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Combinatorial approach to Baker-Campbell-Hausdorff exponents

A survey of problems related to the Baker-CampbellHausdorff formula is given. The technique of effective « addition » of non-commuting exponents is outlined. The first part of the paper reviews the theory of the discrete BCH formula including its most recent developments. The second part presents a new effective approach to the continuous BCH-formula and yields a general algorithm for an arbitrary analytic function of the quantum theoretical evolution operator. The results are expressed in terms of new natural ordering operators containing the chronological and normal orderings as special cases.