Twistor diagram equalities for generalized mass-scattering integrals for Dirac fields

A method which enables one to convert the integral of a holomorphic projective twistor one-form into the integral of a holomorphic projective twistor three-form is used to modify the end-vertex structures of certain twistor diagrams that represent mass-scattering integrals for Dirac fields. Each term of the twistorial diagrammatic expansions recovering the entire fields is then re-expressed appropriately. This gives rise to four sets of explicit twistor-diagram equalities for the mass-scattering formulae.I am most grateful to Professor Roger Penrose for making a key point concerning the introduction of the basic auxiliary-twistor vertices. My warmest thanks go to the World Laboratory for supporting this work financially. I wish to acknowledge the Third World Academy of Sciences for a relevant travel grant. I am grateful, also, to Dr. Asghar Quadir for his invaluable suggestions.     

Einstein-Maxwell Spacetime with Two Commuting Spacelike Killing Vector Fields and Newman-Penrose Formalism

Einstein-Maxwell spacetimes endowed with twocommuting spacelike Killing vector fields areconsidered. Subject to the hypotheses that one of thetwo null geodesic congruence orthogonal to thetwo-surface generated by the two commuting spacelikeKilling vector fields is shearfree and theelectromagnetic field is non null, it is shown that,with a specific choice of null tetrad, theNewman-Penrose equations together with the Maxwell equations for theclass of spacetime considered may be reduced to asecond-order ode of Sturm-Liouville type, from whichexact solutions of the class of spacetimes consideredmay be constructed. Examples of exact solutions arethen given. Exact solutions with distribution-valuedWeyl curvature describing the scattering ofelectromagnetic shock wave with gravitational impulsiveor shock wave of variable polarisation are also constructed.     

Golden Oldie

A form of initial value problem is considered in which the initial hypersurface is not spacelike but null. This approach has the striking advantage over the more usual Cauchy problem that all constraints (initial data equations) are eliminated from the theory, for a wide class of interacting fields in special relativity and also for general relativity. The theory is most naturally described in terms of the two-component spinor calculus, for which an elementary introduction is given here. A general scheme for interacting fields, which holds both in special and general relativity, is presented which describes all fields in terms of sets of irreducible spinors. The concept of an exact set of such spinors is introduced and it is shown that this concept is the appropriate one for an initial value problem on a null cone without constraints. The initial data can be expressed in the form of a complex number, called a null datum, defined at each point of the null cone, one corresponding to each spinor. There is the curious feature of these null data that apparently it is sufficient here, to have onehalf as much information per point as in the corresponding Cauchy problem. The classical Maxwell-Dirac theory and the Einstein-Maxwell theory are two examples that can be put into the form of exact sets. The Einstein empty-space equations are also of particular note, and in this case the null datum describes essentially the intrinsic geometry of the null cone. The argument given here as applied to a general exact set is incomplete in two important respects. Firstly it depends on the null data being analytic, and secondly the initial hypersurface must be a cone. However, both these restrictions are removed in the case of certain elementary fields called basic free fields, examples of which are the Weyl neutrino field, the free Maxwell field, and the linearized gravitational field. For these cases a simple explicit formula is introduced which expresses the field at any point in terms of the null datum, as an integral taken over the intersection of the initial null hypersurface with the null cone of the point.This article originally appeared in 1963 in Aerospace Research Laboratories 63-56 (P.G. Bergmann). It is an important and oft-cited work, but as it has never been published in a widely distributed journal, it is generally inaccessable to the relativity community. This regrettable situation is hereby rectified-Ed.This work was done while the author was at Princeton, Syracuse, and Cornell Universities, visiting under a NATO Fellowship administered by the Department of Scientific and Industrial Research in London. The work at Syracuse was supported by the Aeronautical Research Laboratory and at Cornell by the National Science Foundation.     

The space-time picture of the Wee partons and the AGK cutting rules in perturbative QCD

We discuss the space-time picture of scattering amplitudes with single and double ladder exchange in perturbative QCD. Particular emphasis is given to the Abramovsky Gribov Kanchelli (AGK) rules which describe the relative contributions of the diffractive dissociation and processes with other multiplicities to the elastic scattering amplitude. Inside the Pomeron Pomeron interaction vertices we find a new matrix which describes the transition from one s-cut to another and which goes beyond the original AGK rules.     

