Volume 5, Issue 9, September 1964
Index of content:

Exact Solution of the Schrödinger Equation for Inverse Fourth‐Power Potential
View Description Hide DescriptionThe solution of the Schrödinger equation for the potential and all angular momenta is given in terms of known functions, namely, the solutions to the modified Mathieu equation with complex parameters and complex argument. Scattering solutions for both attractive and repulsive potentials are given and in the latter case the explicit expression for the S matrix is given.

Interaction of Nonrelativistic Particles with a Quantized Scalar Field
View Description Hide DescriptionWe demonstrate the mathematical existence of a mesontheory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on the interaction, we obtain a well‐defined self‐adjoint operator. The solution of the Schrödinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a constant infinite phase shift due to the self‐energy of the particles. In the Heisenberg picture, we obtain a solution in the limit of no cutoff. We use a canonical transformation due to Gross to separate the divergent self‐energy term. It is shown that the canonical transformation is implemented by a unitary operator, and that the transformed Hamiltonian, with an infinite constant subtracted, can be interpreted as a self‐adjoint operator.

On Multiple‐Valued Random Functions
View Description Hide DescriptionA technique is developed for the statistical description of multiple‐valued functions. For the special case of single‐valued functions, the relations developed are shown to reduce in a natural way to the customary ones. Transformation laws are developed relating the statistical descriptions of a multiple‐valued function in two coordinate systems. The relationship of the present work to the Rice formula for the zeros of a random function is shown. The statistical description of homogeneous, isotropic, multiple‐valued functions in two dimensions is developed. Previously known results regarding expected arc length of random functions are obtained using the present technique.

Linear Representation of Spinor Fields by Antisymmetric Tensors
View Description Hide DescriptionA redundant, linear representation of four‐component spinors ψ(x) by antisymmetric tensorsf _{μν}(x) is proposed on the basis of the relation where γ^{μν} is the skew product of gamma matrices and ν is a constant fiducial spinor. The redundant degrees of freedom in f _{μν} as well as the particular choice of ν are both treated as unobservable ``gauges,'' and the appropriate gauge groups are discussed. As part of this analysis, we demonstrate that the possibility that tensors can behave as spinors is intimately connected with the existence of two coordinate‐invariant, gauge‐covariant subsidiary conditions. A linear, tensorial reformulation of the Dirac equation is given, and shown to be the Euler‐Lagrange equation of the conventional action functional for spinor fields. Finally covariance under arbitrary space‐dependent coordinate and gauge transformations is discussed, and a generally covariant, tensorial form of the Dirac equation is proposed differing from the conventional generally covariant equation in the degree of arbitrariness in the (spin) connection.

A Proof that the Free Energy of a Spin System is Extensive
View Description Hide DescriptionThe free energy obtained from the canonical partition function for a finite spin system possesses a certain convexity property, of which theorems by Peierls and Bogoliubov are particular applications. This property is used in proving the following result: Consider a regular lattice of spins in the form of a parallelepiped (in two dimensions a parallelogram, in one dimension a linear chain). The free energy of the system divided by the number of spins approaches a definite limit as the linear dimensions of the system become infinite. The limit is not influenced by certain common types of boundary conditions. A similar result, but with convergence understood in a weaker sense, holds for derivatives of the free energy such as entropy, magnetization, and specific heat. In the proof it is necessary to assume that the Hamiltonian has the translational symmetry of the spin system, and that long‐range interactions decrease sufficiently rapidly with the distance r between spins. (For example, as r ^{−3−ε} with ε > 0 for interactions between pairs of spins in 3 dimensions.)

A Theorem on the Determinantal Solution of the Fredholm Equation
View Description Hide DescriptionFor a Fredholm equation with Green's‐function‐type kernel, a new proof is given of a known theorem relating the Fredholm determinant with the behavior of the solution at the origin. The utility of this theorem in practical calculations is pointed out, as are some of its implications for potential scattering in quantum mechanics. The scattering phase shift is shown to have the property.

