Baker-Campbell-Hausdorff formula

One direction in which improvements on the primitive propagator Eq. (19) can be found is to search for higher-order approximants, which allow one to obtain a given level of accuracy with a lower P. In this respect, an indispensable tool is the general Baker-Campbell-Hausdorff formula for two noncommutable operators A and B [66]... 

Splitting of the operators can be quite tricky. In the case of Eq. (6.113), the splitting is motivated by the observation that it is easy to implement exp(—iv(r)) in real space (at all points on the mesh), but it is difficult to take double derivatives at all points on the mesh. This difficulty can be overcome by using the fact that the derivatives can be implemented trivially in Fourier space. So, if somehow the Laplacian part could be split from w(r) part, then the progress can be made. This is the motivation for splitting the operator in Eq. (6.113) into derivative and nonderivative parts by Baker-Campbell-Hausdorff formula. For most practical purposes, a symmetric decomposition is carried out so that within an error of the order dt ... 

Using the Baker-Campbell-Hausdorff formula leads to the result that only terms corresponding to connected diagrams survive... 

The first term of eq (8) above is manifestly extensive, while the connectivity property of the second term requires a careful treatment, since this involves a product of two matrix-elements and may not have terms with common orbital labels in the two factors. Using the Baker-Campbell-Hausdorff formula for the product of exponentials, the second term can be written as... 

The last expression is the classical or canonical coherent state Iz). The Baker-Campbell-Hausdorff (BCH) formula, yields... 

This is the start of the celebrated Baker-Campbell-Hausdorff series [21] (often referred to as the BCH Lemma for short) which gives an explicit formula for the product of exponentials of non-commuting operators. 

The Baker-CampbeU-Hausdorff formula is a fundamental expansion in elementary Linear Algebra and Lie group theory (J. E. Campbell, Proc. London Math. Soc. 29, 14 (1898) H. F. Baker, Proc. London Math. Soc. 34, 347 (1902) F. Hausdorff, Ber. Verhandl. Saechs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 58, 19 (1906)). 

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