Special-case closed form of the Baker–Campbell–Hausdorff formula

Abstract

The Baker–Campbell–Hausdorff formula is a general result for the quantity , where X and Y are not necessarily commuting. For completely general commutation relations between X and Y, (the free Lie algebra), the general result is somewhat unwieldy. However in specific physics applications the commutator , while non-zero, might often be relatively simple, which sometimes leads to explicit closed form results. We consider the special case , and show that in this case the general result reduces to Z ( X , Y ) = ln ( e X e Y ) = X + Y + f ( u , v ) [ X , Y ] . ?>Furthermore we explicitly evaluate the symmetric function , demonstrating that f ( u , v ) = ( u − v ) e u + v − ( ue u − ve v ) uv ( e u − e v ) , ?>and relate this to previously known results. For instance this result includes, but is considerably more general than, results obtained from either the Heisenberg commutator or the creation-destruction commutator .