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Mathematics 2018, 6(8), 135; https://doi.org/10.3390/math6080135

Explicit Baker–Campbell–Hausdorff Expansions

School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
Current address: Research Institute of Mathematical Sciences, Kyoto University, Sakyo Ward, Kyoto, Kyoto Prefecture 606-8317, Japan
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Received: 20 July 2018 / Revised: 5 August 2018 / Accepted: 6 August 2018 / Published: 8 August 2018
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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)=(uv)eu+v(ueuvev)uv(euev)=(uv)(uevveu)uv(eveu). This result is much more general than those usually presented for either the Heisenberg commutator, [P,Q]=iI, 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+IeLXe+LYIeLXLX+Ie+LYLY[X,Y]. We then indicate some potential applications. View Full-Text
Keywords: Lie algebras; matrix exponentials; matrix logarithms; Baker–Campbell–Hausdorff (BCH) formula; commutators; creation-destruction algebra; Heisenberg commutator Lie algebras; matrix exponentials; matrix logarithms; Baker–Campbell–Hausdorff (BCH) formula; commutators; creation-destruction algebra; Heisenberg commutator
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Van-Brunt, A.; Visser, M. Explicit Baker–Campbell–Hausdorff Expansions. Mathematics 2018, 6, 135.

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