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Volume 8, Issue 9, September 1967

Linear Closure Approximation Method for Classical Statistical Mechanics
View Description Hide DescriptionA description is given of a general closure principle involving the minimization of the mean square error. The procedure based upon this principle can be applied to the truncation of the BBGKY hierarchy at various stages and to the approximation of unwanted terms arising in the equation of motion method by linear combinations of the observables to be retained. On a general level the significance of the closure principle is described in terms of the geometry of function space, and several useful general properties of the principle are derived. A discussion is devoted to the relation between the closure error (i.e., the least mean square error) and the error in the end result (e.g., the free energy, the radial distribution function, etc.); however, the results, while providing some insight, are not sufficiently refined to provide upper bounds to errors in all problems of statistical mechanics where the method is applicable. On the level of specific application it is shown that the principle yields results identical to the random phase approximation and to the linearized version of the Kirkwood superposition approximation in two special cases. Later sections of the paper describe in greater than usual generality, the formalism connecting thermodynamic properties and other equilibrium properties with the microscopic equations of motion in which closure approximations have been introduced. Two illustrative examples of the application of the over‐all method were made to the case of a classical system of electrons in a uniform background of compensating charge, one leading to the well‐known results of Debye and the other to a more accurate and elaborate theory developed in quantitative detail elsewhere.

Expansion of the T Matrix for Resonance Collisions
View Description Hide DescriptionThe eigenfunctions of the kernel of the Lippmann‐Schwinger collision equation, corresponding to outgoing waves in all channels, are used to obtain an expansion of the T matrix valid for multichannel collisions, including rearrangements. From this expansion, the transition amplitude in the general case of overlapping resonances is obtained and a characterization of bound states and resonance states is given.

A Class of Eigenvalues of the Fine‐Structure Constant and Internal Energy Obtained from a Class of Exact Solutions of the Combined Klein‐Gordon‐Maxwell‐Einstein Field Equations
View Description Hide DescriptionThis paper deals with the combined Klein‐Gordon‐Maxwell‐Einstein field equations, which govern completely and self‐consistently the spinless, charged, gravitating matter distribution. One of the theorems that have been proved here states that, from a static, purely gravitational universe, a class of electrogravitational universes containing a stationary matter field can be constructed, provided a single differential equation is satisfied. The construction of the electrogravitational universe from the Schwarzchild solution hinges on the solubility of the ordinary differential equation, where the prime denotes differentiation and α^{2} stands for the fine‐structure constant. Next, the following nonlinear eigenvalue problem related to this differential equation has been posed. Are there some positive values of α corresponding to which solutionsU(x) exist such that (i) Uis analytic and positive in x ∈ (0, ∞] (⇒ the volume element has one sign), (ii) U(0) = 0 (this condition is physically unpleasant but forced by the differential equation itself), (iii) U′(∞) = 0 (⇒ no force at the center of spherically symmetric mass and charge distributions), (iv) U′(0) = α (⇒ the total charge of the material distribution is α)? The answer is ``yes'' and it has been rigorously proved that there exists a unique solution of the problem. The corresponding value of α comes out to be 1.4343(ℏc)^{½}, which, unfortunately, does not agree with the experiment (the discrepancy may be attributed to the neglect of the second quantization). If the restriction in U(x) to be positive is withdrawn, then a countable number of solutions exist with the corresponding eigenvalues for the electronic charge, internal energy, and mass. These solutions give rise to universes which are topologically inequivalent to Euclidean space and contain a finite number of shells. It should be mentioned that the present eigenvalue problem appears as a consequence of the ``Weyl‐Majumdar'' condition on the electrogravitational universe. There may well exist other eigenvalue problems for the fine‐structure constant within the framework of the Klein‐Gordon‐Maxwell‐Einstein field equations without the ``Weyl‐Majumdar'' condition. ``… es kann dann in jedem Punkte das Krümmungsmass in drei Richtungen einen beliebigen Werth haben, wenn nur die ganze Krümmung jedes messbaren Raumtheils nicht merklich von Null verschieden ist …''—Riemann.

Generalized WKB Method with Applications to Problems of Propagation in Nonhomogeneous Media
View Description Hide DescriptionA generalized WKB method is derived for the solution of the general second‐order differential equation. The problem is reduced to the solution of two coupled first‐order differential equations. By an appropriate choice of auxiliary functions, the coupling coefficients may be made sufficiently small to facilitate the solution of the coupled equations. It is shown that these solutions can be used in a range of problems in which the regular WKB solutions fail. These generalized solutions may also be used to derive asymptotic expansions of known functions. Applications of the method to higher‐order differential equations are indicated, and solutions to the nonlinear Riccati equation are considered.

