Showing papers on "Scalar field published in 1979"

Theory of electromagnetic beams

Abstract: A relatively simple method for calculating the properties of a paraxial beam of electromagnetic radiation propagating in vacuum is presented. The central idea of the paper is that the vector potential field is assumed to be plane-polarized. The nonvanishing component of the vector potential obeys a scalar wave equation. A formal solution employing an expansion in powers of $\frac{{w}_{0}}{l}$ is obtained, where ${w}_{0}$ is the beam waist and $l$ the diffraction length. This gives the same result for the lowest-order components of the transverse and longitudinal electric field of a Gaussian beam that was derived by Lax, Louisell, and McKnight using a more complicated approach. We derive explicit expressions for the second-order transverse electric field and the third-order longitudinal field corrections.

Closure of the Transport Equation for the Probability Density Function of Turbulent Scalar Fields

Abstract: The transport equation for the probability density function (pdf)P of a scalar variable in a turbulent field is derived and various closure approximations for the turbulent convective and the molecular transport term are discussed. For the special case of a turbulent diffusion flame, for which the density, temperature and composition can be considered as a function of a scalar variable f, the transport equation for Pf (z) is closed using the conditional mean of the velocity for the turbulent convective term and an integral model for the molecular transport term. Preliminary results are presented and compared with a two-parameter form of the pdf Pf (z). Introduction Probability density functions (pdf) play an increasingly significant role in the theoretical treatment of turbulent flows. Lundgren [ 1 ] devised a method to derive a hierarchy of transport equations for N-point probability density fuctions (pdfs) for fluctuating variables in a turbulent incompressible flow. Fox [12] and Dopazo and O'Brien [5], [7] extended this method to compressible and reacting flows. Dopazo and O'Brien suggested closure assumptions for the onepoint pdf equation based on a quasi-Gaussian property of conditional moments, which was successful for nearly Gaussian problems [6]. The Lundgren approach has been applied to turbulent flames by Pope [4]. This author constructed the transport equation for the pdf of a scalar quantity f describing the instantaneous composition of a turbulent diffusion flame under certain simplifying assumptions. Beside, he suggested an interesting closure-procedure for the unknown term arising from molecular transport that will be discussed below. Paper presented at the Colloquium on "Probability Distribution Function-Methods for Turbulent Flows" in Aachen, Germany, August, 29-30, 1977. 0340-0204/79/0004-0047S02.00 © Copyright by Walter de Gruyter & Co. · Berlin · New York .48 J. Janicka, W. Kolbe, W. Kollmann In this paper the transport equation for the pdf of a scalar quantity f in a turbulent field is considered. Closure assumptions for two different unknown terms occurring in this equation are discussed in the special case of a turbulent diffusion flame, the composition, density and temperature of which can be related to the scalar quantity f, [9]. For each term a new closure approximation is suggested. The physical significance of all approximations involved is investigated. Results of test calculations are presented for both the homogeneous case and the turbulent diffusion flame. 1 . Transport equation for the probability density of a scalar For turbulent flows, a hierarchy of transport equations for a probability density function (pdf) f(x, t) at a finite number of points in x-space can be derived using a method devised by Lundgren [1 ]. The structure of this hierarchy is similar to the well-known BBGKY-hierarchy in statistical mechanics. The closure approximations for the BBGKY-hierarchy, such as the Kirkwood closure for simple liquids [2], do not work satisfactorily for the turbulent case because there essential conditions are not fulfilled (for details cp. [3]). Therefore, a different approach towards constructing closure approximations is suggested. It is based mainly on physical assumptions about the turbulent transport mechanism. Consider a normalized scalar function f(x, t) of space coordinates ÷ = (÷á, α = 1 , 2, 3) and time t, that satifies the following equation Here ñ is the density which is a function of f(x, t), à is the molecular transport coefficient and S0 is the source term. The velocity í = (íá , α. = 1,2,3) satisfies the Navier-Stokes equations 9va dva 8p 8f , (5) which is the simultaneous one-point pdf of the conserved scalar quantity f and the velocity va at (x, t). To derive a transport equation for P* we only need elementary properties of the averaging process (and avoid delicate mathematical questions connected with the probability measure of a turbulent flow field). Differentiation of eq. (5) yields (6) · at at az at aza Inserting (1) and (2) into (6) leads to

On the use of the total scalar potential on the numerical solution of fields problems in electromagnetics

