Volume 8, Issue 11, November 1967
Index of content:

Spin‐s Spherical Harmonics and ð
View Description Hide DescriptionRecent work on the Bondi‐Metzner‐Sachs group introduced a class of functions _{s}Y_{lm} (θ, φ) defined on the sphere and a related differential operator ð. In this paper the _{s}Y_{lm} are related to the representation matrices of the rotation group R _{3} and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the _{s}T_{lm} (θ, φ) to the spherical harmonics of R _{4} is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the _{s}T_{lm} under this latter group is shown to realize a representation of the Lorentz group.

Invariant Transformations and Newman‐Penrose Constants
View Description Hide DescriptionThe relationship between constants of the motion and invariant transformations is discussed. Particular emphasis is placed on strong conservation laws which are of the form . The existence of such a law leads to constants of the motion which are surface integrals and therefore generally do not generate an invariant transformation. However, when there is an associated weak conservation law, such that t ^{μ},_{μ} ≡ − y_{A}L^{A} (L^{A} = 0 are the field equations and y_{A} the invariant change in field variables), a nontrivial invariant transformation exists. These results are applied to the discussion of the Newman‐Penrose constants for the electromagnetic field. The conclusion arrived at is y_{A} = 0, which suggests that the invariant transformation generated by the Newman‐Penrose constants is trivial.

Solution of the Differential Equation
View Description Hide DescriptionA series solution of the above differential equation is presented.

New Form of Characteristic Functional for Cascade Processes
View Description Hide DescriptionThe characteristic functional for an energy‐dependent cascade process is redefined so as to introduce the arbitrary function without using exponentials. An immediate consequence is that the functional becomes the function at t = 0 when the process is multiplicative with one particle at the initial time. A simple one‐component model of a cosmic‐ray shower is used to illustrate how this definition leads directly to (a) a simple form of the Chapman‐Kolmogoroff equation, (b) forward and backward integro‐differential equations for the functional, and (c) the derivation of all probability functions of interest for the process. An example is given of the distributions in number of particles at different depths when the total cross section for multiplication is proportional to the energy. Some general forms of functional are proposed. The extension to electron‐photon showers is outlined.

Expansion Method for Nonlinear Boundary‐Value Problems
View Description Hide DescriptionA homogeneous nonlinear boundary‐value problem, which reduces to the Helmholtz equation when the nonlinearity is removed, is solved by an expansion method using as a basis the eigenfunctions of the linear Helmholtz equation. The nonlinear differential equation is reduced to a nonlinear algebraic system in the expansion coefficients, which can be easily solved with any desired degree of accuracy. It is found that with only three terms in the eigenfunction expansion, a satisfactory agreement with the exact numerical solution of the problem is obtained, even for strongly nonlinear cases. Solutions are presented both in one‐dimensional slab and cylindrical geometries. It is also shown that the method can be applied to inhomogeneous problems.

Interpolation Method for the Many‐Body Problem
View Description Hide DescriptionVariational principles for lower bounds to the energy, or free energy for T > 0°, of many‐body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy—or free energy—for the model Hamiltonian H_{M} . Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H − H_{M} approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs‐Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.

Matrix Products and the Explicit 3, 6, 9, and 12‐j Coefficients of the Regular Representation of SU(n)
View Description Hide DescriptionThe explicit Wigner coefficients are determined for the direct product of regular representations, , of SU(n), where N = n ^{2} − 1. Triple products C_{m}C_{i}C_{m} = αF_{i} + βD_{i} , and higher‐order products, are calculated, where C_{i} may be F_{i} or D_{i} , the N × N Hermitian matrices of the regular representation, and m is summed. The coefficients α, β are shown to be 6‐j symbols, and higher‐order products yield the explicit 9‐j, 12‐j, symbols. A theorem concerning (3p)‐j coefficients is proved.

Internal Multiplicity Structure and Clebsch‐Gordan Series for the Exceptional Group G(2)
View Description Hide DescriptionAn explicit algebraic formula is obtained for the multiplicity M̄(γ) of a vector γ belonging to the fundamental domain of the group G(2). Using this, the internal multiplicity M^{m} (m′) of a weight m′ of the irreducible representation D(m) with the highest weight m is calculated through Kostant's formula for the dominant weights. The Clebsch‐Gordan decomposition of the direct product of two irreducible representations is then obtained.

Unitary Representations of the Group O(2, 1) in an O(1, 1) Basis
View Description Hide DescriptionWe consider the unitary irreducible representations of the group SO(2, 1), belonging to the continuous and the discrete classes. We cast them into a form in which the noncompact generator of an O(1, 1) subgroup is diagonal. We examine some properties of the remaining generators in this basis. We recover the known result that the spectrum of the noncompact generator covers the real line twice for representations of the continuous class, and once for those of the discrete class.

Geometric Theory of the Spin‐Weighted Functions
View Description Hide DescriptionThe spin‐weighted functions introduced recently are shown to be eigenfunctions of the total angular momentum for appropriately defined geometric objects on the sphere, namely the Pensov objects.

