From dynamical systems to renormalization

TL;DR

This paper explores the use of logarithmic derivatives in graded Lie algebras to understand dynamical systems, revealing connections to quantum field theory and offering new methods for normal form construction.

Contribution

It generalizes the exp-log correspondence in Lie algebras using logarithmic derivatives and applies quantum field theory techniques to dynamical systems.

Findings

Logarithmic derivatives relate to normal form theory.

Birkhoff decomposition aids in constructing normal forms.

Analogies with quantum field theory provide new insights.

Abstract

We study in this paper logarithmic derivatives associated to derivations on graded complete 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.