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arXiv · 1501.02506

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

Alexander Van-Brunt·Matt Visser
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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 $[a,a^\dagger]=I$.

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BibTeXRIS

Alexander Van-Brunt, Matt Visser. 2015-01-11. Special-case closed form of the Baker-Campbell-Hausdorff formula. https://doi.org/10.1088/1751-8113%2F48%2F22%2F225207

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