Campbell-Hausdorff formula

From Academic Kids

In mathematics, the Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to

z = ln(e^xe^y)

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

e^Z = e^X e^Y,

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>

[edit]

References

L. Corwin & F.P Greenleaf (1990) Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples,
- (ISBN 052136034X)

[edit]