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Volume 17, Issue 7, July 1976

Diffraction corrections to the equilibrium properties of the classical electron gas. Pair correlation function
View Description Hide DescriptionWe perform a systematic study of the diffraction corrections (h/≠0) in the high‐temperature range (k _{ B } T≳1 Ry) for the pair correlation function of the one‐component classical electron gas with a neutralizing background, up to the third‐order in the plasma parameter Λ=e ^{2}/k _{ B } Tλ_{ D }. This program is achieved through the effective interaction V _{ e e }(r) = (e ^{2}/r)(1−e ^{−c r }) with c ∼ (thermal De Broglie wavelength)^{−1}, allowing for a straightforward and tractable generalization of the one‐component classical plasma model. The nodal expansion of the potential of average force is performed order‐by‐order with finite Mayer‐Salpeter diagrams. The classical results of De Witt (second‐order) and Cohen–Murphy (third‐order) are recovered in the h/→0 limit. The resummation to all orders of the bubble diagrams gives access to the short‐range behavior of the pair correlation function, which is found similar with the Monte‐Carlo results.

New representation of the Tomimatsu–Sato solution
View Description Hide DescriptionWe devise a new representation of the simplest Tomimatsu–Sato solution of Einstein’s vacuum field equations. This permits us to dispose of the previously troublesome ’’directional singularities’’ through the introduction of an advanced (or retarded) time coordinate. In the neighborhood of the locations in question the T–S space is shown to possess a Killing tensor of valence two, which allows us to solve the geodesic problem in this neighborhood completely. Finally, we present for future analysis a plausible toroidal model of the material source for the T–S solution.

Approximate form of the Tomimatsu–Sato δ=2 solution near the poles x=1, y=±1
View Description Hide DescriptionA consideration of the line element for the Tomimatsu–Sato δ=2 solution near the poles x=1, y=±1 reveals the existence of a metric which in nonsingular there. The further study of this metric indicates that it corresponds to a vacuum, type D gravitational field, and, as such, it is among those type D vacuum solutions specified by W. Kinnersley. Reasons are given in support of the belief that the derived metric is a valid approximation to the exact T–S solution close to the poles.

The solution to an inverse problem in stratified dielectric media
View Description Hide DescriptionWe solve the problem of determining the average dielectric constant and thickness of the layers that constitute a stratified dielectric medium from measurements of transmitted power at a single frequency. Each sample of the medium that is available for measurement is modeled as a stack of ’’n’’ layers of dielectrics of thicknesses l _{ i } and dielectric constantsK _{ i } (see Fig. 1). We assume that n, l _{ i }, K _{ i }, i=1,2,..., are all independent random variables and their values, of course, depend on the particular sample and the layer indexed by ’’i’’. Furthermore, it is assumed that the l _{ i } are identically distributed with some exponential distribution and that the K _{ i } are identically distributed. There are no other constraints or assumptions about these distributions except the following which are made more precise in the text: (1) The various averages of interest are finite. (2) Distributions which precipitate certain singular conditions (which is not a problem in ’’almost all cases’’) are excluded. Then the method described in this paper determines uniquely: (1) the average of the thicknesses of the layers E l _{ i }; (2) the average of the dielectric constantsE K _{ i } from measurements of transmitted power and without any further knowledge of the distributions of l _{ i }, K _{ i }, or n. We remark that the theory presented here applies also to acoustic or mechanical systems with the appropriate interpretation of the physical parameters.

On the equivalence of nonrelativistic quantum mechanics based upon sharp and fuzzy measurements
View Description Hide DescriptionStarting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three‐dimensional configuration space by a so‐called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy‐free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.

Remarks on conformal space
View Description Hide DescriptionSome remarks about a recent article dealing with Maxwell equations written on conformal space M ^{4} _{ c } are presented. The relevance of the manifoldM ^{4} _{ c } in conformal physics, the cotransformation of fields under the conformal group, and the presence of external currents in Maxwell equations are discussed. A simple proof of the conformal covariance of a linear gauge condition for Maxwell equations is then exhibited.

Angular momentum of systems of electric and magnetic charges and of singular flux surfaces
View Description Hide DescriptionThe angular momentum of a system consisting of an electric chargee and a magnetic chargeg is, as is well known, (e g). We derive general formulas for systems consisting of arbitrary electric and magnetic charges, dyons and of singular magnetic flux lines or surfaces of arbitrary integrable topological shapes. The total angular momentum is then quantized and related to the quantized flux.

Cluster expansions for fermion fields by the time dependent Hamiltonian approach
View Description Hide DescriptionA cluster expansion is given for a fermion field moving in an external field according to the interaction ψ̄ψφ in one space dimension.

Canonical symmetrization for unitary bases. I. Canonical Weyl bases
View Description Hide DescriptionIt is shown that the Weyl basis formed by the canonical symmetrization of an n‐dimensional, p‐rank tensor space with canonical projection operators of S_{ p } is a Gel’fand basis of U(n). This basis may easily be generated using standard projection operator techniques.

