Volume 15, Issue 9, September 1974
Index of content:

Complex potential formulation of the axially symmetric gravitational field problem
View Description Hide DescriptionSpin‐coefficients and null tetrad components of the Ricci tensor and the Weyl conform tensor are evaluated in terms of a single complex gravitational potential ε, while null tetrad components of the electromagnetic stress energy tensor are evaluated in terms of a second complex potential φ. All the results are expressed elegantly in terms of a differential operator ð, similar to the ``thop'' of Newman and Penrose. The problem of finding physically pertinent stationary axially symmetric Einstein‐Maxwell fields is reduced to the search for a complex solution ξ_{0}(x, y) of one nonlinear differential equation subject to simple subsidiary conditions.

Weak gravitational fields
View Description Hide DescriptionWe consider the set of C^{k} bounded tensor fields of type (r,s) on R ^{4} in the topology of uniform C^{k} convergence. For each k≥2, the map sending a metric to its curvature tensor is shown to be analytic at the Minkowski metric. The same is true of the map sending a metric to its Einstein tensor. The well‐known linearized theory of gravitation amounts to studying the directional derivatives of these maps. An iterative method for solving the full field equations along an analytic curve of Einstein tensors passing through zero is proposed.

Asymptotically simple space‐time manifolds
View Description Hide DescriptionAsymptotic simplicity is shown to be k‐stable (k≥3) at any Minkowski metric on in both the Whitney fine C^{k} topology and a coarser topology (in which the C^{k} twice‐convariant symmetric tensors form a Banach manifold whose connected components consist of tensor field asymptotic to one another at null infinity). This result, together with a sequential method for solving the field equations previously proposed by the authors, allows a fairly straightforward proof that a well‐known result in the linearized theory holds in the full nonlinear theory as well: There are no nontrivial (i.e., non‐Minkowskian) asymptotically simple vacuum metrics on whose conformal curvature tensors result from prescribing zero initial data on past null infinity.

On the three‐body linear problem with three‐body interaction
View Description Hide DescriptionA three‐body potential is introduced for which Schrödinger's equation of the three‐body linear problem with additional harmonic and inverse cube forces is solved exactly.

Exact solution of a one‐dimensional three‐body scattering problem with two‐body and/or three‐body inverse‐square potentials
View Description Hide DescriptionThe exact solution is presented of the scattering problem of three equal particles interacting in one‐dimension via two‐ and/or three‐body inverse‐square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.

Wave propagation in a random medium: A complete set of the moment equations with different wavenumbers
View Description Hide DescriptionPropagation of waves in a random medium is studied under the ``quasioptics'' and the ``Markovrandom process'' approximations. Under these assumptions, a Fokker‐Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the Fokker‐Planck equation of the characteristic functional. The applications of our results to the pulse smearing of the pulsar signal and the frequency correlation function of the wave intensity in interstellar scintillation are briefly discussed.

On the automorphisms of real Lie algebras
View Description Hide DescriptionWe establish some properties of automorphisms of real Lie algebras, which in particular allow us to construct the derivation algebra of a Lie algebra from the derivations of its radical. We apply this construction to some familiar kinematical algebras.

Spectral representation of the pentagon diagram amplitude
View Description Hide DescriptionA method developed in two previous papers is used to derive a double spectral representation with Mandelstam boundary for the pentagon diagram amplitude for the production process A B → C D N. Restrictions on the masses and kinematic invariants for which this representation is valid are found and it is discussed how a representation can be obtained for wider ranges of these variables. Finally, a comparison is made with the results of other authors.

Schrödinger equation with inverse fourth‐power potential, a differential equation with two irregular singular points
View Description Hide DescriptionThe Schrödinger radial equation with inverse fourth‐power potential is treated analytically. Solutions in the form of integral representations of the generalized Laplace type are considered. Standard solutions are defined relative to each of the two irregular singular points of the differential equation. The coefficients in the linear relations persisting between any three of the standard solutions are obtained. The expressions for the coefficients, which contain some Taylor and Laurent series and finite determinants, are suitable for electronic computation. From the coefficients the S matrix and the scattering phase shifts may be obtained immediately.

Dynamical symmetries and constants of the motion for classical particle systems
View Description Hide DescriptionBy formulating the conditions for dynamical symmetry mappings directly at the level of the dynamical equations (which are taken in the form of Newton's equations,Lagrange's equations, Hamilton's equations, or Hamilton‐Jacobi equation), we derive new expressions for dynamical symmetries and associated constants of the motion for classical particle dynamical systems. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (a) x̄^{i} =x^{i} +δx^{i} [δx^{i} ≡ξ^{ i }(x)δa] with associated changes in the independent variable t (path parameter) defined by (b) δt≡{∫2φ[x(t)]dt+c} δa. A generalized form of the related integral theorem (a method for obtaining constants of the motion based upon deformations of a known constant of the motion under dynamical symmetry mappings) is obtained. We take the ``Newtonian form'' of the dynamical equations to have a coordinate‐covariant structure with forces defined by a general polynomial in the velocities and obtain dynamical symmetry conditions for all such systems. For the special case of conservative systems the related integral theorem is applied. Based upon Lagrange's equations with L=L(x^{i},ẋ^{i} ) we find the conditions for dynamical symmetry mappings may be expressed in the form . From this form we obtain a new formula for concomitant constants of the motion: (d) [∂(δL)/∂ẋ^{j} ] ẋ^{j} −δL = k. By use of the related integral theorem such constants of the motion can be expressed as deformations of the energy integral under the dynamical symmetry mappings defined by (c). A short derivation of the Noether identity is given which is independent of the integration processes of Hamilton's variational principle. For mappings of the type (a), (b) ``Noether type'' symmetries and associated constants of the motion are formulated. For a conservative dynamical system with L≡(1/2)g_{ij}ẋ^{i}ẋ^{j} − V(x) we find such Noether symmetries are basically conformal motions, while those derived from (c) are basically projective collineations. For such systems the constants of the motion (d) are evaluated and shown in general to differ from those obtained from the Noether method. We show for conservative dynamical systems that the formulation of dynamical symmetry mappings directly at the level of the Hamilton‐Jacobi equation leads to the Noether symmetry conditions. Dynamical symmetry conditions are formulated for Hamilton's equation in phase space and shown to be more general than canonical transformations. The formulation of the related integral theorem in phase space is found to be a generalization of Poisson's theorem. For systems with H(x^{A} ), A = 1,…,2n, it is an immediate observation that δH induced by a symmetry mapping is a constant of the motion. Application to the isotropic harmonic oscillator shows both symmetric tensor and angular momenta constants of the motion are obtained in this manner. An additional constant of the motion ∂_{ A } ξ^{ A }−2φ(x^{A} ) is shown in general to be a concomitant of a phase space symmetry transformation.

