Volume 13, Issue 2, February 1972
Index of content:

Mapping onto Solutions of the Gravitational Initial Value Problem
View Description Hide DescriptionTwo approaches to the Einstein initial value problem for vacuum gravitational fields are considered. In the first, the metric of a spacelike slice is prescribed arbitrarily and it is shown that momenta satisfying the constraints can be constructed by exploiting the well‐known relation of three of the four constraints to the three‐dimensional coordinate transformation group. Specifically, it is shown that there exists a coordinate mapping of a certain specified set of functions onto momenta satisfying the constraints for a specified 3‐metric. A further interpretation of this procedure is discussed. In the second approach the 3‐geometry of a spacelike slice is specified up to a conformal factor. It is shown that, using a coordinate transformation method similar to the above, transverse traceless momenta can be constructed and that this construction depends essentially only on the conformal geometry of the spacelike hypersurface. The remaining constraint is satisfied by a choice of the conformal factor. As a result, it follows that the initial value equations can be satisfied by mapping certain specified sets of functions onto solutions by using coordinate transformations and a group of scale transformations which include conformal transformation of the metric. This is significant because the unconstrained initial data (gravitational degrees of freedom) are represented by a pair of scale‐invariant transverse, traceless tensors of weight. These objects, in turn, give irreducible representations of the coordinate and scaling groups which are used to effect solutions of the initial value equations.

Operator Treatment of the Gel'fand‐Naimark Basis for SL (2, C)
View Description Hide DescriptionAn operator form is developed to treat the Gel'fand‐Naimark z basis for the homogeneous Lorentz group. It is shown that the operator Z with eigenvaluesz is a definite operator‐valued function of the generators of SL (2, C). A unified formulation of the unitary representations of the Lorentz group is obtained in a Hilbert space endowed with an affine metric operator G whose functional dependence on the generators is derived explicitly. The Dirac bra‐ket formalism is extended by making a distinction between covariant and contravariant state vectors. The matrix elements of G are shown to coincide with the intertwining operator of Gel'fand and co‐workers. The principal series, the supplementary series, and the two kinds of integer point representations are unified by means of a single scalar product involving the metric operator.

Deformable Magnetically Saturated Media. I. Field Equations
View Description Hide DescriptionIn this article we propose a variational approach to the study of nonlinear elastic solids in which magnetization is constant in magnitude. The emphasis is placed upon the application of the different invariances used in modern continuum mechanics: Euclidean invariance, objectivity, and material symmetry. In Part I, a variational treatment is given in the spirit of ``oriented media theory.'' A comparison is made with the results of a direct treatment starting with the postulation of balance laws. Part II is devoted to the development of constitutive equations for a variety of material classes.

Variational Methods in the Wave Operator Formalism: Applications in Variation‐Perturbation Theory and the Theory of Energy Bounds
View Description Hide DescriptionVariational principles of the Lippmann‐Schwinger type are used to develop approximations to eigenenergies and eigenfunctions within the wave‐operator formalism. The present approach starts with exactly soluble ``primary'' eigenvalue equations to give explicit results valid beyond the limits of conventional perturbation theory. The variational functionals are expressed in terms of resolvents of the primary Hamiltonian, and bounds to the functionals are constructed also for cases where the resolvents are only partly known. Approximations to eigenenergies and eigenfunctions are obtained in terms of quantities in the Brillouin‐Wigner perturbation theory. Connections with methods for upper and lower energy bounds are discussed, and the convergence properties of the nonlinear Padé summation is recovered in this way. Closed formulas within the double perturbation theory framework are presented as a logical extension.

One‐Center Two‐Electron Integrals Arising in Electron‐Ion Scattering Calculations
View Description Hide DescriptionThe two‐electron integrals which arise in electron‐atomic ion scatteringtheory are expressed as a finite sum of terms each involving a generalized hypergeometric function. We give what is believed to be the first finite expansion known for the exchange integral. The many well‐known recursive and transformation features of the hypergeometric functions are then utilized to conveniently and accurately evaluate the integrals with few restrictions on the values of the parameters.

