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