Vector Identities Research Articles

A mimetic mass, momentum and energy conserving discretization for the shallow water equations

A spatial semi-discretization is developed for the two-dimensional depth-averaged shallow water equations on a non-equidistant structured and staggered grid. The vector identities required for energy conservation in the continuous case are identified. Discrete analogues are developed, which lead to a finite-volume semi-discretisation which conserves mass, momentum, and energy simultaneously. The key to discrete energy conservation for the shallow water equations is to numerically distinguish storage of momentum from advective transport of momentum. Simulation of a large-amplitude wave in a basin confirms the conservative properties of the new scheme, and demonstrates the enhanced robustness resulting from the compatibility of continuity and momentum equations. The scheme can be used as a building block for constructing fully conservative curvilinear, higher order, variable density, and non-hydrostatic discretizations.

Vector and axial anomaly in the Thirring–Wess model

We study the two dimensional vector meson model introduced by Thirring and Wess, that is to say the Schwinger model with massive photon and massless fermion. We prove, with a renormalization group approach, that the vector and axial Ward identities are broken by the Adler–Bell–Jackiw anomaly; and we rigorously establish three widely believed consequences: (a) the interacting meson-meson correlation equals a free boson propagator, although the mass is additively renormalized by the anomaly; (b) the anomaly is quadratic in the charge, in agreement with the Adler–Bardeen formula; (c) the fermion-fermion correlation has an anomalous long-distance decay.

Magneto-telluric impedance: fundamental models and possibilities of their generalization

A problem of impedance description of magnetic field on the surface of heterogenic Earth is discussed in the article. The concepts of impedance tensor, Vise - Parkinson tensor, generalized impedance tensor, Leontovich's impedance approximation and its generalization are analyzed. It is noticed that these models are of particular character and are real only in cases of special forms of giving primary field and do not always correspond to conditions of conducting real magnet ot elluric experiment. Generalized system of vector identities of impedance type is given for harmonic electromagnetic field on the closed interfaces and generated on their base generalized accurate differential equation of impedances and the system of scalar equations which determine this surface. It is noticed that in case of necessity of using their incomplete, truncated forms these models in contrast to classical ones, have interior criteria of their application and do not demand for their substantiation any additional empirical or heuristic considerations.

Improved bilinears in lattice QCD with nondegenerate quarks

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

A Vector Derivation of the Semigeostrophic Potential Vorticity Equation

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.

Cylindrical vector wave function representation of Green's dyadics for uniaxial bianisotropic media

This paper presents a novel and rigorous eigenfunction expansion of electric‐type dyadic Green's function for an unbounded uniaxial bianisotropic medium in terms of the cylindrical vector wave functions. The Green's dyadic is obtained on the basis of the well‐known Ohm‐Rayleigh method together with some newly developed vector and tensor identities formed by the differential, curl, and dot product of the constitutive dyadics and the cylindrical vector wave functions. The above identities greatly simplify the process of finding the vector expansion coefficients of the dyadic Green's function of the uniaxial bianisotropic media. The dyadic Green's function derived is expressed in terms of the contribution from the irrotational solenoidal types of vector wave functions, with the λ integrals removed using the residue theorem.

Theoretical and computational aspects of the self-induced motion of three-dimensional vortex sheets

Theoretical and computational aspects of the self-induced motion of closed and periodic three-dimensional vortex sheets situated at the interfaces between two inviscid uids with generally different densities in the presence of surface tension are considered. In the mathematical formulation, the vortex sheet is described by a continuous distribution of marker points that move with the velocity of the fluid normal to the vortex sheet while executing an arbitrary tangential motion. Evolution equations for the vectorial jump in the velocity across the vortex sheet, the vectorial strength of the vortex sheet, and the scalar circulation field or strength of the effective dipole field following the marker points are derived. The computation of the self-induced motion of the vortex sheet requires the accurate evaluation of the strongly singular Biot-Savart integral whose existence requires that the normal vector varies in a continuous fashion over the vortex sheet. Two methods of computing the principal value of the Biot-Savart integral are implemented. The first method involves computing the vector potential and the principal value of the harmonic potential over the vortex sheet, and then differentiating them in tangential directions to produce the normal or tangential component of the velocity, in the spirit of generalized vortex methods developed by Baker (1983). The second method involves subtracting off the dominant singularity of the Biot-Savart kernel and then accounting for its contribution by use of vector identities. Evaluating the strongly singular Biot-Savart integral is thus reduced to computing a weakly singular integral involving the mean curvature of the vortex sheet, and this allows the routine discretization of the vortex sheet into curved elements whose normal vector is not necessarily continuous across the edges, and the computation of the self-induced velocity without kernel desingularization. Numerical simulations of the motion of a closed or periodic vortex sheet immersed in a homogeneous fluid confirm the effectiveness of the numerical methods for a limited time of evolution. Numerical instabilities arise after a certain evolution time due to the ill-posedness of vortex sheet dynamics. The motion may be regularized by desingularizing the Biot-Savart kernel using either Krasny's (1986 b ) method or spectrum truncation. Depending, however, on the physical mechanism that drives the motion, the instabilities may persevere.

