Variant of Baker-Campbell-Hausdorff Formula

In summary, the conversation discusses the desire to find a clean/closed form version of the expression $$e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}$$ where ##X,Y,Z## are matrices that do not commute with each other. The use of the BCH identity to expand each term in the expression is mentioned, but there is difficulty in simplifying the result due to the lack of control over how the expressions in ##X## and ##Y## commute with ##Z##. Suggestions are made to approach the problem using nilpotent matrices of low degree.
  • #1
thatboi
127
18
TL;DR Summary
I want to evaluate ##e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}##
Hi all,
I was wondering if there was a clean/closed form version of the following expression: $$e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}$$
where ##X,Y,Z## are matrices that don't commute with each other. I know of the BCH identity ##e^{X}Ye^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] + \frac{1}{3!}[X,[X,[X,Y]]] + ...##
and I have used the identity to expand each term in my expression, but I cannot see a good way of cleaning up the result, I'm just left with a bunch of nested operators.
Any help would be appreciated, thanks!
 
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  • #2
thatboi said:
TL;DR Summary: I want to evaluate ##e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}##

Hi all,
I was wondering if there was a clean/closed form version of the following expression: $$e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}$$
where ##X,Y,Z## are matrices that don't commute with each other. I know of the BCH identity ##e^{X}Ye^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] + \frac{1}{3!}[X,[X,[X,Y]]] + ...##
and I have used the identity to expand each term in my expression, but I cannot see a good way of cleaning up the result, I'm just left with a bunch of nested operators.
Any help would be appreciated, thanks!
No way. What you are basically asking for is how to move the garbage on the left through ##Z## in order to annihilate the garbage on the right. Since you allow anything to happen by changing the sides of ##Z##, there is no way to make predictions.
 
  • #3
fresh_42 said:
No way. What you are basically asking for is how to move the garbage on the left through ##Z## in order to annihilate the garbage on the right. Since you allow anything to happen by changing the sides of ##Z##, there is no way to make predictions.
Could you elaborate on what you mean by "Since you allow anything to happen by changing the sides of ##Z## "?
 
  • #4
thatboi said:
Could you elaborate on what you mean by "Since you allow anything to happen by changing the sides of ##Z## "?
You have certain combined expressions in ##X## and ##Y## on the left and no control over how they commutate with ##Z##. ##e^X## and ##Z## aren't even in the same space.

I would approach this problem with nilpotent matrices of low degree, say the three-dimensional Heisenberg algebra for instance. Then with matrices of a bit increased degree of nilpotency. Maybe you can find a pattern. I will see if I can find something in the books.
 

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