Abstract: SUMMARY The paper summarizes the formulation of a set of computer algorithms for the solution of the three-dimensional non-linear Poisson field problem. Results are presented that were obtained by applying algorithms to the analysis of two-dimensional magnetostatic fields. Scalar and vector potentials were used, and it is shown that the convenient single valued scalar potential associated with the induced sources gives severe accuracy problems in permeable regions. The results become as good as those obtained using vector potential if the scalar potential associated with the total field is used for permeable regions. The combination of two scalar potentials has a significant advantage for three-dimensional problems. The non-linear Poisson equation occurs in many areas of physics and engineering. The equation is relatively easy to solve compared to other defining equations, but for many applications the solutions must be very accurate. In magnetostatic problems, for example, the geometry of boundaries and surfaces separating differing media is often complicated and field accuracies of the order of 0.1 per cent and higher are essential. These conditions are frequently encountered in the wide range of electromagnets associated with the design of charged particle accelerators, spectrometers, detectors, focusing devices and plasma containment experiments used in physics and also in the broad spectrum of machines, transformers, etc. used in electrical engineering. Although the methods discussed in this paper are of general applicability they are looked at with particular reference to electromagnetics. Both differential and integral operator formulations have been used to solve the magnetostatic problem. Many well established programs solve the two-dimensional cases using differential formulations based directly on the defining equation usually in terms of the single component vector potentiaI. 1 - 3 These programs are capable of giving high accuracy although the position of the far field boundary can have a significant effect on the results. This latter difficulty has been overcome at the expense of increased computational cost by solving the integral form of the equations in terms of field components directly. An added advantage of this approach is that only regions containing material media (e.g., iron) are discretised. This is very useful in extending to three dimensions where the need to have a mesh of elements connecting many different regions of complex shape is seen as a limitation of the differential approach. This difficulty is avoided in programs based on integral formulations 4-6 and at the present time these provide a general technique for solving non-linear three-dimensional magnetostatic problems providing high accuracy is not required.

Construction of Green's functions from an exact S matrix

Abstract: The bootstrap program for determining Green's functions from an exact $S$ matrix is carried out for the simplest soliton field theory of a scalar field with $S$-matrix operator $S={(\ensuremath{-}1)}^{\frac{N(N\ensuremath{-}1)}{2}}$, where $N$ is the total number operator. Despite the formal simplicity of the $S$ matrix, the Green's functions derived have a rich structure. The results can be checked since this field theory is none other than that of the order variable of the Ising model in the scaling limit above the critical temperature.

Constructive procedure for perturbations of spacetimes

Abstract: The method of Debye scalar superpotentials has previously been extended by the authors to curved spaces to yield a constructive procedure for neutrino, electromagnetic, and gravitational perturbations of algebraically special spacetimes. The solution of a decoupled scalar wave equation is differentiated to give the solution of the corresponding spinor or tensor perturbation field equations. In this paper covariant formulations and proofs are given. The results are derived in a general spinor formalism framework which extends earlier exterior differential form and tensor treatments of the electromagnetic case.

Quantum topology of molecular charge distributions. II. Molecular structure and its change

Abstract: This paper illustrates how the concepts of atoms and bonds may be given definite expression in terms of the topological properties of the charge density, ρ (r), and how, as a consequence of these identifications, one is led to a definition of structure and to a phenomenological analysis of structural stability. This approach finds its natural expression in Rene Thom’s general analysis of structural stability as it applies to a system whose behavior is describable in terms of the gradient of some scalar field. Chemical observations are made in real space, and thus chemical behavior is determined by the morphology of a system’s charge distribution and its evolution with time. The analysis of the topological properties of ρ (r) via the associated gradient vector field ∇ρ (r), reduces to the identification of the critical points in ρ (r). Two types of critical points assume special roles in the analysis. A (3,−3) critical point, a maximum in ρ (r), is an attractor and is identified with the position of a nucleus in the molecular system under study. The basin of the attractor defines the atom associated with the corresponding nucleus. A (3,−1) critical point defines the interatomic surface separating two neighboring atoms, and the bond path linking their nuclei, the line along which the charge density is maximum with respect to lateral displacements. Hence, neighboring atoms are defined to be bonded to one another and the network of bond paths, for a given nuclear configuration, determines its molecular graph. Structure is defined as that set of molecular graphs which contain the same number of bond paths, linking the same nuclei. Thus a change in structure necessitates a change in the number and/or arrangement of bond paths. The making and/or breaking of chemical bonds associated with such a change is topologically a discontinuous process, and the associated change in structure is therefore, abrupt: a continuous change in the nuclear coordinates, the parameters which control the behavior of the system, can lead to a discontinuous change in molecule’s behavior. A point in control space defining the nuclear configuration for which such discontinuous behavior is observed, is called a catastrophe point. The set of catastrophe points thus partitions nuclear configuration space into regions of different structure. The breaking or making of bonds is a catastrophe of the bifurcation type, resulting from the formation of a singularity in ρ (r), whereas the switching of a bond from one nucleus to another is a catastrophe of the conflict type. It is shown that the analytical description of the formation of a three‐membered ring structure from all possible neighboring structures (as illustrated for H+3 and H2O) is provided by the unfolding of a particular type of catastrophe, the elliptic umbilic.

