Variant of Baker-Campbell-Hausdorff Formula

[thatboi]

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!

 

[fresh_42]

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.

 

[thatboi]

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 $Z$ "?

 

[fresh_42]

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.