Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

Noncommutative parameters of quantum symmetries and star products

The star product technique translates the framework of local fields on noncommutative spacetime into nonlocal fields on standard spacetime. We consider the example of fields on κ-deformed Minkowski space, transforming under κ-deformed Poincaré group, with noncommutative parameters. By extending the star product to the tensor product of functions on κ-deformed Minkowski space and κ-deformed Poincaré group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincaré group.     

振动筛系统的两类余维三分岔与非常规混沌演化

建立了振动筛系统的动力学模型,推导出了其周期运动的Poincaré 映射,基于Poincaré 映射方法着重研究了系统Flip-Hopf-Hopf余维三分岔、三次强共振条件下的Hopf-Hopf余维三分岔以及三种非常规的混沌演化过程.研究结果表明,此两类余维三分岔点附近的动力学行为变得更加复杂和新颖,在分岔点附近出现了三角形吸引子、3T²环面分岔以及“五角星型”、“轮胎型”概周期吸引子,揭示了环面爆破、环面倍化以及T²环面分岔向混沌演化的过程,这些结果对于振动筛系统的动力学优化设计提供了理论参考. 关键词： 余维三分岔 非常规混沌演化 T²环面分岔')" href="#">T²环面分岔     

Unified treatment: analyticity,Regge trajectories,Veneziano amplitude,fundamental regions and Moebius transformations

In this paper we present a unified treatment that combines the analyticity properties of the scattering amplitudes, the threshold and asymptotic behaviors, the invariance group of Moebius transformations, the automorphic functions defined over this invariance group, the fundamental region in (Poincaré) geometry, and the generators of the invariance group as they relate to the fundamental region. Using these concepts and techniques, we provide a theoretical basis for Veneziano type amplitudes with the ghost elimination condition built in, related the Regge trajectory functions to the generators of the invariance group, constrained the values of the Regge trajectories to take only inverse integer values at the threshold, used the threshold behavior in the forward direction to deduce the Pomeranchuk trajectory as well as other relations. The enabling tool for this unified treatment came from the multi-sheet conformal mapping techniques that map the physical sheet to a fundamental region which in turn defines a Riemann surface on which a global uniformization variable for the scattering amplitude is calculated via an automorphic function, which in turn can be constructed as a quotient of two automorphic forms of the same dimension. Received: 22 October 2002 / Revised version: 19 February 2003 Published online: 23 May 2003 RID="a" ID="a" e-mail: arc@lucent.com     

Elementary Systems of (1 + 1) Kinematical Groups: Contraction and Quantization

We present the (algebra) group contraction chain SU(1, 1) → P(1, 1) → G(1, 1), where P(1, 1) and G(1, 1) are the Poincaré and the Galilei groups, respectively, in (1 + 1) dimensions. We have paid attention to the contraction of the pseudo-extended Poincaré group to the central extended Galilei group. Objects like group laws, coadjoint orbits and representations of the contracted groups have been obtained in terms of their noncontracted counterparts. As an application we study the Moyal quantization of classical systems, having those groups as symmetry groups, by means of the contraction of the so called Stratonovich-Weyl kernels which provide such quantization.     

Nonlinear and quantum origin of doubly infinite family of modified addition laws for fourmomenta

We show that infinite variety of Poincaré bialgebras with nontrivial classicalr-matrices generate nonsymmetric nonlinear composition laws for the fourmomenta. We also present the problem of lifting the Poincaré bialgebras to quantum Poincaré groups by using e.g. Drinfeld twist, what permits to provide the nonlinear composition law in any order of dimensionfull deformation parameterλ (from physical reasons we can putλ=λ _p whereλ _p is the Planck length). The second infinite variety of composition laws for fourmomentum is obtained by nonlinear change of basis in Poincaré algebra, which can be performed for any choice of coalgebraic sector, with classical or quantum coproduct. In last Section we propose some modification of Hopf algebra scheme with Casimir-dependent deformation parameter, which can help to resolve the problem of consistent passage to macroscopic classical limit. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. Supported by KBN grant 5PO3B05620     

Local state probabilities for solvable restricted solid-on-solid models:A n,D n,D n (1) , andA n (1)

The local state probabilities (LSPs) are exactly computed for four hierarchies of solvable lattice models. They are restricted solid-on-solid (RSOS) models whose local states and their adjacent conditions are specified by Dinkin diagrams of typesA _n,D _n,D _n ⁽¹⁾ andA _n ⁽¹⁾ . The LSPs are expressed in terms of modular functions characterized by branching identities among the theta functions. Their automorphic properties are used to study the critical behaviors. Some fine structures are found in the spectrum of the critical exponents.     

