Maths - "Baker-Campbell-Hausdorff formula

There is a very useful identity, which is,

e(a+b) = ea * eb

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:

ec = ea * eb

where:

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

where:

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.

 


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