''Relaxation'' of initial probability density functions in the turbulent convection of scalar fields

Abstract: The evolution of an initially binary (zero unity) scalar field undergoing turbulent and molecular mixing is studied in terms of conservation equations for the probability density function of the scalar property. Attention is focused on the relaxation of the dynamic system to a state independent of the intial conditions. A few existing methods are discussed and evaluated and a new mechanistic model is proposed. Classical iteration techniques are used to obtain an equation for the single point probability density and the unperturbed Green’s function. It is suggested that use of the true Green’s function or perturbed propagator of the system might be necessary in order to obtain the correct evolution of the probability density function.

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=μ

Abstract: This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=μ, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.

Gauge invariant perturbations on most general spherically symmetric space-times

Abstract: The Einstein field equations and the conservation equations linearized around a most general (astrophysically relevant) spherically symmetric space-time are reduced to a set of equations for gauge-invariant geometrical objects defined on the two-dimensional timelike submanifold spanned by the radial and time coordinates. Odd-parity metric and matter perturbations are each expressed in terms of a vector field, matter perturbations in terms of an additional scalar field on this submanifold. Even-parity perturbations are expressed in terms of a symmetric tensor field and a scalar field for the metric and in terms of two scalars, a vector, and a symmetric tensor field for matter. The odd-parity vectorial perturbations are derivable from a single master scalar equation, and their junction conditions across the surface of a collapsing star are given.

Solitons of Coupled Scalar Field Theories in Two Dimensions

Abstract: This work offers a method for finding some exact soliton solutions to coupled relativistic scalar field theories in 1+1 dimensions. The method can yield static solutions as well as quasistatic "charged" solutions for a variety of Lagrangians. Explicit solutions are derived as examples. A particularly interesting class of solutions is nontopological without being either charged or time dependent.

On the Euclidean approach to quantum field theory in curved spacetime

Abstract: Quantum field theory in curved spacetime is examined from the Euclidean approach, where one seeks to define the theory for metrics of positive (rather than Lorentzian) signature. Methods of functional analysis are used to give a proof of the heat kernel expansion for the Laplacian, which extends the well known result for compact manifolds to all non-compact manifolds for which the Laplacian and its powers are essentially self-adjoint on the initial domain of smooth functions of compact support. Using this result, precise prescriptions of the zeta-function, dimensional, and point-splitting type are given for renormalizing the action of a Klein-Gordon scalar field. These prescriptions are shown to be equivalent up to local curvature terms. It is also shown that for static spacetimes, the Euclidean prescription for defining the Feynman propagator agrees with the definition of Feyman propagator obtained by working directly on the spacetime.

{Hertz-Bromwich-Debye-Whittaker-Penrose} Potentials in General Relativity

Abstract: The problem of deriving scalar potentials governing electromagnetic and gravitational perturbations of vacuum space-times is discussed. For the case of an algebraically special vacuum background space-time, explicit formulae for all quantities of physical interest are given in terms of derivaties of the scalar potential.

Quantum effects in the early universe. II. Effective action for scalar fields in homogeneous cosmologies with small anisotropy

Abstract: The one-loop contributions of conformally invariant scalar fields to the effective action are calculated for homogeneous cosmological models with small anisotropy. The dynamical equations which determine the classical geometry are displayed and the matrix elements of the stress-energy tensor between the initial and final vacuums are determined.

Evolution of scalar perturbations near the Cauchy horizon of a charged black hole

Abstract: We describe the evolution of a scalar test field on the interior of a Reissner-Nordstroem black hole. For a wide variety of initial field configurations the energy density in the scalar field is shown to develop singularities in a neighborhood of the geometry's Cauchy horizon, suggesting that for a stellar collapse curvature singularities will develop prior to encountering the Cauchy horizon. The extension to the interior of stationary perturbations due to exterior sources is shown not to disrupt the Cauchy horizon.

Conformal anomalies for interacting scalar fields in curved spacetime

Abstract: The occurrence of anomalies in the trace of the energy-momentum tensor for scalar field theories in curved space-time is discussed. For the special case of spherical space-time, an O(n + 1) -covariant formalism is used to rederive the standard free-field anomaly in four dimensions, and to calculate the anomaly in six dimensions. It is then shown that for an interacting scalar field theory there is a further contribution to the trace anomaly proportional to the renormalization group ..beta.. function. This assertion is then checked by explicit calculations in phi/sup 4/ theory in four dimensions and phi/sup 3/ theory in six dimensions and values for the anomaly found to fourth order in the renormalized coupling constants lambda and g. Finally, these results are generalized to the case of an arbitrary background space-time, where it is shown that the introduction of a position-dependent coupling constant lambda (x) enables the relation between the trace anomaly and the ..beta.. function to be expressed in the form T/sup ..mu..//sub I/..mu.. = -..beta.. (lambda) deltaW/sub I//deltalambda (x) vertical-bar /sub lambda//sub( x/)/sub lambda/ where W/sub I/ is the sum over vacuum bubble diagrams with interactions.