The Poincaré Group in a Demisemidirect Product with?a?Non-associative Algebra with Representations that?Include Particles and Quarks—II

The quarks and particles’ mass and mass/spin relations are provided with coordinates in configuration space and/or momentum space by means of the marriage of ordinary Poincaré group representations with a non-associative algebra made through a demisemidirect product, in the notation of Leibniz algebras. Thus, we circumvent the restriction that the Poincaré group cannot be extended to a larger group by any means (including the (semi)direct product) to get even the mass relations. Finally, we will discuss a connection between the phase space representations of the Poincaré group and the phase space representations of the associated Leibniz algebra.     

Non-Completeness of Zograf–Takhtajan's Kähler Metric for Teichmüller Space of Punctured Riemann Surfaces

Zograf and Takhtajan introduced a new K?hler metric on the Teichmüller space T _g , _n (n>0), in calculating the first Chern form of the Quillen metric for families of -operators. The metric is described in terms of the Eisenstein–Maass series. We prove that it is incomplete. And we also give an alternative proof of non-completeness of the Weil–Petersson metric. For that, we use the pinching family, constructed by Wolpert, whose tangent vectors are always represented by using the relative Poincaré series associated with the pinched geodesic. Received: 18 September 1995 / Accepted: 7 March 1999     

Hamilton’s theory of turns and a new geometrical representation for polarization optics

Hamilton’s theory of turns for the group SU(2) is exploited to develop a new geometrical representation for polarization optics. While pure polarization states are represented by points on the Poincaré sphere, linear intensity preserving optical systems are represented by great circle arcs on another sphere. Composition of systems, and their action on polarization states, are both reduced to geometrical operations. Several synthesis problems, especially in relation to the Pancharatnam-Berry-Aharonov-Anandan geometrical phase, are clarified with the new representation. The general relation between the geometrical phase, and the solid angle on the Poincaré sphere, is established.     

Quasi-invariant measures on sets of piecewise smooth homeomorphisms of closed intervals and circles and representations of diffeomorphism groups

Families of measures are constructed that are quasi-invariant with respect to the action of C ¹-diffeomorphisms, for a closed interval and a circle, and have bounded Borel measurable second derivatives. A series of pairwise nonequivalent irreducible representations of the group of C ³-diffeomorphisms of the circle is introduced on the space of functions that are square integrable against these measures.     

Hamiltonian Structure of Poincaré Gauge Theory and Separation of Non-Dynamical Variables in Exact Torsion Solutions

The canonical Hamiltonian of the Poincaré gauge theory of gravity is reanalyzed for generic Lagrangians. It is shown that the time components e₀^α and Γ₀^αβ of the tetrad and the linear connection fields of a Riemann-Cartan space-time U₄ constitute gauge degrees of freedom which remain non-dynamical during the time evolution of the system. Whereas the e₀^α are to be identified with the lapse and shift functions N^α known from the ADM formalism in Einstein's theory, the additional Lorentz degrces of freedom Γ₀^αβ are pertinent to Poincaré gauge models. These non-dynamical variables are instrumental in the derivation of exact torsion solutions obeying modified double duality conditions for the U₄-curvature. Thereby, in the case of spherical symmetry and for the charged Taub-NUT metric, we obtain the most general torsion configuration for a large class of quadratic Lagrangians. Previously found solutions are contained therein and can be recovered after fixing special “gauge”.     

Off-shell d=10, N=1 Poincaré supergravity and the embeddibility of higher-derivative field theories in superspace

The constraints on the supertorsion are presented which lead to off-shell d=10, N=2 Poincaré supergravity and the superfluids which are known to enter the superspace action are identified. It is pointed out that Poincaré off-shell constraints have to be used in a superspace formulation of the low-energy effective actions of N=1 superstrings. However, current formulations of N=1 superstrings propagating in arbitrary supergravity backgrounds are incompatible with these off-shell constraints.     

Nonlinear Maxwell Equations

Abstract

The infinite series of Lorentz and Poincaré-invariant nonlinear versions of the Maxwell equations are suggested. Some properties of these equations are considered.     

General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains

A construction of relativistic wave equations on the homogeneous spaces of the Poincaré group is given for arbitrary spin chains. Parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski space time and two-dimensional complex sphere is the most suitable homogeneous space for the physical applications. The Lagrangian formalism and field equations on the Poincaré and Lorentz groups are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this problem are expressed via an expansion in hyperspherical functions defined on the complex two-sphere. PACS numbers: 02.30.Gp, 02.60.Lj, 03.65.Pm, 12.20.-m