Volume 11, Issue 3, March 1970
Index of content:

Classical Green's Functions in Response Theory
View Description Hide DescriptionThe present paper is concerned with the retarded many‐time thermodynamicGreen's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many‐time Poisson‐bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many‐time thermodynamicGreen's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many‐time Green's functionequations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov‐Sadovnikov (B‐S) hierarchy from the lowest‐order many‐time Green's functionequation of motion.

Global and Infinitesimal Nonlinear Chiral Transformations
View Description Hide DescriptionThe problem of determining arbitrary nonlinear representations of a given compact Lie group is studied with the object of constructing Lagrangians invariant under the group. To achieve this, expressions for the covariant derivatives are obtained and it is shown how previous treatments based on global and on infinitesimal considerations are related. The noncompact case, the relationship between non‐linearity and zero‐mass particles, and the possibility of embedding the representation manifold in a higher‐dimensional space are all discussed.

Some Remarks on Classical Scalar Radiation from Point Sources
View Description Hide DescriptionThe radiation from a point source with proper time‐dependent coupling strength to a real classical scalar field is worked out. The adiabatic switching limit reproduces the results of the physical case of constant coupling for the radiation. The problem of defining a ``radiated'' field is discussed.

TwoGroup Neutron Transport Theory in Spherical Geometry
View Description Hide DescriptionA set of normal modes for the two‐group steady‐state neutron transport equation in spherical geometry is constructed. The singular eigenfunction‐expansion technique is then used to develop a rigorous solution to the isotropically emitting spherical shell‐source problem in an infinite medium.

Technique for Finding the Moment Equations of a Nonlinear Stochastic System
View Description Hide DescriptionA technique is described for deriving the moment equations of a nonlinear stochastic system with a random forcing term. The nonlinear term is then linearized by means of minimizing the mean‐square error between nonlinear and linear terms. The traditional derivation of the Fokker‐Planck equation for the conditional probability density, and hence the moment equations, is bypassed.

On the Solution of a Generalized Wiener‐Hopf Equation
View Description Hide DescriptionThis paper deals with the generalized Wiener‐Hopf equation,where α (= σ + iτ) is a complete variable, G(α), H(α), and Ψ^{(i)}(α) are known functions, and X _{+}(α) and Y _{−}(α) are unknowns, analytic in upper and lower half‐planes, respectively, as indicated by their respective subscripts. This type of equation arises in a class of boundary‐value problems in electromagnetic theory, the geometries of which may be described as modified Wiener‐Hopf type. The method of approach, which is fundamentally different than those currently available in the literature, is based on a pairing of singularities in the complex α plane. This leads to a functional equation which is exactly solvable in its asymptotic form. The knowledge of this solution permits one to employ one of several rapidly converging numerical procedures available in the literature for a more accurate solution. Two examples illustrating the application of the procedure are included in the paper.

Colorings of a Hexagonal Lattice
View Description Hide DescriptionThe number of ways W^{L} of coloring the bonds of a hexagonal lattice of L sites (L large) with three colors so that no adjacent bonds are colored alike is calculated exactly, giving W = 1.20872 …. This is equivalent to counting the number of 4‐colorings of the faces of the lattice and can also be regarded as a multiple‐dimer problem. If one introduces activities corresponding to certain vertex configurations, then the system is found to have an infinite‐order phase transition between two ordered states.

Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field
View Description Hide DescriptionAn inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external fieldH for H > 0. The concavity of magnetization, which holds not only for spin‐½ but also for arbitrary‐spin Ising ferromagnets, provides a basis for certain thermodynamic inequalities near the ferromagneticcritical point, including one involving the ``high temperature'' indices α and γ.

Generalized Anharmonic Oscillator
View Description Hide DescriptionThe generalized anharmonic oscillator is defined by the Hamiltonian H_{N} , which in the coordinate space representation is given by H_{N} = −d ^{2}/dx ^{2} + ¼x ^{2} + g(½x ^{2})^{ N }. The analytic properties of the energy levels of H_{N} as functions of complex coupling g are derived and described. Zeroth‐order WKB techniques are used in the mathematical analysis. For all N, the results are qualitatively similar to those for the ordinary anharmonic oscillator in which N = 2 and, thus, the results are model independent for this wide class of models. The limiting case N → ∞ is solved exactly without using WKB techniques. The exact solution agrees with the WKB solution to zeroth order. This agreement is most impressive and testifies to the accuracy and utility of WKB methods.

