Multivectorial representation of Lie groups

In vector spaces of dimensionn=p+q a multivector (Clifford) algebraC(p, q) can be constructed. In this paper a multivectorC(p, q) representation, riot restricted to the bivector subalgebraC ²(p, q), is developed for some of the Lie groups more frequently used in physics. This representation should be especially useful in the special cases of (grand) unified gauge field theories, where the groups used do not always have a simple tensor representation.     

Haag-Ruelle-Hepp scattering formalism for essentially local non-localizable fields

We generalize the axiom of locality to the non-localizable case by an axiom of essential locality which means a natural local continuity property of the field commutator on the light-cone. For localizable fields essential locality coincides with ordinary locality while localizable local fields restricted to non-localizable test spaces still remain essentially local. Essential locality is proved to be sufficient for a derivation of the usual Haag-Ruelle-Hepp scattering formalism. Functions of the free field are examples of essentially local fields which may not be localizable.     

Einstein-Maxwell null fields with a null current distribution

Null Einstein-Maxwell charge-free fields such that the propagation vector is a scalar multiple of a gradient are not determined uniquely by the geometry of the space-time. The metric for space-times admitting such exceptional fields can always be transformed to the Wyman-Trollope form. This same result follows if there is a non-vanishing null current density associated with the field.     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Propagation of vectorial nonparaxial Gaussian beams through an annular aperture

Starting from the vectorial Rayleigh diffraction integrals, the nonparaxial propagation of vectorial Gaussian beams through an annular aperture is studied. The analytical propagation expressions are derived, which permit us to treat the on-axis field and far field of vectorial nonparaxial Gaussian beams diffracted at the annular aperture, the nonparaxial diffraction at a circular aperture and a circular disc as our special cases in a unified way. The validity of our treatment is confirmed by direct numerical integration of the Rayleigh formulae. It is shown that the f-parameter and annular obscuration affect the beam nonparaxiality in the case of diffraction at the annular aperture.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

双椭圆纳米金属棒构成的表面等离子体波导的传输特性分析

设计了一种双椭圆纳米金属棒表面等离子体波导,采用频域有限差分法,对这种波导所支持的基模的能流密度分布、有效折射率和传播长度随几何结构参数和工作波长的依赖关系进行了分析.结果表明,沿纵向的能流主要分布在两个椭圆金属棒所形成的中间区域,且越靠近金属棒的弧形边,沿纵向的能流越大.通过调节两个金属棒的中心距离以及它们的两个半轴的大小,可以调节模式的有效折射率和传播长度.在工作波长确定的条件下,相对于a=b的情形来说,在a<b时,场与金属表面接触的面积较大,场 关键词： 集成光学 光波导 表面等离子体波导     

Around Polygons in ℝ3 and S3

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ³ and S ³, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ¹ and the symplectic quotient of SO(3) acting diagonally on (S ²) ⁿ . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given. Received: 20 September 1999 / Accepted: 28 November 2000     

Logicoalgebraic structures II. Supports in test spaces

Test spaces are mathematical structures that underlie quantum logics in much the same way that Hilbert space underlies standard quantum logic. In this paper, we give a coherent account of the basic theory of test spaces and show how they provide an infrastructure for the study of quantum logics. IfL is the quantum logic for a physical systemL, then a support inL may be interpreted as the set of all propositions that are possible whenL is in a certain state. We present an analog for test spaces of the notion of a quantum-logical support and launch a study of the classification of supports.     

Integrable sigma-models and Drinfeld–Sokolov hierarchies

Local commuting charges in sigma-models with classical Lie groups as target manifolds are shown to be related to the conserved quantities appearing in the Drinfeld–Sokolov (generalized mKdV) hierarchies. Conversely, the Drinfeld–Sokolov construction can be used to deduce the existence of commuting charges in these and in wider classes of sigma-models, including those whose target manifolds are exceptional groups or symmetric spaces. This establishes a direct link between commuting quantities in integrable sigma-models and in affine Toda field theories.     

Stochastically excited waves in a 1D random medium – the mean field and power spectrum

Scalar wave propagation is examined when both the wave source and the propagation speed are random. Results are derived for the mean field and the power spectrum using the second-order Born approximation. The results depend on whether the source S(x, t) and the propagation speed c(x, t) are correlated or not. When they are uncorrelated, the mean field is zero. When they are correlated, the mean field is non-zero only when the source is non-stationary. The power spectrum is incoherent to leading order. There is a transfer of energy from lower to higher frequencies owing to wave scattering. The corresponding frequency upshift of the power profile in the (k, ω) domain is mainly caused by the cross power between the direct and the twice scattered field, which represents a second-order incoherent power contribution. The results are confirmed using a numerical solution of the wave equation where the scattered field is expanded to fifth order.     

