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Events
Mini-Workshop on renormalization and number theory

MPI für Mathematik in den Naturwissenschaften Leipzig

A3 01 (Sophus-Lie room)

conference

20/06/2005 20/06/2005

Mini-Workshop on renormalization and number theory

Recent progress in the understanding of the combinatorial and mathematical aspects of renormalization of quantum fields have been achieved by new methods using algebraic tools as Hopf algebras, Rota-Baxter algebras and alike. At this workshop, the focus is on such approaches with an emphasis on recent work in relation to number theoretical aspects of renormalization. Motivic Galois theory may provide an interesting tool studying the relation between number theoretic and quantum field theoretic problems.

Program

09:00 - 10:00	Matilde Marcolli (MPI für Mathematik, Bonn, Germany) Renormalization and motivic Galois theory In this joint work with Alain Connes we investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain "motivic Galois group", which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, it is a semi-direct product by the multiplicative group of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over the field of rational numbers, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.
10:30 - 11:30	Alessandra Frabetti (Universite' Lyon 1, Villeurbanne, France) Renormalization "Ã la Dyson" and Hopf algebras This talk has two aims: on one side to sketch the long way from elementary particles to combinatorial Hopf algebras, through an overview accessible to non-specialists. On the other side, to present a very personal mathematical interpretation of the renormalization formulas proved by Freeman Dyson in '49 for quantum electrodynamics, which fully justify the appearance of Faà di Bruno and Connes-Kreimer Hopf algebras in this context.
14:00 - 15:00	Jaques Magnen (Ecole Polytechnique, Palaiseau, France) Perturbation theory and the renormalization group The talk consists in a description of the perturbation expansion in terms of the effective parameters and where the coefficients are regularized diagramms. It will be argued that this expansion is the most satisfactory both physically and for computationnal purposes.
15:30 - 16:30	Kurusch Ebrahim-Fard (Rheinische Friedrich-Wilhelms Universität, Bonn, Germany) Rota-Baxter Algebras: Factorizing Feynman Rules A Rota-Baxter algebra is an algebra with a linear operator R satisfying the identity $R(x)R(y)=R(R(x)y)+R(xR(y))-qR(xy)$, where q is a constant. It first occurred in the work of Glen Baxter in probability, and was popularized by the work of Gian-Carlo Rota and Pierre Cartier. They recently appeared in connection with the seminal work of Connes-Kreimer on renormalization theory in pQFT, Loday's dendriform operads, and associative analogs of the (modified) classical Yang-Baxter equation. We will review these results in some detail with an emphasis on the factorization problem underlying the Birkhoff decomposition of Hopf algebra characters.
17:00 - 18:00	Bertfried Fauser (MPI für Mathematik in den Naturwissenschaften, Leipzig, Germany) (Co)Arithmetic: Can numbers help with the theory of quanta We start with the arithmetic of natural numbers (eventually completed to form the integers) and introduce the concept of coaddition, comultiplication, counit, antipode, etc. of arithmetic. A careful study of graded duality unveils relations which are fruitfully used on index sets in the theory of symmetric functions, quantum mechanics, and quantum field theory (QFT). The amazing occurance of number theoretic functions in QFT becomes natural. A pathway is opened to transport number theoretic methods into QFT and vice versa. We give arguments, that renormalization has to do with multiplicativity versus complete multiplicativity of (arithmetic number theoretical) functions, and that hard number theoretic problems and hard QFT problems emerge from the same structural source. Our main conjecture is, that the arithmetic structure developed is functorial and can be established in a large class of categories.

Mini-Workshop on renormalization and number theory

Program

Scientific Organizers

Bertfried Fauser

Eberhard Zeidler

Administrative Contact

Bertfried Fauser