On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

TL;DR

This paper explores convergence and prolongation issues of the Baker-Campbell-Hausdorff series in Banach algebras, providing counterexamples and analyzing conditions under which the series converges or can be extended.

Contribution

It offers new insights into the non-convergence and prolongation problems of the BCH series, including counterexamples and relations to recent physics results.

Findings

BCH series can fail to converge in Banach algebras.

Closed formulas can extend the BCH series beyond convergence.

Analytic prolongations can be singular even when the series converges.

Abstract

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent.