Baker–Campbell–Hausdorff formula

In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {displaystyle Z} to the equation In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {displaystyle Z} in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X {displaystyle X} and Y {displaystyle Y} and iterated commutators thereof. The first few terms of this series are: where ' ⋯ {displaystyle cdots } ' indicates terms involving higher commutators of X {displaystyle X} and Y {displaystyle Y} . If X {displaystyle X} and Y {displaystyle Y} are sufficiently small elements of the Lie algebra g {displaystyle {mathfrak {g}}} of a Lie group G {displaystyle G} , the series is convergent. Meanwhile, every element g {displaystyle g} sufficiently close to the identity in G {displaystyle G} can be expressed as g = e X {displaystyle g=e^{X}} for a small X {displaystyle X} in g {displaystyle {mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {displaystyle G} —written as e X e Y = e Z {displaystyle e^{X}e^{Y}=e^{Z}} —can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence. If X {displaystyle X} and Y {displaystyle Y} are sufficiently small n × n {displaystyle n imes n} matrices, then Z {displaystyle Z} can be computed as the logarithm of e X e Y {displaystyle e^{X}e^{Y}} , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that Z := log ⁡ ( e X e Y ) {displaystyle Z:=log(e^{X}e^{Y})} can be expressed as a series in repeated commutators of X {displaystyle X} and Y {displaystyle Y} . Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall. The formula is named after Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff who stated its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. An earlier statement of the form was adumbrated by Friedrich Schur in 1890 where a convergent power series is given, with terms recursively defined. 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. Following Schur, it was noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902); and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947). The history of the formula is described in detail in the article of Achilles and Bonfiglioli and in the book of Bonfiglioli and Fulci. For many purposes, it is only necessary to know that an expansion for Z {displaystyle Z} in terms of iterated commutators of X {displaystyle X} and Y {displaystyle Y} exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, where the precise coefficients play no role in the argument.) A remarkably direct existence proof was given by Martin Eichler.; see also the 'Existence results' section below. In other cases, one may need detailed information about Z {displaystyle Z} and it is therefore desirable to compute Z {displaystyle Z} as explicitly as possible. Numerous formulas exist; we will describe two of the main ones (Dykin's formula and the integral formula of Poincaré) in this section. Let G be a Lie group with Lie algebra g {displaystyle {mathfrak {g}}} . Let