Characters of Irreducible Representations of the Simple Groups. I. General Theory
View Description Hide DescriptionNew formulas for the characters of irreducible representations of simple groups are presented. They yield the character directly as a sum instead of a quotient as does Weyl's formula. The procedure is purely geometrical and is based on the properties of the regular lattice associated with every simple group in the ``global'' theory of Lie groups due to Hopf and Stiefel.

Nonrelativistic Coulomb Green's Function in Momentum Space
View Description Hide DescriptionThe nonrelativistic Coulomb Green's function in momentum space is obtained in closed form by Fourier transforming the known expression for the coordinate‐space Green's function. Also, an integral representation for the momentum‐space Green's function is obtained which looks rather attractive from the point of view of applications.

A Model of Interacting Radiation and Matter
View Description Hide DescriptionWe investigate the long‐time behavior of a model consisting of N two‐level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long‐time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is where is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self‐consistent field approximation. In the next order, particle‐field correlations appear. We explicitly solve the equations of motion for the particle‐field correlations in terms of the average quantities that appear in the self‐consistent field approximation. We show the self‐consistent field approximation consists of five first‐order differential equations. Next we show the equations of motion for the density matrix of the system correct to order ()^{2} are equivalent to eight first‐order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest‐order microscopic equations are equivalent to semiphenomenological theories and our higher‐order equations contain only the measurable second‐order moments of the electromagnetic field in addition to the variables that appear in semi‐phenomenological theories.

Comments on Nonlinear Wave Equations as Models for Elementary Particles
View Description Hide DescriptionIt is shown that for a wide class of nonlinear waveequations there exist no stable time‐independent solutions of finite energy. The possibility is considered whether elementary particles might be oscillating solutions of some nonlinear waveequation, in which the wavefunction is periodic in the time though the energy remains localized.

On Some Topological Properties of Feynman Graphs and Their Application to Formulas Related to the Feynman Amplitudes
View Description Hide DescriptionThe purpose of this note is to demonstrate the usefulness of the topological concept of the tree sets introduced into any Feynman graph. We demonstrate here a relationship between certain functions appearing in two different forms of parametrized Feynman amplitudes by using the properties of the tree sets and some purely determinantal manipulations. As another exposition, we also present an alternative proof of a theorem due to Nakanishi by using only the concept of tree sets with almost no algebraic manipulation involved; the proof in this case is seen to be particularly simple and lucid.

Scattering of Electromagnetic Waves by a Ferrite in a Waveguide
View Description Hide DescriptionAs for any multichannel scattering problem, variational techniques can be utilized in the determination of the elements of the scattering matrix or of the equivalent network elements for a gyromagnetic obstacle in a waveguide. As always, however, it can be quite difficult to interpret numerical results which in general are neither upper nor lower bounds. A variational bound originally developed for the determination of the phase shift for a given angular momentum in a quantum mechanical central potential scattering problem is here adapted to the solution of a transversally magnetized, lossless ferrite slab in a rectangular waveguide propagating only one mode, the TE_{10} mode. With a simple trial function and with the aid of a comparison scattering problem which need not be tensor in character (so that the determination of upper and lower bounds is not really difficult), close bounds are obtained on cot η_{ e } and cot η_{0}, the cotangent of the real uncoupled phase shifts associated with the even and odd standing waves, respectively. The bounds obtained on cot η_{ e } and cot η_{0} determine bounds on the equivalent π network. A second variational bound, which can be simpler to apply and which can be applied to a wider class of problems, is also developed. This too is an adaption of a formalism originally introduced in quantum mechanical scattering problems, and depends upon a consideration of the spectrum of the fundamental operator of the theory, the Hamiltonian in the quantum mechanical case and an analogue thereof in the electromagnetic case.

