General Coordinate Transformations Research Articles

Nonperturbative evolution equation for quantum gravity

A scale-dependent effective action for gravity is introduced and an exact nonperturbative evolution equation is derived which governs its renormalization group flow. It is invariant under general coordinate transformations and satisfies modified Becchi-Rouet-Stora Ward identities. The evolution equation is solved for a simple truncation of the space of actions. In $2+\ensuremath{\varepsilon}$ dimensions, nonperturbative corrections to the \ensuremath{\beta} function of Newton's constant are derived and its dependence on the cosmological constant is investigated. In 4 dimensions, Einstein gravity is found to be ``antiscreening;'' i.e., Newton's constant increases at large distances.

Lie Symmetries of Einstein’s Vacuum Equations in N Dimensions

We investigate Lie symmetries of Einstein’s vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein’s equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein’s equations), we set to zero the coefficients of derivatives that do not appear in Einstein’s equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein’s equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.

The Most General Corrdinate Transformation Between the Hydrogen Atom and Harmonic Oscillator in Arbitrary Dimensions

In this paper, we find out the most general coordinate transformation between a d-dimensional hydrogen atom and a D-dimensional harmonic oscillator by means of the SU (1,1) algebra. In other words, the generalized Kustaanheimo-Stiefel transformation is established.

Complexity in quantum gravity

There are two difficulties with quantum gravity. One is that the theory has to be quantum mechanical, and the other is that it has to be invariant under general coordinate transformations. Presumably the resolution of these difficulties will lie in certain delicate revisions that will have to be considered for both disciplines. Often, the fact that perturbative quantum gravity is in excellent shape, is not sufficiently appreciated. The theory is admittedly nonrenormalizable, which means that at every order in the perturbative expansion, new uncalculable “constants of nature” emerge. The number of uncalculable constants at every order is quite small however, certainly in comparison with the amount of information that the calculations could provide. There are two reasons why this theory is usually completely dismissed. One is its tremendous complexity, as already at low orders the number of algebraical manipulations needed in the calculations is gigantic. Secondly, of course, the emergence of uncomputable numbers (nonrenormalizability) renders the theory useless at the Planck scale. It is clear that a nonperturbative formulation of quantum gravity will have to be entirely different, but it is important to observe that most of the proposed alternatives to perturbative quantum gravity are actually much less predictive than the simple perturbation theory. (Super)string theory is making rather vociferous claims for the status of “only consistent theory of quantum gravity,” but this theory, too, is formulated perturbatively; the expansion, here, is one in topological complexity of string diagrams, and this expansion is as hopelessly divergent as the ordinary perturbative theory. However, the fact that, at each given order, there are no unknown counter terms strongly suggests a more powerful nonperturbative underlying system waiting to be uncovered. Yet there is a danger in such expectations, which can be illustrated by observing that there are many ordinary quantum field theories that have unique perturbative expensions but show completely new physics at the nonperturbative level. Therefore, what is needed is a fundamentally nonperturbative formulation of a theory.

A novel implicit difference algorithm of primitive variable flow equations with higher-order extra terms

A stable scheme for solving the Euler and Navier-Stokes compressible primitive variable equations by adding higher-order extra terms to the flux-vector splitting method has been developed. The approach eliminates approximate factorization and matrix operation. This gives an improvement in computational efficiency as compared to the standard Beam-Warming and Steger implicit factored schemes. The method is spatially second-order accurate, fully conservative and implemented with general coordinate transformations for treating the complex geometries. It shows good convergence rate and acceptable accuracy in capturing the shock waves. Results computed from the code include transonic flows through turbine stator, compressor rotor with large separate turbulent flow and supersonic compressor staggered wedge cascade. Comparisons with well-documented experimental data are included and the agreements are good.

