Volume 9, Issue 7, July 1968
Index of content:

Cauchy Problem in the Scalar‐Tensor Gravitational Theory
View Description Hide DescriptionWe show that the scalar‐tensor gravitational field equations can be split up into a set of initial‐value equations and a set of time‐evolution equations and that the initial‐value equations preserve their form as they propagate forward in time. The proper amount of freedom for the stress‐energy tensor is maintained, and the Cauchy problem is thus posed reasonably well.

New Methods for Reduction of Group Representations. II
View Description Hide DescriptionA better algorithm for reduction of a group representation than that given in an earlier paper [J. R. Gabriel, J. Math. Phys. 5, 494 (1964)] is derived.

Tamm‐Dancoff Method in the 2V Sector of the Lee Model
View Description Hide DescriptionThe 2V sector of the Lee model with boson sources at zero separation is considered from the Tamm‐ Dancoff point of view. This involves three singular integral equations which are solved by appealing to results previously gained in the Lehmann‐Symanzik‐Zimmermann method of solution. Two scattering amplitudes and a production amplitude are rederived via the state vectors and an equation for the determination of the two‐meson exchange potential is discussed.

Derivation of Newton's Law of Gravitation from General Relativity
View Description Hide DescriptionA static situation of two objects held apart by a strut is considered within the framework of general relativity, and the gravitational attraction between the objects is inferred from the stress in the strut. In the Newtonian limit—the objects are well separated and the field weak everywhere except in their immediate vicinity—Newton's law of gravitation is reproduced. This check goes well beyond verifications of the Newtonian limit based on considerations of ``test particles,'' since arbitrarily strong self‐fields are not excluded for either object. It is also an explicit verification of the equality of active and passive gravitational masses.

Analyticity Properties of the Scattering Amplitude for Singular Potentials
View Description Hide DescriptionA modified Sommerfeld‐Watson transformation is established for the singular potential V(r) = g ^{2} r ^{−4}. The positions of the Regge poles and their residues are obtained for a general class of singular potentials and the analyticity properties of the scattering amplitude f(k, cos θ) are discussed.

Particlelike Solutions to Nonlinear Complex Scalar Field Theories with Positive‐Definite Energy Densities
View Description Hide DescriptionIt is shown that a self‐interacting complex scalar field theory with a positive‐definite energy density can admit spatially localized singularity‐free particlelike solutions. A condition on the self‐interaction energy density, sufficient to guarantee the existence of such solutions, is that its derivative should be nonincreasing and not identically constant as the squared absolute value of the field increases from zero.

Charged Particlelike Solutions to Nonlinear Complex Scalar Field Theories
View Description Hide DescriptionIt is shown that spatially localized singularity‐free particlelike solutions exist for Lorentz‐covariant complex scalar field theories with minimal gauge‐invariant electromagnetic coupling, a positive‐definite energy density, and suitably prescribed nonlinear self‐interaction. Such a theory provides a perfectly consistent structural model on the classical level for a chargedelementary particle of finite positive energy.

Variational Bounds for the Potential in Terms of the S‐Wave Phase Shift
View Description Hide DescriptionWe establish a variational expression involving the S‐wave phase shift and bound‐state parameters and the potential, which appears particularly suited to evince from the S‐wave‐scattering data information (i.e., bounds or variational approximations) on the potential. An application of this approach to solvable potentials yields a number of inequalities involving special functions.

Weak‐Graph Method for Obtaining Formal Series Expansions for Lattice Statistical Problems
View Description Hide DescriptionA unified exposition of the weak‐graph method for obtaining formal series expansions for lattice statistical problems is presented. The prototype of this method is the derivation of the hyperbolictangent high‐temperature expansion for the spin‐½ Ising model. Also, recent expansions of the monomer‐dimer problem and various hydrogen‐bonded problems have been treated by essentially the same method. In this paper the method is further illustrated by obtaining series expansions for the low‐temperature spin‐½ Ising problem, the low‐density hard‐core lattice‐gas problem, the high‐temperature spin‐1 Ising problem, the k‐color problem, and two new model problems, the ramrod model and a special ternary model. The weak‐graph method enables one to obtain especially useful series expansions for a certain class of problems, including the spin‐½ Ising problem and the monomer‐dimer problem, which have essentially a binary nature.

Combinatorial Theorem for Graphs on a Lattice
View Description Hide DescriptionSeveral problems in lattice statistical mechanics, such as the spin‐½ Ising problem and the monomer‐dimer problem, can be formulated in terms of the p‐generating function for the weak subgraphs of a regular lattice. This paper presents an algebraic transformation theorem which allows the p‐generating function for the weak subgraphs of a lattice to be determined from the p‐generating function for the far less numerous subset consisting of the closed weak subgraphs. This result will be especially useful in reducing the labor required to obtain exact finite series for various problems. The theorem also enables one to give a straightforward proof of the Ising susceptibility graph theorem due to Sykes.

