There are various ways of writing the formula, but all ultimately yield an expression for
in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in
The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.
Modern expositions of the formula can be found in, among other places, the books of Rossmann[1] and Hall.
An earlier statement of the form was adumbrated by Friedrich Schur in 1890[3] where a convergent power series is given, with terms recursively defined.
[4] 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[5] (1897); elaborated by Henri Poincaré[6] (1899) and Baker (1902);[7] and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906).
[8] The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947).
(See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book,[2] where the precise coefficients play no role in the argument.)
The following general combinatorial formula was introduced by Eugene Dynkin (1947),[13][14]
A complete elementary proof of this formula can be found in the article on the derivative of the exponential map.
the Lie algebra is the tangent space of the identity I, and the commutator is simply [X, Y] = XY − YX; the exponential map is the standard exponential map of matrices,
using the series expansions for exp and log one obtains a simpler formula:
is expressible as a combination of commutators was shown in an elegant, recursive way by Eichler.
[21]) This simple example illustrates that the various versions of the Baker–Campbell–Hausdorff formula, which give expressions for Z in terms of iterated Lie-brackets of X and Y, describe formal power series whose convergence is not guaranteed.
Thus, if one wants Z to be an actual element of the Lie algebra containing X and Y (as opposed to a formal power series), one has to assume that X and Y are small.
This is the degenerate case used routinely in quantum mechanics, as illustrated below and is sometimes known as the disentangling theorem.
This result is behind the "exponentiated commutation relations" that enter into the Stone–von Neumann theorem.
[This infinite series may or may not converge, so it need not define an actual element Z in
(The definition of Δ is extended to the other elements of S by requiring R-linearity, multiplicativity and infinite additivity.)
is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated by X and Y.
where the exponents of higher order in t are likewise nested commutators, i.e., homogeneous Lie polynomials.
The following identity (Campbell 1897) leads to a special case of the Baker–Campbell–Hausdorff formula.
Denote with AdA for fixed A ∈ G the linear transformation of g given by AdAY = AYA−1.
This variation is commonly used to write coordinates and vielbeins as pullbacks of the metric on a Lie group.
A special case of the Baker–Campbell–Hausdorff formula is useful in quantum mechanics and especially quantum optics, where X and Y are Hilbert space operators, generating the Heisenberg Lie algebra.
Specifically, the position and momentum operators in quantum mechanics, usually denoted
This "exponentiated commutation relation" does indeed hold, and forms the basis of the Stone–von Neumann theorem.
A related application is the annihilation and creation operators, â and â†.
The degenerate Baker–Campbell–Hausdorff formula is frequently used in quantum field theory as well.