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 . Let

exp:

be the exponential map, defining

The general formula is given by:

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 .

The first few terms are well-known:

There is no expression in closed form.

For a matrix Lie algebra 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,

When we solve for Z in

e^Z = e^X e^Y,

we obtain a simpler formula:

.

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

• z₁ = X + Y

References

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

External link

• MathWorld page