Volume 10, Issue 8, August 1969
Index of content:

Transverse Dielectric Tensor for a Free‐Electron Gas in a Uniform Magnetic Field
View Description Hide DescriptionAn elementary method of calculating that part of the tensordielectric coefficient which determines the propagation of transverse electromagnetic radiation through a free‐electron gas in a uniform external magnetic field is presented. The method presented here is based on a particle‐orbit analysis and is somewhat analogous to a generalized version of the Kramers‐Heisenberg quantum theory of gaseous dispersion. It is shown that the elements of the transverse dielectrictensor can be obtained from a knowledge of the quantum‐mechanical transition probabilities for emission and absorption of photons (that is, from a knowledge of the Einstein A and B coefficients). The formal expression for the dielectrictensor thus obtained is shown to be valid for both the degenerate and the nondegenerate system of electrons. The dielectrictensor thus obtained is shown to reduce in the classical limit to the familiar results of the conventional classical hot plasma kinetic theory. The first quantum correction to the classical hot plasma dielectrictensor is explicitly given and it is shown that, under suitable conditions, this quantum correction will play a significant role in the analysis of the experimental studies of the electrodynamic behavior of ``classical electron‐hole plasmas'' in a uniform external magnetic field.

Singularities of Conformal‐Invariant Scattering Amplitudes
View Description Hide DescriptionIt is shown that conformal invariance alone, with no specific dynamics, provides severe constraints on the location of singularities of the off‐mass‐shell amplitude for elasticscattering of spinless particles. Kinematical and dynamical singularities are correlated in an interesting fashion with the singular and general solutions of differential equations which follow from the momentum‐space requirements of conformal invariance. Explicit expressions are found for the location of singularities which are independent of t, the square of the momentum transfer. The possibility of having an asymptotic behavior of t ^{α} for large t together with singularities specified by the function α is discussed.

Exact Static‐Model Bootstrap Solutions for Arbitrary 2 × 2 Crossing Matrices
View Description Hide DescriptionExplicit, exact solutions, satisfying a bootstrap criterion in the form of Levinson's theorem and having real coupling constants, are exhibited for the two‐channel Low equation for arbitrary real values of the parameter appearing in the 2 × 2 crossing matrix. This disproves a claim made recently in the literature that the bootstrap criterion will restrict values of the crossing‐matrix parameter to those corresponding to the internal‐symmetry group SU _{2}. It is demonstrated that the previous alleged proof is inconclusive.

Asymptotic Fields of the Lee Model
View Description Hide DescriptionThe field equations for the Lee model are solved by constructing eigenfields from the most general possible combination of bare fermion and boson operators. These solutions are found to require an infinite number of boson terms, the coefficients of which obey the integral equations of scattering theory. It is discovered that the algebra of the Lee‐model fermion eigenfields is not that of free particle operators. The anticommutator of the ith fermion eigenfield has the form , where the sum is over all other fermion eigenfields. An algebra of this type is not peculiar to the Lee model. It will hold for all fields which obey a general Pauli exclusion principle and which are orthogonal to one another. The explanation for this algebra lies in the fact that these eigenfields represent operators which create states and not particles. Several other models exhibiting this same behavior (including the harmonic oscillator) are presented; for these models boson fields may also be constructed which have the above algebra. The asymptotic convergence of these eigenfields is not examined. However, it is found that all such Lee‐model fermion fields, including those constructed by the Yang‐Feldman method, must satisfy the above algebra and do not enjoy a free‐fermion canonical algebra.

Diffraction of Electromagnetic Waves by a Right‐Angle Dielectric Wedge
View Description Hide DescriptionThe problem of diffraction of a plane‐polarized electromagnetic wave incident on a right‐angle dielectric wedge is formulated as a singular integral equation in k space. A solution of the singular integral equation is constructed as a power series in the index of refraction. This series converges when the index of refraction is near unity. Using this solution, the electric‐field amplitude at the tip of the wedge is examined. We also prove as incorrect a closed‐form analytic expression claimed in the literature to be a global solution of the problem considered here.