Direct Integration of Field Equations

Summary form only given. The integral equations of the field equations in electrostatics, magnetostatics, and electrodynamics are derived with the aid of a scalar-vector Green's theorem which can be obtained by several different approaches. One of them is to start with the vector Green's theorem first formulated by Stratton. The steps lead to the scalar-vector Green's theorem is outlined. The integration of the field equations can then be executed readily using that theorem. The results can be interpreted in terms of potential functions if so desired but they are not necessary. In fact, these interpretations are not unique. The second part of this work consists of a review of Stratton and Chu's (1939) famous formula. It is shown that the diffraction integrals obtained by Franz (1948) with the aid of the dyadic Green's functions can be derived readily from Stratton and Chu's formula by means of some simple vector identities. Schelkunoff's (1951) claim that his formulas are stronger than Stratton and Chu's formula is without foundation. In the first place he never demonstrated where his strength is. In fact, it can be shown that his expressions are equivalent to Franz's expressions which are derivable from Stratton and Chu's formula. Finally, it should be pointed out that the line integral deliberately added by Stratton and Chu to compensate for the discontinuity of the fields across the edge of an aperture is really not necessary because this line integral as a result of Meixner's edge condition is a null integral.

Two-dimensional chiral anomaly in differential regularization

The two-dimensional chiral anomaly is calculated using differential regularization. It is shown that the anomaly emerges naturally in the vector and axial Ward identities on the same footing as the four-dimensional case. The vector gauge symmetry can be achieved by an appropriate choice of the mass scales without introducing the seagull term. We have analyzed the reason why such a universal result can be obtained in differential regularization.

Finite quark mass effects in the improved ladder Bethe-Salpeter amplitudes

We study the finite quark mass effects of the low-energy QCD using the improved ladder Schwinger-Dyson and Bethe-Salpeter equations which are derived in the manner consistent with the vector and axial-vector Ward-Takahashi identities. The non-perturbative mass-independent renormalization allows us to calculate the quark condensate for a non-zero quark mass. We explicitly show that the PCAC relation holds. The key ingredients are the Cornwall-Jackiw-Tomboulis effective action, the generalized Noether current and the introduction of the regularization function to the Lagrangian. The reasonable values of the pion mass, the pion decay constant and the quark condensate are obtained with a rather large $\Lambda_{QCD}$. The pion mass square and the pion decay constant are almost proportional to the current quark mass up to the strange quark mass region. It suggests that the chiral perturbation is applicable up to the strange quark mass region. We study the validity of the approximation often used in solving the Bethe-Salpeter equations too.

Non-perturbative renormalization constants using Ward identities

We extend the application of vector and axial Ward identities to calculate b A , b P and b T , coefficients that give the mass dependence of the renormalization constants of the corresponding bilinear operators in the quenched theory. The extension relies on using operators with non-degenerate quark masses. It allows a complete determination of the O( a) improvement coefficients for bilinears in the quenched approximation using Ward Identities alone. Only the scale dependent normalization constants Z 0 P (or Z 0 S ) and Z T are undetermined. We present results of a pilot numerical study using hadronic correlators.

Vector difference calculus for physical lattice models

A vector difference calculus is developed for physical models defined on a general triangulating graph G, which may be a regular or an extremely irregular lattice, using discrete field quantities roughly analogous to differential forms. The role of the space ${\ensuremath{\Lambda}}^{p}$ of $p$-forms at a point is taken on by the linear space generated at a graph vertex by the geometrical $p$-simplices which contain it. The vector operations divergence, gradient, and curl are developed using the boundary \ensuremath{\partial} and coboundary $d.$ Dot, cross, and scalar products are defined in such a way that discrete analogs of the vector integral theorems, including theorems of Gauss-Ostrogradski, Stokes, and Green, as well as most standard vector identities hold exactly, not as approximations to a continuum limit. Physical conservation laws for the models become theorems satisfied by the discrete fields themselves. Three discrete lattice models are constructed as examples, namely a discrete version of the Maxwell equations, the Navier-Stokes equation for incompressible flow, and the Navier linearized model for a homogeneous, isotropic elastic medium. Weight factors needed for obtaining quantitative agreement with continuum calculations are derived for the special case of a regular triangular lattice. Green functions are developed using a generalized Helmholtz decomposition of the fields.

