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
Author to whom correspondence should be addressed.
Received: 20 July 2018 / Revised: 5 August 2018 / Accepted: 6 August 2018 / Published: 8 August 2018
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
, BCH expansion reduces to the tractable closed-form expression
is explicitly given by the the function
This result is much more general than those usually presented for either the Heisenberg commutator,
, or the creation-destruction commutator,
. 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
as usual, we obtain the explicit result
We then indicate some potential applications.
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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MDPI and ACS Style
Van-Brunt, A.; Visser, M. Explicit Baker–Campbell–Hausdorff Expansions. Mathematics 2018, 6, 135.
Van-Brunt A, Visser M. Explicit Baker–Campbell–Hausdorff Expansions. Mathematics. 2018; 6(8):135.
Van-Brunt, Alexander; Visser, Matt. 2018. "Explicit Baker–Campbell–Hausdorff Expansions." Mathematics 6, no. 8: 135.
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