# Anti-commutator version of Zassenhaus formula

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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

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