Closed forms of Zassenhaus formula

Léonce Dupays

Published 2021-07-02Version 1

Zassenhaus formula is used in a wide range of fields in physics and mathematics, from fluid dynamics to differential geometry. The non commutativity of the elements of the algebra implies that the exponential of a sum of operators cannot be splitted in the product of exponential of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which form is generally an infinite product of exponentials. However, for some special commutators, closed forms can be found. We propose a closed form for Zassenhaus formula when the commutator of the operators $X$ and $Y$ satisfy the relation $$[X,Y]=uX+vY+c\mathbb{1}.$$ Zassenhaus formula then reduces to the closed form \begin{eqnarray} e^{X+Y}=e^{X}e^{Y}e^{g_{r}(u,v)[X,Y]}=e^{X}e^{g_{c}(u,v)[X,Y]}e^{Y}=e^{g_{l}(u,v)[X,Y]}e^{X}e^{Y}, \end{eqnarray} a left-sided, centered and right-sided formula are found, with respective arguments, \begin{eqnarray} g_{r}(u,v)=g_{c}(v,u)e^{u}=g_{l}(v,u)=\frac{u\left(e^{u-v}-e^{u}\right)+v\left(e^{u}-1\right)}{vu(u-v)}. \end{eqnarray}