3,635 research outputs found
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
Braided groups and braided matrices are novel algebraic structures living in
braided or quasitensor categories. As such they are a generalization of
super-groups and super-matrices to the case of braid statistics. Here we
construct braided group versions of the standard quantum groups . They
have the same FRT generators but a matrix braided-coproduct \und\Delta
L=L\und\tens L where , and are self-dual. As an application, the
degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices
; it is a braided-commutative bialgebra in a braided category. As a
second application, we show that the quantum double D(\usl) (also known as
the `quantum Lorentz group') is the semidirect product as an algebra of two
copies of \usl, and also a semidirect product as a coalgebra if we use braid
statistics. We find various results of this type for the doubles of general
quantum groups and their semi-classical limits as doubles of the Lie algebras
of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)
Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support
Introduction to GPDs and TMDs
Generalised parton distributions (GPDs) and transverse momentum dependent
parton distributions (TMDs) describe complementary aspects of the
three-dimensional structure of hadrons. We discuss their relation to each other
and recall important theory results concerning their properties and their
connection with physical observables.Comment: 12 pages, 8 figures. v2: reference correcte
- …