Poisson Brackets of Normal-Ordered Wilson Loops

We formulate Yang-Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of $\hbar$) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum operators, we find, using ideas closly related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them.Comment: 34 pages, 4 eps figures, LaTeX2.09; citations adde

Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space dimensions with mass resonance, exhibits long range scattering phenomena. Modified wave operators and solutions are constructed for these equations. We also show that the modified wave operators can be chosen such that they linearize the non-linear representation of the Poincar\'e group defined by the NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy

From Classical to Quantum Mechanics: "How to translate physical ideas into mathematical language"

In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the question of a "true quantization" with "h" must be seen as an independent problem not directly related with quantum formalism. Moreover, this non-standard formulation of Classical Mechanics exhibits a new kind of operation with no classical counterpart: this operation is related to the "quantization process", and we show why quantization physically depends on group theory (Galileo group). This analytical procedure of quantization replaces the "correspondence principle" (or canonical quantization) and allows to map Classical Mechanics into Quantum Mechanics, giving all operators of Quantum Mechanics and Schrodinger equation. Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We find also that this approach gives a natural semi-classical formalism: some exact quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehensible way both logical and analytical connection between classical and quantum pictures.Comment: 47 page

Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

Improved Implementation of Point Location in General Two-Dimensional Subdivisions

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in CGAL, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph G is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(log n) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.Comment: 21 page

S^1 \times S^2 as a bag membrane and its Einstein-Weyl geometry

In the hybrid skyrmion in which an Anti-de Sitter bag is imbedded into the skyrmion configuration a S^{1}\times S^{2} membrane is lying on the compactified spatial infinity of the bag [H. Rosu, Nuovo Cimento B 108, 313 (1993)]. The connection between the quark degrees of freedom and the mesonic ones is made through the membrane, in a way that should still be clarified from the standpoint of general relativity and topology. The S^1 \times S^2 membrane as a 3-dimensional manifold is at the same time a Weyl-Einstein space. We make here an excursion through the mathematical body of knowledge in the differential geometry and topology of these spaces which is expected to be useful for hadronic membranesComment: 9pp in latex, minor correction

Anti de Sitter Holography via Sekiguchi Decomposition

In the present paper we start consideration of anti de Sitter holography in the general case of the (q+1)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for q=3 the new element in the present paper is the presentation of the bulk space as the homogeneous space G/H = SO(q,2)/SO(q,1), which homogeneous space was studied by Sekiguchi.Comment: 10 pages, to appear in the Proceedings of the XI International Workshop "Lie Theory and Its Applications in Physics", (Varna, Bulgaria, June 2015

Symplectic connections and Fedosov's quantization on supermanifolds

A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.Comment: Submitted to J. of Phys. Conf. Se

Conformal Maxwell theory as a singleton field theory on AdS_5, IIB three-branes and duality

We examine the boundary conditions associated with extended supersymmetric Maxwell theory in 5-dimensional anti-De Sitter space. Excitations on the boundary are identical to those of ordinary 4-dimensional conformal invariant super electrodynammics. Extrapolations of these excitations give rise to a 5-dimensional topological gauge theory of the singleton type. The possibility of a connection of this phenomenon to the world volume theory of 3-branes in IIB string theory is discussed.Comment: 19 pages, TeX, no figures; v2: misprints corrected, references added, discussion on Chern-Simons couplings revised. v3: References added, misprints corrected and a discussion in section 2 revised. v4: Typos corrected and reference adde