Index of content:
Volume 19, Issue 6, June 1978

Lie theory and the wave equation in space–time. 4. The Klein–Gordon equation and the Poincaré group
View Description Hide DescriptionA detailed classification is made of all orthogonal coordinate systems for which the Klein–Gordon equation in space–time, ψ_{ t t }−Δ_{3}ψ=λψ, admits a separation of variables. We show that the Klein–Gordon equation is separable in 261 orthogonal coordinate systems. In each case the coordinate systems presented are characterized in terms of three symmetric second order commuting operators in the enveloping algebra of the Poincaré group. This paper also consitutes an important step in the study of separation of variables for the wave equation in space–time ψ_{ t t }−Δ_{3}ψ=0, and its relation to the underlying conformal symmetry group O(4,2) of this equation.

Lie theory and the wave equation in space–time. 5. R‐separable solutions of the wave equation ψ_{ t t }−Δ_{3}ψ=0
View Description Hide DescriptionA detailed classification is made of all orthogonal coordinate systems for which the wave equation ψ_{ t t }−Δ_{3}ψ=0 admits an R‐separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have already been found in previous articles. We find 106 coordinates to given a total of 368 conformally inequivalent orthogonal coordinates for which the wave equation admits an R separation of variables.

A variational principle for transport theory
View Description Hide DescriptionA maximal variational principle is used to construct an infinite medium Green’s function for treating the boundary value problems of the linear transport theory (neutron and radiative). For the neutrons we consider the one‐speed case and correspondingly for the radiative transfer the monochromatic case. The scattering properties of the medium are presumed to be dependent on the relaxation length. Thus, for the neutrons the secondary production function depends on the neutron’s relaxation length and for the radiative transfer the albedo for single scattering is dependent on the optical depth. These two functions are kept arbitrary so that a large class of problems can be covered. The basic principle involves a functional which is an absolute maximum when the trial function is an exact solution of an integral equation of the Fredholm type. The kernel of the integral equation is required to satisfy certain symmetry and boundedness properties. We also exhibit an interesting relation between the absolute maximal and Schwinger’s stationary variational principles, which in general is neither a maximum nor a minimum.

A new formula for the spinor norm
View Description Hide DescriptionA formula is established, valid for all real pseudo‐orthogonal groups, permitting an easy computation of the spinor norm of any pseudo‐orthogonal matrix. It is a generalization of the well‐known criterion used for O(3,1) according to which the spinor norm is determined by the sign of the last element of a Lorentz matrix.

Boundary condition solutions of the generalized Feller equation
View Description Hide DescriptionSolutions of the generalized one‐dimensional autonomous parabolic Feller equation for given boundary conditions are established. Since a basic solution is known, the Green–Riemann technique is used. It leads to two Green–Riemann limit functions relative to subsets of the space of two parameters connected with the equation. Properties of these limit functions are discussed. These functions are then used to establish boundary conditionsolutions in the form of unilateral convolutions involving as boundary conditions those functions which are summable over every nonnegative compact interval. Finally, it is shown that, relative to two subsets of the space of two equation parameters, there exist initial and boundary conditionsolutions.

The structure of the many‐body wavefunction for scattering
View Description Hide DescriptionWe show that the scattered part of the many‐body wavefunction initiated by two incoming clusters is given by a fully connected operator acting on the initial channel state. The structure of this operator suggests a division of the full wavefunction into two‐cluster components. A set of coupled equations in both the differential and integral form is then derived for these components. These equations have structure and properties similar to the three‐body equations of Faddeev. We demonstrate that each component has outgoing waves in a unique two‐cluster partition. The transition amplitude for any final arrangement can therefore be extracted directly from the outgoing waves in the relevant components.

A class of multidimensional nonlinear Langmuir waves
View Description Hide DescriptionNonlinear Langmuir waves in a plasma governed by the dimensionless equations i∂E/∂t=−∇^{2}E+nE, ∂^{2} n/∂t ^{2} =∇^{2}[n+g (‖E‖^{2})] are studied, where E is the complex amplitude of the high‐frequency electric field,n is the low frequency perturbation in the ion density from its constant equilibrium value, and g is a given function of ‖E‖^{2}. General conditions for the existence or nonexistence of a class of multidimensional solitary‐wave and nonlinear periodic travelling‐wave solutions in the form E(t,x) =h(k⋅x−v t) and n (t,x) =s (k⋅x −v t) are established. The results are applied to the special cases: (i) g (‖E‖^{2}) =‖E‖^{2} corresponding to the usual pondermotive force, and (ii) g (‖E‖^{2}) =K[1−exp(−‖E‖^{2})], K is a positive constant, representing ion density saturation.

Optical analysis of resonances in a velocity‐dependent potential
View Description Hide DescriptionResonances in a velocity‐dependent square well potential are analyzed in terms of multiple reflections of the incident wave on the internal walls of the potential well. The results differ considerably from those obtained for static potentials with a similar analysis.

Null infinity is not a good initial‐data surface
View Description Hide DescriptionAn example is given of a space–time which is asymptotically flat and globally well‐behaved, and yet which admits a nonzero Maxwell field, with zero source, having no incoming radiation from past null infinity.

A particle model based on stringlike solitons
View Description Hide DescriptionThis paper deals with two scalar fields ϑ (x,y,z,t) and φ (x,y,z,t) which are governed by two coupled nonlinear differential equations. Some of the spatial field distributions of ϑ and φ are topologically stable and represent solitons in three‐dimensional space. The simplest stable solitons are identified with the electron and the positron. The asymptotic solutions of the fields are studied. ϑ is shown to fall off asymptotically as ±a/r, where a is a constant related to the elementary charge and r the distance from the ’’site’’ of the soliton.