Instantaneous Action‐at‐a‐Distance Formulation of Classical Electrodynamics
View Description Hide DescriptionThe possibility of formally translating the interaction of charges from charge ↔ field ↔ charge to charge ↔ charge, where the orbits satisfy Newtonian (second order in t), yet covariant, equations of motion, is exploited for the Wheeler‐Feynman interaction. A method for computing the forces on the charges correct to second order in the coupling constant e ^{2} is presented, and ten constants of the motion correct to e ^{2} are found. The integration is effected via the Noether theorem with the inhomogeneous Lorentz group as symmetry transformations. An important result is that a well‐known correction to the Coulomb interaction which accounts for the uniform motion of charges is revealed to be, to first order in e ^{2}, a frame‐invariant expression. The consequent corrected Coulomb dynamics admits first‐order integrals identical to those of the Wheeler‐Feynman dynamics.

Motion of a Charged Particle in a Spatially Periodic Magnetic Field
View Description Hide DescriptionDegenerate perturbation theory is employed to discuss the motion of a charged particle in a constant magnetic field on which is superimposed a weak, transverse, spatially periodic magnetic field. A first‐order solution of the equations of motion is presented. It is shown that the secular motion is periodic in time. The significance of this result with respect to the stability of protons in the inner Van Allen belt is discussed.

Perturbation Method for a Nonlinear Wave Modulation. I
View Description Hide DescriptionIn this paper we consider a system of nonlinear waveequations which admits, in a linear approximation, a planewave solution with high‐frequency oscillation. Then, for the wave of small but finite amplitude, we investigate how slowly varying parts of the wave such as the amplitude are modulated by nonlinear self‐interactions. A stretching transformation shows that, in the lowest order of an asymptotic expansion, the original system of equations can be reduced to a tractable, single, nonlinear equation to determine the amplitude modulation.

Critical Behavior of Several Lattice Models with Long‐Range Interaction
View Description Hide DescriptionWe consider a one‐dimensional model with infinite‐range interaction, a two‐dimensional model, and a three‐dimensional model, whose free energies can be expressed in terms of the largest eigenvalue of an integral equation. High‐ and low‐temperature expansions in powers of the reciprocal of the range of the exponential part of the interaction, with the classical Curie‐Weiss theory as leading term, are developed and studied in the critical region. We find that to leading order in the critical region the resummed high‐ and low‐temperature expansions are analytic at the classical critical point but are nonanalytic at a displaced critical point. The modified singularities, which are no longer of Curie‐Weiss type, give critical exponents which are identical with those obtained by Brout and others, and are almost surely not the true exponents. The technique, however, suggests a possible general method of successive approximation to true critical behavior.

On Perturbation Expansions for Real‐Time Green's Functions
View Description Hide DescriptionA proof by Craig that the perturbation expansion for the real‐time self‐energy of a particle in a many‐particle system has the same form in any statistical state is shown to be invalid and the stated theorem is shown to be untrue.

On Next‐Nearest‐Neighbor Interaction in Linear Chain. I
View Description Hide DescriptionGround‐state properties of the Hamiltonian(σ_{ N+1} ≡ σ_{1}, σ_{ N+2} ≡ σ_{2}) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest‐neighbor interactions only (α = 0) in the presence of the next‐nearest‐neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground‐state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest‐ as well as the next‐nearest‐neighbor interaction.

On Next‐Nearest‐Neighbor Interaction in Linear Chain. II
View Description Hide DescriptionContinuing our work on the ground‐state properties of the Hamiltonian,we have completed the study of 10 spins. The results of short‐chain calculations provide better upper and lower bounds of the ground‐state energy per particle as N → ∞, but no simple formula can be fitted to the data to get this limit for all α. For J > 0 and α = ½, however, this is exactly found to be −¾J. Some upper and lower bounds for the free energy are also derived.

