K. Saito's Duality for Regular Weight Systems and Duality for Orbifoldized Poincaré Polynomials

We will show that the duality for the regular weight system introduced by K. Saito can be interpreted as the duality for the orbifoldized Poincaré polynomial . Received: 2 March 1998 / Accepted: 7 March 1999     

Poincaré vortices

Traditional interferometric methods for measuring the vortex structure of complex wave fields suffer from many intrinsic problems and seldom yield results of any accuracy. Using the unique properties of what I call Poincaré vortices, I develop a radically different method based on Stokes parameters that offers many practical advantages. The theory of this new method is discussed, and its unique capabilities are illustrated by reconstruction with high accuracy of the vortex structure of a simulated random field containing numerous vortices, including several closely spaced vortex pairs that would be difficult, if not impossible, to resolve by traditional means.     

Polyvector Super-Poincaré Algebras

A class of ₂-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W₀+W₁ is the maximal solvable ideal of W₀ is generated by W₁ and commutes with W. Choosing W₁ to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W₀ consists of polyvectors, i.e.all the irreducible submodules of W₀ are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.     

Quadratic Deformations of Lie–Poisson Structures

In this letter, first we give a decomposition for any Lie–Poisson structure associated to the modular vector. In particular, splits into two compatible Lie–Poisson structures if . As an application, we classified quadratic deformations of Lie– Poisson structures on up to linear diffeomorphisms. Research partially supported by NSF of China and the Research Project of “Nonlinear Science”.     

Complete Gauge theory for the whole Poincaré group

Gauge theories for nonsemisimple groups are examined. A theory for the Poincaré group with all the essential characteristics of a Yang-Mills theory necessarily possesses extra equations. Inonü-Wigner contractions of gauge theories are introduced which provide a Lagrangian formalism, equivalent to a Lagrangian de Sitter theory supplemented by weak constraints.Work supported by FINEP under Contract No. B/76/80/146/00/00.Fellow of the CNPq, Brasilia.     

The geometry of minimal replacement for the Poincaré group

The constructs of this paper rest on two elementary facts: (1) the Poincaré group P₁₀ is the maximal group of isometries of Minkowski space-time M₄; (2) P₁₀ has a faithful matrix representation as a subgroup of GL(5, R) that maps an affine set into itself. Local action of P₁₀ and Yang-Mills minimal replacement are shown to induce a well-defined minimal replacement operator that maps the tensor algebra over M₄ onto the tensor algebra over a new space-time U₄. The natural frame and coframe fields of M₄ go over into a canonical system of frame and coframe fields of U₄ with both translation and Lorentz-rotation parts. The coframe fields define soldering 1-form fields for U₄ that give rise to the standard geometric quantities through the Cartan equations of structure. This leads to unique determinations of all relevant connection coefficients and the associated 2-forms of curvature and torsion that involve the compensating 1-forms for local action of both the translation and the Lorentz-rotation sectors. The metric tensor of U₄, that is induced by the minimal replacement operator, is shown to satisfy the Ricci lemma; U₄ is necessarily a Riemann-Cartan space. This space admits gauge covariant constant basis fields for the Lie algebra of the Lorentz group and for the Dirac algebra. The induced basis for the Dirac algebra evaluates the images of Dirac operators under minimal replacement, while the induced basis for the Lie algebra of L(4, R) serves to show that the holonomy group of U₄ is the Lorentz group. The minimal replacement operator is extended to include the case of a total gauge group that is the direct product of the Poincaré group and a Lie group of internal symmetries of matter fields. This provides a precise method of lifting any action integral of the matter fields from M₄ up to U₄ so that invariance properties are retained when the total group acts locally. The natural representations afforded by minimal replacement result in curvature being evaluated in terms of first order derivatives of the compensating fields that share many properties in common with the Dirac derivation algebra for spin fields. Direct interpretations of the compensating fields are obtained from the geodesic equations.     

