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Niveau: Supérieur, Doctorat, Bac+8

Renormalization Hopf algebras and combinatorial groups Alessandra Frabetti Universite de Lyon, Universite Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. May 5, 2008 Abstract These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory , held in Villa de Leyva (Colombia), July 2–20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the La- grangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar ?3 theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series.

Renormalization Hopf algebras and combinatorial groups Alessandra Frabetti Universite de Lyon, Universite Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. May 5, 2008 Abstract These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory , held in Villa de Leyva (Colombia), July 2–20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the La- grangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar ?3 theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series.

- algebra over
- group
- hopf algebra
- lectures given
- g?g ??
- hopf algebras
- euler-lagrange equation
- functions
- interacting quantum fields

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Published by | mijec |

Published | 01 November 1918 |

Reads | 36 |

Language | English |

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