On the Inverse Scattering Problem at Fixed Energy for Potentials Having Nonvanishing First Moments
View Description Hide DescriptionIt has been shown previously by Newton that his solution (and its extension by Sabatier) of the problem of finding a central potential from a knowledge of all phase shifts at fixed energy yields a series whose expansion coefficients converge slowly unless the first moment of the potential vanishes. In particular, any truncation of the series after a finite number of terms necessarily results in potentials which have vanishing first moments. In this paper we propose a new, but formally somewhat similar, series for the potential which, for such truncations, does not suffer from this physically rather severe restriction. The series also furnishes new exact solutions of the Schrödinger equation at fixed energy. A closed‐form expression for the scattering amplitude is obtained for a specific example. The problem of constructing the new series from the phase shifts is not discussed.

Note on the Uniqueness of the Solution of an Equation of Interest in the Inverse Scattering Problem
View Description Hide DescriptionIt is proved that the solutions recently obtained by the authors [J. Math. Phys. 11, 805 (1970)] of the Regge‐Newton integral equation (of interest in connection with the inverse scattering problem at fixed energy) are, for a given kernel, inhomogeneity, and boundary condition, uniquely determined.

Curvature Collineations in Empty Space‐Times
View Description Hide DescriptionIt is shown that the only curvature collineations admitted by an empty space‐time, not of Petrov type N, are conformal motions. The curvature collineations admitted by the plane‐fronted gravitational waves are found.

General Relativistic Thin‐Sandwich Theorem
View Description Hide DescriptionThe configuration variables for the gravitational fieldg_{mn} are assigned arbitrarily on two infinitesimally neighboring spacelike hypersurfaces. We then investigate the extent to which a solution of the vacuum Einstein field equations can be found consistent with the given assignment. A local approach, employing Dirac's Hamiltonian formalism, reveals that solutions can be found locally which are nonunique and highly unstable.

Summability Methods in Perturbation Theory
View Description Hide DescriptionThe Mittag‐Leffler summability method is applied to operator‐valued analytic functions and a corresponding procedure for perturbation theory is derived, which has a bigger region of convergence. This region is explicitly described.

Analytical Solution of the Boundary‐Value Problem for the Nonlinear Helmholtz Equation
View Description Hide DescriptionA rigorous analytical solution in terms of the elliptic integral function of the first kind is derived for the nonlinear inhomogeneous boundary‐value problem δ(d ^{2} f/dx ^{2}) = αf ^{2} − εf − σ, f(s) = f̃.

Calculation of Lattice Green's Functions
View Description Hide DescriptionA proof is given that, in the numerical evaluation of lattice Green's functions, it is possible to restrict the wave‐vector summation to an irreducible section of the Brillouin zone. It is shown that, in order to use this simplification, the Green's functions must be appropriately symmetrized.

On the Solution of the Differential Equation
View Description Hide DescriptionThe following solution is obtained:,where P _{0}(x, y, t) is the solution when the constants a, b, and c are zero, and where the time functions α, …, ε are given explicitly.

Relativistically Invariant Classical Hamiltonian Mechanics with Extra Variables
View Description Hide DescriptionIt is shown that one can construct a relativistically invariant classical mechanics in Hamiltonian form by utilizing as independent dynamical variables the position of each particle, together with its canonical conjugate variable, and the velocity of each particle, together with its canonical conjugate variable. The ten generators of the Poincaré group, obeying the correct Poisson bracket relations between themselves and the position variables, are constructed in their most general form. By restricting this form, it is possible to construct a Hamiltonian theory where the equations of motion for the particle's positions and velocities depend only upon the positions and velocities (and not upon their conjugate variables): these turn out to be the most general relativistically invariant classical equations of motion of this type. This is not useful as a starting point for constructing a physically interesting quantum theory, since the probability density in the quantized theory obeys the classical Liouville equation. This approach is also applied to the Galilean‐invariant classical mechanics.

Exact Finite Method of Lattice Statistics. III. Dimers on the Square Lattice
View Description Hide DescriptionThe 2‐dimensional square‐lattice model of hard dimeric molecules has been studied over the entire range of dimer densities. The effects of an external field have been included by allowing different activities for the two possible molecular orientations. Thermodynamic properties have been obtained by the transfer matrix technique for lattices of increasing finite circumference and infinite length. The convergence with increasing size has been the most rapid of all systems yet studied by this technique, satisfactory convergence usually being realized with a circumference of only ten sites. The results are in agreement with previous zero‐field studies employing other methods. There is no phase transition in the presence or absence of an external field, and the equation of state is remarkably insensitive to the field. The limited information obtained regarding the nature of the approach to the close‐packed limit is consistent with recent predictions based on the series method.

Toward a General Theory of Measurement. I
View Description Hide DescriptionA self‐consistent measurement theory is set forth in this paper, which can be applied to the problem of measurement in field theory. Following the formulation of Birkhoff and von Neumann, we construct a system of axioms sufficient to ensure that it is possible to find an implicit mechanism within each theory for the experimental procedure corresponding to each observable proposition of that theory. It is shown that any theory of a type similar to quantum mechanics cannot consistently describe a closed system from the point of view of a single observer.