Volume 6, Issue 3, March 1965
Index of content:

Some Constants of the Linearized Motion of Vlasov Plasmas
View Description Hide DescriptionA new class of nonlocal constants for the linearized Vlasov equations is presented. These are constants which, in general, are not expressible as simple phase‐space integrals of a locally defined integrand. The relevance of these constants to stability considerations is indicated.

Mathematical Methods for Evaluating Three‐Body Interactions Between Closed‐Shell Atoms or Ions
View Description Hide DescriptionThe mathematical methods involved in the evaluation of three‐body interactions between rare‐gas atoms or alkali‐halide ions with Gaussian type electron wavefunctions are outlined in first and second orders of perturbation theory. These methods are of general applicability for the analysis of many‐center coulombic and exchange integrals occurring in problems of atomic and solid‐state physics in which Gaussian functions can be used as a basis.

Quantum Statistics of Multicomponent Systems
View Description Hide DescriptionMethods of quantum statistics are applied to the general multicomponent system and explicit rules for the calculation of thermodynamic quantities and distribution functions in terms of the basic particle interactions are derived. Particular attention is given to the nonrelativistic system of charged particles, and several inherent difficulties are investigated and treated. Included are the renormalization, by the Λ transformation, of charged‐particle energy contributions due to the emission and reabsorption of photons and the formal sum over all long‐range Coulomb effects wherever (diagrammatically) they give divergent expressions. The latter sum is shown to result in the familiar Debye‐Hückel theory in the high‐temperature, low‐density limit.

Quantum Mechanics of a Many‐Boson System and the Representation of Canonical Variables
View Description Hide DescriptionWe present a method of treating the assembly of interacting bosons under Bose‐Einstein condensation. Without applying the Bogoliubov approximation in which the creation and the annihilation operators of zero‐momentum particles are replaced by a c number, we keep the quantum nature of these operators—thus the title, ``Quantum Mechanics.'' The method is a quantum mechanical adaptation of the theory of small oscillation. The oscillation means the fluctuation of the number of condensed particles. The interaction between particles determines the stability of this oscillation. When it is stable and its amplitude is not macroscopic, the Bogoliubov approximation is valid. In this way, our method provides a validity criterion for the Bogoliubov approximation as well as an estimation of the errors thereby committed. We have to note that the excitations associated with the fluctuation of condensed particles can never be obtained within that approximation. Our method is applied to the Huang model, the assembly of bosons interacting through a hard core plus weak attractive potential. Having found that, within the physically accessible range of the particle density, the above‐mentioned oscillation is stable, we can conclude that Huang's treatment is well founded. We have discussed the mathematical background of our approximation by invoking the representation theory of canonical variables of an infinitely large system.

Dynamics of a Simple Many‐Body System of Hard Rods
View Description Hide DescriptionGeneral formulas are given for the exact calculation of the nonequilibrium properties of the one‐dimensional system of equal‐mass hard rods both for a finite but large system and in the limit of infinite size. Only properties which depend upon labeling one or more of the particles are nontrivial in this system. Various results are obtained on Poincaré cycles, delocalization of a particle with time and electrical conductivity when one particle is charged.

Lepton Scattering Amplitudes in Two Model Field Theories
View Description Hide DescriptionTwo lepton‐leptonscattering amplitudes are considered within the context of a no‐recoil Bloch‐Nordsieck model, with emphasis on the singularities in that configuration‐space variable conjugate to momentum transfer. For the interaction , renormalizable in four dimensions, and in the approximation of including only the exchange of all possible bosons between a pair of leptons, a light cone singularity no worse than that of the one‐boson‐exchange graph is found. Similar statements may be made for the same interaction, nonrenormalizable in six dimensions, provided certain continuations in the center‐of‐mass energy variable are employed; otherwise, an essential singularity appears. A remark illustrating the formation of bound states is made for the renormalizable interaction. No argument is given to establish the relevance of these models to the actual field‐theoretic situations.

Asymptotic Expansion of the Intensity of the Small‐Angle X‐Ray Scattering from Cylinders of Arbitrary Cross Section
View Description Hide DescriptionFrom a previously developed expression for the intensity of the small‐angle x‐ray scattering from randomly oriented right cylinders of arbitrary cross section and with uniform electron density, an asymptotic expansion is developed which can be used to approximate the scattered intensity at relatively large angles of the small‐angle region. The asymptotic expansion is useful for numerical calculations at these large angles, where calculations by the usual techniques of numerical integration and series expansion are most difficult. The terms in the asymptotic expansion are found to be determined by the behavior of the characteristic function β(r) of the cross section at certain values of r. The general asymptotic expansion is used to calculate asymptotic expressions for the scattering from right circular cylinders and rectangular parallelepipeds.

Interacting Fermions in One Dimension. I. Repulsive Potential
View Description Hide DescriptionThe exact energies and wavefunctions for the ground state and low‐lying excited states of a system of N − 1 one‐dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large‐volume, finite‐density limit when the particles interact via a repulsive delta function potential. A number of properties of the system such as pair correlation functions and the effective mass of a certain class of excitations are also discussed.

