Exact solutions and particle creation for nonconformal scalar fields in homogeneous isotropic cosmological models

We solve the problem of describing scaling factors of a homogeneous isotropic space-time such that the exact solution for the scalar field with a nonconformal coupling to curvature can be obtained from solutions for the conformally coupled field by redefining the mass and momentum. We give explicit expressions in the form of Abelian integrals for the dependence of time on the scaling factor in these cases. We obtain an exact solution for the scalar field coupled to the Gauss-Bonnet-type curvature and show that the corresponding nonconformal contributions can dominate in particle creation by the gravitational field.     

Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo-Martin-Schwinger and Wigner-Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov-Kubo-Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the SU(2) algebra of observables in the simplest model of an ideal quantum spin paramagnet.     

Critical point equation and closed conformal vector fields

In this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non-trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.     

CP Violation in the models of fermion localization on a domain wall (brane)

We briefly review the mechanism of fermion localization on a domain wall (“thick brane”) generated by a topologically nontrivial vacuum configuration of scalar fields. We propose an extension of the scalar field coupling to fermions that endows the fermions with an axial mass. In the case of several flavors and fermion generations, this extension can lead to the appearance of the Standard Model Cabibbo-Kobayashi-Maskawa matrix. We also consider a model with two scalar doublets that ensures an additional mechanism of CP-parity violation.     

Unit Tangent Sphere Bundles with Constant Scalar Curvature

As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.     

An electrically conducting sphere in a three-dimensional, alternating magnetic field

The electrical vector and scalar potential as well as the underlyinginduced eddy current density of a conducting sphere exposedto an external, time-varying magnetic field are calculated.The external magnetic field is assumed to be generated by sinusoidallyalternating current density distributions. Admitting arbitrarythree-dimensional current density fields, this paper extendsprevious results on this subject. Among other things, thesecalculations prove that in a sphere there is no radial componentof the induced eddy current density field. As an applicationof these results the impedance and power absorption of the sphereare derived. The impedance is the key for the contactless, inductivemeasurement of the electrical conductivity of metallic melts.     

A complementary theory of light scattering by homogeneous spheres

A theory of the scattering of electromagnetic waves by homogeneous spheres, the so-called Mie theory, is presented in a unique and coherent manner in this paper. We begin with Maxwell's equations, from which the vector wave equations are derived and solved by means of the two orthogonal solutions to the scalar wave equation. The transverse incident electric field is mapped in spherical coordinates and expanded in known mathematical functions satisfying the scalar wave equation. Determination of the unknown coefficients in the scattered and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of a sphere. Far-field solutions for the electric field are then given in terms of the scattering functions. Transformation of the electric field to the reference plane containing incident and scattered waves is carried out. Extinction parameters and the phase matrix are derived from the electric field perpendicular and parallel to the reference plane. On the basis of the independent-scattering assumption, the theory is extended to cases involving a sample of homogeneous spheres.     

Calcul symbolique de Dirac

We construct a symbolic calculus of observables on a Dirac particle. Some basic classical observables correspond to operators arising from representation/theoretical considerations, the discussion of position operators, or Clifford analyses on the shell mass.     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

Compact manifolds of constant scalar curvature

We consider a compact non-negatively curved Riemannian manifold M of constant scalar curvature and obtain a sufficient condition for it to be isometric to a sphere.     

Some compact hypersurfaces of constant scalar curvature in a sphere

We construct some compact hypersurfaces of constant scalar curvature in a sphere with cohomogeneity 1([2]).The author is grateful to Professor K. Ogiue for his useful criticism.     

Cosmological Applications of a Wavelet Analysis on the Sphere

The cosmic microwave background (CMB) is a relic radiation of the Big Bang and as such it contains a wealth of cosmological information. Statistical analyses of the CMB, in conjunction with other cosmological observables, represent some of the most powerful techniques available to cosmologists for placing strong constraints on the cosmological parameters that describe the origin, content and evolution of the Universe. The last decade has witnessed the introduction of wavelet analyses in cosmology and, in particular, their application to the CMB. We review here spherical wavelet analyses of the CMB that test the standard cosmological concordance model. The assumption that the temperature anisotropies of the CMB are a realisation of a statistically isotropic Gaussian random field on the sphere is questioned. Deviations from both statistical isotropy and Gaussianity are detected in the reviewed works, suggesting more exotic cosmological models may be required to explain our Universe. We also review spherical wavelet analyses that independently provide evidence for dark energy, an exotic component of our Universe of which we know very little currently. The effectiveness of accounting correctly for the geometry of the sphere in the wavelet analysis of full-sky CMB data is demonstrated by the highly significant detections of physical processes and effects that are made in these reviewed works.     

Massive vector gauge theory and comparison with Higgs—Connes—Lott

A massive vector gauge theory constructed from the matrix derivative approach of non-commutative geometry is compared with the Higgs—Connes—Lott theory. In the massive vector gauge theory, a new extra shift-like symmetry which is due to the one form constant matrix derivative allows the theory to have a mass term while keeping the gauge symmetry intact. In the Higgs—Connes—Lott theory, the transformation of scalar field makes up the deficiency of symmetry due to the mass term. Thus, when the scalar field is absent there remains no gauge symmetry just like the Proca model. In the massive vector gauge theory, the shift-like symmetry makes up the deficiency of symmetry due to the mass term even in the absence of the scalar field.     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

Scattering by moving bodies: The quasi stationary approximation

The scattered field produced by a rotating or vibrating object in the presence of a time harmonic plane wave is usually approximated by a time dependent sequence of stationary fields, called a quasi stationary field. Recently the mathematical theory of scattering of scalar fields of Lax and Phillips has been extended to cover moving bodies. Within this framework the quasi stationary approximation is shown to be valid in each sphere of finite radius which contains the body. Far fields for the exact field and the quasi stationary field are defined and the difference is estimated in terms of a small parameter.     

Maslov complex germ method for systems with second-class constraints

We study the semiclassical mechanics of systems with second-class constraints. We assume that the quantum mechanics of such a system is constructed by the Batalin–Fradkin–Tyutin method, where some additional coordinates and momenta are introduced and second-class constraints are converted into firstclass constraints. We also assume that the algebraic quantization method is used to quantize the extended system. To construct the semiclassical approximation, we use the Maslov complex germ theory. We study the semiclassical states of a system with second-class constraints, their scalar product, and the action of semiclassical observables in the first order of the semiclassical expansion. We consider the transformation of semiclassical states in the course of evolution.     

On the Webster Scalar Curvature Problem on the CR Sphere with a Cylindrical-Type Symmetry

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

单位球面的三维紧致极小子流形

本文得到高维球面上三维紧致极小子流形的若干刚性定理．在一个整体拚挤条件下，数量曲率的拚挤常数被改进了，也对李奇曲率的拚挤问题进行了讨论．     

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.