On the Interpolation of a Vector Field

Abstract: By its very nature, interpolation in a vector field is ambiguous, owing to the somewhat arbitrary nature of the vector norm. Since a two-dimensional vector field cm be specified by two scalar quantities. which can be separately interpolated, the ambiguity can be resolved by forcing the interpolated wind field to preserve the vorticity and divergence fields associated with the raw data. A method to calculate divergence and vorticity directly from randomly spaced wind observations is developed and, using analytically generated data, shown to produce more accurate results than conventional computations. Two methods of retrieving the wind field from the analysed scalar fields are presented and also tested on the analytic field. Finally, total analysis, from wind observations to gridded wind fields, is demonstrated on real meteorological data.

The structure of two-dimensional scalar fields with applications to vision

Abstract: Two-dimensional scalar fields (e.g. pictures) are often described by way of a linear superposition of simple base functions. It is argued that such decompositions are often unnatural in the sense that the decomposition takes no regard of the structure of the field and it may happen that the parts are more complicated than the whole. Moreover, such decompositions are not invariant with respect to even small topological deformations of the dimensions or the grey scale of the picture, whereas such deformations do not affect the perceptual structure. We present a method to decompose two-dimensional scalar fields in the following way: the whole is a hierarchically structured superposition of parts, such that these parts are featureless (do not contain local extrema or saddle points). The hierarchical structure can be considered a generative grammar for smooth pictures. The concept is extended towards pictures that are sampled with a collection of graded apertures. We introduce the concept of the aperture spectrum, this construct describes the structure of a picture sampled with any aperture. This kind of description is likely to be important for the analysis of visual functions.

The Vacuum Stress Tensor for Automorphic Fields on Some Flat Space-times

Abstract: The authors calculate the vacuum stress-energy tensor for scalar fields having an explicit U(2) symmetry and for neutrinos in three flat space-times whose constant-t hypersurfaces are a 3-torus, Klein bottle and twisted 3-torus. For special values of the parameters, previously calculated values are regained.

Path integrals for a particle in curved space

Abstract: We consider a particle obeying the Schr\"odinger equation in a general curved $n$-dimensional space, with arbitrary linear coupling to the scalar curvature of the space. We give the Feynman path-integral expressions for the probability amplitude, $〈x,s|{x}^{\ensuremath{'}},0〉$, for the particle to go from ${x}^{\ensuremath{'}}$ to $x$ in time $s$. This generalizes results of DeWitt, Cheng, and Hartle and Hawking. We show in particular, that there is a one-parameter family of covariant representations of the path integral corresponding to a given amplitude. These representations are different in that the covariant expressions for the incremental amplitudes, $〈{x}_{l+1},{s}_{l}+\ensuremath{\epsilon}|{x}_{l},{s}_{l}〉$, appearing in the definition of the path integral, differ even to first order in $\ensuremath{\epsilon}$ (after dropping common factors). Finally, using the proper-time representation, we give the corresponding generally covariant expressions for the propagator of a scalar field with arbitrary linear coupling to the scalar curvature of the spacetime.

The scalar wave equation in a Schwarzschild space-time

Abstract: This paper studies the asymptotic behaviour of solutions of the zero rest mass scalar wave equation in the Schwarzschild space-time in a neighbourhood of spatial infinity which includes parts of future and past null infinity. The behaviour of such fields is essentially different from that which occurs in a flat space-time. In particular fields which have a Bondi-type expansion in powers of $`r^{-1}\text{'}$ near past null infinity do not have such an expansion near future null infinity. Further solutions which have physically reasonable Cauchy data probably fail to have Bondi-type expansions near null infinity.

Scalar field and gravitational instability

Abstract: We study monopole perturbations of spherically symmetric scalar-vacuum and scalar-electrovacuum fields in general relativity and reduce the problem to a one-dimensional Schrodinger-like equation with a certain effective potential. Imposing certain boundary conditions, we select physically meaningful perturbations. Some of them grow exponentially and we conclude that the background system is unstable.

Properties of a general-relativistic kink solution

Abstract: Properties of the general-relativistic kink solution of a nonlinear scalar field recently obtained are discussed. It has been shown that the kink solution is stable against radial perturbations. Possible applications to hadron physics from the geometrodynamic point of view are suggested.

Decay of a Yang-Mills field coupled to a scalar field

Abstract: Consider a gauge fieldF and a scalar field φ with a self-couplingV(φ) as well as the standard coupling betweenF and φ. If 0≦2V(φ)≦φ·V′(φ), there are no classical lumps. IfV(φ)=|φ|4 the system is conformally invariant and all the energy radiates out along the light cone.