Approach to Stochastic Lagrangian Integrals and Their Asymptotic Evaluation for Sound Propagation in Continuous Stochastic Media
View Description Hide DescriptionPreviously the formal solution of the Eulerian‐Lagrangian problem for sound propagation in continuous stochastic media was reframed so that the emphasis on the need for complete knowledge of the statistical nature of the Lagrangianfunctional of interest is shifted to the need for knowledge of the asymptotic behavior of certain stochastic Lagrangian integrals which result from the application of a central limit theorem for stochastic functionals. In this paper, the asymptotic evaluation of these Lagrangian stochastic integrals is developed and illustrated for both ensemble and subensemble expectations in a statistically isotropic medium. In addition, the relationship between this method of analysis and the comparable Wiener integral is discussed.

SU (n − 2) × SU (2) × U (1) Bases for SU (n)
View Description Hide DescriptionExplicit basis dates for the general IR of SU(n) are defined in the scheme SU(n − 2) × SU(2) × U(1); the approach makes use of elementary multiplets.

On the Asymptotic Behavior of Certain Dynamical Systems
View Description Hide DescriptionThe asymptotic behavior at large times for certain dynamical systems arising in the Hamiltonian formulation of classical mechanics is investigated. It is shown that for potentials which die out sufficiently fast at large distances the unbounded states of the system are asymptotically free. This result complements the corresponding result for quantum mechanical systems, and is obtained by analogous methods. In addition, the existence, differentiability, and asymptotic completeness of the associated wave mappings is established under appropriate further assumptions by classical methods.

Contractions of the Low‐Dimensional Real Lie Algebras
View Description Hide DescriptionA complete and detailed classification and analysis of all the Inonu‐Wigner contractions of all the real Lie algebras of dimension 1, 2, and 3 is presented. Starting with a more natural classification of the algebras, many corrections and completions are made to the earlier results of Sharp. Among other things, a proper contraction is constructed between two three‐dimensional Lie algebras both of which have two‐dimensional derived algebras‐a phenomenon previously claimed impossible.

Particle Transport in Spherical Media with a Central Black Cavity
View Description Hide DescriptionA transform procedure is applied to the integral form of the Boltzmann transport equation to obtain the particle density in a spherically symmetric medium surrounding a central black absorber. The singular eigenfunction expansion technique is applied to provide a general solution, and boundary conditions are derived for the general equation for arbitrary multiplication and source distributions. Specific applications to the spherical Milne problem and a uniform source distribution are presented to demonstrate the application of the general transform technique.

Decomposition of the Principal Series of Unitary Irreducible Representations of SU(2, 2) Restricted to the Subgroup
View Description Hide DescriptionAn explicit form is given for the unitary irreducible representations of the principal nondegenerate series of the group SU(2, 2). These representations are then restricted to the subgroup and are found to be equivalent to the regular representation of

Necessary Conditions for N‐Representability of Reduced Density Matrices
View Description Hide DescriptionIn this work a large new class of extreme points of the set of N‐representable 2‐matrices is obtained. They correspond to the exact ground state of an operator B(g) which generalizes the BCS Hamiltonian. Further, three hitherto unpublished necessary conditions for N‐representability are obtained.

A Statistical Mechanical Approach to the Problem of a Fluid in an External Field
View Description Hide DescriptionA rigorous, equilibrium statistical mechanical treatment of a fluid in a weak external field is given. The technique involves a cell division which leads to upper and lower bounds for the free energy density. Under a suitable double limiting procedure these limits coalesce, yielding a free energy consisting of a field‐free term plus a field‐dependent term. The cell division allows a direct physical definition of the local pressure p(s) and the local density ρ(s). This treatment provides a rigorous derivation of the thermodynamics of a fluid in a weak external field and, in particular, the hydrostaticequation gradp = − ρ gradφ.