Special relativity in general relativity?
View Description Hide DescriptionStarting from the metric in harmonic coordinates for a test particle m _{1} around a heavy particle at rest, the EIH Lagrangian is recovered by making a Lorentz transformation, followed by a canonical transformation and an appropriate symmetrization in the two masses. This raises the question of a special relativity content in general relativity, a feature not directly implied by the general covariance.

Stability of stochastic functional differential equations
View Description Hide DescriptionA system of functional differential equations with random retardation, ẋ(t) = f(t, x_{t} ), is studied, where x_{t} (θ) = x(t + θ), η(t, ω) ≤ θ ≤ 0, − r ≤ η(t, ω) ≤ 0, and η(t, ω) is a stochastic process defined on some probability space (Ω, μ, P). Some comparison theorems are stated and proved in details under suitable assumptions on f(t, x_{t} ). Sufficient conditions for stability in the mean for the trivial solution then follow. The usefulness of the sufficient conditions is illustrated by an example with two different Lyapunov functions.

Application of cumulant techniques to multiplicative stochastic processes
View Description Hide DescriptionThe use of cumulant techniques for analyzing time dependent, stochastic matrix expressions of the form is explained. Because cumulants are complicated expressions when B̃(t) does not commute with itself at unequal times, we explicitly work out cumulant expressions up to fourth order. The fourth order terms can be used to demonstrate that noncommutivity prevents the generalization, to time‐dependent, stochastic matrices which do not commute with themselves at unequal times, of the result which applies to commuting stochastic processes that states: If the stochastic process is Gaussian, then its cumulant expansion truncates after the second cumulant. Furthermore, it is argued that if the stochastic matrix process is both Gaussian and purely random then the cumulant expansion does truncate after the second cumulant, after all. The significance of this result with respect to the application of approximation involving cumulants is mentioned.

A classification of second‐order raising operators for Hamiltonians in two variables
View Description Hide DescriptionWe develop a group theoretic method based on results of Winternitz et al. to compute and classify all first‐ and second‐order raising and lowering operators admitted by Hamiltonians of the form The key to our results, which generalize to higher dimensions, is a proof that admits a second‐order raising operator only if the Schrödinger equation separates in Cartesian, polar, or elliptic coordinates.

Structure of the 12j and 15j coefficients in the Bargmann approach
View Description Hide DescriptionGenerating functions of the 12j and 15jangular momentum recoupling coefficients are computed explicitly in the Bargmann formalism. Symmetry properties are deduced therefrom. A geometrical Möbius strip representation (originally due to Ord‐Smith for the 12j case), which can be generalized to all n, suggests a 4n‐fold symmetry for the 3nj coefficients (n ≥ 4).

Variational formulation of the relativistic theory of microelectromagnetism
View Description Hide DescriptionBy adjoining a set of adequate potentials to the classical electromagnetic potential, it is possible to formulate a variational principle that yields the equations of the micromorphic EM theory proposed by Eringen and Kafadar [J. Math. Phys. 11, 1984 (1970)]. The energy‐momentum law for micromorphic EM fields is obtained and constitutive equations are derived for relativistic EM‐elastic fields.

Lattice Green's function of the body‐centered cubic lattice at arbitrary points
View Description Hide DescriptionLattice Green's function for the body‐centered cubic lattice at arbitrary points outside and inside the band is evaluated by the method of analytic continuation using Mellin‐Barnes type integral.

Multigroup replication property for external, spherically symmetric problems of transport theory
View Description Hide DescriptionThe replication property for multigroup spherically symmetric external problems in the transport theory is derived and applied to reduce the system of multigroup integral transportequations to a system of planelike singular integral equations, which can be solved by means of well‐known methods.

Existence and uniqueness of solutions to Low's problem
View Description Hide DescriptionIn the framework of S‐matrix theory, the partial scattering amplitudes are sought as a solution of a certain problem involving analyticity, unitarity, and crossing symmetry. This problem, with a condition of analyticity which is weaker than the usual one, is called Low's problem in this paper. By means of the fixed‐point theorems of Schauder and Banach‐Cacciopoli, conditions for the existence and uniqueness of solutions to Low's problem are given.

Properties of two‐phase ``cell materials''
View Description Hide Description``Symmetric cell materials'' and ``asymmetric cell materials'' were defined by Miller in connection with the physical properties (such as the effective dielectric constant) of two‐phase solid mixtures. It is shown here that while the ``symmetric cell material'' is self‐consistent, the ``asymmetric cell material'' is not: The postulated three‐point probabilities do not add up to the correct one‐point probabilities. A self‐consistent generalization of the ``symmetric cell material'', based on the requirement that a certain integral must reduce to an integral over a finite region, is developed, and one construction procedure for producing such a material is described.