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

TL;DR

This paper derives a closed-form expression for the Baker-Campbell-Hausdorff formula when the commutator of two operators is a linear combination of the operators plus a scalar, simplifying calculations in specific physics contexts.

Contribution

It provides an explicit closed-form solution for the BCH formula under a specific linear commutation relation, extending previous known results.

Findings

Explicit formula for Z(X,Y) in the special case [X,Y]=uX+vY+cI

Derivation of the symmetric function f(u,v) for this case

Connection to known results like Heisenberg and creation-destruction commutators

Abstract

The Baker-Campbell-Hausdorff formula is a general result for the quantity $Z(X,Y)=\ln( e^X e^Y )$, 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 $[X,Y]$, while non-zero, might often be relatively simple, which sometimes leads to explicit closed form results. We consider the special case $[X,Y] = u X + vY + cI$, 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 $f(u,v)=f(v,u)$, demonstrating that \[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)}, \] 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 $[P,Q]=-i\hbar I$ or the creation-destruction commutator $[a,a^\dagger]=I$.