Recursion Relations for Coulomb Matrix Elements
View Description Hide DescriptionIt is pointed out that, by using the identity (ψ, (HW − WH′)ψ′) = (E − E′)(ψ, Wψ′), it is possible to derive useful relations among physically interesting matrix elements.

Analytic Functionals in Quantum Field Theory
View Description Hide DescriptionA class of Lorentz invariant generalized functions can be defined as analytic functionals, i.e., as continuous linear functionals which are contour integrals over a suitable space of test functions. These generalized functions include in particular the invariant functions of quantum field theory, but also include ``propagators with higher order poles.'' The analysis shows exactly which of these are well defined, especially in the important special case of zero mass. Various applications to quantum field theory are indicated.

Canonical Formulation of Relativistic Mechanics
View Description Hide DescriptionInstantaneous action‐at‐a‐distance relativistic mechanics of the type considered by Currie and by Hill is cast into a Hamiltonian form wherein the transformations of the inhomogeneous Lorentz group are canonical. The Currie‐Jordan‐Sudarshan zero‐interaction theorem is circumvented by renouncing the demand that physical positions be canonical; the implications for measurement theory of renouncing this demand are discussed. Examples are given.

Expansions in Spherical Harmonics. IV. Integral Form of the Radial Dependence
View Description Hide DescriptionA function f(r_{AB} ) (θ_{ AB }, φ_{ AB }) of a vector r _{ AB } = Σr _{ i } can be expanded in spherical harmonics (θ, φ) of the directions of the individual vectors. The radial coefficients satisfy simple differential equations which, in three previous papers, were solved in terms of series in ; these were different in various regions, depending on the relative magnitudes of the r_{i} . In this paper the solutions are found as multiple integrals over the product of Legendre polynomials and of a function G(w), where w depends linearly on the r_{i} . The kernel G(w) is independent of the number of constituent vectors, their relative sizes, and the orders of their harmonics; it contains the Heaviside step function H(w) as a factor which takes care of the various regions. The precise form of G can be found from f and L by an integral equation which for L = 0, 1 is solved for arbitrary f, and for L > 1 for sufficiently large positive powers. The expressions of Milleur, Twerdochlib, and Hirschfelder for the bipolar angle average can be obtained simply by repeated integration of G(w) or directly from the differential equations. For the inverse distance between two points, G(w) becomes Dirac's delta function; the number of integrations is thereby reduced by one. Possible applications of the new approach to the evaluation of molecular many‐center integrals are outlined. Some corrections are given for the results of the previous papers in the series.

Relation between Creeping Waves and Lateral Waves on a Curved Interface
View Description Hide DescriptionDiffraction effects at a gently curved interface between two media are investigated. Particular attention is paid to the behavior of the field on the diffracted rays which propagate along the interface into the shadows. It is found that far from the launching point of such a ray the field comprises of a series of modes which decay exponentially, due to the continuous leakage of energy away from the interface. At moderate distances, in the penumbra region, this series is poorly convergent. It can be converted into an integral, which can be evaluated asymptotically there, yielding a field with an algebraic decay. The field is like that diffracted along a plane interface, the so‐called lateral wave, and reduces to it when the radius of curvature becomes infinite. The regions of transition from one representation to the other are determined, and uniform asymptotic expressions, valid across those regions, are given. All our results apply to a two‐dimensional scalar problem, but results for three dimensions and for vector problems can be derived in a similar way.

Electric Fields in a Semi‐Infinite Medium Whose Conductivity Varies Laterally
View Description Hide DescriptionThe electric field induced in a semi‐infinite medium whose conductivity varies laterally is calculated when the inducing field is chosen to approximate a vertically incident magnetic wave which is polarized in the direction of the conductivity variation. The specific form of the variation of conductivity is σ = σ_{0} + σ_{1}(y/d)^{2}, where y is a coordinate parallel to the surface. It is shown for several specific cases that the magnitude of the electric field is less than the electric field in a solid whose conductivity is σ_{0}. In addition, the electric field is calculated for several values of σ_{0}/σ_{1} at y = 0.

