Maths - "Baker-Campbell-Hausdorff formula

There is a very useful identity, which is,

e^(a+b) = e^a * e^b

However this only applies when 'a' and 'b' commute, so it applies when a or b is a scalar for instance.

The more general case where 'a' and 'b' don't necessarily commute is given by:

e^c = e^a * e^b

where:

c = a + b + a×b + 1/3(a×(a×b)+b×(b×a)) + ...

where:

• × = vector cross product

This is a series known as the Baker-Campbell-Hausdorff formula.

This shows that if when a and b become close to becoming parallel then a×b approaches zero and c approaches a + b so the rotation algebra approaches vector algebra.

Derivation

We will use the formula for an infinite series of an exponent:

e(q) =

∞ ∑ n=0

(q)n n!

So multiplying two of these together we get:

e(a) *e(b) =

∞ ∑ n=0

(a)n n!

∞ ∑ m=0

(b)m m!

Now we want to move the second summation outside the first part, but since the first part has already been summed 'n' times we need to substitute n with n-m to correct for that:

e(a) *e(b) =

∞ ∑ n=0

∞ ∑ m=0

(a)(n-m) (n-m)!

(b)m m!

I don't think what we have done so far has implied any commutation because the operands are still in the same order.

 

EuclideanSpace

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Prerequisites

On this page we discussed how to represent and evaluate functions where the domain and codomain are complex numbers.