Volume 27, Issue 4, April 1986
Index of content:

Structural analysis and elementary representations of SL(4, R) and GL(4, R) and their covering groups
View Description Hide DescriptionThe structure of the groups SL(4,R) and GL(4,R) and their universal cover− ing groups SL(4,R) and GL(4,R), respectively, and Lie algebras sl(4,R) and gl(4,R), respectively, are studied. The parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out. The Iwasawa and Bruhat decompositions are given explicitly. All elementary representations (ER) of SL(4,R) are explicitly given in two equivalent realizations. Using the preceding detailed structural analysis the SL(4,R) constructions are used for the explicit realization of all ER of SL(4,R), GL(4,R), and GL(4,R). The results shall be applied (among other things) elsewhere for the construction of all irreducible representations of the above groups.

The symmetric group: Algebraic formulas for some S _{ f } 6j symbols and S _{ f }⊇S _{ f 1 }×S _{ f 2 } 3j m symbols
View Description Hide DescriptionExplicit rank‐dependent expressions have been obtained for some symmetric group (S _{ f }) 6j symbols and some S _{ f }⊇S _{ f 1 }×S _{ f 2 } 3j m symbols using Butler’s recursion method. A key point in deriving these results is the use of the reduced notation introduced by Murnaghan to label irreps. Various symmetries of the 6j and 3j m symbols have been imposed. These include the complex conjugation, permutation, and transpose conjugation. We incorporate a new symmetry that arises from the occurrence of the two isomorphic direct product groups S _{ f 1 }×S _{ f 2 } and S _{ f 2 }×S _{ f 1 } as subgroups of S _{ f }. In relation to the tables of 6j and 3j m symbols presented, a discussion is given of the symmetric group‐unitary group duality.

Extension and replacement bases for semisimple Lie algebras
View Description Hide DescriptionTwo simple prescriptions are given for obtaining sets of orthogonal bases for semisimple Lie algebras. The first method allows one to obtain the irreducible representations of all the simple Lie algebras, starting from SU(2) and Dynkin’s method for constructing representations from simple roots and highest weights. The second method relates algebras of the same rank. Several examples are discussed. A method is given for listing the bases that may be obtained from the prescriptions.

Boson realization of sp(4, R). II. The generating kernel formulation
View Description Hide DescriptionIn a previous paper of this series, the matrix elements were discussed with respect to boson states of an operator K ^{2} required for the boson realization of the sp(4, R) Lie algebra. In the present paper, it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp(4, R) coherent states. The results have relevance for the determination of the matrix elements of the generators of the sp(4, R) Lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group, and are, in principle, generalizable to symplectic algebras of higher dimensions.

The Fourier transform of confining potentials
View Description Hide DescriptionThe precise meaning of the Fourier transform of ‖x‖^{ν} is examined. A general expression is given for real positive ν. For odd ν, derivatives of principal value integrals are obtained, while even ν gives rise to derivatives of the delta function.

A class of sums of Gegenbauer functions: Twenty‐four sums in closed form
View Description Hide DescriptionSeries of the type ∑^{∞} _{ n=0} (n+λ) [C ^{λ} _{ n } (1)]^{2−k−l−m }∏^{ k } _{ h=1} C ^{λ} _{ n } (x _{ h }) ⋅ ∏^{ l } _{ i=1} D ^{λ} _{ n } (z _{ i }) ⋅ ∏^{ m } _{ j=1}D^{λ} _{ n } (u _{ j }) are studied. Here C ^{λ} _{ n } is Gegenbauer’s polynomial, also called the ‘‘ultraspherical polynomial,’’ and D ^{λ} _{ n } and D^{λ} _{ n } are Gegenbauer functions of the second kind. Interrelationships and analytic properties are discussed, and closed forms for 24 of these sums are given, more than half of which are new.

Partial wave decomposition of the Glauber amplitude for the elastic scattering of structureless charged particles by atomic hydrogen
View Description Hide DescriptionThe partial wave amplitudes for the conventional Glauber approximation to the elasticscattering of structureless charged particles by hydrogen atoms are evaluated in closed form. The asymptotic behavior of these amplitudes in various limits is described. The large‐r asymptotic behavior of the Glauber effective interaction and the logarithmic divergence of the elastic Glauber amplitude as θ → 0 are discussed. Some numerical results for these partial amplitudes are presented.

