The algebra of complex 2 × 2 matrices and a general closed Baker–Campbell–Hausdorff formula

Abstract

We derive a closed formula for the Baker–Campbell–Hausdorff series expansion in the case of complex matrices. For arbitrary matrices A and B, and a matrix Z such that , our result expresses Z as a linear combination of A and B, their commutator , and the identity matrix I. The coefficients in this linear combination are functions of the traces and determinants of A and B, and the trace of their product. The derivation proceeds purely via algebraic manipulations of the given matrices and their products, making use of relations developed here, based on the Cayley–Hamilton theorem, as well as a characterization of the consequences of and/or its determinant being zero or otherwise. As a corollary of our main result we also derive a closed formula for the Zassenhaus expansion. We apply our results to several special cases, most notably the parametrization of the product of two matrices and a verification of the recent result of Van-Brunt and Visser (2015 J. Phys. A: Math. Theor. 48 225207) for complex matrices, in this latter case deriving also the related Zassenhaus formula which turns out to be quite simple. We then show that this simple formula should be valid for all matrices and operators.