Degree of Higher‐Order Optical Coherence
View Description Hide DescriptionIn this paper we define the normalized coherence function of arbitrary order (m, n), in a manner which seems to be a natural generalization of that defined for the second‐order coherence function. Both classical and quantized optical fields are considered and the results are compared. It is shown that for classical fields and also for quantized optical fields having nonnegative definite diagonal coherent state representations of the density operator, the modulus of these normalized coherence functions is bounded by the values 0 and 1. This definition differs from the one recently given by Glauber for quantized optical fields, where the normalized coherence functions may take arbitrarily large values even for fields having nonnegative definite diagonal representations of the density operator. Conditions for ``complete coherence,'' i.e., those under which the modulus of the normalized coherence function attains the limiting value 1, are discussed. Some consequences of stationarity and quasi‐monochromaticity are also discussed.

Energy‐Momentum Conservation Implies Translation Invariance: Some Didactic Remarks
View Description Hide DescriptionWe discuss from a rigorous viewpoint two more‐or‐less familiar cases where energy‐momentum conservation implies invariance under space‐time translations. First, if a closed linear operator on a Hilbert space has a domain that is invariant under spectral projections belonging to the four‐momentum operators, and if it ``conserves energy‐momentum,'' it necessarily commutes with the appropriate representation of the translations. (Bounded operators, such as the S matrix, are a special case.) At least for separable spaces, the domain restriction characterizes the closed operators for which the theorem is true. Second, if a bounded bilinear form between momentum states of m and n particles in a Fock space (or more generally, a bounded multilinear form) conserves energy momentum, the corresponding tempered distribution has a conservation delta function at points where the mass shell is a C _{∞}manifold; but no derivatives of delta functions can occur. In this connection, we are led to a result that seems to be new: the cluster parameters (``connected amplitudes'') of a family of bounded bilinear forms, labeled by (m, n), are also bounded bilinear forms. The two systems, of course, mutually conserve energy momentum.

Angular Coefficients of Atomic Matrix Elements Involving Interelectronic Coordinates
View Description Hide DescriptionA method for determining the angular coefficients of atomic matrix elements is illustrated. The angular coefficients of matrix elements for , and are evaluated using single‐particle states of definite angular momentum. The use of tensor operators enables a separation into angular and radial parts. The atomic matrix elements are then expressed as sums over products of n‐j symbols and radial integrals. These sums are restricted by the values of the single‐particle state angular momenta, and in all cases the effects of single‐particle couplings disappear. The calculation does not require the use of a particular coordinate system, as is the case for multiple products of spherical harmonics.

High‐Energy Phase Shifts Produced by Repulsive Singular Potentials
View Description Hide DescriptionAn expansion in inverse powers of the energy (usually fractional powers) of the WKB phase shifts δ_{ l } produced by repulsive potentials, singular at the origin, has been derived. In most cases this expansion is valid for angular momenta l < l _{max}, where l _{max} increases with energy. For large l a power series expansion in the ``coupling constant'' g is developed. The two regions of validity complement each other and sometimes even overlap. The potentials considered have an r^{−p}, p ≥ 1, singularity and a k^{m}, m < 2, energy dependence. Define q = p + m − 2; then for q > 0, the case of strong interaction, we get δ_{ l } ∼ −(gk^{q} )^{1/p }, when k → ∞. The constant of proportionality is independent of l. However, the next term in the expansion depends on l. For q = 0, the case of intermediate interaction, δ_{ l } becomes independent of energy when k → ∞, but it depends in a complicated way on both g and l. Finally, for q < 0, the case of weak interaction, δ_{ l } ∼ −gk^{q} (l + ½)^{1−p }, p > 1.

Initial‐Value Problem for Relativistic Plasma Oscillations
View Description Hide DescriptionThe solution of the collisionless, relativistic Vlasov‐Maxwell set of equations is given for the case where neither ambient electric nor magnetic fields are present and a smeared‐out negative charge background preserves over‐all space‐charge neutrality with a relativistic protonplasma. The method of solution is based upon the eigenfunction expansion method invented by van Kampen. It is shown that, besides the relativistically modified modes of Case and Zelazny, there exists a new discrete set of modes whose phase velocities are greater than the speed of lightin vacuo. The solution also exhibits the coupling of electrostatic and electromagnetic modes and the existence of complex phase velocities. This initial‐value problem (or the problem with the electron and proton roles reversed) is of interest because of the existence of cosmic rays, relativistic electrons emitting synchrotron radiation in nonthermal radio sources, solar noise generation, where electron velocities may approach c (Bailey, 1951), and high‐energy electrons in laboratory machines (Post, 1960). In these cases the nonrelativistic treatment is clearly inadequate.

