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

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 $[a,a^\dagger]=I$.

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