Volume 5, Issue 12, December 1964
Index of content:

Theorem Relating the Eigenvalue Density for Random Matrices to the Zeros of the Classical Polynomials
View Description Hide DescriptionA theorem due to Stielties shows that the problem of locating the zeros of the classical polynomials is equivalent to finding the electrostatic equilibrium positions for a set of interacting point charges. Defining a density function for these same charges we find that the density of zeros for the nth Hermite polynomial is the same as the density of eigenvalues for an ensemble of n‐dimensional Hermitian matrices. Similarly the location of the zeros of the nth Laguerre polynomial determines the density of eigenvalues for an ensemble of n‐dimensional positive matrices, and the zeros of the ½nth Tchebichef polynomial determine the density for the real part of the eigenvalues for an ensemble of n‐dimensional unitary matrices.

If Microns Were Fermis
View Description Hide DescriptionEvidence is rapidly accumulating that elementary particle states are simply the rotational levels of a symmetrical top, as derived from relativistic quantum mechanics. Some of this evidence is presented for the case of zero‐strangeness bosons. An analogy is drawn between this point of view and the nonrelativistic theory of the rotational levels of a rigid molecule.

Electromagnetic Radiation in the Presence of Moving Simple Media
View Description Hide DescriptionThe radiation pattern of an arbitrary source immersed in a moving simple medium is calculated by deducing the differential equation for the potential 4‐vector in the rest frame of the source and then solving the equation in terms of a Green's function. As an illustrative example, the case where the source is an oscillating dipole is worked out in detail.

High‐Field Magnetoresistance of Inhomogeneous Semiconductors and Plasmas. II. Two‐Dimensional Inhomogeneity Distributions
View Description Hide DescriptionWe extend in this paper the treatment of the high field magnetoresistance of a previously described classical model of a semiconductor(plasma) containing a two‐dimensional distribution of inhomogeneities. The basic assumptions on the classical model are that the scale of the inhomogeneities is large compared to the mean thermal wavelength of an electron and the Landau level spacing is large compared to kT. The magnetic fieldH is taken parallel to the z coordinate and the inhomogeneity distribution is characterized by a sufficiently smooth potential φ(x, z). The 4‐moment equations are solved asymptotically for large H, and an equivalent asymptotic solution is obtained, subject to certain mathematical assumptions, for the transport equation. The magnetoresistance is shown, in general, not to saturate, but to increase, as H^{2}, with increasing H.

High‐Field Magnetoresistance of Inhomogeneous Semiconductors and Plasmas. III. Two‐Dimensional Inhomogeneity Distributions
View Description Hide DescriptionIn previous papers we have studied the high magnetic fieldmagnetoresistance of a classical modelsemiconductor (or a plasma) produced by the presence in the sample of more or less random spatial inhomogeneities on a scale small compared to the size of the sample. We extend our computations of the leading terms of the components of the resistivity tensor to the two‐dimensional case in which the external magnetic fieldH is inclined (but not perpendicular) to the plane of variation of the spatial inhomogeneity distribution. The field H is applied along the z axis, and the inhomogeneity distribution is characterized by a sufficiently smooth potential φ(x, z + λy). While the magnetoresistance ratios saturate in the y and z directions, one finds that this ratio is proportional to H^{2} (as H → ∞) in the x direction.

Numerical Method for the Exact Expansion of Generating Functions
View Description Hide DescriptionA new method is developed for the evaluation of coefficients in multinomial generating functions. This method requires solely numerical techniques to derive the coefficients in the expansion of the generating function, and thus avoids the organizational problems that occur when algebraic methods are used to expand these functions. With the use of a digital computer this method enables the number of coefficients to be extended considerably. As an example of its application some of the coefficients in a four‐variable generating function are calculated.

Generalization of the Variational Method of Kahan, Rideau, and Roussopoulos and Its Application to Neutron Transport Theory
View Description Hide DescriptionThe variational method of Kahan, Rideau, and Roussopoulos (KRR) frequently used in neutron transport theory to estimate weighted averages is extended and generalized. In the KRR variational method a first variation in the trial functions produces a second variation in the estimate of the weighted average. Two generalized variational functionals which depend on trial operators instead of trial functions are given. A first variation in the trial operators produces an nth variation in the estimate of the weighted average when an nth order generalized variational functional is used. Both perturbation theory and the KRR variational method are derived as special cases of the generalized variational method. Several examples including calculations of transport equation spatial moments using diffusionequationsolutions as trial operators are studied with good results.

