Simplifying the Reinsch algorithm for the Baker–Campbell–Hausdorff series

The Goldberg version of the Baker–Campbell–Hausdorff series computes the quantity Z(X,Y)=lneXeY=∑wg(w)w(X,Y), where X and Y are not necessarily commuting in terms of “words” constructed from the {X, Y} “alphabet.” The so-called Goldberg coefficients g(w) are the central topic of this article. This Baker–Campbell–Hausdorff series is a general purpose tool of very wide applicability in mathematical physics, quantum physics, and many other fields. The Reinsch algorithm for the truncated series permits one to calculate the Goldberg coefficients up to some fixed word length |w| by using nilpotent (|w| + 1) × (|w| + 1) matrices. We shall show how to further simplify the Reinsch algorithm, making its implementation (in principle) utterly straightforward using “off the shelf” symbolic manipulation software. Specific computations provide examples which help to provide a deeper understanding of the Goldberg coefficients and their properties. For instance, we shall establish some strict bounds (and some equalities) ...