Volume 17, Issue 10, October 1976
Index of content:

Formal solutions of inverse scattering problems. II
View Description Hide DescriptionThe formal solutions of inverse scattering problems presented in Paper I [J. Math. Phys. 10, 1819 (1969)] are shown here to converge in certain cases of potential scattering for sufficiently weak potentials, and in certain cases of refractive scattering for sufficiently weak variations in the index of refraction. The solutions for the cases of boundary scattering, on the other hand, are not likely to converge, because there is no way to make the effect of the boundary sufficiently weak.

On induced representations for finite groups
View Description Hide DescriptionTwo new notions are introduced as tools for the study of representations of finite groups. First, in the spirit of duality, a basis set of class orientated characters is shown to possess nice properties with respect to induction and subduction, which lead to simple proofs of some well‐known theorems. Secondly, as a useful device in constructive representation theory, a study is made of the subrepresentations which often naturally occur when a representation is induced from a subgroup to a supergroup.

Markovian subdynamics in quantum dynamical systems
View Description Hide DescriptionSubdynamical systems induced from a given quantum dynamical system are studied in the framework of operator algebras. Sufficient conditions are shown for induced subdynamics to be Markovian. It is also proved that the ergodicity of states in Markovian subsystems is preserved.

Perfect fluids and symmetry mappings leading to conservation laws
View Description Hide DescriptionSome of the results recently obtained by Glass relating to shear‐free perfect fluids are extended and further interpreted. In particular, it is pointed out that certain symmetry methods are fundamental to this type of investigation. This approach of using symmetry mappings (e.g., characterizing vanishing shear) at the level of the matter tensor is seen to naturally lead to considerations of an important family of symmetry properties which include Ricci collineations as a special case. These considerations are used to obtain new conservation expressions holding for perfect fluids.

Principal null directions without spinors
View Description Hide DescriptionThe method used by Petrov to obtain the first classification of vacuum space–times has since been overshadowed by methods which yield a finer classification based on principal null directions. It is shown that this finer classification can be expressed in Petrov’s terms and a simple algorithm is obtained for finding all repeated principal null directions by matrix methods.

Extended inertial frames and Lorentz transformations. II
View Description Hide DescriptionIt is shown that there is a unique ten‐parameter group of projective (fractional‐linear) transformations of space–time preserving free‐particle motion and containing the inhomogeneous Lorentz group as a limiting case when a characteristic fundamental length of the projective group (whose structure comes from the group of rotations in the five‐dimensional space of homogeneous space–time coordinates) is made infinite. The basic differential geometry, free‐particle dynamics, and Lie groupalgebra going with the projectivity extended space–time structure are developed, and it is remarked that the Hamiltonian for free particles most generally will have eight branches, owing to the fundamental space–time metric being that for a Finsler space. The means to projectively extending electrodynamics are also briefly noted, and the need for characterizing electrodynamics by two intrinsic quantities (charge, and a purely electromagnetic length) are pointed out.

The high conductivity limit in mean field electrodynamics
View Description Hide DescriptionAn attempt is made to resolve conflicting views on the reliability of first order smoothing theory when applied to electromagnetic induction of magnetic field by a turbulently moving conductor, in the astrophysically most interesting case of large microscale magnetic Reynolds numbers.

On the analyticity of certain stationary nonvacuum Einstein space–times
View Description Hide DescriptionIt is shown that certain nonvacuum solutions of Einstein’s general relativistic field equations are analytic space–times, i.e., an analytic atlas exists with respect to which the components of the metric tensor, and all material fields are analytic functions. The two specific cases discussed here are interesting from an astrophysical point of view. The first is the class of space–times containing a source free electromagnetic field: the exterior of a charged black hole, for example. The second is the class of space–times filled with a rigidly moving perfect fluid, often used to describe the interior of a rotating star.

Steady, one‐dimensional multigroup neutron transport with anisotropic scattering
View Description Hide DescriptionThe solution of steady, one‐dimensional half‐space multigroup transport problems with degenerate anisotropicscattering is obtained for L _{1}sources and incident distributions. The solution is expressed in terms of contour integrals of the resolvent operator (λI−K)^{−1}, where K is the ’’separated’’ transport operator. The connection between this method and the ’’Case eigenfunction’’ method is briefly discussed, and the half‐space albedo problem is treated in detail. This problem reduces to obtaining the Wiener–Hopf factorization of the dispersion matrix, hence to solving two coupled nonlinear, nonsingular matrix integral equations.