Canonical symmetrization for the unitary bases. II. Boson and fermion bases
View Description Hide DescriptionThe canonical Weyl basis described in Paper I is generalized to give a boson and fermion calculus which generates the symmetric and antisymmetric bases of U(n m) respectively contained in the irreducible bases of U(n) ×U(m). The boson calculus may be used to find the multiplicity free Clebsch–Gordan coefficients of U(n).

Uniform bounds of the Schwinger functions in boson field models
View Description Hide DescriptionWe study the lattice and space cutoff boson field models with periodic boundary condition in d‐dimensional space–time (d‐N ^{+}). We prove that if the pressures of the interaction (and also the pressures of the interaction with linear external fields) under consideration are bounded uniformly in the cutoffs, the corresponding Schwinger functions are also bounded uniformly in the cutoffs. By applying the above result we prove the uniform bounds of the space cutoff Schwinger functions for the (λφ^{4}−σφ^{2}−μφ)_{3} model and the lattice and space cutoff Schwinger functions for the exponential type interactions in d‐dimensional space–time.

Critique of the generalized cumulant expansion method
View Description Hide DescriptionThe method of ordered cumulants is presented for the solution of multiplicative stochastic processes. The relationship between this method and the irreducible cluster integral method of Mayer, which is used in the theory of the imperfect gas, is elucidated. The cluster property of ordered cumulants is proved. A critique of the literature in this area is presented which exposes some errors in the formulas. Several examples are indicated for the application of the ordered cumulant method.

Exponential decay and regularity properties of the Hartree approximation to the bound state wavefunctions of the helium atom
View Description Hide DescriptionThe exponential decay and regularity properties of the Hartree approximation to the bound statewavefunctions of the helium atom are proved.

Iterative solution of the Hartree equations
View Description Hide DescriptionAn iterative scheme based on eigenpairs of Hilbert–Schmidt operators obtained from a Green’s function representation for solutions of a linearization of the Hartree equations,d ^{2} y _{ i }(r)/d r ^{2} −[l _{ i }(l _{ i }+1)/r ^{2}]y _{ i }(r)+(2z/r) y _{ i }(r) −(2/r) Σ^{ N } _{ j=1,j≠i } Y _{ j }(r) y _{ i }(r) =λ^{2} _{ i } y _{ i }(r), Y _{ i }(r) =F^{ r } _{0} y ^{2} _{ i }(s) d s+rF^{∞} _{0} s ^{−1} y ^{2} _{ i }(s) d s, y _{ i }(0) =0, y _{ i }(∞) =0, F^{∞} _{0} y ^{2} _{ i }(s) d s=1, i=1,2,...,N, establishes existence of solutions to the Hartree equations and the sequence of eigenpairs generated converge subsequentially to a solution. In the case of the helium atom, for which we show some computational results, sequential convergence is obtained. Due to the Hilbert–Schmidt nature of the operators involved, the iterative method is implementable with Galerkin methods.

On the extrapolation of optical image data
View Description Hide DescriptionIn this paper we show that the extrapolation of an image’s piece as well as the object‐reconstruction problem are improperly posed in the sense that the solutions do not depend continuously on the data. We try to restore the stability for these problems introducing suitable additional constraints. In the present work we treat in detail only the extrapolation of the image data. At this purpose we use and illustrate two numerical methods, which are based on the doubly‐orthogonality of the linear‐prolate‐spheroidal‐functions. Finally a probabilistic approach to these questions is outlined.

Integration formula for Wigner 3‐j coefficients
View Description Hide DescriptionAn integration formula is given for Wigner 3‐j coefficients having 72 different forms associated directly with the symmetry properties of the Regge square. The integration formula permits a direct derivation of the integral of a product of three rotation matrix elements, and is shown to include an integration formula for spin projection coefficients as a special case.

Self‐gravitating fluids with cylindrical symmetry
View Description Hide DescriptionIn a recent paper [P. S. Letelier, J. Math. Phys. 16, 1488 (1975)], the problem of self‐gravitating fluid with cylindrical symmetry with p = ρc ^{2} has been reduced to a single equation. The present note solves the equation and also rectifies an oversight in the above‐mentioned paper.

Eigenvalues of invariants of U(n) and SU(n)
View Description Hide DescriptionIn this paper, we present four distinct ways of obtaining the eigenvalues of invariants of unitary groups, in any irreducible representation. The invariants are defined according to a different contraction convention. Their eigenvalues can be given in terms of special partial hooks different from those found by other authors.

Fundamental canonical realizations of connected Lie groups
View Description Hide DescriptionAn important class of transitive canonical realizations of connected Lie groups is studied by means of a general formalism. We give a simple method for the classification and the construction of these realizations.

Nth order perturbation theory for hydrogen
View Description Hide DescriptionThis paper presents a simple reformulation of second order perturbation theory for hydrogen. We calculate first the perturbed wavefunction and then the perturbed matrix element. This procedure is repeated to obtain the nth order matrix element in terms of n−1 integrals. The parameters in the nth order matrix element are defined recursively. Schwinger’s representation of the Coulomb Green’s function follows immediately from our expression for the second order matrix element.