A simple expression for the terms in the Baker–Campbell–Hausdorff series

Abstract

A simple expression is derived for the terms in the Baker-Campbell-Hausdorff series. One formulation of the result involves a finite number of operations with matrices of rational numbers. Generalizations are discussed.

The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin

“…-to Physics, especially with the consolidation of Quantum and Statistical Mechanics-see the early applications from the 1960s and 1970s: Białynicki-Birula et al (1969), Dragt and Finn (1976), Eriksen (1968), Friedrichs (1953), Gilmore (1974), Kumar (1965), Mielnik and Plebański (1970), Murray (1962), Suzuki (1977), Wei (1963), Weiss and Maradudin (1962), Wichmann (1961), and Wilcox (1967); see also the numerous recent papers in Physics journals related to CBHD: Blanes et al (1998), Blanes et al (2009), Bose (1989), Casas (2007), Casas and Murua (2009), Klarsfeld and Oteo (1989a), Klarsfeld and Oteo (1989b), Kobayashi et al (1998), Kolsrud (1993), Moan (1988), Moan and Noteo (2001), Oteo (1991), and Reinsch (2000); and finally see the investigations concerning the relevant convergence and optimization problems: Blanes and Casas (2004), Day et al (1991), Moan and Niesen (2008), Newman et al (1989), Richtmyer and Greenspan (1965), and Thompson (1982Thompson ( , 1989; -to Group Theory: Magnus (1950), Magnus et al (1966), and Michel (1976); -to the Analysis of Linear PDEs: See the seminal works from the 1970s and 1980s Folland (1975), Folland and Stein (1982), …”

Computing the Baker–Campbell–Hausdorff series and the Zassenhaus product

“…The present paper addresses this problem in some detail. To this end, in Sections 2 and 3, we first review various methods given in the literature [4][5][6][7][8][9][10][11][12] and compare their efficiency using our own computer implementations (see Section 4). In Section 3 a new efficient method is presented for the Zassenhaus terms.…”

Robust transmission of non-Gaussian entanglement over optical fibers

“…This occurs because of the following identities. First, it is simple to show, using the Baker-Campbell-Hausdorff (BCH) formula [26] that…”