Refraction and geometry in Maxwell's equations

Abstract Computational studies of Maxwell's equations in complex geometries encountered in photonic band structure calculations run into difficulties when several length scales occur, such as the wavelength of light in free space and the skin depth in metal. These problems are remedied by using an adaptive co-ordinate system which expands or contracts length scales as necessary. Here we show that moving to a general co-ordinate transformation is equivalent to renormalizing ε and μ. This is an huge simplification because now we need only write one computer code in a Cartesian system, and we can use this same code to handle any co-ordinate system by adjusting the ε and μ we feed into the calculation.

Refraction and geometry in Maxwell's equations

Abstract Computational studies of Maxwell's equations in complex geometries encountered in photonic band structure calculations run into difficulties when several length scales occur, such as the wavelength of light in free space and the skin depth in metal. These problems are remedied by using an adaptive co-ordinate system which expands or contracts length scales as necessary. Here we show that moving to a general co-ordinate transformation is equivalent to renormalizing ε and μ. This is an huge simplification because now we need only write one computer code in a Cartesian system, and we can use this same code to handle any co-ordinate system by adjusting the ε and μ we feed into the calculation.

Operator Ordering Problem and Coordinate-Free Nature of Geometric Quantization11The project supported by NNSF of China and the Grant LWTZ-1298 of the Chinese Academy of Sciences

Operator ordering problem in geometric quantization is investigated. While the quantum operator for given by geometric quantization is found to be ordered via the rule invariant under general coordinate transformations, it is shown that the quantum operators for where cannot be well-defined by geometric quantization. This is considered as a natural consequence of the coordinate-free nature of geometric quantization and the Van Hove's theorem.

Ideas and Methods of Supersymmetry and Supergravity, or a Walk Through Superspace

Preface. Mathematical background: The Poincar^D'e group, the Lorentz group Finite-dimensional representations of ^ISpin(3,1) The Lorentz algebra Two-component and four-component spinors Representations of the Poincar^D'e group Elements of differential geometry and gravity The conformal group The mass-shell field representation Elements of algebra with supernumbers Elements of analysis with supernumbers The supergroup of general coordinate transformations on R^Tp/q. Supersymmetry and superspace: Introduction: from R^Tp/q to supersymmetry Superalgebras, Grassmann-shells and super Lie groups The Poincar^D'e superalgebra Unitary representation of the Poincar^D'e superalgebra Real superspace R^T4/4 and superfields Complex superspace C^T4/2, chiral superfields and covariant derivatives The on-shell massive superfield representations The on-shell massless superfield representations From superfields to component fields The superconformal group. Field theory in superspace: Supersymmetric field theory Wess-Zumino model Supersymmetric nonlinear sigma-models Vector multiplet models Supersymmetric Yang-Mills theories Geometric approach to super Yang-Mills theories Classically equivalent theories. Quantized superfields: Picture-change operators Equivalence of component field and superfield perturbation theories Effective action (super) funtional The Wess-Zumino model: perturbative analysis Note about gauge theories Feynman rules for super Yang-Mills theories Renormalization Examples of counterterm calculations: an alternative technique Superfield effective potential. Superspace geometry of supergravity: Gauge group of supergravity and supergravity fields Superspace differential geometry Supergeometry with conformal supergravity constraints Prepotentials Einstein supergravity Prepotential de formations Supercurrent and supertrace Supergravity in components. Dynamics in supergravity: Pure supergravity dynamics Linearized supergravity Superg

Generators of Internal Lorentz Transformations and of General Affine Coordinate Transformations in Teleparallel Theory of (2+1)-Dimensional Gravity: Cases with Static Circularly Symmetric Space-Times

In a teleparallel theory of (2+1)-dimensional gravity developed in a previous paper, we examine generators of internal Lorentz transformations and of general affine coordinate transformations for static circularly symmetric exact solutions of gravitational field equation. The \lq \lq spin" angular momentum, the energy-momentum and the \lq \lq extended orbital angular momentum" are explicitly given for each solution. Also, we give a critical comment on Deser's claim that neither momentum nor boosts are definable for finite energy solutions of three-dimensional Einstein gravity.

Gravitational Lorentz anomaly from the overlap formula in two dimensions.

