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In mathematics, the Baker–Campbell–Hausdorff formula is the solution to ::: for possibly noncommutative and in the Lie algebra of a Lie group. This formula tightly links Lie groups to Lie algebras by expressing the logarithm of the product of two Lie group elements as a Lie algebra element using only Lie algebraic operations. The solution on this form, whenever defined, means that multiplication in the group can be expressed entirely in Lie algebraic terms. The solution on another form is straightforward to obtain; one just substitutes the power series for and in the equation and rearranges.〔 See equation (2) in section 1.3.〕 The point is to express the solution in Lie algebraic terms. This occupied the time of several prominent mathematicians. The formula is named after Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff who discovered its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of the Lie correspondence and in quantum field theory. It was first noted in print by Campbell〔J. Campbell, ''Proc Lond Math Soc'' 28 (1897) 381–390; ibid 29 (1898) 14–32.〕 (1897); elaborated by Henri Poincaré〔H. Poincaré, ''Compt Rend Acad Sci Paris'' 128 (1899) 1065–1069; ''Camb Philos Trans'' 18 (1899) 220–255.〕 (1899) and Baker (1902);〔H. Baker, ''Proc Lond Math Soc'' (1) 34 (1902) 347–360; ibid (1) 35 (1903) 333–374; ibid (Ser 2) 3 (1905) 24–47.〕 and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906).〔F. Hausdorff, "Die symbolische Exponentialformel in der Gruppentheorie", Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.〕 The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947).〔 p. 23〕 ==The Campbell–Baker–Hausdorff formula: existence== The Campbell–Baker–Hausdorff formula implies that if ''X'' and ''Y'' are in some Lie algebra defined over any field of characteristic 0, then :: , can, possibly with conditions on , , and ,〔For an explicit set of convergence criteria, see Matrix Lie group illustration below.〕 be written as a formal infinite sum of elements of . For many applications, one does not need an explicit expression for this infinite sum, but merely assurance of its existence, like, for instance, in this〔 Formulae 3.1, 3.2 and 3.3 modified for physics convention are used.〕 construction of a Lie group representation from a Lie algebra representation. Existence can be seen as follows. The ring :: of all non-commuting formal power series in non-commuting variables ''X'' and ''Y'' has a ring homomorphism Δ from ''S'' to the completion of ::, called the coproduct, such that :: and likewise for ''Y''. (The definition of the coproduct is extended recursively by the rule ). This has the following properties: *exp is an isomorphism (of sets) from the elements of ''S'' with constant term 0 to the elements with constant term 1, with inverse log * is grouplike (this means ) if and only if ''s'' is primitive (this means ). *The grouplike elements form a group under multiplication. *The primitive elements are ''exactly the formal infinite sums of elements of the Lie algebra'' generated by ''X'' and ''Y''. (Friedrichs' theorem 〔N. Jacobson, ''Lie Algebras'', John Wiley & Sons, 1966.〕) The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows:〔 The elements ''X'' and ''Y'' are primitive, so exp(''X'') and exp(''Y'') are group like; so their product is also group like; so its logarithm is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated by ''X'' and ''Y''. The universal enveloping algebra of the free Lie algebra generated by ''X'' and ''Y'' is isomorphic to the algebra of all non-commuting polynomials in ''X'' and ''Y''. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring ''S'' used above is just a completion of this Hopf algebra. An alternative, remarkably direct and concise, recursive proof that all homogeneous polynomials in are in the Lie algebra is due to Eichler.〔Eichler, M. (1968). "A new proof of the Baker-Campbell-Hausdorff formula", ''J. Math. Soc. Japan'' 20, 23-25. (online ) open access.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Baker–Campbell–Hausdorff formula」の詳細全文を読む スポンサード リンク
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