Volume 13, Issue 12, December 1972
Index of content:

Coordinate systems associated with asymptotically shear‐free null congruences
View Description Hide DescriptionWe present here the asymptotic coordinate transformation between a coordinate system associated with null hypersurfaces and one associated with an asymptotically shear‐free (but twisting) null congruence. The general asymptotically flat metric is expressed in this new coordinate system. Special cases of this are the algebraically special metrics in their ``natural'' coordinate system.

Relativistic dynamical symmetries and dilatations
View Description Hide DescriptionCharacterizing the state of a relativistic particle by a pair (x _{μ},ξ_{μ}) of 4‐vectors, we are led, in a natural way, to a group of canonical transformations which includes the Poincaré group and dilatations. The structure of the group and its induced irreducible unitary representations are explored. It is shown that has a semisimple noncompact subgroup which permits a systematic treatment of exact and of broken dilatation symmetry. The relevance of these ideas to scale dimension and to a new symmetry, scale conjugation, is discussed. As an application, a mass formula is derived from broken dilatation symmetry.

Converging bounds for the free energy in certain statistical mechanical problems
View Description Hide DescriptionWe present sequences of converging upper and lower bounds to the partition function per spin specifically for a ferromagneticIsing model which are valid in the entire finite, magnetic‐field, inverse‐temperature plane. They are based on the exact high‐temperature expansions for a finite system, properties of generalized Padé approximants, and the Villani limit theorem. The results depend only on the general structure of the partition function and certain monotonicity with system size properties which hold fairly generally.

A spectral representation for Coulomb wavefunctions
View Description Hide DescriptionWe obtain a simple spectral representation for the momentum space wavefunctions of the Schrödinger equation with a Coulomb potential in the form of a contour integral. Both the bound state and the scattering solutions are evaluated as residues at the poles enclosed by this contour.

Erroneous bound state conditions from an algebraic misrepresentation of spin wave theory
View Description Hide DescriptionA recently derived condition [C. J. Liu and Yutze Chow, J. Math. Phys. 12, 2144 (1971)] for the existence of bound states in the Heisenberg ferromagnet is shown to be the erroneous result of an improper construction of the Hilbert space for an algebraic representation of the spin operators.

The nonstandard : model. I. The technique of nonstandard analysis in theoretical physics
View Description Hide DescriptionThe methods of nonstandard analysis are demonstrated as a preliminary step for the construction of the nonstandard : model. Elementary quantum mechanical problems are solved and the renormalization of the scalar field (Yukawa interaction) is investigated.

The nonstandard : model. II. The standard model from a nonstandard point of view
View Description Hide DescriptionAs a second step in the construction of the nonstandard : model we analyze Glimm and Jaffe's work from the nonstandard point of view.

Applications of the intertwining operators for representations of the restricted Lorentz group
View Description Hide DescriptionSome properties of the intertwining operations, which carry over certain infinite‐dimensional, reducible (but not completely reducible) representations of the restricted Lorentz group into finite‐dimensional (irreducible) representations are studied.

Equations of motion for the sources of asymptotically flat spaces
View Description Hide DescriptionWe present an exact calculation that leads to the equations of motion (which naturally contain gravitational radiation reaction terms) of a system subject to no external forces. The novelty of our approach lies in the fact that the system is to be considered as the source of an asymptotically flat space and that all the revelant physical quantities such as the velocity ν^{μ}, 4‐momentum p ^{μ}, angular momentum‐center of mass tensorS ^{μν} (as well as higher moments) are then defined in terms of surface integrals taken at infinity. A subset of the Einstein equations (equivalent to Bondi's supplemantary conditions) then yields the time‐evolution equations for these variables.

Irreducible tensor operators for finite groups
View Description Hide DescriptionNormalized tensor operators for a finite group are defined by means of coefficients U which formalize the descent in symmetry from to . The properties of these coefficients are demonstrated and tables given. Some examples of application show their use and utility.

Solutions of a nonlinear integral equation for high energy scattering. III. Analyticity of solutions in a parameter, explored numerically
View Description Hide DescriptionSolutions of the Ball‐Zachariasen equation, discussed in two previous papers, depend analytically on a parameter c which measures the strength of particle production. Numerical experiments, designed to elucidate the structure of the Riemann surface, are reported. The results are consistent with a very pretty hypothesis which describes the Riemann surface completely.

Scattering of a scalar wave from a random rough surface: A diagrammatic approach
View Description Hide DescriptionThe solution to the problem of a scalar wave scattered from a rough surface is given under the conditions that the normal derivative of the field vanish at the surface and that the surface height be a single‐valued function with Gaussian statistics. The solution is in terms of a series with a diagram representation. Partial summation of the series in terms of linear integral equations is briefly discussed.