Rotation and Lorentz Groups in a Finite Geometry
View Description Hide DescriptionThe introduction in physics of a finite geometry approximating the ordinary Euclidean one poses the problem of studying the relativity groups over such a geometry. We present a detailed analysis of the structure and irreducible representations of the rotation, Lorentz, and Poincaré groups. It is found that, besides the usual quantum numbers, a new two‐valued label is necessary to specify the representations.

Model for Converging Detonations in Solid Explosives
View Description Hide DescriptionThe fluid‐dynamic equations which describe converging detonation fronts in high explosives are solved over a coordinate system that changes from circular to a pleated pattern as the detonation converges. There are three reasons for introducing such a coordinate system rather than making the usual assumption that when a spherical or cylindrical explosive charge is initiated simultaneously on its curved surface the detonation front maintains its symmetry as the detonation converges to the center: (1) Calculations made elsewhere show that a converging circular shock wave is unstable. (2) Calculations in this paper show that if spherical or cylindrical symmetry is maintained, density and velocity become infinite, but if the detonation front folds these quantities remain finite except at a few points which can be removed by cuts. The average fluid velocity remains almost constant over 99% of the detonation front. (3) Experiments indicate that a cylindrical detonation front becomes rather symmetrically deformed.

Plethysm and the Theory of Complex Spectra
View Description Hide DescriptionNotable progress in developing the theory of complex spectra has come from the exploitation of the symmetry properties of atomic wavefunctions using the theory of continuous groups. Many seemingly simple results have previously been obtained by what would appear to be unreasonably complex methods. Many of these complexities may be removed, and new results derived, using the algebra of plethysm first developed by Littlewood. Applications to three central problems in the theory of complex spectra are discussed: (1) the classification of the atomic states of n‐electron configurations; (2) the analysis and classification of the N‐particle operators that arise in the application of perturbation theory to atomic problems; (3) the derivation of selection rules for the matrix elements of operators.

Type‐Null Vacuum Solutions in General Relativity
View Description Hide DescriptionIt is shown that the necessary and sufficient conditions for the vacuum solutions of a certain metric to be of type null according to the Pirani‐Petrov classification are that a certain other metric be conformally flat. The general solution is obtained.

Radiating Spheres in the Scalar‐Tensor Theory of General Relativity
View Description Hide DescriptionUsing Bondi's ``radiation coordinates,'' the Brans‐Dicke field equations describing the interior of a sphere consisting of a perfect fluid and a radiation field, part of which is purely isotropic and the remainder of which is purely radial, are stated. A method for seeking solutions to these equations in the radiation‐filled space outside the sphere is given.

Bases for the Representations of U _{4} in the Chain
View Description Hide DescriptionThe highest‐weight polynomials of irreducible representations of U _{3} occurring in the reduction of the direct product of two irreducible representations are constructed from Young diagrams. The different irreducible representations in the product are labeled by a parameter which distinguishes their multiplicities also. The method of this paper can be easily extended to any U_{n} .

Spherical Model as an Instance of Eigenvalue Degeneracy
View Description Hide DescriptionIt is shown that the free energy of the spherical model can be expressed in terms of the largest eigenvalue of an integral equation. In three dimensions the spectrum of the integral equation becomes degenerate as the critical point is approached from the high‐temperature side, thus heralding the onset of long‐range order.

3‐Sphere ``Backgrounds'' for the Space Sections of the Taub Cosmological Solution
View Description Hide DescriptionThe cosmological solution due to Taub resembles a radiation filled Robertson‐Walker solution which contains long‐wavelength gravitational radiation rather than electromagnetic radiation. This paper presents several schemes for obtaining 3‐sphere ``backgrounds'' for the t = const space slices in the Taub solution. [ is characterized by being homogeneous for each t.] The reason for finding such backgrounds is to present models for defining wave‐background separation in general. Unspecified averaging methods as suggested by Isaacson work for short‐wavelength gravitational radiation. We attempt here to specify a method which works even when the radiation is of long wavelength. One background is obtained by averaging the metric by Lie transport along certain invariantly defined vector fields on . This background is compared with two other (different) plausible definitions for a background, and reasons are given to suggest that the Lie‐transport average is the preferable definition.

Construction of Orthonormal Angular‐Momentum Operators
View Description Hide DescriptionThe problem of constructing an orthonormal basis for the (2j + 1)^{2}‐dimensional space of angular‐momentum operators is reduced to the problem of constructing a sequence of orthogonal polynomials of a discrete real variable. A number of properties of the polynomials are obtained, including a pure recurrence formula and several mixed recurrence relations. Three explicit representations of the polynomials are given, together with a table of orthonormal angular‐momentum operators.

Solutions of Einstein's Equations for a Fluid Which Exhibit Local Rotational Symmetry
View Description Hide DescriptionAll solutions of Einstein's equations for pressure‐free matter which exhibit local rotational symmetry were classified in an earlier paper by one of us. This paper extends the earlier theory to the case of a general fluid, with an electromagnetic field possibly present. A classification of these solutions for a perfect fluid is given, and assuming a physically reasonable equation of state, some exact solutions of cosmological interest are obtained. Finally, the difficulties encountered when extending the treatment to a general fluid are discussed; the same general classification can be made.