The Baker–Campbell–Hausdorff formula via mould calculus

Abstract

The well-known Baker–Campbell–Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product $$\text {e}^X \text {e}^Y$$ can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction to Ecalle’s mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin (Dokl Akad Nauk SSSR (NS) 57:323–326, 1947) explicit formula for the logarithm, as well as another formula recently obtained by Kimura (Theor Exp Phys 4:041A03, 2017) for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.