Closed forms of the Zassenhaus formula

Abstract

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 X ˆ and Y ˆ satisfy the relation [ X ˆ , Y ˆ ] = u X ˆ + v Y ˆ + c 𝟙 . Such an expression boils down to three equivalent versions, a left-sided, a centered and a right-sided formula: e X ˆ + Y ˆ = e X ˆ e Y ˆ e g r ( u , v ) [ X ˆ , Y ˆ ] = e X ˆ e g c ( u , v ) [ X ˆ , Y ˆ ] e Y ˆ = e g ℓ ( u , v ) [ X ˆ , Y ˆ ] e X ˆ e Y ˆ , with respective arguments, g r ( u , v ) = g c ( v , u ) e u = g ℓ ( v , u ) = u e u − v − e u + v e u − 1 v u ( u − v ) for u ≠ v and g r ( u , u ) = u + 1 − e u u 2 w i t h g r ( 0 , 0 ) = − 1 / 2. With additional special case g r ( 0 , v ) = − e − v − 1 + v v 2 , g r ( u , 0 ) = e u ( 1 − u ) − 1 u 2 .