Campbell-Hausdorff formula
|
|
In mathematics, the Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to
- z = ln(exey)
for non-commuting x and y. It is named for John Edward Campbell (1862-1924), H. F. Baker and Felix Hausdorff.
Specifically, let G be a simply-connected Lie group with Lie algebra <math>\mathfrak g\ <math>. Let
- exp: <math>\mathfrak g\rightarrow G <math>
be the exponential map, defining
- <math>Z = X * Y = \mbox{ln(exp}X\cdot\mbox{exp}Y\mbox{)}, \ X, Y\in\mathfrak g. <math>
The general formula is given by:
- <math>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.
<math>
Here
- ad(A)B = [A,B]
is the adjoint endomorphism.
In terms in the sum where <math>s_n = 0<math>, the last three factors should be interpreted as <math>(\mbox{ad} X)^{r_n - 1} X<math>.
The first few terms are well-known:
- <math>X*Y = X + Y + \frac {1}{2}[X,Y] - \frac {1}{12}[X,[Y,X]] - \frac {1}{12}[Y,[X,Y]] - \frac
{1}{48}[Y,[X[X,Y]]] - \frac{1}{48}
[X,[Y,[X,Y]]] + \mbox{(commutators of five and greater terms)}.<math>
There is no expression in closed form.
For a matrix Lie algebra <math> G\sub GL(n,\mathbb{R}), <math> 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,
- <math>\mbox{exp}\ X = e^X = \sum_{n=0}^{\infty}{\frac
{X^n}{n!}}. <math>
When we solve for Z in
- eZ = eX eY,
we obtain a simpler formula:
- <math> 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!}<math>.
We note that the first, second, third and fourth order terms are:
- <math>z_1 = X + Y<math>
- <math> z_2 = \frac
{1}{2} (XY - YX)<math>
- <math>z_3 = \frac
{1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY)<math>
- <math>z_4 = \frac
{1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX).<math>
References
- L. Corwin & F.P Greenleaf (1990) Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples,
External link
- MathWorld page (http://mathworld.wolfram.com/Baker-Campbell-HausdorffSeries.html)