I Variant of Baker-Campbell-Hausdorff Formula

AI Thread Summary
The discussion centers on the challenge of simplifying the expression e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}, where X, Y, and Z are non-commuting matrices. The original poster seeks a closed form for this expression but struggles with the complexity of nested operators after applying the Baker-Campbell-Hausdorff (BCH) identity. Responses highlight the difficulty in manipulating the terms due to the lack of control over how the matrices commute with Z, suggesting that the problem may not have a straightforward solution. One suggestion involves exploring nilpotent matrices to identify potential patterns. Overall, the consensus is that simplifying the expression is highly complex due to the properties of the matrices involved.
thatboi
Messages
130
Reaction score
20
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!
 
Mathematics news on Phys.org
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.
 
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## "?
 
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.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top