2015/01/31 by Alexander Van-Brunt, Matt Visser · 1 citation
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP #quant-ph
paper · pdf · doi:10.1088/1751-8113/48/22/225207
published as Journal of Physics A 48 (2015) 225207 · V1: 5 pages. V2: 4 references added, some minor typos fixed, some discussion added. No change in conclusions. Now 6 pages. This version accepted for publication in Journal of Physics A: Mathematical and Theoretical
arxiv created 2015/04/22 · arxiv updated 2015/05/19
The Baker-Campbell-Hausdorff formula is a general result for the quantity Z(X,Y)=ln( eX eY ), 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( eX eY ) = 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)eu+v-(ueu-vev)\over u v (eu - ev), 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]=-iℏ I or the creation-destruction commutator [a,a^†]=I.