Proper Dirac quantization of a free particle on a D-dimensional sphere

We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to Abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically incorrect.     

Extended two-parameter squeezed states

In a recent paper, a new λ-dependent squeezed states was proposed [J. Beckers, N. Debergh and F.H. Szafraniec, Phys. Lett. A 243 (1998) 256–260] in which the usual Fock-space creation operator has been modified by introducing a simple translational term. In this paper, we generalize this approach by adding a new term proportional to the creation operator using exponential quadratic operators.     

On fermion masses in a dimensional reduction scheme

A candidate model for Grand Unification, arising from a Coset Space Dimensional Reduction scheme based on anE(7) gauge theory, is found to have a promising set of fermionic quantum numbers. Unfortunately, these fermions all develop large (geometric) masses. We derive formulae for the square of the Dirac operator and for fermion masses for a large class of CSDR schemes, revealing this as a general feature.     

Varying the Unruh temperature in integrable quantum field theories

A computational scheme is developed to determine the response of a quantum field theory (QFT) with a factorized scattering operator under a variation of the Unruh temperature. To this end a new family of integrable systems is introduced, obtained by deforming such QFTs in a way that preserves the bootstrap S-matrix. The deformation parameter β plays the role of an inverse temperature for the thermal equilibrium states associated with the Rindler wedge, β = 2π being the QFT value. The form factor approach provides an explicit computational scheme for the β ≠ 2π systems, enforcing in particular a modification of the underlying kinematical arena. As examples deformed counterparts of the Ising model and the sinh-Gordon model are considered.     

Extension of Weinberg's quasi-particle theory to standing waves

In the context of a potential, a résumé of scattering theory is developed along two alternative lines, either (a) by the resolvent of the Lippmann-Schwinger equation, leading to the reaction matrixT, or (b) by a modified resolvent associated with standing waves, leading to the reactance matrixK. On this basis it is shown that Weinberg's theory of quasi-particles, which was originally appropriate to the scheme (a), has a counterpart in the scheme (b), in which the quasi-particle states are standing waves, and the result is a quasi-Born expansion ofK. The equations for passing between schemes (a) and (b) are developed. It is argued that the quasi-particle theory of scheme (b) possesses some advantages of simplicity.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Order α 2 s massive quark contribution to the vacuum polarization of massles quarks

We calculate to order α ₂ ^s in the QCD coupling the effect of massive quarks on the vacuum polarization of massless quarks both in a momentum subtraction scheme and in the (modified) minimal subtraction scheme. We also illustrate how decoupling of heavy quarks takes place in this context in the minimal subtraction scheme.     

Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

The influence of cell geometry on the Godunov scheme applied to the linear wave equation

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier–Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.     

Detection of topological symmetry in the one-dimensional Affleck-Kennedy-Lieb-Tasaki model of integer spin with ultracold spinor atoms in optical lattices

A scheme is proposed for detection of the topology in the one-dimensional Afeck-Kennedy-Lieb-Tasaki model,based on ultracold spinor atomic gas in an optical lattice.For this purpose,a global operation O(θ)is introduced with respect to the breaking of spinrotational symmetry.Consequently,the topology can be manifested unambiguously by identifying the special values ofθwhere the expectation value of the global operator with degenerate ground states is vanishing.Furthermore,experimentallyθcan be detected readily by the interference of ultracold atomic gases.This scheme can be implemented readily in experiment since it does not need the addressing of individual atoms or the probing of a boundary.     

Spectrum generating algebra and no-ghost theorem for fermionic massive string

The covariant operator quantization of the ordinary free spinning BDH string modified by adding the supersymmetric Liouville sector is analysed in the even target space dimensions d = 2, 4, 6, 8. The spectrum generating algebra for this model is constructed and a general version of the no-ghost theorem is proven. The counterpart of the GSO projection leads to a family of tachyon-free unitary free string theories. One of these models is equivalent to the non-critical Ramond-Neveu-Schwarz spinning string truncated in the Neveu-Schwarz sector to the tachyon-free eigenspace of the fermion parity operator.     

Effect of the Pauli principle on collective states in transitional Xe,Ba and Ce nuclei

It is shown how the modified harmonic method can be used to correct the quadrupole operator of the interacting boson model for the effect of the Pauli principle on states with several d-bosons. This gives a possible explanation for the observed strange reduction of ground-state band B(E2) values in the xenon-barium region.     

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.     

