From dynamical systems to renormalization

In this paper we study logarithmic derivatives associated to derivations on completed graded Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a Lie algebra and its Lie group. Such correspondences occur na...

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Bibliographic Details
Published in:Journal of mathematical physics 2013-09, Vol.54 (9), p.1
Main Author: Menous, Frédéric
Format: Article
Language:English
Subjects:
Algebra
DECOMPOSITION
DIFFERENTIAL EQUATIONS
INTERACTIONS
LIE GROUPS
LOCALITY
MATHEMATICAL METHODS AND COMPUTING
PERTURBATION THEORY
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
QUANTUM FIELD THEORY
RENORMALIZATION
VECTOR FIELDS
Vector space
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Summary:In this paper we study logarithmic derivatives associated to derivations on completed graded Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a Lie algebra and its Lie group. Such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms. These concepts, stemming from the theory of dynamical systems, can be rephrased in the abstract setting of Lie algebra and the same difficulties as in perturbative quantum field theory (pQFT) arise here. Surprisingly, one can adopt the same ideas as in pQFT with fruitful results such as new constructions of normal forms with the help of the Birkhoff decomposition. The analogy goes even further (locality of counter terms, choice of a renormalization scheme) and shall lead to more interactions between dynamical systems and quantum field theory.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4820375