Baker-Campbell-Hausdorff formula

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In mathematics, the Baker-Campbell-Hausdorff formula is the solution to

$z = \log(e^x e^y)\,$

for non-commuting x and y. It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff. It was first noted in print by Campbell, elaborated by Henri Poincaré and Baker, and systematized by Hausdorff. The formula below was introduced by Eugene Dynkin.

1 Baker-Campbell-Hausdorff formula
2 See also
3 References
4 External links

[edit] Baker-Campbell-Hausdorff formula

Specifically, let G be a simply-connected Lie group with Lie algebra $\mathfrak g$ . Let

$\exp : \mathfrak g\rightarrow G$

be the exponential map, defining

$Z = X * Y = \log(\exp X\exp Y),\qquad X,Y,Z\in\mathfrak g.$

The general formula is given by:

$X * Y = \sum_{n>0}\frac {(-1)^{n-1}}{n}$

$\sum_{ \begin{matrix} & {r_i + s_i > 0} \\ & {1\le i \le n} \end{matrix}} \frac{(\sum_{i=1}^n (r_i+s_i))^{-1}}{r_1!s_1!\cdots r_n!s_n!} \times(\mbox{ad} X)^{r_1}(\mbox{ad} Y)^{s_1}\cdots (\mbox{ad} X)^{r_n}(\mbox{ad} Y)^{s_n - 1}Y.$

Here ad(A)B = [A,B] is the adjoint endomorphism. In terms in the sum where $s n = 0$ , the last three factors should be interpreted as $(\mbox{ad} X)^{r_n - 1} X$ .

The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie Algebra):

$X*Y = \,\!$ $X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]]$

$- \frac {1}{48}[Y,[X,[X,Y]]] - \frac{1}{48}[X,[Y,[X,Y]]] + \cdots$

There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications.

For example, note that if [X,Y] vanishes, then the above formula manifestly reduces to X +Y. If the commutator [X,Y] is a constant (central), then all but the first three terms on the right-hand-side of the above vanish.

If the one of the Lie algebra elements X maps the kernel of ad Y into itself, other forms of the Campbell-Baker-Hausdorff formula might serve well:

$X*Y = X + \frac{ad X e^{ad X}}{e^{ad X}-1} ~Y + O(Y^2)$ , as is evident from the second of the

integral formulas below. So, if the commutator is [X,Y]=s Y, for some non-zero s, this formula reduces to just Z = X + Y s/ (1−exp (−s)) , which then leads to braiding identities such as

$e^{X} e^{Y} = e^{\exp (s) ~Y} e^{X}$ .

There are numerous such well-known expressions applied routinely in physics (cf. Magnus).

For a matrix Lie group $G \sub \mbox{GL}(n,\mathbb{R})$ 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,

$\mbox{exp}\ X = e^X = \sum_{n=0}^{\infty}{\frac{X^n}{n!}}.$

When we solve for Z in

$e^Z = e^X e^Y,\,\!$

we obtain a simpler formula:

$Z = \sum_{n>0} \frac{(-1)^{n-1}}{n} \sum_{\begin{matrix} &{r_i+s_i>0} \\ & {1\le i\le n}\end{matrix}} \frac{X^{r_1}Y^{s_1}\cdots X^{r_n}Y^{s_n}}{r_1!s_1!\cdots r_n!s_n!}.$

We note that the first, second, third and fourth order terms are:

$z_1 = X + Y\,\!$

$z_2 = \frac{1}{2} (XY - YX)$

$z_3 = \frac{1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY)$

$z_4 = \frac{1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX).$

[edit] See also

Dyson series

[edit] References

Wolf Rossmann "Lie Groups: An Introduction through Linear Groups". Oxford University Press, 2002.
Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9
W. Miller, "Symmetry Groups and their Applications", Academic Press, New York, 1972, pp 159–161.
W. Magnus, Comm Pur Appl Math VII (1954) 649–673.
L. Corwin & F.P Greenleaf, Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples, Cambridge University Press, New York, 1990, ISBN 0-521-36034-X.
H. Baker, Proc Lond Math Soc (1) 34 (1902) 347–360; ibid (1) 35 (1903) 333–374; ibid (Ser 2) 3 (1905) 24–47.
J. Campbell, Proc Lond Math Soc 28 (1897) 381–390; ibid 29 (1898) 14–32.
F. Hausdorff, Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.
H. Poincaré, Compt Rend Acad Sci Paris 128 (1899) 1065–1069; Camb Philos Trans 18 (1899) 220–255.