Statistical Ensembles of Complex, Quaternion, and Real Matrices
View Description Hide DescriptionStatistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the real case, under the restriction that all eigenvalues are real). We also determine, in the complex case, all the correlation functions of the eigenvalues, as well as their limits when the order N of the matrices becomes infinite. In particular, the limit of the eigenvalue density as N → ∞ is constant over the whole complex plane.

Self‐Consistent Approximations in Many‐Body Systems
View Description Hide DescriptionBaym's definition of a self‐consistent approximation is rephrased in a more diagrammatic way and compared with formulations of the exact many‐body problem of Balian, Bloch, and DeDominicis. It is shown that many of the results of these authors are valid for any self‐consistent approximation as defined by Baym.

Some Recent Developments in Invariant Imbedding with Applications
View Description Hide DescriptionA summary of some recent developments in the theory of invariant imbedding is presented. Applications to some simple problems in wave propagation,diffusion theory, and transport theory demonstrate some of the advantages of the new and almost ``mechanistic'' approach. In addition, a slightly different attack is applied to nonlinear difference equations and to the classical phase‐shift problem.

Poles in Feynman Diagrams with Several Loops
View Description Hide DescriptionA formula is given that extracts the pole contribution to a Feynman diagram having an internal line. The result is more complicated than earlier results for diagrams with only a single loop since the result is not directly expressible as the pole multiplied by a numerator that is the sum of the reduced diagrams. The apparent discrepancy with Cutkosky's formula for the residue of the pole is reconciled. The algebraic techniques employed are in principle applicable to the problem of determining the Landau surface when the diagram has several loops. A derivation of Stokes' formula for Feynman parametric integrals is given.

Construction of the Charge Operator for Higher Symmetry Schemes
View Description Hide DescriptionThe problem of the construction of an additively conserved operator with integral eigenvalues, to be identified with the electric charge is solved in complete generality for the groups locally isomorphic to U(1) SU(n). It is found that the representations fall into classes on which different charge operators may be defined. Several results previously obtained for particular classes of representations are found here as special cases. We specialize the results to N = 4 and discuss several models presently in the literature.

Transformation Having Applications in Quantum Mechanics
View Description Hide DescriptionBy properly ordering functions of noncommuting operators, a one‐to‐one transformation between operator functions and corresponding functions of commuting algebraic variables can be made. With this transformation, boson operator equations such as the Schrödinger equation can be converted to differential equations for the transformed functions, the resulting equations containing solely commuting variables. Once the solution to the transformed equation is obtained, the inverse transformation may be applied to yield the solution to the original operator equation. The method is extended to include angular momentum operators.

Isoperimetric Problem with Application to the Figure of Cells
View Description Hide DescriptionBiological applications suggest the following geometrical problem. Consider n three‐dimensional cells, touching or not, and assume that the free energy of their figure is the sum H = A + αB of the area A of the cell walls adjacent to the ambient fluid plus an adjustable constant 0 ≤ α ≤ 2 times the area B of the walls separating two cells. Given the partial volumes of the cells, the problem is to describe the shape of the (optimal) figure that renders H as small as possible; the analogous problem for two‐dimensionalcells is the subject of this paper. Geometrical proofs of the following features of optimal two‐dimensional figures are presented below: (a) the edges bounding the cells are circular arcs; (b) at an inside corner, three edges meet at angles 2π/3; (c) at an outside corner, three edges meet with outside angle 2 cos^{−1} α/2; (d) pressures can be ascribed to the cells so that the pressure drop across an edge is proportional to its curvature; (e) bubbles appear at each inside corner as α passes 3^{½}. All these facts have three‐dimensional analogues with similar proofs.

Calculation of Certain Phase‐Space Integrals
View Description Hide DescriptionWe give the result of an approximate calculation of the phase‐space integral,which appears when one takes into account the fact that only a part of the kinematically allowed phase‐space is accessible to particles produced in high‐energy collisions, because of the observed smallness of their transverse momenta.

Composite Particles in a Relativistic Model of Two‐Body Scattering
View Description Hide DescriptionThe relativistic model of two‐particle scattering recently described by Jordan, Macfarlane, and Sudarshan is extended to include bound states. The projection operator associated with the boundstate manifold is explicitly given. This relates the structure of the composite particle to the scattering amplitude. There is a discussion of the conditions the phase shifts have to satisfy in order that they can be fitted with the model. Explicit solutions are derived for a large class of phase shifts.

Fourier Series Expansion for the General Power of the Distance between Two Points
View Description Hide DescriptionThe general power r^{n} = (r _{1} ^{2} − 2r _{1} r _{2} cos ω + r _{2} ^{2})^{ n/2} of the distance between two points is expressed as a Fourier series Σ R_{n, l} (r _{1}, r _{2}) cos lω. Following Sack's method, the radial functions R_{n, l} are obtained as power series in r _{<}/r _{>}. Symmetrical expressions in r _{1} and r _{2} and recurrence relations are found for R_{n, l} .