In this Rapid Communication we show that the overlap formulation of chiral gauge theories correctly reproduces the gravitational Lorentz anomaly in two dimensions. This formulation has been recently suggested as a solution to the fermion doubling problem on the lattice. The well-known response to general coordinate transformations of the effective action of Weyl fermions coupled to gravity in two dimensions can also be recovered.

Novel Coordinate System and Rotation Transformations for Antenna Applications

ABSTRACT General coordinate system transformations and rotation transformations for antenna, applications are presented in a novel coherent and unified fashion. In coordinate system transformations, the Cartesian transformation matrix (CTM) is used to relate the representations of a vector in two Cartesian coordinate systems. Useful CTMs such as the Cartesian–spherical transformation and the Eulerian transformation are formulated based on three elementary CTMs. In rotation transformations, the rotation transformation matrix (RTM) is used to describe the rotation of an object, which may be a point, a vector, a coordinate system, or any physical entity. The relationship between the representations of an RTM in different Cartesian coordinate systems art: developed. Three examples: (i) Scanning dual–reflector antennas, (ii) Map generation for satellite antennas, and (iii) Coordinate systems for antenna measurement (EL–over–AZ and AZ–OVER–EL) are given to demonstrate the versatility of the combined application...

Standard and generalized Newtonian gravities as `gauge' theories of the extended Galilei group: I. The standard theory

Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant action principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an action principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an action principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contraction, for , of the ADM-De-Witt action of general relativity. The Galilean action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.

Poincaré gauge theory of (2+1)-dimensional gravity.

A Poincar\'{e} gauge theory of (2+1)-dimensional gravity is developed. Fundamental gravitational field variables are dreibein fields and Lorentz gauge potentials, and the theory is underlain with the Riemann-Cartan space-time. The most general gravitational Lagrangian density, which is at most quadratic in curvature and torsion tensors and invariant under local Lorentz transformations and under general coordinate transformations, is given. Gravitational field equations are studied in detail, and solutions of the equations for weak gravitational fields are examined for the case with a static, \lq \lq spin"less point like source. We find, among other things, the following: (1)Solutions of the vacuum Einstein equation satisfy gravitational field equations in the vacuum in this theory. (2)For a class of the parameters in the gravitational Lagrangian density, the torsion is \lq \lq frozen" at the place where \lq \lq spin" density of the source field is not vanishing. In this case, the field equation actually agrees with the Einstein equation, when the source field is \lq \lq spin"less. (3)A teleparallel theory developed in a previous paper is \lq \lq included as a solution" in a limiting case. (4)A Newtonian limit is obtainable, if the parameters in the Lagrangian density satisfy certain conditions.

Canonical analysis of Poincare gauge theories for two-dimensional gravity

Following the general method discussed by us earlier, Liouville gravity and the two-dimensional model of non-Einsteinian gravity L approximately curv2 + torsion2+cosm const can be formulated as ISO (1,1) gauge theories. In the first order formalism the models present, besides the Poincare gauge symmetry, additional local symmetries, the K-symmetries. These, related to general coordinate transformations on-shell, have canonical generators Ja satisfying a complicated constraint algebra. One can replace the Ja by means of a simple Dirac procedure, with constraints satisfying an Abelian algebra but still generating the K-symmetry. This can be done without altering the symmetry content and the equations of motion of the models. One then remarkably simplifies the canonical structure, as the constraints now satisfy only the ISO (1, 1) algebra plus an Abelian algebra. Moreover, one shows that the Poincare group can always be used consistently as a gauge group for gravitational theories in two dimensions.

Landau-level ground-state degeneracy and its relevance for a general quantization procedure.