Correlations in an N‐mode laser
View Description Hide DescriptionAnalytic expressions for the correlations of the light emitted by an N‐mode laser are given. We study in particular a special case of strongest couplings and show, for example, that at steady state, the second order cross intensity correlation tends toward a negative value, −1/(N−1), for large pump parameters. We also study the time dependent solutions, and express the second order amplitude and intensity correlations of the emitted light in terms of the eigenvalues and eigenfunctions of a Schrödinger‐type differential equation which reduces to that studied by Risken for N=1 and to that studied by M‐Tehrani and Mandel for N=2. Analytic expressions which approximate the eigenvalues of this differential equation for a general N are given.

On the generation of new solutions of the Einstein–Maxwell field equations from electrovac spacetimes with isometries
View Description Hide DescriptionWe present transformation formulas which facilitate the determination of the metrics, electromagnetic fields, connections and Weyl tensors of those electrovac spacetimes which result when a given solution of the Einstein–Maxwell equations with an isometry is subjected to the transformations of the Kinnersley group. Several applications of our calculational procedures are given as illustrations, and a number of general theorems are presented. In particular, we infer that when we apply such techniques to the only known solution of Petrov type N with twisting principal null rays, the new solutions which result will be algebraically general.

Approximate symmetry groups of inhomogeneous metrics: Examples
View Description Hide DescriptionBy definition, an N‐dimensional positive‐definite inhomogeneous metric is not invariant under any N‐parameter, simply‐transitive continuous group of motions. Nonetheless, it is possible to construct a group (simply‐transitive and of N parameters) that comes closest to leaving the given metric invariant. We call this group the approximate symmetry group of the metric. In an earlier paper, we described a technique for constructing the approximate symmetry group of a given metric. Here, we briefly review that technique and then present some examples of its application. All two‐dimensional metrics are analyzed, and simple criteria are given for determining their approximate symmetry groups. Three three‐dimensional metrics are investigated: the invariant hypersurfaces of the Kantowski–Sachs space–times and two families of hypersurfaces in the Gowdy T ^{3} space–times. The approximate symmetry group of the former is found to be of Bianchi Type I and those of the latter may be I or VI_{0}. Defining, via our technique, a measure I of the magnitude of a metric’s inhomogeneity, we study the time dependence of I for the hypersurfaces in the Gowdy metric. We find it is possible in some cases for these hypersurfaces to approach homogeneity (I→0) both in the asymptotic future and near the initial singularity. Finally, we constant a homogeneous background metric for these hypersurfaces.

Nonlinear realizations of the direct product of two Lorentz groups on a skew‐symmetric tensor space
View Description Hide DescriptionIn this paper, we present a study of exact, infinitesimal, nonlinear realizations of the direct product (L _{1}×L _{2}) of two Lorentz groups on an antisymmetric tensor space η. Included in this study is a detailed discussion of the second order (L _{1}×L _{2})/L ^{+} coset realizations where L ^{+} is the symmetric Lorentz subgroup. Invariant metric forms defined on η are exhibited and compared with invariant metric forms obtained in some previous studies of coset realizations.

The distribution function for stretched dipoles in an applied electric field
View Description Hide DescriptionAn expression is derived for the distribution function of an assemblage of ion pairs in the presence of a uniform applied electric field, for the most interesting and important case, namely when the interaction force between the components of such stretched dipoles is unshielded Coulombic. Our derivation makes use of results obtained previously when determining, by means of a perturbation technique, the relative increase in the dissociation constant of a weak electrolyte due to an applied electric field. The expression we obtain for the ion‐pair distribution function agrees with that given previously by Onsager [J. Chem. Phys. 2, 599 (1934)], whose derivation has never been published in full.

Linear response theory revisited. I. The many‐body van Hove limit
View Description Hide DescriptionA critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time‐dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,B ^{ R } _{α}(t) =[exp(−Λ_{ d } t)]B _{α}, where B _{α} is a Schrödinger operator and Λ_{ d } is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.

A no‐go theorem for polarization structure
View Description Hide DescriptionIt would be extremely advantageous, from the point of view of the planning of experiments and of the testing of theoretical models by such experiments, if a representation of the reaction matrix could be found such that the differential cross section is given by the absolute value squared of a single amplitude, and the simplest type of polarizationexperiments are expressed in terms of functions involving only two amplitudes each. It is shown that, within the framework of a set of plausible and very weak constraints, such a representation does not exist.

On the charged Kerr–Tomimatsu–Sato family of solutions
View Description Hide DescriptionThe charged Kerr–Tomimatsu–Sato family of solutions with arbitrary integer distortion parameter δ for gravitational fields of spinning masses is presented. The Bonnor–Misra–Pandey–Srivastava–Tripathi–Wang family of solutions is also referred to.

Simple analytic expressions for the Coulomb off‐shell Jost functions
View Description Hide DescriptionThe off‐shell Jost functions have been introduced by Fuda and Whiting. We give simple closed expressions for f _{ C,l }(k,q), the off‐shell Jost functions for the Coulomb potential, and we derive their connection with the ordinary Coulomb Jost functions f _{ C,l }(k).

Topological cohesion
View Description Hide DescriptionIt is shown that suitably regular, finite‐energy solutions of the Yang–Mills–Higgs equations are nondissipative whenever their initial data are topologically significant.