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Volume 10, Issue 3, March 1969

Physical‐Region Discontinuity Equations for Many‐Particle Scattering Amplitudes. I
View Description Hide DescriptionThe discontinuity equations are derived for all singularities of multiparticle scattering functions that enter the portion of the physical region lying below the four‐particle threshold. These equations, which are the precise statements of the Cutkosky formulas, are calculated directly from the unitarity equations. The only analyticity properties used are those obtained from the S‐matrix macroscopic‐causality condition. That is, the scattering functions are taken to be analytic in the physical region except on positive‐α Landau surfaces, around which they continue in accordance with the well‐defined plus‐iε rule.

Normal and Abnormal Diffusion in Ehrenfests' Wind‐Tree Model
View Description Hide DescriptionThe Ehrenfest wind‐tree model—a special case of the Lorentz model—where noninteracting point particles move in the plane through a random array of square scatterers, is used to study the divergences previously discovered in the density expansions of the transport coefficients. Two cases for which the results are qualitatively different are discussed. When the scatterers are not allowed to overlap, the diffusion of particles through the array of scatterers is normal, characterized by a diffusion constant D. The calculation of D ^{−1} is carried to the second order in the density of the scatterers, and involves a discussion of the above‐mentioned divergences and a resummation of all most‐divergent terms in the straightforward expansion of D ^{−1}. If, however, the trees are allowed to overlap, the growth of the mean‐square displacement with time is slower than linear, so that no diffusion coefficient can be defined. The origin and possible relevance of this new phenomenon to other problems in kinetic theory is discussed. The above results have stimulated molecular dynamics calculations by Wood and Lado on the same model. Their preliminary results seem to confirm the theoretical predictions.

Functional Integration Theory for Incompressible Fluid Turbulence. II
View Description Hide DescriptionTwo iterative solutional procedures are reported for the nonlinear integro‐differential dynamical equation obtained previously by the stationary functionality method and supplemented here by a subsidiary dynamical condition. Rapidly convergent for grid Reynolds numbers between about 100 and 300, both methods yield general expressions for the two‐point equal‐time velocity‐correlation tensor in approximate agreement with experiment for the initial period of decay.

Spectral Theory of the Difference Equation f(n + 1) + f(n − 1) = [E − φ(n)]f(n)
View Description Hide DescriptionIn this work, the spectrum of the second‐order difference equationin the l ^{2}Hilbert space is studied for the case in which the limit of the sequence exists. By means of a simple representation the problem is transferred to one about the spectrum of an abstract operator in a separable Hilbert space. This operator T has a form analogous to the Schrödinger operator, namely T = T _{0} + A, where T _{0} is self‐adjoint with a purely continuous spectrum but bounded, while A depends on the sequence {φ(n)}. In fact, A is of Hilbert‐Schmidt type for any {φ(n)} in l ^{2}, and of trace class if the series converges. Sufficient conditions for the existence of a discrete spectrum and more generally, of proper values, are found. Using the theory of the wave operators range exp (−iT _{0} t), results on the existence of a mixed spectrum are obtained.

Invariants to All Orders in Classical Perturbation Theory
View Description Hide DescriptionThe method of averaging is applied to systems having Hamiltonians of the form,where H _{1} is periodic in each of the components of Ψ. When the system is nondegenerate it is shown that corresponding to each component of Ψ there is a quantity K_{i} which is invariant to all orders in ε. When the system has an m‐fold degeneracy, somewhat weaker results are obtained. In this case it is shown that the Hamiltonian, when expressed in terms of the average variables, depends on the angle variables only through their m degenerate combinations. This is true to all orders in ε. Thus, if Ψ has s components there are s ‐ m invariants provided that the average variables can be made canonical. However, the conditions under which degenerate perturbation theory can be made canonical are not known. The invariants which arise when the Hamiltonian has an adiabatic or a harmonic time dependence are also discussed. The techniques are applied to the simple case of a harmonic oscillator whose frequency varies slowly with time.

Isoperimetric Solutions Related to the Thermodynamics of Plasmas
View Description Hide DescriptionMore solutions of isoperimetric problems are obtained which lead to several extensions and refinements of previous results in the thermodynamics of plasmas: (1) Energy requirement for density fluctuations in a relativistic plasma is determined. At the typical relativistic temperature of kT = mc ^{2} it becomes as compared with the previous nonrelativistic value of An optimum expression for the free energy in a plasma is derived in terms of familiar thermodynamic variables. (3) For a collisional plasma, the energy requirement again assumes the same general form and is equal to .

