, some interesting preliminary study of Killing spinors of type D (0,k) in heavens is presented.

The quantum mechanical Poincaré and Galilei groups
View Description Hide DescriptionSecond countable locally compact representation groups for the Poincaré group (resp. for the Galilei group) and for some of its subgroups are constructed in the ’’quantum mechanical case,’’ i.e., when time and space–time inversions are assumed to be represented by antiunitary operators. The question of their uniqueness up to topological group isomorphisms is investigated.

Some classes of exact solutions of the nonlinear Boltzmann equation
View Description Hide DescriptionWe derive and discuss some exact solutions of the full (nonlinear) Boltzmann equation. One of these is the similarity solution recently found by Krook and Wu for the velocity relaxation problem. Other similarity solutions do exist, and we point out their usefulness in the search for exact solutions of the spatially inhomogeneous Boltzmann equation.

No‐interaction theorem of Currie, Jordan, and Sudarshan. Expansions in c ^{−1}
View Description Hide DescriptionBy means of an expansion in c ^{−1} (c being the velocity of light) it is shown that the no‐interaction theorem of Currie, Jordan, and Sudarshan is valid only when terms of order of at least c ^{−6} are included. To confirm this result, we derive the most general family of approximate Lagrangians up to order c ^{−4}, whose limit is Newtonian.

A method for calculation of Regge poles in atomic collisions
View Description Hide DescriptionA method for solving the Schrödinger equation is given. It is specially developed for applications in atomic (short wavelength) collisions. The method is also useful for calculating Regge poles, without having to define the potential for complex coordinate. The stability of the method is discussed.

An exact solution for a derivative nonlinear Schrödinger equation
View Description Hide DescriptionA method of solution for the ’’derivative nonlinear Schrödinger equation’’ i q _{ t }=−q _{ x x }±i (q*q ^{2})_{ x } is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

Absence of ordering in a class of real liquids—Application to nematic liquid crystals
View Description Hide DescriptionA classical liquid, the constituants of which are characterized by an internal structure described by a compact connected Lie group, is studied. The order parameter of such systems is shown to vanish for dimension smaller than three. The example of nematic liquid crystals is considered.

Time‐dependent scattering theory for infinite delta function potentials in one dimension
View Description Hide DescriptionExistence of the Mo/ller wave operators is proved for a system of n quantum mechanical particles interacting through infinite delta function potentials in one dimension.

A theory of the electromagnetic two‐body interaction
View Description Hide DescriptionA theory of the electromagnetic two‐body interaction is described which leads to equations of motion solvable by local (numerical) integration.

Error bounds for the complex‐coordinate method
View Description Hide DescriptionIn this paper we derive calculable bounds for the error in two‐body s‐wave scattering calculations done by the complex‐coordinate method. In the process we derive a bound on the error in the Born approximation. The utility of the results is discussed in light of simple examples.

Lower hybrid solitary waves
View Description Hide DescriptionIt is found that, in a magnetized plasma, finite amplitude lower hybrid waves can propagate as solitons consisting of localized density cavities together with doublets of electric field spikes. An analytical expression, as well as the corresponding evolution equation, are derived for the small amplitude limit. Such solitons can also exist for many other waves having similar dispersive and nonlinear characteristics.

High‐precision determination of the critical screening length for the static screened Coulomb potential
View Description Hide DescriptionBy a combination of analytical and numerical methods, it is determined that the critical screening length for two particles interacting through an attractive static screened Coulomb potential lies between 0.8399032a _{0} and 0.8399039a _{0}, where a _{0} is the Bohr radius.

Summation of partial wave expansions in the scattering by long range potentials. I
View Description Hide DescriptionPunctual Padé approximants are considered as a summation method of the slowly convergent partial wave expansions associated with the scattering by long range potentials. The asymptotic behavior of the family of sequences [n,n+m], with fixed n, of the Padé table, is studied. A set of theorems are proven, which show that their rate of convergence increases rapidly with n. It is noted that these approximants may be computed by means of the recurrent ε and η algorithms.

Pseudoscalar transition between a spin‐J and a spin‐5/2 baryon
View Description Hide DescriptionThe decay width for a spin‐J baryon decays into a spin‐5/2 baryon, and a pseudoscalar meson is expressed in terms of different types of decay amplitudes. The overall ratio among decay amplitudes and the approximate decay width formula are derived by ignoring the higher partial waves.

Electromagnetic radiation near black holes and neutron stars
View Description Hide DescriptionAnalytic solutions to Maxwell’sequations in the Schwarzschild geometry are given. These are obtained by differentiating a single superpotential, which is valid (and bounded) at and everywhere outside the gravitational radius. The results have application to black‐hole and neutron‐star electrodynamics.