A simple proof of the angular momentum Helmholtz theorem and the relation of the theorem to the decomposition of solenoidal vectors into poloidal and toroidal components
View Description Hide DescriptionVector spherical harmonics are used in a simple proof of the angular‐momentum Helmholtz theorem. The decomposition of vectors defined on a sphere into two components which this theorem gives is carried out explicitly. Furthermore, the potentials which occur in the theorem are given explicitly in terms of the original vector. The decomposition of solenoidal vectors into poloidal and toroidal components is also carried out explicitly. It is shown how these components are related to the components given by the angular‐momentum Helmholtz theorem.

Complete extension of the symmetry axis of the Tomimatsu–Sato solution of the Einstein equations
View Description Hide DescriptionThe symmetry axis of the simplest Tomimatsu–Sato field is considered. Since this manifold is not geodesically complete for every value of the parameters occurring in the metric, a complete extension is given, and it is shown that its causal structure is very similar to that of the symmetry axis of the Kerr field.

Gauge theories for space–time symmetries. I
View Description Hide DescriptionThe general formulation of gauge invariant field theories based upon space–time symmetries is developed and given its geometrical interpretation. The consequences of gauge invariance, in the form of identities and conservation laws, are derived and the field equations are obtained from a class of gauge invariant Lagrangians. This is the first paper of a series, the subsequent work treating specific cases, in particular, conformal invariance.

Diffraction by a half‐plane perpendicular to the distinguished axis of a gyrotropic medium
View Description Hide DescriptionThe Wiener–Hopf–Hilbert method is used to obtain an exact solution to the problem of diffraction by a perfectly conducting half‐plane in a gyrotropic medium, when the distinguished axis of the medium is perpendicular to the half‐plane. The incident field is a plane wave whose direction of propagation is perpendicular to the edge of the half‐plane. The problem has not previously been solved exactly. The answer is given in terms of Fourier transforms of the field components; these turn out to be simple algebraic functions. But the field quantities themselves are not, in general, expressible in terms of known functions. A few special cases are investigated and possible generalizations of the problem are mentioned.

Bogoliubov inequality for unbounded operators and the Bose gas
View Description Hide DescriptionWe provide the mathematical arguments which are needed to obtain a rigorous proof of the absence of condensation in a one‐ and two‐dimensional Bose gas of particles having superstable interactions.

Feynman path integrals and quantum mechanics as h/→0
View Description Hide DescriptionWhen the space of paths is a certain Hilbert spaceH, we show how to extend the Feynman path integral F of DeWitt and Albeverio and Hoegh‐Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of F and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.

Quantization problem and variational principle in the phase‐space formulation of quantum mechanics
View Description Hide DescriptionThe problem of quantization in the phase‐space formulation of quantum mechanics is considered. An integral equation for the phase‐space eigenfunctions is derived which is equivalent to the standard eigenvalueequation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ‐representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.

Prolongation structures and a generalized inverse scattering problem
View Description Hide DescriptionThe prolongation structure method of Wahlquist and Estabrook is used to determine a generalized inverse scattering problem for the equation u _{ t t }=u _{ x x }+6(u u _{ x })_{ x }+u _{ x x x x } which describes the motion of shallow‐water waves under gravity. The relevant Gel’fand–Levitan equation is solved for the single soliton solutions.

Prolongation structures and nonlinear evolution equations in two spatial dimensions
View Description Hide DescriptionThe prolongation structure approach of Wahlquist and Estabrook is used to determine nonlinear evolution equations in two spatial dimensions for which an inverse scattering formulation exists. The equations of nonlinear wave–envelope interactions and the Kadomtsev–Petviashvilli–Dryuma equation are considered in detail.

Path integrals and ordering rules
View Description Hide DescriptionIt is shown again that path‐integral quantization has no preference for any particular ordering rule for the Hamiltonian.

Octonionic Hilbert spaces, the Poincaré group and SU(3)
View Description Hide DescriptionA formalism based on real octonions is developed in order to construct an octonionic Hilbert space for the description of colored quark states. The various possible forms of scalar products and related scalars are discussed. The choice of a direction in the space of octonion units leads naturally to a representation of the Poincaré group in terms of complex scalar products and complex scalars. The remaining octonion directions span the color degrees of freedom for quarks and anti‐quarks. In such a Hilbert space, product states associated with color singlets are shown to form a physical quantum mechanical Hilbert space for the description of hadrons. Color triplets, on the other hand, correspond to unobservable parafermion states of order three.