2015/05/18 by Alexander Van-Brunt, Alexander Van–Brunt, Matt Visser
Mathematics · Physics and Astronomy · #Algebra over a field #Algorithm #BCH code #Combinatorics #Commutator #Decoding methods #Exposition (narrative) #Fractional Differential Equations Solutions #Function (biology) #Hausdorff space #Mathematical physics #Mathematics #Numerical methods for differential equations #Physics #Pure mathematics #Quantum chaos and dynamical systems #Series (stratigraphy) #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.3390/math6080135
published as Mathematics 6 (2018) 135 · 15 pages
arxiv created 2015/05/18 · openalex publication_date 2018/08/08 · arxiv updated 2018/08/16 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/30
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)=(u−v)eu+v−(ueu−vev)uv(eu−ev)=(u−v)−(ue−v−ve−u)uv(e−v−e−u). This result is much more general than those usually presented for either the Heisenberg commutator, [P,Q]=−iℏI, 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+Ie−LX−e+LYI−e−LXLX+I−e+LYLY[X,Y]. We then indicate some potential applications.