The Baker-Campbell-Hausdorff formula via mould calculus

TL;DR

This paper introduces mould calculus as a new approach to derive the Baker-Campbell-Hausdorff formula, providing concise proofs and exploring generalizations to multiple exponentials.

Contribution

It offers a novel, simplified proof of the BCH formula using mould calculus and connects it with recent formulas for exponential products, extending to multiple exponentials.

Findings

Short proof of BCH formula via mould calculus

Connection between BCH and Kimura's formula

Generalization to products of multiple exponentials

Abstract

The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{\'o} Ecalle's mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] 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.