Explicit Baker-Campbell-Hausdorff formulae for some specific Lie algebras

Abstract

In a previous article, [arXiv:1501.02506, JPhysA {\bf48} (2015) 225207], we demonstrated that whenever $[X,Y]=uX+vY+cI$ the Baker-Campbell-Hausdorff formula reduces to the tractable closed-form expression \[ Z(X,Y)=\ln( e^X e^Y ) = X+Y+ f(u,v) \; [X,Y], \] where $f(u,v)=f(v,u)$ is explicitly given by \[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)} = {(u-v)-(ue^{-v}-ve^{-u})\over u v (e^{-v} - e^{-u})}. \] This is much more general than the results usually presented for either the Heisenberg commutator $[P,Q]=-i\hslash I$, or the creation-destruction commutator $[a,a^{†}]=I$. In the current article we shall further generalize and extend this result, primarily by relaxing the input assumptions. We shall work with the structure constants $f_{ab}{}^c$ of the Lie algebra, (defined by $[T_a,T_b]=f_{ab}{}^c{T}_c$), and identify suitable constraints one can place on the structure constants to make the Baker--Campbell--Hausdorff formula tractable. We shall also develop related results using the commutator sub-algebra $[\mathfrak{g},\mathfrak{g}]$ of the relevant Lie algebra $\mathfrak{g}$. Under suitable conditions, and taking $L_{A}B=[A,B]$ as usual, we shall demonstrate that \[ \ln( e^X e^Y ) = X + Y + {I \over e^{-L_X} - e^{+L_Y} } \left( {I-e^{-L_X}\over L_X} + {I-e^{+L_Y}\over L_Y} \right) [X,Y]. \]

15 pages

References in corpus (9)

• An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
• Special-case closed form of the Baker-Campbell-Hausdorff formula
• An algorithm for the Baker-Campbell-Hausdorff formula
• Closed Form of the Baker-Campbell-Hausdorff Formula for the Generators of Semisimple Complex Lie Algebras
• Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas
• Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra
• Simplifying the Reinsch algorithm for the Baker-Campbell-Hausdorff series
• The Baker-Campbell-Hausdorff formula in the free metabelian Lie algebra
• Wilson Loops on Riemann Surfaces, Liouville Theory and Covariantization of the Conformal Group

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