Systematics of strongly self‐dominant higher‐order differential equations based on the Painlevé analysis of their singularities
View Description Hide DescriptionThis paper presents a simple way of classifying higher‐order differential equations based on the requirements of the Painlevé property, i.e., the presence of no movable critical points. The fundamental building blocks for such equations may be generated by strongly self‐dominant differential equations of the type (∂/∂x)^{ n } u =(∂/∂x ^{ m })[u ^{(m−n+p)/p }] in which m and n are positive integers and p is a negative integer. Such differential equations having both a constant degree d and a constant value of the difference n−m form a P a i n l e v é c h a i n; however, only three of the many possible Painlevé chains can have the Painlevé property. Among the three Painlevé chains that can have the Painlevé property, one contains the Burgers’ equation; another contains the dominant terms of the first Painlevé transcendent, the isospectral Korteweg–de Vries equation, and the isospectral Boussinesq equation; and the third contains the dominant terms of the second Painlevé transcendent and the isospectral modified (cubic) Korteweg–de Vries equation.Differential equations of the same order and having the same value of the quotient (n−m)/(d−1) can be mixed to generate a new hybrid differential equation. In such cases a hybrid can have the Painlevé property even if only one of its components has the Painlevé property. Such hybridization processes can be used to generate the various fifth‐order evolution equations of interest, namely the Caudrey–Dodd–Gibbon, Kuperschmidt, and Morris equations.

Symmetry transformations, isovectors, and conservation laws
View Description Hide DescriptionA system of second‐order partial differential equations is considered. It is shown that a conservation law may be associated with any pair consisting of (i) a symmetry transformation, (ii) a symmetry transformation if the system is self‐adjoint, a solution to the adjoint of the equations of variation otherwise. Such conservation laws continue to hold when symmetry transformations are replaced by isovectors. It is also proved that isovectors identify additional conservation laws by deformation of a given one, whence it follows that there exists a natural conserved current associated with every isovector. Also an application is made to find conserved currents for the Navier–Stokes equations.

Killing spinors on spheres and hyperbolic manifolds
View Description Hide DescriptionProperties of Killing spinors on spheres and hyperbolic manifolds are investigated with an emphasis on the relations to Killing vectors, conformal Killing vectors, and solutions of Maxwell’s equations.

The problem of ‘‘global color’’ in gauge theories
View Description Hide DescriptionThe problem of ‘‘global color’’ (which arose recently in monopole theory) is generalized to arbitrary gauge theories: a subgroup K of the ‘‘unbroken’’ gauge group G is implementable iff the gauge bundle reduces to the centralizer of K in G. Equivalent implementations correspond to equivalent reductions. Such an action is an internal symmetry for a given configuration iff the Yang–Mills field reduces also. The case of monopoles is worked out in detail.

Stochastic averaging by functional differentiation
View Description Hide DescriptionThe stochastic average of exponential expressions that depend at most quadratically upon a Gaussian random function is evaluated using a functional method. The average can be obtained formally for arbitrary forms of the correlation function. The result, however, depends upon a functional determinant and the inverse of a linear operator. Cases are discussed, in particular the Ornstein–Uhlenbeck process, where the latter quantities can be evaluated explicitly.

Statistical filter theory and the O’Doherty–Anstey effect: Dependence on offset
View Description Hide DescriptionThis paper considers the effects of multiples, generated at plane bed interfaces, on the characteristics of a mean seismic wave propagating at an angle to the bedding plane. It is found that(a) the multiples produce frequency‐ and angle‐dependent phase shifts to the coherent wave as well as providing an effective attenuation, which is also frequency and angle dependent; (b) a slim angular pencil of monochromatic waves rapidly loses information about its original angular width due to the multiples as the pencil propagates; (c) a seismic pulse, initially traveling at a fixed angle, has both its envelope amplitude and its phase distorted by multiples, and after a short distance of transmission into the medium, the wave shape is nearly completely determined by the generated multiples and is only slightly beholden to the initial pulse shape; (d) the phase and group directions of the mean seismic wave are different than the incident wave’s direction and are canted closer to the horizontal with the group direction being the most highly canted; and (e) lateral spreading information of the mean seismic wave is contained in the cross‐correlated response of separated geophones and, in principle, can be extracted from cross‐correlated measurements. How the generic response depends on the power spectrum of the reflectivity sequence is illustrated by comparing and contrasting results for a transitional sedimentation pattern with those from a cyclic sedimentation pattern; the former produces both a frequency‐dependent time delay and attenuation while the latter produces a pure time delay except in the local vicinity of isolated, but periodic frequencies. Numerical estimates, using parameters believed representative of typical seismic conditions, indicate that all of the effects uncovered are large—in the sense that they fall squarely in the regime where they can be expected to have a significant impact both on the subsurface evolution of seismic waves and on interpretations of subsurface conditions made using surface‐received signals.

