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

Two disjoint aspects of the deformation programme: quantizing Nambu mechanics; singleton physics

We present briefly the deformation philosophy and indicate, with references, how it was applied to the quantization of Nambu mechanics and to particle physics in anti De Sitter space.Comment: 4 pages; to be published with AIP Press in Proceedings of the 1998 Lodz conference "Particles, Fields and Gravitation". LaTeX (compatibility mode) with aipproc styl

Masslessness in nn-dimensions

We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincar\'e'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De Sitter'' subgroups ${\bar{SO}}_0(1, n)$ and ${\bar{SO}}_0(2, n-1)$ (they remain irreducible or decompose into a sum of two) and the contraction of the latter to ``Poincar\'e''. Then we discuss the notion of masslessness in $n$ dimensions and compare the situation for general $n$ with the well-known case of 4-dimensional space-time, showing the specificity of the latter.Comment: 34 pages, LaTeX2e, 1 figure. To be published in Reviews in Math. Phy

Scaling transformation and probability distributions for financial time series

The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.Comment: 10 pages, 4 figures, latex and ps file

Nambu mechanics, nn-ary operations and their quantization

We start with an overview of the "generalized Hamiltonian dynamics" introduced in 1973 by Y. Nambu, its motivations, mathematical background and subsequent developments -- all of it on the classical level. This includes the notion (not present in Nambu's work) of a generalization of the Jacobi identity called Fundamental Identity. We then briefly describe the difficulties encountered in the quantization of such $n$-ary structures, explain their reason and present the recently obtained solution combining deformation quantization with a "second quantization" type of approach on ${\Bbb R}^n$. The solution is called "Zariski quantization" because it is based on the factorization of (real) polynomials into irreducibles. Since we want to quantize composition laws of the determinant (Jacobian) type and need a Leibniz rule, we need to take care also of derivatives and this requires going one step further (Taylor developments of polynomials over polynomials). We also discuss a (closer to the root, "first quantized") approach in various circumstances, especially in the case of covariant star products (exemplified by the case of su(2)). Finally we address the question of equivalence and triviality of such deformation quantizations of a new type (the deformations of algebras are more general than those considered by Gerstenhaber).Comment: 23 pages, LaTeX2e with the LaTeX209 option. To be published in the proceedings of the Ascona meeting. Mathematical Physics Studies, volume 20, Kluwe