Lower Bounds of the Energy Eigenvalues of Systems Containing Identical Particles
View Description Hide DescriptionLower bounds of the energy eigenvalues of systems containing identical particles were obtained by a generalization of the method of Calogero and Marchioro [J. Math. Phys. 10, 562 (1969)].

Upper and Lower Bounds of the Eigenvalues of a Second‐Order Linear Self‐Adjoint Differential Equation
View Description Hide DescriptionUpper and lower bounds of the eigenvalues of a second‐order linear self‐adjoint differential equation were obtained by a generalization of the methods of Nordtvedt [J. Math. Phys. 8, 1406 (1967)].

Reduction of a Class of Nonlinear Integral Equations to a Cauchy System
View Description Hide DescriptionIt is shown that a wide class of nonlinear integral equations can be transformed into a Cauchy system. Then it is shown that a solution of the Cauchy system provides a solution of the original nonlinear integral equation. Such reductions are important because modern computers can solve initial value problems with speed and accuracy. There are intended applications in the theories of multiple scattering, optimal filtering, and lateral inhibition of neural systems. This new approach makes no use of successive approximations or series expansions.

Operator Valued Measures in Quantum Mechanics: Finite and Infinite Processes
View Description Hide DescriptionIn this work, operator valued measures are used to study finite and infinite sequences of measurements. It is shown that to each such process q ^{Δ} there is uniquely associated a probability operator measureO ^{ qΔ} which contains all the statistical properties of the process. In order to make this association for infinite processes, the operator valued equivalent of the Kolmogorov extension theorem is needed. This theorem is given and proved. It is then shown that for each q ^{Δ} and each set E of possible outcome sequences, there are two ways to find the probability that carrying out q ^{Δ} on a system in state ρ gives an outcome sequence φ in E. The usual method of repeating q ^{Δ} on ρ over and over again generates a sequence α of outcome sequences φ. The probability is obtained as the limit relative frequency that α(j) is in E, for j = 0, 1, …. The other, new, method is the repeated measurement of on ρ. The remarkable aspect of this equivalence is that the mathematical procedures of the usual method for determining if α(j) is in E or not `disappear' into the operators of the new method. This is discussed in some detail and examples are given.

Occupation Statistics for Parallel Dumbbells on a 2 × N Lattice Space
View Description Hide DescriptionIt is shown that occupation statistics for parallel dumbbells on a 2 × N lattice obeys the central limit theorem. On the basis of this conclusion, a modification of the maximum term method is utilized to enumerate for large N the number of distinguishable arrangements arising when indistinguishable, parallel dumbbells are placed on a 2 × N array. The coverage at which the maximum number of arrangements occurs and the pseudovariance for the distribution are also determined.

Perturbation Calculation of an Algebraic Realization of Spin Waves
View Description Hide DescriptionVia the results of a previous paper on an algebraic realization of spin‐wave theory [J. Math. Phys. 12, 2144 (1971)], the Hamiltonian of the Heisenberg model of ferromagnetism is regrouped into a main part and a perturbed part. By introducing a generalized Bethe‐Salpeter equation it is shown that the main part can be diagonalized by the single‐particle states, two‐particle states, and two‐particle bound states. To study the scatterings of the perturbed part, an equation for the amplitude of the two‐particle scattering states is derived. At low temperatures this equation is solved by power series expansion. The modification of thermodynamic free energy and spontaneous magnetization due to perturbation is equal to the first Born approximation multiplied by a function Q(S).

Nonlinear Initial‐Boundary‐Value Problem for Convection, Diffusion, Ionization, and Recombination Processes
View Description Hide DescriptionThe time‐dependent convection‐diffusion equation with ionization and recombination reactions is reduced by means of a nonlinear transformation to a differential equation, in which the nonlinear term represents a small perturbation. The general procedure of solution for the corresponding nonlinear initial‐boundary‐value problem is then established by means of the method of successive approximations. Uniqueness and convergence of the analytical solution are discussed. As applications, the temporal change of an initial distribution of electrons and ions is discussed for a finite box system and an infinitely extended system, respectively.