Closed forms of the Zassenhaus formula

Abstract

The Zassenhaus formula finds many applications in theoretical physics or 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 expressed as the product of exponentials of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which takes generally the form of an infinite product of exponentials. Such a procedure is often referred to as ‘disentanglement’. However, for some special commutators, closed forms can be found. In this work, we propose a closed form for the Zassenhaus formula when the commutator of operators and satisfy the relation . Such an expression boils down to three equivalent versions, a left-sided, a centered and a right-sided formula: with respective arguments, for and With additional special case