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Campbell-Hausdorff formula

The Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to z = ln(exey) for non-commuting x and y.

Specifically, let G be a simply-connected Lie group with Lie algebra . Let the map exp: be an exponential map, defining,

The general formula is given by: .

Here ad(A) B = [A,B] 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 eZ = eX eY, we obtain a simpler formula:

.

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

References and external links

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