Scattering forces and electromagnetic momentum density in crossed circularly polarized standing waves

We analyze the forces on a small dipolar particle and the electromagnetic momentum density in a configuration consisting in two perpendicular circularly polarized stationary waves. The field distribution shows regions in which the electric and magnetic fields are parallel corresponding to a null Poynting vector. Although the average value of the momentum density, proportional to the Poynting vector, is zero in these regions, there are scattering forces acting on small particles due to light's spin force. The total scattering force suggests a new definition of the average value of the momentum density for free propagating electromagnetic fields.     

Scattering forces and electromagnetic momentum density in crossed circularly polarized standing waves

We analyze the forces on a small dipolar particle and the electromagnetic momentum density in a configuration consisting in two perpendicular circularly polarized stationary waves. The field distribution shows regions in which the electric and magnetic fields are parallel corresponding to a null Poynting vector. Although the average value of the momentum density, proportional to the Poynting vector, is zero in these regions, there are scattering forces acting on small particles due to light's spin force. The total scattering force suggests a new definition of the average value of the momentum density for free propagating electromagnetic fields.     

On geometrodynamics and null fields

The possibility of describing null electromagnetic fields by purely metric concepts has recently been subject to some doubt. Following a method devised by Hlavatý, we here investigate the relations that a Riemannian manifold must satisfy in order to correspond to a null electromagnetic field. It is shown that in most cases the fulfilment of five geometrical relations is a necessary and sufficient condition for the existence of a null electromagnetic field. The latter is unique, except for an arbitrary constant phase factor (as in the case of non-null fields). However, in some exceptional cases, there is a larger degree of arbitrariness in the null electromagnetic field that corresponds to a given metric. Such fields (which always possess wave fronts) are not reducible to metric concepts. We then turn to examine how it can occur that null electromagnetic fields require the fulfilment of five relations, rather than three, as non-null ones. In order to settle this question, we make an attempt to consider null fields as a limiting case of non-null ones, by superimposing an arbitrary infinitesimal non-null field on a finite null one. It is then shown that the Rainich vector of such a field does not have a well defined limit, when the perturbing non-null field tends to zero. It is thereby inferred that null electromagnetic fields really have a special status within the frame of geometrodynamics.     

Scattering from non-overlapping potentials. I. General formulation

The problem of scattering from an assembly of non-overlapping spherical potentials is solved in partial-wave basis for each of the constituent potentials. The resulting scattering operator is a quotient of two infinite matrices and depends on “on-shell” partial wave amplitudes of the individual potentials. It suggests in general a truncation scheme which essentially considers only those partial waves effective for each collision at the given energy. The multiple-scattering series is recovered and limiting cases of low energy and high energy are considered. Applications to high-energy scattering of elementary particles on nuclei are briefly discussed.     

Analysis of one-dimensional rough surface scattering based on bidirectional reflectance distribution function model

Through in-depth analysis of the scattering properties of a one-dimensional random rough surface, a method to calculate the spatial bidirectional reflectance distribution function (BRDF) characteristics of a randomly rough surface is proposed. The scattering characteristics of an electrically large rough surface can be substituted by combination of subsection electromagnetic fields. The radar cross-section (RCS), which is transformed from the BRDF, is compared with numerical results from the Method of Moment (MoM). The agreement of the results verifies that this method could calculate the electromagnetic scattering field efficiently and straightforwardly. Moreover, the resulting combined fields from the BRDF of subsections in the rough surface show a good match with the entire scattering field of a whole surface, which indicates that a large electrically rough surface could be feasibly and effectively substituted with the combination of scattering elements using the BRDF method and the statistical method.     

Multiphoton stimulated compton scattering

The processes of multiphoton Compton scattering of relativistic electrons are considered in a special one-dimensional geometry. The multiphoton parameter and the energy distribution of scattered electrons are found. The nonlinear behaviour of the gains of the two external electromagnetic waves is investigated in the saturation region. The multiphoton parameter is shown to be of order of one and the perturbation theory is shown to be inapplicable at very weak fields. For saturation of the gains much stronger fields are required which however are much weaker than those which are necessary for appearance of multiphoton processes in the spontaneous Compton scattering.     

