3
$\begingroup$

I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.

Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$

$$ \log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2),$$

but it doesn't provide a reference for this formula. Who discovered this representation of the BCH formula and does it hold in the general case for general $X$ and $Y$ or is this only valid in the case $[X,Y] = sY$?

Another question related to this, is there a name for when $[X,Y] = sY$?

A third question is, what is the proper formal Taylor expansion for the $\frac{\ad_X}{1-e^{-\ad_X}}$ term? Is this convergence of this formal Taylor expansion limited by any particular spectral radius? The wiki article does not seem to go into detail about that.

$\endgroup$
5
  • 1
    $\begingroup$ There is a nice discussion of this, not answering your questions perhaps, on terrytao.wordpress.com/tag/baker-campbell-hausdorff-formula $\endgroup$
    – Ben McKay
    Sep 26 at 15:38
  • 1
    $\begingroup$ @BenMcKay, re, there are three articles in that tag; did you mean Tao - The $C^{1, 1}$ Baker–Campbell–Hausdorff formula? $\endgroup$
    – LSpice
    Sep 26 at 16:21
  • 1
    $\begingroup$ I don't know any name for it, but the situation $[X, Y] = s Y$ is precisely the case when there is a representation of the Lie algebra of the so called “$a x + b$ group” $\left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\right\}$ sending $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ to $X$ and $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ to $Y$ … so perhaps that could be used to make up a name? $\endgroup$
    – LSpice
    Sep 26 at 16:24
  • 1
    $\begingroup$ See also the 2020 paper "Notes on the theorem of Baker-Campbell-Hausdorff-Dynkin" by Michael Mueger (math.ru.nl/~mueger/PDF/BCHD.pdf). $\endgroup$ Sep 26 at 18:32
  • 1
    $\begingroup$ And w.r.t. numerical integration of diff eqs, see the discussion surrounding eqns. 2.45 and 3.2 on pp. 33 and 38 of "Lie-group methods" by Iserles, Munthe-Kaas, Nørsett, and Zanna (damtp.cam.ac.uk/user/na/NA_papers/NA2000_03.pdf). $\endgroup$ Sep 26 at 18:56

1 Answer 1

4
$\begingroup$

$\DeclareMathOperator\ad{ad}$The identity $$\log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2)\tag{$*$}\label{star}$$ is due to Poincaré [1], who showed that $W(t)=\log (e^X e^{tY})$ solves the ODE $$W'(t)=\frac{\ad_{W(t)}}{1-e^{-\ad_{W(t)}}}Y,\;\;W(0)=X.$$ The identity \eqref{star} is the solution to first order in $t$. It holds generally, you don't need $[X,Y]=sY$.

[1] H. Poincaré, Sur les groupes continus, Trans. Cambridge Phil. Soc., 18, 220–255 (1900). JFM 31.0386.01

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.