The asymptotic behaviour of form factors

Similar to the method of S.A. Anikin and O.I. Zavialov a modified renormalization procedure is applied to the already renormalized current operator. This allows the derivation of an identity for the coefficient functions of this operator and the proof of new renormalization group equations. These equations are applied to the theory and the electromagnetic form factor of quarks in QCD. In the last case the one-loop approximation gives the leading behaviour exp(?clnlnQ²lnQ²) for large euclidean values of the momentum transfer Q² = ?q² at the external electromagnetic vertex. If the leading logarithmic terms (g²ln²Q²)ⁿ are extracted from this result, then the behaviour exp(?g²c′ln²Q²) is obtained. Whereas the first result is modified by contributions from two-loop calculations, the result concerning the leading logarithmic terms does not change.     

Convergence properties of methods for effective operator calculations studied in a simple many-body model

Several methods for effective interaction and operator calculations based on the Rayleigh-Schrödinger and Brillouin-Wigner perturbation expansions are studied. Special emphasis is given to the use of Padé approximants for effective operator calculations. The convergence properties of the methods are studied numerically in a Lipkin many-body model. Among the various methods using Padé-approximants, the method based on a variational approach in the BW scheme is found to give most encouraging results for the present model.     

Approximation of off-shell scattering operators for non-lorentzian line shape calculations

A previously derived approximation for the matrix elements of the Fano relaxation operator m(ω), which gives rise to non-Lorentzian line wings, is examined. This approximation is expressed in terms of an off-the-energy shell scattering operator. Calculation of this operator by a second-order Born expansion is unsuitable because this procedure limits the radiation frequency dependence to the same order. The main part of this paper is devoted to expressing the off-shell scattering operator in the time-ordered formalism. In this form, a scattering operator approximation suggested by Dillonet al.⁽⁶⁾ can be applied which is not restrictive to the frequency dependence.     

A new compact difference scheme for second derivative in non-uniform grid expressed in self-adjoint form

A single-parameter family of self-adjoint compact difference (SACD) schemes is developed for discretizing the Laplacian operator in self-adjoint form. Developed implicit scheme is formally second-order accurate (with respect to truncation error) with a free parameter, α which helps control the numerical properties in the spectral plane. The SACD scheme is analyzed in the spectral plane for its resolution properties for both periodic and non-periodic problems using the matrix spectral analysis [T.K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind schemes, J. Comput. Phys. 192 (2) (2003) 677–694]. The major objective here is to identify the advantages of the new scheme over the traditional explicit second order CD2 scheme, in discretizing the Laplacian operator in self-adjoint form. This appears in Navier–Stokes equation and in other transport equations and boundary value problems (bvp) expressed in orthogonal co-ordinate systems, either in physical or in transformed plane. We also compare the developed method with the higher order compact schemes for non-uniform grids. To demonstrate the accuracy of SACD scheme we have tested it for: (i) bi-directional wave propagation problem, given by the second order wave equation and (ii) an elliptic bvp, as in the Stommel ocean model for the stream function. These examples help infer the properties of SACD scheme when solving different types of partial differential equations. Most importantly the effects of grid-stretching and choice of value of the free parameter (α) are investigated here. We also compare the present implicit compact method with explicit compact method known as the higher order compact (HOC) method.Also, the practical applications of the SACD scheme are explored by solving the Navier–Stokes equation for the cases of: (a) a flow inside a lid-driven cavity and (b) the receptivity and instability of an external adverse pressure gradient flow over a flat plate. In the former, unsteadiness of the flow is captured and in the latter, the receptivity of the flow is studied in causing flow instability by triggering Tollmien–Schlichting waves. The new scheme shows a marked improvement over the explicit scheme for low Reynolds number steady flow in lid driven cavity. Whereas for the flow in the same geometry at higher Reynolds numbers, efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures. Presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer.     

Factorization method for Schrödinger equation in relativistic configuration space and q-deformations

Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q—oscillator.     

Elimination of fast variables

First nonlinear ordinary differential equations for the evolution of a physical system are considered. When they involved a small (or large) parameter the evolution occurs on two time scales. A general scheme is developed for extracting equations for the slow evolution, including higher order corrections. The scheme unifies the multifarious methods that exist in the literature. It leads to a subdivision in three categories, determined by the structure of the equations.Although linear equations are merely a specialization, they can readily be generalized to the case of linear partial differential equations, such as the so amply studied Fokker-Planck equation. These are treated with special attention to the definition of the required projection operator.Throughout, numerous examples and applications are given.     

Classical V-A theory and non-conservation of the dynamical parity

The V-A theory with and without cut-off is studied. The existence of finite-energy global solutions and scattered solutions is proved, for certain regularizations. The scattering operator turns out to be C^∞. Moreover, finite-charge global “solutions” are defined. The dynamical parity operator is discussed. Finally some remarks about the continuous spin modified V-A theory are made.