Combinatorics of n-point functions via Hopf algebra in quantum field theory

We use a coproduct on the time-ordered algebra of field operators to derive simple relations between complete, connected and 1-particle irreducible n-point functions. Compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations. It may also be used to efficiently perform tree level computations.Comment: 26 pages, LaTeX + AMS + eepic; minor corrections and modification

Tree expansion in time-dependent perturbation theory

The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.Comment: 22 pages, 2 figure

Visualizing networks defined by links in Wikipedia articles

A Wikipedia é um dos portais mais populares da internet, contendo mais de 40 milhões de artigos em qualquer uma das línguas em que está disponível. Os artigos da Wikipedia referenciam outros artigos por meio de hiperligações. As hiperligações traduzem a ligação e interdependência entre artigos. Neste contexto, este trabalho apresenta uma aplicação para a visualização de um grafo de conhecimento definido por hiperligações entre artigos da Wikipedia Inglesa, partindo de um artigo inicial. Dado que, em geral, o número de hiperligações dos artigos da Wikipedia é muito elevado, a aplicação baseia-se num critério natural de seleção em função da sua relevância. Os nodos do grafo obtido têm hiperligações para artigos da Wikipedia, o que proporciona um modo alternativo de navegar na Wikipedia por meio de um grafo.Wikipedia is one of the most popular websites over the Internet with more than 40 million articles in any of the languages in which it is available. Links in Wikipedia articles target related articles. Links translate connections and dependencies upon Wikipedia articles. In this context, this work presents an application to visualize a knowledge graph defined by links in English Wikipedia articles, starting from a base one. Since, in general, the number of links in Wikipedia articles is very large, the application uses a natural criterion for selecting links in terms of their relevance. Moreover, the graph nodes have hyperlinks to Wikipedia articles which gives an alternative way to browse Wikipedia.info:eu-repo/semantics/publishedVersio

Combinatorics of 1-particle irreducible n-point functions via coalgebra in quantum field theory

We give a coalgebra structure on 1-vertex irreducible graphs which is that of a cocommutative coassociative graded connected coalgebra. We generalize the coproduct to the algebraic representation of graphs so as to express a bare 1-particle irreducible n-point function in terms of its loop order contributions. The algebraic representation is so that graphs can be evaluated as Feynman graphs

Generating loop graphs via Hopf algebra in quantum field theory

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.Comment: 22 pages, LaTeX + AMS + eepic; new section with alternative recursion formula added, further minor changes and correction

Generating connected and 2-edge connected graphs

We focus on the algorithm underlying the main result of [6]. This is an algebraic formula to generate all connected graphs in a recursive and e cient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequently to further classes of connected graphs, namely, 2-edge connected, simple and loopless graphs. Our method con- sists of basic graph transformations only