从动力系统到重整化
From dynamical systems to renormalization
摘要：
本文研究了分次完备李代数上导子的对数导数及其逆的存在性。这些对数导数，当可逆时，推广了李代数和它的李群之间的exp-log对应。这种对应关系在动力系统的研究中很自然地出现，当处理向量场的线性化时，共振向量场的非线性化对应于对数导数的不可逆性和规范形的存在性。这些概念，源于理论的动力系统，可以改写的抽象设置的李代数和相同的困难，在微扰量子场论（pQFT）出现在这里。令人惊讶的是，人们可以采用与pQFT相同的思想，并取得丰硕的成果，例如在Birkhoff分解的帮助下构造新的规范形式。这个类比甚至更进一步（反项的局部性，重整化方案的选择），并将导致动力系统和量子场论之间更多的相互作用。
英文摘要：
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.
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