Imaging deviation through non-uniform flow fields around high-speed flying vehicles

The non-uniform flow field around high-speed flying vehicle “bends” the ray and imposes a deviation at the end of the propagation path. This imaging deviation is a kind of aero-optic effect. In this paper, we catalog the factors that influence the deviation into two classes: the vehicle-related factors and the flowfield-related factors. Flow density computation and density–refractive index conversion are discussed. A backward ray-tracing scheme is proposed. The deviations, the propagation path distances in the non-uniform flow field, and the density distributions along propagation paths for two different flying cases are computed. Three flowfield-related factors should be considered in order to reduce the deviation: the propagation path distance in the non-uniform flow field, which should be as short as possible; the angle of incidence at the freestream boundary, which should be as small as possible; and the density distribution along propagation path in the non-uniform flow field, which should be as flat as possible.     

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D ₂(222), D ₄(422) and D ₆(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.     

Interfacial wave propagation due to an interfacial line source

Interfacial wave propagation parallel to a dielectric interface has been studied by considering an electric current line source present at the interface. The first order asymptotic evaluation of field components shows a null of the electric field at the interface. An amplitude null represents an unstable structure in the phase map and a phase front discontinuity across the interface. Higher order asymptotic evaluation has been employed to gain further insight into this propagation problem. The results show that the wavefronts need not be discontinuous. The continuity of the phase fronts is preserved with the help of interesting and stable structures such as saddle points and center points in the phase map of the electric field in both half spaces.      

Reversible charge transfer in solids; binary approximation

A common theory of reversible charge transfer (RCT) kinetics for low and high excitation power with the use of the conditional concentration of acceptors is presented. A comparison with recently published RCT theory is given. The results are similar or identical only in restricted or fractal spaces of low dimension, and in the case of low concentration of acceptors, i.e. in all the cases when the conditions of the binary approximation are fulfilled. The deviation from the binary approximation results in a physically incorrect picture, the cause being due to the different space averaging procedure. The probability of the donor's cation state is an increasing function of the concentration of the acceptor and of the characteristic parameters of the rate constants. Received 26 May 1999     

Green's functions associated with the edth operators

An integral representation for the inverses of the differential operators called edth is given in the two cases of particular interest. This operator plays a large role in the study ofH spaces and of self-dual solutions to the Yang-Mills field equations in the asymptotic domain.     

稀疏光学相控阵设计方法研究

为克服光学相控阵单元间隔必须小于工作波长二分之一的限制,构建了一种稀疏光学相控阵模型。分析了一维稀疏光学相控阵在近场和远场条件下的扫描原理,并提出了一种设计方法。对其相关参数进行仿真的结果表明：稀疏光学相控阵所用的单元数目较少,单元间隔远大于工作波长,扫描范围较大,波束宽度较窄,且在整个扫描空间内没有栅瓣。因此,稀疏光学相控阵单元间隔不受工作波长的限制,同时具有较好的扫描性能。     

Symmetric spaces of exceptional groups

We address the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1 + 2 cases for G₂ and F₄, respectively, are easily explained from the octonionic nature of these groups. The 4 + 3 + 2 cases on the E_6,7,8 series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in 5, 4, and 3 space—time dimensions.     

Electromagnetic soliton in an anisotropic ferromagnetic medium under nonuniform perturbation

We study the propagation of electromagnetic wave (EMW) in a linear as well as in a nonlinear anisotropic ferromagnetic medium which are assumed to be free from electric charges by making a nonuniform perturbation analysis. It is found that as the EMW propagates through the linear anisotropic ferromagnetic medium, the magnetic induction and hence the magnetic field component of the EMW are being modulated in the form of solitons. Also, the magnetization of the ferromagnetic medium is excited in the form of solitons. While the magnetic induction soliton is restricted to the plane normal to the direction of propagation, the magnetization excitations are not restricted to any particular plane. Unsaturated nonlinear ferromagnetic media is also found to give similar results.     

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces G₂,F₄ is re-examined and for E_6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.