The quantum dynamics of a two-dimensional charged spin-1/2 particle is studied for general, symmetry-free curved surfaces and general, nonuniform magnetic fields that are, when different from zero, orthogonal to the defining two surface. Although higher Landau levels generally lose their degeneracy under such general conditions, the lowest Landau level, the ground state, remains degenerate. Previous discussions of this problem have had less generality and/or used supersymmetry, or else have appealed to very general mathematical theorems from differential geometry. In contrast our discussion relies on simple and standard quantum-mechanical concepts. The mathematical similarity of the physical problem at hand and that of a phase-space path-integral quantization scheme of a general classical system is emphasized. Adopting this analogy in the general case leads to a general quantization procedure that is invariant under general coordinate transformations---completely unlike any of the conventional quantization prescriptions---and therefore generalizes the concept of quantization to hitherto inaccessible situations. In a complementary fashion, the so-obtained picture of general quantization helps to derive useful semiclassical formulas for a Hall current in the case of a filling factor equal to one for a general surface and magnetic field.

A primitive equation ocean circulation model using a general vertical coordinate transformation: 1. Description and testing of the model

The ocean general circulation model of the Geophysical Fluid Dynamics Laboratory has been modified to accommodate a generalized vertical coordinate transformation. The model with a slightly modified σ transformation (σ = z/H) is applied to three simple test cases. A clear improvement of barotropic and baroclinic topographic Rossby waves has been achieved compared with the z‐coordinate model. A somewhat more typical time‐independent problem for the overflow of dense water masses over a zonal ridge exhibits the potential benefits as well as the flaws of the new model. Vertically integrated mass transport is very sensitive to changes in bottom slope in regions where planetary and topographic β are of comparable magnitudes. This translates into large errors in the z model due to the crude approximation to the actual topography. The deep flow in the z model does not follow the bottom exactly. Horizontal overshoots of dense water lead to convective mixing on the southern side of the ridge. This can be a serious problem for modeling more realistic overflow situations with the z model. Increased diapycnal diffusion has been identified as the most serious problem with the terrain‐following coordinate system for this specific application.

A primitive equation ocean circulation model using a general vertical coordinate transformation: 2. Application to an overflow problem

The version of the ocean general circulation model of the Geophysical Fluid Dynamics Laboratory with a generalized vertical coordinate transformation introduced in part 1 is used to study the overflow of a zonal ridge in a coarse‐resolution grid typical for climate applications. The configuration roughly resembles the Greenland‐Scotland Ridge with respect to scale, latitude, and height of the ridge. Reasonable qualitative agreement with observed conditions over the Iceland‐Faroe section has been achieved, especially when a deeper passage was included at the eastern end of the idealized model ridge. Upwelling of deep water occurs at the eastern boundary, where friction and, more importantly, diapycnal diffusion accomplish the necessary changes in potential vorticity. The overflow consists of intermediate water when the renewal time for the deep water is made much larger than that for intermediate water. The meridional mass transport is most sensitive to geometric factors like the height of the ridge. Frictional and diapycnal processes are crucial for the potential vorticity conversion at the ridge and have a strong effect on the water mass distribution. Bottom friction increases the near‐bottom flow across f/H lines, resulting in a large amount of northern basin deep water south of the ridge. This effect can be represented only when the frictional bottom boundary layer is adequately resolved by the numerical grid.

Investigation on the multidimensional nuclear fission dynamics

The expression of multidimensional fission rate with quantum effect at finite temperature has been obtained by means of the approach of path integral expressed as Fourier series. The covariant form of the fission diffusion equations and the quantities related to it under general coordinate transformations in phase space are studied. The differences between two or three-dimensional and one-dimensional cases are analyzed carefully by using the {c, h, α} parametrization. The realistic Langevin calculations of two-dimensional transient rate are compared with the one-dimensional case and generalized Kramers rate formula.

Treatment of constraints in the stochastic quantization method and covariantized Langevin equation

We study the treatment of the constraints in the stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking into account the Ito calculus. Then we obtain an improved Langevi equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to an O( N) non-linear α model and it is shown that singular terms appearing in the improved Langevin equation cancel out the σ n(O) divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of idependent variables, is actually equivalent to the one in the general-coordinate transformation covariant and vielbein-rotation invariant formalish.