Explicit Baker–Campbell–Hausdorff Expansions

Abstract

The Baker–Campbell–Hausdorff (BCH) expansion is a general purpose tool of use in many branches of mathematics and theoretical physics. Only in some special cases can the expansion be evaluated in closed form. In an earlier article we demonstrated that whenever [X,Y]=uX+vY+cI, BCH expansion reduces to the tractable closed-form expression Z(X,Y)=ln(eXeY)=X+Y+f(u,v)[X,Y], where f(u,v)=f(v,u) is explicitly given by the the function f(u,v)=(u−v)eu+v−(ueu−vev)uv(eu−ev)=(u−v)−(ue−v−ve−u)uv(e−v−e−u). This result is much more general than those usually presented for either the Heisenberg commutator, [P,Q]=−iℏI, or the creation-destruction commutator, [a,a†]=I. In the current article, we provide an explicit and pedagogical exposition and further generalize and extend this result, primarily by relaxing the input assumptions. Under suitable conditions, to be discussed more fully in the text, and taking LAB=[A,B] as usual, we obtain the explicit result ln(eXeY)=X+Y+Ie−LX−e+LYI−e−LXLX+I−e+LYLY[X,Y]. We then indicate some potential applications.

Keywords

• Lie algebras
• matrix exponentials
• matrix logarithms
• Baker–Campbell–Hausdorff (BCH) formula
• commutators
• creation-destruction algebra
• Heisenberg commutator