Is there action‐at‐a‐distance linear confinement?
View Description Hide DescriptionThe possilibity of constructing an action‐at‐a‐distance form of linear confinement is demonstrated. Using the Fokker–Wheeler–Feynman action principle, known from classical action‐at‐a‐distance electrodynamics, with an action containing the relativistically invariant two‐particle Heaviside step function, equations of motion and appropriate potentials exhibiting the linearity of their behavior are derived. The plausibility of the generators of motion describing dynamics with the linear potentials is verified on the simple circular‐orbit model of a two‐component system, and the expected energy spectrum in terms of semiclassical quantization is obtained.

Action‐at‐a‐distance linear potentials and conformal conservation laws
View Description Hide DescriptionAs a consequence of invariance under the full conformal group of transformations, 15 proper conserved quantities are derived for the system of two massive particles interacting via classical action‐at‐a‐distance linear potentials that have been found in the preceding paper [J. Weiss, J. Math. Phys. 2 7, 1015 (1986)] in terms of the step θ function Lagrangian.

Potential envelopes and the large‐N approximation
View Description Hide DescriptionIf E is an eigenvalue of the quantum‐mechanical Hamiltonian H= 1/2 Δ+V(r) in Nspatial dimensions, then large‐Ntheory, the potential‐envelope method, and scale‐optimized variational energies all lead to quasiclassical approximations having the same form given by E(α) = min_{ r>0}[ 1/2 αr ^{−} ^{2}+V(r)], where α depends on the quantum numbers and on N. Energy bounds provided by the envelope method allow us to prove that in many cases the large‐N results are lower energy estimates. For pure power‐law potentials all these energies approach the exact eigenvalue in either of the limits l → ∞ or N → ∞.

Energy eigenvalues of d‐dimensional quartic anharmonic oscillator
View Description Hide DescriptionOn the basis of a radial generalization of the JWKB quantization rule, which incorporates higher orders of the approximation, an explicit analytical formula is derived for the energy levels of the three‐dimensional quartic anharmonic oscillator. The formula exhibits the scaling property of the exact eigenvalues, and is readily generalized to any dimension. Together with the Hellmann–Feynman theorem, it yields the values of the diagonal moments of r ^{2k }. The predicted energies and moments are in excellent agreement with known numerical results.

Massless fields in curved space‐time: The conformal formalism
View Description Hide DescriptionA conformally invariant theory for massless quantum fields in curved space‐time is formulated. We analyze the cases of spin‐0, ‐ 1/2 , and ‐1. The theory is developed in the important case of an ‘‘expanding universe,’’ generalizing the particle model of ‘‘conformal transplantation’’ known for spin‐0 to spins‐ 1/2 and ‐1. For the spin‐1 case two methods introducing new conformally invariant gauge conditions are stated, and a problem of inconsistency that was stated for spin‐1 is overcome.

Compatibility of weak rigidity with some types of elastic schemes
View Description Hide DescriptionThe behavior of the hypoelastic‐Synge, hypoelastic‐Maugin, and hypoelastic‐Carter and Quintana almost‐thermodynamic material schemes, under weak rigidity hypotheses, is studied. In every case, the absence of principal transverse shock waves (or the vanishing of the corresponding speeds) is obtained. The same result follows for the longitudinal shock waves when the Lamé coefficient μ does not vanish. A definition of an elastic almost‐thermodynamic material scheme based on the Fermi–Walker transport is proposed and compared with the above elastic schemes. The speeds of the principal shock waves associated are attained and its compatibility with the Ferrando–Olivert incompressibility condition is proved. In the presence of weak rigidity the elastic schemes here defined lead (assuming μ≠0) to the Born‐rigidity condition.

A Kerr object embedded in a gravitational field
View Description Hide DescriptionThe exact expression for the Ernst potential for a Kerr object embedded in a gravitational field is derived using a formalism developed by Kramer and Neugebauer.