A kinetic theory for power transfer in stochastic systems
View Description Hide DescriptionIn the asymptotic limit of weak inhomogeneities and long times or distances, we obtain a system of kinetic equations governing the power transfer among the modes of oscillation of certain stochastic dynamical systems. We include applications to coupled oscillators, waveguides, beams, the quantized motion of a particle in a random potential, and the Klein‐Gordon equation with random plasma frequency.

Noumenon: Elementary entity of a new mechanics
View Description Hide DescriptionIf the postulate of symmetry on which special relativity is built is rejected, a generalization of the relativistic notions of event and space‐time can be proposed. The generalization leads to the notion of noumenon. The noumenon possesses a handedness; it is a seven‐parameter entity obtained by associating with an event the angular momentum corresponding to a particular evolution of that event. The noumenon is defined in the complex extension C_{4} of space‐time R_{4}. The main purpose of the paper is to prove the acceptability of the new concepts when confronted with experimental results which are generally regarded as supporting the classical theory of relativity. The potential fruitfulness of the new concepts is shown by a short review of similar ideas developed independently by several authors in different fields of physics: A Maxwelliantheory of gravitation is developed; interactions between gravitation and electromagnetism appear, which have common characteristics with weak interactions; and it is suggested that the extra degrees of freedom of the noumenon are related to the quantum numbers of elementary particles.

Diffraction by an infinite array of parallel strips
View Description Hide DescriptionThe problem of diffraction of a plane wave by an infinite array of parallel strips is attacked by the newly developed modified residue calculus method. The solution is found in terms of an infinite set of zeros of an analytic function. The asymptotic behavior of the set of zeros is specified by the edge condition, while the first several zeros are determined from a matrix equation. The rapid convergence of these zeros to their asymptotic values is demonstrated through numerical examples. For a given array of strips, it is shown that there exists a total reflection phenomenon at a critical frequency and incident angle. This fact suggests the possibility of constructing an open resonator with an extremely sparse resonance frequency.

The inverse scattering problem at fixed energy for L ^{2}‐dependent potentials
View Description Hide DescriptionThe problem of finding central and L ^{2}‐dependent potentials, acting among spinless particles, from the knowledge of the S matrix as a function of angular momentum at a fixed energy is studied. The Newton method for central potentials is generalized to this case, and it is shown that phase shift information at fixed energy is not enough to give us both the central and the L ^{2}‐dependent potential.

Elastic general relativistic systems
View Description Hide DescriptionA theory of elastic deformations of general relativistic systems is presented. The theory is derived from a generalized Hooke's law. An important feature of this theory is that its classical limit corresponds to the classical elasticity theory of prestressed materials. A perturbation description of small deformations is developed and applied to the test body case. For the first time, the strain‐curvature equation for an elastic test body interacting with a gravitational wave has been derived from a complete theory. The semi‐classical work of Dyson, showing the interaction of a gravitational wave with the inhomogeneities of the shear modulus, is rederived and placed within the framework of general relativity. The theory presented is quite comprehensive in scope and applicable to fully relativistic situations such as the elastic behavior of neutron stars.

On the possibility of observing first‐order corrections to geometrical optics in a curved space‐time
View Description Hide DescriptionGravity's effect on the polarization of test electromagnetic fields is presented. It is shown that under ordinary circumstances the effect is not measurable.

A new representation of the solution of the Ising model
View Description Hide DescriptionIt is shown that the transfer matrices for various Ising lattices in two dimensions commute with certain linear operators. The problem of finding an explicit form for the largest eigenvector is considerably simplified. The expansion coefficients appearing in the eigenvectors found as the solution of a set of nonlinear difference equations are Pfaffians. The connection between this type of solution and other solutions is clarified. This form for the eigenvector also simplifies the calculation of correlation functions. Some geometrical aspects of the Ising model are discussed.

On the inverse problem for a hyperbolic dispersive partial differential equation
View Description Hide DescriptionThe inverse problem for a two‐dimensional (space‐time) hyperbolic partial differential equation, with coefficients, functions of the spatial variable only, is considered. Exterior to a region of compact support in the spatial variable, the equation reduces to the wave equation, and, from knowledge of the solution in the exterior region (namely in terms of reflected and transmitted waves for a prescribed incident wave), the problem is to deduce the coefficients in the interior region. This is achieved by treating the problem as a Cauchy initial value problem and using the Riemann function to deduce a dual set of integral equations. The coefficients or linear combinations of them are deduced from the solutions of the integral equations. The question of uniqueness is partially answered, by estimating the domain of convergence of the Neumann series. The application of the analysis to electromagnetic scattering from a slab of varying conductivity and permitivity is indicated.