Useful Operator in Plasma Kinetic Theory
View Description Hide DescriptionAn operator which facilitates the derivation of the plasma kinetic equation is introduced and discussed. The operator determines the first integral of the pair correlation function without the necessity for knowledge of the pair correlation function itself. To break through the lengthy algebraic effort which is usually encountered in solving the truncated BBGKY hierarchy equations, the present operator method is found to be far superior to the singular integral equation technique. The mathematical simplification which can be gained from the use of this operator is demostrated by several examples.

Representation of Fields in a Two‐Dimensional Model Theory
View Description Hide DescriptionThe two‐dimensional model recently suggested by Schroer is examined. The free scalar massless field in two dimensions is discussed in detail. The infrared‐myriotic representations of this field which arise in the model are described; it is shown that, up to unitary equivalence, the required representation depends only on the net total charge of the fermions. Discussion is also given of distribution‐theoretic aspects of the fields, and in particular, of possible restrictions on test functions.

On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_{n}
View Description Hide DescriptionThe analog of the SU _{2} (1 − j) symbol is defined and discussed in detail for SU_{n} . An appropriate generalization to SU_{n} of the Condon‐Shortley phase convention is explicitly given.

On the Representations of the Semisimple Lie Groups. IV. A Canonical Classification for Tensor Operators in SU _{3}
View Description Hide DescriptionIt is shown that the multiplicity structure of the general SU_{n} operators may be put in a one‐to‐one correspondence with the multiplicity structure of the corresponding states. This result allows a convenient labeling scheme to be devised for the general SU_{n} Wigner operator and leads in a natural way to the concept of a reduced Wigner operator. The problem of multiplicity in tensor operators is shown to have a canonical resolution in the conjugation classification which is discussed in detail for the SU _{3} case.

Applications of the Lorentz Transformation Properties of Canonical Spin Tensors
View Description Hide DescriptionSome applications of the Lorentz transformations of relativistic spin tensors in the canonical representation are discussed. The problem of precession of polarization is discussed in Sec. 2. It is shown that the kinematical equation, obtained quite simply, already contains the ``Thomas factor.'' In Sec. 3, applications to polarization analysis of decay products are considered. The canonical form of S‐matrix elements are used and multipole parameters for successive decays of the type a → b + c are obtained in an arbitrary frame in a relatively simple way. The exact relativistic way in which the multipole parameters depend, in an arbitrary frame, on the particle momenta are discussed for decays of the type a → b + c + d. While the canonical representation is used mainly, the corresponding technique in the spinor representation is discussed.

Reduction of the N‐Particle Variational Problem
View Description Hide DescriptionA variational method is presented which is applicable to N‐particle boson or fermion systems with two‐body interactions. For these systems the energy may be expressed in terms of the two‐particle density matrix: . In order to have the variational equation: δE/δΓ = 0 yield the correct ground‐state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N‐particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2  1′, 2′) and γ(1  1′) are the two‐particle and one‐particle density matrices of an N‐particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]

Effect of Small Irregularities on Electromagnetic Scattering from an Interface of Arbitrary Shape
View Description Hide DescriptionA perturbation analysis is used to determine the effect of small irregularities on electromagnetic scattering from an interface between two media. The interface irregularities are replaced by approximate equivalent surface currents, and the field in space can then be found using the dyadic Green's function of the unperturbed problem. The approach is valid when the irregularity has small slope and amplitude small compared to the wavelengths and local radii of curvature. To facilitate applications, the theory of dyadic Green's functions is developed, and the necessary functions are given for some important geometries.

Comparison of Two Methods for Lower Bounds to Eigenvalues
View Description Hide DescriptionIt is shown that two procedures given by Bazley and Fox for the calculation of lower bounds to eigenvalues of self‐adjoint operators are essentially equivalent.

Generalized Perturbation Expansion for the Klein‐Gordon Equation with a Small Nonlinearity
View Description Hide DescriptionA previously given method for deriving secularity‐free perturbation expansions for the Klein‐Gordon equation with a ``small'' nonlinear term is generalized to include situations in which the lowest‐order solution is not restricted to be a monochromatic wave.

S Theorem and Construction of the Invariants of the Semisimple Compact Lie Algebras
View Description Hide DescriptionAn infinitesimal proof of the Stheorem, which states that the invariants of a compact semisimple Lie algebra are symmetric with respect to the discrete Weyl group of the algebra, is given. The complete set of invariants of the various compact semisimple Lie algebras found by Racah are rederived in a somewhat different and explicit way, the Stheorem being used to establish their completeness.

Addendum: One‐Speed Neutron Transport in Two Adjacent Half‐Spaces
View Description Hide DescriptionThe interface current for the problem of two half‐spaces with a constant source in one half‐space is obtained in closed form.