- #1

thatboi

- 127

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- 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!

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!