Finite‐Range Effects in Distorted‐Wave Born‐Approximation Calculations of Nucleon Transfer Reactions
View Description Hide DescriptionA method for doing distorted‐wave Born‐approximation calculations for nucleontransfer reactions is presented. This method is designed to be used when the zero‐range approximation cannot be made. The method has the advantage that only simple quadratures need be performed. When recoil effects are negligible, our method leads to a particularly simple form. The technique developed here can be applied to the evaluation of the six‐dimensional integrals that result when the general two‐body interaction is considered. Thus it may be usefully applied to nuclear‐structure calculations when it is desired to use wavefunctions which are not harmonic‐oscillator eigenfunctions.

Partial Group‐Theoretical Treatment for the Relativistic Hydrogen Atom
View Description Hide DescriptionThe Lie algebra of SU (1, 1) and its Hermitian representations are used together with spherical harmonics to solve the wave equations for the nonrelativistic q‐dimensional oscillator and the relativistic Kepler problem.

Discontinuities in Nonideal Magnetogasdynamics
View Description Hide DescriptionAn evolutionary condition is derived from generalized jump relations including dissipation terms and applied to the basic equations of magnetogasdynamics. Two contact discontinuities existing as solutions of these equations are found not to be evolutionary.

Effective Dielectric Tensor and Propagation Constant of Plane Waves in a Random Anisotropic Medium
View Description Hide DescriptionRandom anisotropic media are assumed to be characterized by dielectrictensors in which the components are random functions of position. A turbulent plasma in a static magnetic field is one example of such media. In this paper wave propagation in turbulent magnetoactive plasma is studied. The averages of electric field and dielectric displacement vectors over an ensemble of these media are found, assuming these average quantities are time‐harmonic plane waves. The effective dielectrictensor is defined as the proportionality factor between the two average quantities. When this effective dielectrictensor is applied to the wave equation, a general dispersion relation for plane waves is derived. Expressions for propagation constants are obtained and some special cases are considered in detail. It is found that, because of random scattering, there are attenuations for both the ordinary and extraordinary waves for the average fields. The results reduce to those obtained by J. B. Keller and F. C. Karal when the anisotropy of the background media is removed

Properties of Velocity‐Dependent Potentials
View Description Hide DescriptionProperties of the solutions of the Schrödinger equation with a velocity‐dependent potential are studied. Particular attention is given to the examination of the singularities of the differential equation. In the particular cases of one dimension and of the l = 0 partial wave of a spherically symmetric problem, a simple correspondence is found between the velocity‐dependent problem and a static one.

Substitution Group and Mirror Reflection Symmetry in Special Unitary Groups
View Description Hide DescriptionThe substitutions leaving the character of the representation of the group SU_{n} invariant are considered. The phases induced by these substitutions on the basis functions are established. The substitution giving the contragrediency transformation has been found. This transformation is interpreted as the reflection of the subspace of commuting operators and the corresponding coordinate systems with respect to the rest subspace. The application of the substitution group to the resolution of multiplicity problem in the case of SU _{3} is demonstrated.

Canonical Realizations of the Rotation Group
View Description Hide DescriptionA general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics, proposed in a preceding paper, is applied to the rotation group. A number of significant physical examples corresponding to nonirreducible realizations is treated in detail: specifically, the mass point, the rotator, and the rigid body with a fixed point. The explicit form of the possible irreducible realizations is worked out. Such realizations do not directly correspond to any realistic physical model but play a relevant role for the introduction of the spin in classical mechanics.

Connection between the Marchenko Formalism and N/D Equations: Regular Interactions. I
View Description Hide DescriptionIn this paper and in the following, we study, in potential scattering, the existence and meaning of the solutions of the N/Dequations in the equivalent formulation f/f. For S waves, considering only regular discontinuity μΔ(x), such that the resulting integral equation is of the Fredholm type, we study the corresponding Fredholm determinant D(μ). We remark that Marchenko formalism gives exactly the same resulting equation and then we have the possibility to interpret in terms of local potentials. We show the connection between the nth trace of the kernel of the resulting integral equation and the nth term of the potential reconstructed from the discontinuity. The connection between dispersion relation and the corresponding potential reconstructed from the discontinuity is given by the relation.In the present paper we limit our study to μ less than the smallest modulus root of D(μ) where a perturbation expansion of the solution exists and we show that V(r, μ) is regular at r = 0. On the other hand, for Yukawa‐type potentials where the inverse Laplace transform is λC(α) the Fredholm determinant is exp and cannot vanish such that the corresponding solutions of the resulting integral equation exist always.

Existence of Solutions of Crossing‐Symmetric N/D Equations
View Description Hide DescriptionA fixed‐point theorem is applied to the N/Dequations of ππ scattering, for which crossing is satisfied exactly by the absorptive part of the amplitude, up to a finite, but arbitrary cut off. It is found that ghost‐free solutions exist, if the subtraction constants are not too large, but that these solutions are not unique, since the Castillejo‐Dalitz‐Dyson ambiguity is not resolved by the requirement of crossing symmetry.

Geometrization of a Complex Scalar Field. II. Analysis
View Description Hide DescriptionThe necessary and sufficient conditions that a Ricci tensor should represent the energy tensor of a complex scalar field are given for the general non‐null case. The complexion gradient of the field is determined only to within a sign by the Ricci tensor, unlike the case of the electromagnetic problem. The physical content of a field which corresponds to massless ``pions'' is expressed entirely in geometric terms within the Rainich scheme.