Phase Transition in Zero Dimensions: A Remark on the Spherical Model
View Description Hide DescriptionIt is shown that the spherical model consisting of N spins with nonzero interaction between two spins only has a phase transition in the limit N → ∞. This is a counterexample to a suggestion of Kac which states that an Ising model will have a transition if the corresponding spherical model has a transition. Possible modifications of Kac's conjecture are suggested and discussed.

Dynamics in the Diagonal Coherent‐State Representation
View Description Hide DescriptionThe problem of following the dynamical behavior of a quantum‐mechanical system in the diagonal coherent‐state representation is examined for those systems whose time evolution is specified by equations of motion for the coherent‐state weight functional which resemble Fokker‐Planck equations but have non‐positive‐definite diffusion matrices. A particular equation of this type describing a linear parametric process is considered in detail and several proposed generalizations of the diagonal representation, which include dynamical effects of the atomic system coupled to the electromagnetic field in simple models of a unimodal laser, are also briefly discussed.

Periodic Small‐Amplitude Solutions to Volterra's Problem of Two Conflicting Populations and Their Application to the Plasma Continuity Equations
View Description Hide DescriptionThe coupled set of first‐order nonlinear differential equations describing a generalized form of Volterra's problem of two conflicting populationsare solved by an approximate method which gives y(t) for the particular case in which the variables x and y vary periodically, the coefficients C_{i} and A_{i} are real, and the peak‐to‐peak amplitude of x is small compared with the mean value of x. The peak‐to‐peak amplitude of y, however, is not necessarily small compared with the mean value of y. When these conditions are satisfied, the functional form of y(t) is approximated by Jacobian elliptic functions. The solutions obtained in this analysis are relevant to special cases of the classical problem of predator and prey, and also to certain low‐frequency oscillations in partially ionized plasmas that arise from periodic solutions to the neutral and charged‐particle continuity equations.

On the Growth of the Ground‐State Binding Energy with Increase in Potential Strength
View Description Hide DescriptionWe study the asymptotic behavior of the ground‐state binding energy G(λ) of −Δ + λV as λ → ∞. Unlike the number of bound states,G(λ) does not have a universal power growth as λ → ∞. It is shown, however, that as λ → ∞ for Kato potentials.Examples are presented for which G ∼ λ^{β} for any 1 < β < 4. Other examples are presented which obey no power growth. We also prove theorems which reflect the close connection between the large λ behavior of G and the small r behavior of V for potentials with a single attractive singularity at r = 0. These can be roughly phrased as follows: If V ∼ −r ^{−α} for r → 0, then G(λ) ∼ λ^{β} with β = 2/(2 − α) as λ → ∞.

Summation over Feynman Histories in Polar Coordinates
View Description Hide DescriptionUse of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central‐force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse‐square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.

Inverse Functions of the Products of Two Bessel Functions
View Description Hide DescriptionSpecial cases of the inverse function of the product of two spherical Bessel functions have been found recently by other writers as _{1} F _{2} hypergeometric functions. We give the general expression as the derivative of a product of spherical Bessel functions. These results have also been found in the classical literature.

Group Theory and Mixed Atomic Configurations
View Description Hide DescriptionA group‐theoretical scheme is introduced to classify the states of an atomic system having two open shells. States labeled according to this scheme may be written,where the quasispins Q_{A} and Q_{B} are coupled together to form a total quasispin Q. Although these states are, in general, mixtures of different configurations , it is found that they serve as a convenient basis for the calculation of matrix elements in (l_{A} + l_{B} )^{ N }. The matrix elements of operators between the states of two configurations are obtained from these matrix elements by means of a unitary transformation. As an example matrix elements of the Coulomb interaction within (f + p)^{ N } are calculated.

On a Condition for Completeness
View Description Hide DescriptionIt is shown that a completeness relation for the eigensolutions of a non‐Hermitian operator H can be derived even if the resolvent operator R(H) of H is allowed to have poles of higher order than just simple poles, as required by Fonda, Ghirardi, Weber, and Rimini. A class of operators satisfying the requirements of this note is cited.