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

Abstract

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) on the number of non-zero Goldberg coefficients. Unfortunately, we shall see that the number of nonzero Goldberg coefficients often grows very rapidly (in fact exponentially) with the word length |w|. Furthermore, the simplified Reinsch algorithm readily generalizes to many closely related but still quite distinct problems—we shall also present closely related results for the symmetric product S(X,Y)=lneX/2eYeX/2=∑wgS(w)w(X,Y). Variations on such themes are straightforward. For instance, one can just as easily consider the “loop” product L(X,Y)=lneXeYe−Xe−Y=∑wgL(w)w(X,Y). This “loop” type of series is of interest, for instance, when considering either differential geometric parallel transport around a closed curve, non-Abelian versions of Stokes’ theorem, or even Wigner rotation/Thomas precession in special relativity. Several other closely related series are also briefly investigated.