# Anti-commutator version of Zassenhaus formula

+ 0 like - 0 dislike
203 views

The Zassenhaus formula goes like $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \cdots,$$ where $X$ and $Y$ are operators may not commute.

Do people know and derive the anti-commutator version of Zassenhaus formula that expresses in terms of anti-commutator $\{X,Y\}$ in a very compact form?

I haven't found it on the literature (yet).

(1) Let us consider Grassmann-parity of $X$ and $Y$ are both even, so that $X$ and $Y$ both contain even number of fermionic operators $f/f^\dagger$, where $$\{f_i,f_j^\dagger\}=\delta_{ij}$$

(2) What if $X$ or $Y$ contain an odd number of fermionic operators?

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user annie marie heart
Such an expansion is (in general) impossible.

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user AccidentalFourierTransform
There cannot be any such general formula unless you specify some special properties of $X$ and $Y$. The formula for a commutator exists because it is associated with adjoint action of a group.

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user Prahar
What's the Grassmann-parity of $X$ and $Y$?

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user Qmechanic
Let us consider Grassmann-parity of X and Y are both even.

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user annie marie heart
$\{f_i,f_j^\dagger\}=\delta_{ij}$ is then not relevant for $\{X,Y\}$.

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user Qmechanic
In that even parity case, one can expand in terms of the sub-operators.

This post imported from StackExchange Physics at 2020-10-30 22:42 (UTC), posted by SE-user annie marie heart

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.