Inversion of penetrable obstacles from far-field data on narrow angular apertures

Reported in this paper are reconstructions of shape and material parameters of two-dimensional, homogeneous, acoustic, penetrable obstacles of arbitrary cross-sections which are immersed in an infinite, homogeneous ambience. Reconstructions are based on the far-field scattering patterns of multiple incident plane acoustic waves. For remotely acquired data, practical necessities require not only that the entire data collection region be less than 2 pi, but also that each received "sees" the object over as narrow an angular aperture as possible. The inversions presented here were obtained under such conditions. Two types of data were used for each incidence namely, "near-monostatic" fields of narrow angular apertures (the narrowest aperture reported is 2 degrees), and a "duostatic" geometry consisting of backscatter plus one other receiver angle. These data sets were acquired for a series of incident angles. The theoretical formalism for inversion is algebraic in nature, requires no integral equation, and possesses a number of advantages for the implementation of a Gauss-Newton type of inversion that was used in this study. Moreover, the algorithm is shown to be inheritently parallelizable.     

The solution of Dirac's equation in algebraically special space-times

A procedure for obtaining solutions to Dirac's equation in algebraically special space-times which admit a shear-free congruence of null geodesies along the repeated principal null direction of the Weyl tensor, is presented. By aligning one of the Dirac spinors to the congruence the problem is reduced to solving one second-order linear partial differential equation for a scalar potential. The solution of the massless field equations for null fields of arbitrary spin s>1/2 aligned to the congruence is also given.     

The isobar-doorway theory for pion-nucleus interactions

Pion-nucleus scattering and reactions are treated in a theory which explicitly introduces the pion-nucleon resonances. Using a separation in Hilbert Space, doorway states of isobar-nuclear systems are introduced and nonresonant processes are clearly separated from resonance interactions. With one choice of doorway states a multiple scattering series is derived which corresponds to the conventional theory with binding energy and other corrections included. When another choice the isobar-doorway model is derived, with parameterization explicitly related to specific dynamic effects, our framework provides a phenomenological model for treating meson-nucleus interactions to all orders. Moreover, the parameters of the model have clear theoretical significance which can extend our knowledge of strong interactions physics. A numerical study is given for elastic scattering.     

Complex paths and uniform approximations in a semi-classical description of direct reactions

The semi-classical (WKB) approximation in three dimensions is considered for scattering processes. Complex solutions of the classical equations of motion are included. They represent contributions to the quantal wave in classically forbidden regions, and in particular describe diffractive effects in the cases where the scattering potential varies appreciably within a wavelength. Special emphasis is given to inelastic and transfer processes, including multistep transitions within a coupled channel frame. Uniform approximation techniques are discussed and widely generalized. Especially, it is shown that when the semi-classical approximations are done consistently at every step of the calculation, there is no need for uniform treatments in the intermediate stages of the calculation, but only at the far end. This simplifies considerably the calculations. Applications to nuclear physics are considered by comparing quantum mechanical and semi-classical cross sections for the scattering of two heavy ions. Excellent agreement is obtained between the two calculations. The new insight brought by the complex semi-classical approximation is used in order to obtain phase rules for the angular distribution, and matching rules. Transition probabilities in their dependence on energy and angle are discussed. They turn out to depend dominantly on ΘE and not on both variables separately.     

Behavior of zero rest-mass fields including gravitation in the vicinity of zeros

For free nonradiation zero rest-mass fields, as for example the nonnull Maxwell and the Weyl tensor fields in vacuum, a uniform treatment of the behavior of the principal null vectors in the vicinity of boundary zeros is given. It is shown that a boundary zero of a field is a singular point of at least one of its principal null vectors. In the special case of a Maxwell field for which the Poynting three-vector vanishes for a certain observer-field, a boundary zero of the Maxwell field is a singular point of the corresponding electric and magnetic fields. The results are valid in flat and curved spacetimes independently of whether the zero rest-mass fields act as source of the spacetime curvature or not.     

Using the T‐Matrix Method for Light Scattering Computations by Non‐axisymmetric Particles: Superellipsoids and Realistically Shaped Particles

Light scattering by non‐axisymmetric particles is commonly needed in particle characterization and other fields. After much work devoted to volume discretization methods to compute scattering by such particles, there is renewed interest in the T‐matrix method. We extended the null‐field method with discrete sources for T‐matrix computation and implemented the superellipsoid shape using an implicit equation. Additionally, a triangular surface patch model of a realistically shaped particle can be used for scattering computations. In this paper some exemplary results of scattering by non‐axisymmetric particles are presented.