Kinetic Theory of a Weakly Coupled Gas
View Description Hide DescriptionUsing the multiple‐time‐scale method on the BBGKY hierarchy, the weak coupling expansion is carried out to higher orders. It is found that there are two local breakdowns of the expansion. One occurs at small relative velocities between particles. The correct asymptotic representation for the small relative velocity region is given. The second breakdown occurs for particles having a large separation at t with their relative velocity oriented in such a way that they were in collision at t = 0. Such a breakdown indicates that in contrast to the Bogoliubov functional assumption, the higher‐order correlation functions should vary on the kinetic time scale in their own right. A sufficient condition on the smoothness of the initial correlation functions is given such that one obtains the Fokker‐Planck equation at the lowest‐order approximation in the expansion. The connection between irreversibility and the requirement of nonsecularity in the multiple‐time‐scale formulation is also indicated.

The Uniqueness of Weak Solutions of the One‐Dimensional Scalar Analog to the Navier‐Stokes Equation
View Description Hide DescriptionA uniqueness theorem for solutions of the one‐dimensional scalar analog to the Navier‐Stokes equation is stated and rigorously established.

Exact Quantization Rules for the One‐Dimensional Schrödinger Equation with Turning Points
View Description Hide DescriptionIt is pointed out that, for a number of problems, exact quantization rules exist which closely resemble that of Bohr‐Wilson‐Sommerfeld. In some cases it is shown how these rules may be derived mathematically from the Schrödinger equation.

Coupled Magnetomechanical Equations for Magnetically Saturated Insulators
View Description Hide DescriptionThe differential equations and boundary conditions governing the macroscopic behavior of nonconducting magnetically saturated media undergoing large deformations, are derived by means of a systematic and consistent application of the laws of continuum physics to a model consisting of an electronic spin continuum coupled to a lattice continuum. The macroscopic effect of the quantum mechanical exchange interaction is included as are dissipation and the associated thermodynamics. The resulting nonlinear equations are specialized to the important case of a small dynamic field superposed on a large static biasing field. Only the linear approximation in the small‐field variables is obtained. This final system of linear equations permits the solution of a variety of magnetomechanical boundary‐value problems.

Quantization of Electrodynamics in the Axial Gauge
View Description Hide DescriptionThe so‐called axial gauge condition A _{3}(x) = 0 is shown to be inconsistent with the condition of Lorentz invariance. This inconsistency is resolved herein.

Spin and Statistics with an Electromagnetic Field
View Description Hide DescriptionBecause of the impossibility of simultaneously satisfying the requirements of manifest Lorentz covariance and positive‐definite (finite) norm in the Hilbert space, no simple proof of the connection between spin and statistics with an electromagnetic field has been given. This note is to point out that it is indeed not necessary to have manifest Lorentz covariance in the full 3 + 1 space to show such a connection. Using the axial gauge A _{3} = 0, we have succeeded in constructing a simple straightforward proof.

The Wave Equation and the Green's Dyadic for Bounded Magnetoplasmas
View Description Hide DescriptionIn studies of electromagnetic wave propagation and radiation in magnetoplasmas, the wave equation takes the form of a dyadic‐vector Helmholtz equation. The investigation here shows that the dyadic‐vector Helmholtz equation is solvable by the separation method in four cylindrical coordinate systems. Solutions in the form of complete sets of eigenfunctions are possible when boundary surfaces are present. For problems involving current sources in the plasma, the Green's dyadics for finite or semifinite domains can be constructed from the complete sets of eigenfunctions which are solutions to the homogeneous equation. The Green's dyadic for infinite domain is also shown to be obtained from that for a semifinite domain through a limiting process.

Triality Type and its Generalization in Unitary Symmetry Theories
View Description Hide DescriptionWithin the context of an extension of the SU _{3}‐symmetry theory recently suggested by Gell‐Mann and further developed by the authors, certain aspects of the theory of the special unitary groups are examined. The plurality type of a given representation is introduced as the generalization of the triality concept to SU _{ n+1} and is shown to be associated with a multiplicative conservation law. Theorems for the reduction of representations of SU _{ n+1} with respect to SU_{n} U _{1} ^{(n)} are derived which are subsequently used to relate plurality type to the existence of fractional eigenvalues for the generator Y _{1} ^{(n)} of U _{1} ^{(n)}.