Negative Finding for the Three‐Dimensional Dimer Problem
View Description Hide DescriptionThe dimer problem can be solved if one can evaluate the permanent of P = (p_{ij} ), the incidence matrix of the lattice. All known methods of solving the two‐dimensional case consist (explicitly or implicitly) in finding another matrix Q = (q_{ij} ), such that p_{ij} = q_{ij}  and per P = det Q, and then computing the determinant of Q. We show that in the three‐dimensional case no such matrix Q exists for any choice of elements q_{ij} , whether real or complex numbers, or quaternions. A stronger negative result of an asymptotic character seems to be true, but this rests upon a plausible but unproved conjecture.

Approach to the Mode Conversion Problem in Nonuniform Acoustic Waveguides
View Description Hide DescriptionA method is described for treating wave propagation in a waveguide structure whose cross section varies in the direction of propagation. Special attention is given to the conversion of energy between waveguide modes of different order. For simplicity, the waveguide is bounded by impedance‐type walls and the lateral height variation is assumed to have circular symmetry. This is considered to be an idealized model of an atmospheric waveguide for acoustic‐wave propagation in the case when there is a localized depression.

Ray Velocity and Exceptional Waves: A Covariant Formulation
View Description Hide DescriptionIn a nonlinear field an accelerated wave sooner or later turns into a shock. When this is not the case, the wave is exceptional (e.g., Alfvén waves of magnetohydrodynamics). Then the normal speed of the wave remains undisturbed. This criterion is given a convenient covariant form in terms of the ray velocity. As an example a thermodynamical relativistic fluid is studied.

General Approach to Fractional Parentage Coefficients
View Description Hide DescriptionThe purpose of this paper is to achieve a clearer understanding of the problems involved in the determination of a closed formula for fractional parentage coefficients (fpc). The connection between the fpc and one‐block Wigner coefficients of a unitary group of dimension equal to that of the number of states is explicitly derived. Furthermore, these Wigner coefficients are decomposed into ones characterized by a canonical chain of subgroups (for which an explicit formula is given) and transformation brackets from the canonical to the physical chain. It is in the explicit and systematic determination of the states in the latter chain where the main difficulty appears. We fully analyze the case of the p shell to show that a complete nonorthonormal set of states in the physical chain can be derived easily using Littlewood's procedure for the reduction of irreducible representations (IR) of SU(3) with respect to the subgroup R(3). This procedure gives a deeper understanding of the free exponent appearing in the polynomials in the creation operators defining the states in the chain. As Littlewood's procedure applies to the chain, and probably can be generalized to other noncanonical chains of groups, it opens the possibility of obtaining general closed formulas for the fpc in a nonorthonormal basis.

Hermite's Reciprocity Law and the Angular‐Momentum States of Equivalent Particle Configurations
View Description Hide DescriptionHermite's reciprocity law is applied to the calculation of the angular‐momentum states of equivalent particle configurations. Connections between boson and fermion states are considered and a method is given for determining the number of times a given term appears in l ^{3} without requiring a complete term analysis. A succinct expression for the number of S‐states arising in the [2^{2}] representation of R _{2l+1} under R _{2l+1} → R _{3} is developed.

Complementary Bounds for Ground‐State Energies
View Description Hide DescriptionComplementary upper and lower bounds are derived for the ground‐state energies of Hamiltonians H = h + V, where V is positive‐definite, by considering the Schrödinger equation in an integral form. Some simple applications are presented.

Frequency Spectrum and Momentum Autocorrelation Function for a Simple Lattice
View Description Hide DescriptionCalculations are made on the dynamics of a familiar lattice model: the simple cubic lattice with central and noncentral harmonic forces between nearest neighbors only. For the case of equal force constants, natural extensions are made of existing analytical approximations for the spectrum of squared frequency. The behavior of the spectrum near its singular points is described more accurately than before and expressions are derived which make it easy to obtain the density of modes at any frequency to about one part in a thousand. The new description of the spectrum is used to improve existing approximations for the classical momentum autocorrelation function for the infinite lattice ρ(τ), and for the function X _{3}(τ′) used by Goodman in calculations on the response of surface atoms in a semi‐infinite lattice. Good agreement with numerical results of Goodman for τ′ = 20, 25, and 30 is obtained. The results for the spectrum also apply to the density of states of electrons in a simple cubic lattice in the tight‐binding approximation.