QCD chiral Lagrangian on the lattice, strong coupling expansion, and Ward identities with Wilson fermions

We discuss a general strategy to compute the coefficients of the QCD chiral Lagrangian using lattice QCD with Wilson fermions. This procedure requires the introduction of a lattice chiral Lagrangian as an intermediate step in the calculation. The QCD chiral Lagrangian is then obtained by expanding the lattice effective theory in increasing powers of the lattice spacing and the external momenta. In order to investigate the general structure of the lattice effective Lagrangian, we perform an analytical calculation at the leading order of the strong-coupling and large-$N$ expansion. We find that the explicit chiral symmetry breaking, introduced on the lattice by the Wilson term, is reproduced in the effective theory by a set of additional terms, which do not have direct correspondence in the continuum chiral Lagrangian. We argue that these terms can be conveniently reabsorbed by a suitable renormalization procedure. This is shown explicitly at the leading order of the strong-coupling and large-$N$ expansion. In fact, we find that at this order, as is known to be the case in the opposite weak-coupling limit, the vector and axial Ward identities of the continuum theory are reproduced on the lattice provided that the bare quark mass and the lattice operators are properly renormalized.

Résolution numérique du problème divergence-rotationnel en coordonnées curvilignes généralisées

We present a numerical algorithm for solving the divergence—curl problem for which the vector field is split using a Helmholtz decomposition. Consequently, the solution method proceeds in two steps: first, find a vector field that has the desired curl; then solve a Poisson equation with Neumann-type boundary conditions. The definition of the discrete differential operators allows us to satisfy the usual vector identities of the continuum. A discrete equivalence between the original problem and the proposed method is demonstrated.

Teaching electromagnetic field theory using differential forms

The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory. First, films clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Ampere's and Faraday's laws obtain graphical representations that are as intuitive as the representation of Gauss's law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations. We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.

Ultraviolet and infrared aspects of the axial anomaly I

This is the first part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly, described in terms of ultraviolet and infrared behaviour of the famous VVA triangle graph. Apart from a general overview of the diversified role played by the anomaly in quantum field theory and particle physics, we present here an elementary introduction to the subject of the anomaly, which hopefully should be comprehensible to an uninitiated reader with only a basic background in quantum field theory. In this article we stress the ultraviolet aspects of the anomaly and the topics covered here are the following: Vector and axial-vector Ward identities for the VVA triangle graph, the anomaly and several ways to derive it, namely the symmetric momentum cut-off and shifting the integration variables in linearly divergent integrals, the Adler-Rosenberg argument, the Pauli-Villars method and dimensional regularization.

Mode coupling in corrugated waveguides with varying wall impedance and diameter change

Mode conversion due to variation of wall impedance and diameter change in circumferentially corrugated circular waveguides is investigated in this paper. Starting from Maxwell's equations and using vector identities, general simplification integral formulae for coupling coefficients in circumferentially corrugated circular waveguides with both wall impedance and diameter changes are derived. The explicit formulae of corresponding coupling coefficients for any coupled modes are obtained. As a special case of general results, explicit expressions for the coupling coefficients of a tapered smooth-wall waveguide are deduced. The signs of the coupling coefficients in corrugated and smooth-wall waveguides are worked out.

Symmetries and anomalies in quantum field theory in the bogolyubov-parasyuk renormalization scheme

Regular and dimensionally regularized values of three types of quantities that occur in the Ward vector and axial-vector identities for triangular spinor amplitudes with arbitrary Lorentz characteristics of the vertices and with distinct masses are found. It is established that the dimensionally regularized values of the quantities satisfy the Ward canonical identities, and that their regular values satisfy certain analogs of the Ward canonical identities. Quantum corrections of the Ward canonical identities are found. The general formulas are applied to the AVV- and AAA-amplitudes in a four-dimensional world. It is shown that the chiral limits for them are determined by the discrete symmetry of amplitudes that are present under the conditions of these limits.

A systematic treatment of vector analysis

A systematic method of treating vector analysis is presented. A general definition of an operational expression containing a new symbolic operator, independent of the choice of coordinate system, is the foundation of the subsequent developments which include an algebraic method of deriving vector identities and the formulation of a generalized Gauss theorem. Vector analysis on a surface is formulated in a similar manner by introducing a symbolic operator for a surface. A generalized Gauss theorem for a surface is formulated that enables the deduction of all the important theorems involving the surface gradient, the surface divergence, and the surface curl. The relationship between the present formulation and the classic work of Weatherburn is pointed out.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Molecular dynamics simulations of polymers: Methods for optimal Fortran programming

AbstractThe structure of the general classical trajectory FORTRAN code is discussed for use in molecular dynamics simulations of polymer processes. Substantial reductions in the number of mathematical operations in the code are achieved by using statement functions, modifying the ususal chain rule derivatives, applying appropriate vector identities, and using closure relationships among the required derivatives of the potential.