Coordinate and Momentum Observables in Axiomatic Quantum Mechanics
View Description Hide DescriptionAn axiomatic model for quantum mechanics is formulated using physically significant axioms. The model contains a slight strengthening of Mackey's first six axioms, together with two axioms which ensure the existence of coordinate and momentum observables. The symmetries or rigid motions are an essential part of the structure, and a link is constructed between these and the quantum proposition system. Coordinate and momentum observables are defined in terms of abstract coordinate systems and one‐parameter groups of motions. It is then shown that as far as the statistical properties of these observables in certain canonical states are concerned, the abstract model may be represented by the usual Hilbert space formulation. Spectral properties of σ‐homomorphisms are also investigated. Also included are two appendices of a technical nature: the first considers one‐parameter groups of derivables, and the second absolutely continuous σ‐homomorphisms.

Algebraic Calculation of Nonrelativistic Coulomb Phase Shifts
View Description Hide DescriptionIt is observed that the in and out states of a particle of energy k ^{2}/2m, entering or leaving a Coulomb field along the +z direction, are eigenstates of L _{3} with eigenvalue 0, and of A _{3}, the third component of the Runge‐Lenz vector, with eigenvalues ±α + ik/m, respectively. From this characterization and the commutation relations of the symmetry group, the phase shifts are easily obtained algebraically.

Refractive Index, Attenuation, Dielectric Constant, and Permeability for Waves in a Polarizable Medium
View Description Hide DescriptionA new method is presented for calculating the refractive index,attenuation,dielectric constant, and permeability for electromagnetic waves in a medium of polarizable particles. It is similar to the method of Yvon and Kirkwood for finding the static dielectric constant. The main merit of the method is that it avoids the statistical hypotheses used in such calculations by Lorentz, Reiche, Hoek, Rosenfeld, and other authors. In addition, it permits the calculations to be continued to any degree of accuracy. We first use the method to obtain the dispersion equation as a power series in the molecular polarizability. The nth term in this series involves the distribution function of n + 1 particles. The terms of first and second degree are written out explicitly in terms of the two‐ and three‐particle distribution functions. When terms of second and higher degree are omitted and the result specialized to particles with a scalar electric polarizability and zero magnetic polarizability, the dispersion equation agrees with that of Rosenfeld. When terms of second degree are retained and the static limit considered, the result reduces to that of Yvon. We next use the method to obtain the dispersion equation as a power series in the particle number density, which seems to be new. To obtain it we introduce ``pure'' n‐particle scattering functions, which are analogous to the Ursell functions of statistical mechanics. This permits us to obtain the density expansion directly in a form simpler than is obtained by resumming the polarizability series.

Gauge Field of a Point Charge
View Description Hide DescriptionThe problem and treatment of integration ambiguities in the conventionally defined Yang‐Mills charges is demonstrated explicitly, using a non‐Abelian solution of the Yang‐Mills equations for a point charge. The internal holonomy group for this solution is noncompact and nonsemisimple, and the solution is not expected to have a direct physical meaning. However, it provides a convenient example showing important and quite unexpected features of gauge theories of the Yang‐Mills type, before quantization. It is found that the number of unambiguously definable and comparable charges is less than the dimension of and less than the rank of as well. If a gauge group is present in the conventional manner, i.e., , this number of charges is less than the rank of the gauge group. Other interesting features of the solution found are: discreteness of certain components of the gauge field, as a result of regularity conditions together with the condition that the Yang‐Mills charge density vanishes outside a sphere of finite radius, and a harmonic oscillation of the other gauge components, while the observable charges are steady. Higher‐order charges are all found to be zero. No action principle is used and no a priori particle fields are introduced. Use is made of the differential‐geometric properties of gauge fields.

Three‐Particle States and Green's Function
View Description Hide DescriptionThe six‐dimensional spherical harmonics are specified in connection with SO _{6} and particular subgroups isomorphic to SO _{3} and SO _{2}; three‐particle states are then labeled by the grand‐angular momentum n, the usual angular momentumj, its projection on a fixed axis m, an integer μ related to SO _{2}, and a degeneracy number ω. Also, we derive the plane‐wave and free Green's function expansions in terms of these spherical harmonics.