Theoretical Study of Finite Dielectric‐Coated Cylindrical Antenna
View Description Hide DescriptionA finite cylindrical antenna which is imbedded in a concentric dielectric rod has been investigated by employing a rigorous formulation. When the antenna is relatively short, a numerical method is used; when the antenna is long, the Wiener‐Hopf technique is applied. In both cases the input admittance and the current distribution are obtained. It is found that the input conductances are larger than for the corresponding free‐space antennas, the field patterns tend to be more broadside and, as the antenna gets longer and longer, the locus of the input admittance becomes a circle instead of converging to one point as it does for a bare cylindrical antenna. The first method is applicable regardless of the thickness of the antenna and the dielectric rod; the second method can be applied only to a sufficiently long antenna. The minimum length is determined by the thickness of the dielectric rod. This study is limited to thin antenna in rather thick dielectric cylinders. However, the dielectric rod is still not thick enough to support a transverse magnetic (T.M.) mode.

S‐Matrix Singularity Structure in the Physical Region. I. Properties of Multiple Integrals
View Description Hide DescriptionWe obtain the real singularities and corresponding discontinuities of a class of multiple integrals over real contours. Our aim is to give a unified treatment, obtaining, by elementary mathematical methods, both previously known results and some new generalizations. In a subsequent paper the results are applied to unitarity integrals.

High‐Energy Scattering Cross Section for Singular Potentials at Large Angles
View Description Hide DescriptionFor scattering off singular repulsive potentials at high energy, the large‐angle scattering is shown to be classical. For nonrelativistic singular potentials of the form V(r) = gr ^{ ‐n } (g > 0, n > 2), we give the explicit form of the differential cross section for large angles at high energy as a power series in (π ‐ θ). The coefficients of the linear through cubic terms in (π ‐ θ) are obtained.

Mean Ergodic Theorems in Quantum Mechanics
View Description Hide DescriptionAn application of the abstract mean ergodic theorems to quantum systems is described, which is rather closely analogous to the application of these same theorems in classical statistical mechanics.

Bound States of a New Kind in Quantum Field Theory
View Description Hide DescriptionIt is the purpose of this article to study the various kinds of bound states and their properties that can occur in quantum field theory. The ability of quantum field theory to create and destroy particles makes the picture of a bound state very complex. However, for this very reason it allows for a larger variety of bound states than can occur in nonrelativistic quantum mechanics. Not only do we consider those bound states that appear as poles in the tau functions, but also as branch points. The bound states that appear as branch points offer many new possibilities in constructing composite theories. An example of this type of bound state is given in which the fermion is a bound state of a boson. The many ambiguities that exist with this kind of bound state are illustrated. In discussing the bound states that appear as poles we have broken them into three subgroups: N‐body bound states, mixed bound states, and self‐interacting bound states. The N‐body bound states are the most analogous to those occurring in nonrelativistic quantum mechanics. The mixed bound states have the peculiar property of not being composed of any unique group of elementary particles. The self‐interacting bound states are those that are directly coupled with themselves. The characteristics of these subgroups of bound states are discussed and illustrated.

Partial‐Wave Analysis in Terms of SL(2, C)/SU(2) Harmonic Functions
View Description Hide DescriptionA new scheme for analyzing the scattering amplitude in the S channel of a two‐body elastic collision in terms of the representation functions of the covering group of the homogeneous Lorentz group is presented. The scheme uses representation functions defined on the homogeneous space SL(2, C)/SU(2), which for the equal‐mass case considered here is the same as the hyperquadric , where m is the particle mass, q is the relative center‐of‐mass momentum (in the initial or final state), and 2q _{0} = S ^{½} is the center‐of‐mass energy. The corresponding representations are multivalued and belong to the so‐called degenerate (but not most‐degenerate) series. The scattering amplitude derived has the correct threshold behavior in the S variable.

Analytic Continuation of Appell's Hypergeometric Series F _{2} to the Vicinity of the Singular Point x = 1, y = 1
View Description Hide DescriptionThe infinite series, absolutely convergent if x + y < 1, for Appell's F _{2} (α, β, β′, γ, γ′; x, y) is analytically continued into a linear combination of four infinite series in powers of (x ‐ 1) and (y ‐ 1); each of the latter four series is absolutely convergent if x − 1 + y − 1 < 1. The analytic continuation is carried out by manipulation of the Mellin‐Barnes integral representations for the hypergeometric functions appearing in the course of the calculation.