Poincaré invariance for the master field on quantum chromodynamics

I construct allSU(N _c ) gauge fields with the property that Euclidean Poincaré transformations can be compensated by gauge transformations. Linear Abelian components are shown to be forbidden by Lorentz invariance. In a suitable gauge, the result is a set of constant potentials parametrized by Lorentz scalars. These scalars are constrained by the equation of motion atN _c =. A special solution is exhibited.Work supported in part by Schweizerischer Nationalfonds.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I thank H. Leutwyler for drawing my attention to the configuration (35), and M. Lüscher, P. Schwab, P. Sorba and J. Stern for their comments.     

Poincaré Algebra and Space-Time Critical Dimensions for Paraspinnings Strings

In this paper, we paraquantize the spinning string theory in the Neveu-Shwarz model. Unlike the Ardalan and Mansouri work [Phys. Rev. D, Vol. 9, (1974) 3341], the paraquantum system is such that both the center of mass variables and the excitation modes of the string verify paracommutation relations. The commutators of the Poincaré algebra are satisfied, except the [p ,p ] one, since one can only write [p ,p ]= 0, for Q1. Because of the relation [x ,x ] =,0 and with the sole use of the trilinear relations, we find existence possibilities of spinning strings defined in a noncommutative space-time at space-time dimensions other than D=10.     

Curvilinear Poincaré vector beams

We develop a method for completely shaping optical vector beams with controllable amplitude, phase, and polarization gradients along three-dimensional freestyle trajectories. We design theoretically and demonstrate experimentally curvilinear Poincaré vector beams that exhibit high intensity gradients and accurate state of polarization prescribed along the beam trajectory.     

Tensor Gauge Fields in Arbitrary Representations of GL(D, ℝ). Duality and Poincaré Lemma

Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we have studied the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D, ). We have proven a generalization of the Poincaré lemma which enables us to solve the above-mentioned problems in a systematic and unified way.     

Poincaré Invariant Three-Body Scattering

Relativistic Faddeev equations for three-body scattering are solved at arbitrary energies in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated withing the framework of Poincaré invariant quantum mechanics. Based on a Malfliet–Tjon interaction, observables for elastic and breakup scattering are calculated and compared to non-relativistic ones.     

Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition.     

Poincaré recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincaré recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows us to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincaré exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincaré recurrences.     

Mechanical systems with Poincaré invariance

Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincaré algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural quantum version. We follow this early work finding and classifying mechanical systems with such an action. New solutions are found together with a new class of models exhibiting an action of the Galilean algebra. These are related to the functional identities underlying the various Hirzebruch genera. The quantum mechanics is also discussed.     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Direct gauging of the Poincaré group

Realization of the Poincaré group as a subgroup ofGL(5,R) that maps an affine set into itself is shown to lead to a well-defined minimal replacement operator when the Poincaré group is allowed to act locally. The minimal replacement operator is obtained by direct application of the Yang-Mills procedure without the explicit introduction of fiber bundle techniques. Its application gives rise to compensating 1-formsW , 1 6, for the local action of the Lorentz groupL(4,R), and to compensating 1-forms ^k , 1k4, for the translation groupT(4). When applied to the basis 1-formsdx ⁱ of Minkowski space, distortion 1-formsB ^k result that define a canonical anholonomic coframe that contains both theT(4) and theL(4,R) compensating fields. When the canonical coframe is considered as a differential system onM ₄, it gives rise to gauge curvature expressions and Cartan torsion, but the latter has important differences from that usually encountered in the associated literature in view of the inclusion of the compensating fields forL(4,R). The standard Yang-Mills minimal coupling construct is used to obtain a total Lagrangian. This leads to a system of field equations for the matter fields, theT(4) compensating fields, and theL(4,R) compensating fields. Part of the current that drives theT(4) compensating fields is the 3-form of gauge momentum energy that obtains directly from the momentum-energy tensor of the matter fields onM ₄ under minimal replacement. Introduction of the Cartan torsion in the free-field Lagrangian is shown to lead to a direct spin decoupling in the sense that the gauge momentum energy (orbital) contribution of the matter fields to the spin current is eliminated. Explicit conservation laws for total momentum energy current and total spin current are obtained.     

Poincaré Map Based on Splitting Methods

Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p ＋ I for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method （a second-order Runge-Kutta method）. Computation illustrates that the results by splitting methods can properly represent systems＇ chaotic phenomena.