Talk:Baker–Campbell–Hausdorff formula

Alternate form
This article should also provide the first few terms of W for the left-multiplication variant


 * $$e^{A+B}=e^A e^B e^W$$

as I'm too lazy to perform the inversion myself. I suppose, for completeness, the right-multiplication version as well. linas 16:20, 1 March 2006 (UTC)

I think in the formula for $$X*Y$$ the power of (-1) should be n-1 and not n

Misprint
I think there is misprint in the formula because ad(Y)(Y)=0


 * It looks strange, yes, but I looked it up and it is intentional. I slightly rewrote the formula to get something which is clearer in my opinion, and added an explanation. -- Jitse Niesen (talk) 05:06, 22 May 2007 (UTC)

Is there a misprint in the adjoint formula in the "selected tractable cases" section? I.e. isn't

[X,Y]=sY --> e^X Y e^{-X} = e^s Y,

rather than

e^X Y e^{-X} = Exp[e^s Y]

as written now? Similarly the equation just prior to this one. — Preceding unsigned comment added by 185.132.137.28 (talk) 18:05, 16 January 2018 (UTC)


 * You should have written your query as a separate question at the bottom of the page. No, the adjoint action is fine, but you misread it. Your expression for just Y is correct and trivial. But the l.h.s. of the equation you are misquoting has the exponential of Y. Had you  concatenated  out the Hadamard rule  for all powers of Y, i.e. all powers of the trivial expression you wrote, and summed over the expansion of the exponential, you would have readily derived the braiding identity given. Cuzkatzimhut (talk) 20:25, 16 January 2018 (UTC)

Thanks for the clarification. — Preceding unsigned comment added by 185.132.137.28 (talk) 13:50, 18 January 2018 (UTC)

Explicit formula
While the first few terms of the BCH formula...

X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] - \frac {1}{48}[Y,[X,[X,Y]]] - \frac{1}{48}[X,[Y,[X,Y]]] + \cdots $$ ...are correct, they're a bit misleading, because $$[Y,[X,[X,Y]]]=[X,[Y,[X,Y]]]$$. You can check this by working them both out in the universal enveloping algebra, where $$[X,Y] = XY-YX$$ (or by typing the whole thing into GAP). See [] for example. I'll change this in the article unless anyone comes up with a good reason not to —Preceding unsigned comment added by 163.1.148.158 (talk) 23:13, 10 December 2007 (UTC)
 * In case someone wanted to have GAP check, this is quite quick:

gap&gt; F:=FreeLieAlgebra(Rationals,["x","y"]);x:=F.1;y:=F.2; &lt;Lie algebra over Rationals, with 2 generators&gt; (1)*x (1)*y gap&gt; y*(x*(x*y))=x*(y*(x*y)); true
 * One could also use free associative algebras and actually compute commutators, but it is easier just to use a Lie algebra. Thanks for the interesting link.  For free groups there are nice bases for the lower central factors given by Hall's commutator formulas, and for free metabelian Lie algebras your link gives a basis.  Is there a nice basis for free Lie algebras?  I sort of thought Hall's method worked for free Lie algebras too.  If there is such a basis, it might be nice to expand in it. JackSchmidt (talk) 00:49, 11 December 2007 (UTC)
 * I believe Hall's basis for the free group lower central factors transfers directly to the free Lie algebra case, but I'm not sure of the details. It's interesting that the online Magma at [] answers "false" to both of the following

L:=FreeLieAlgebra(Rationals,2); (y,(x,(x,y)))eq (x,(y,(x,y))); z:=(x,y)+2*x; (x,(y,z))+(z,(x,y))+(y,(z,x)) eq 0; ...I think there is something wrong with it. 79.67.200.228 (talk) 01:31, 11 December 2007 (UTC)
 * This bug may be by design:

 The free Lie algebra LF(X) is spanned by M(X). However, the elements of this set are not linearly independent. It is a nontrivial problem to describe a basis of the free Lie algebra. One of several possibilities is the well-known Hall basis. Currently Magma does not support calculations involving bases of the free Lie algebra, as they are of little use for our main problem: the construction of a basis and multiplication table for a finitely-presented Lie algebra. ... Thus, mathematically speaking, in Magma rather than work in the free Lie algebra, we actually work in the free nonassociative anticommutative algebra.
 * There are other oddly named functions which have similar caveats. JackSchmidt (talk) 04:06, 11 December 2007 (UTC)

Definitions
What is adX? It appears to be the adjoint representation? If so, the definition (that comes well after the first usage) isn't consistent with that article, as far as I can tell? Petrelharp (talk) 05:03, 23 February 2011 (UTC)
 * I gather it is the Adjoint endomorphism, adX (Y) = [X,Y]. Cuzkatzimhut (talk) 12:19, 23 February 2011 (UTC)

Hadamard's(?) Lemma
Anyone know why this is called Hadamard's lemma? I can find references for the lemma itself, but nowhere that it's called Hadamard's. (Except for one dubious paper that was written after the name Hadamard's was introduced into this article...)  For example, the reference in that section doesn't call it Hadamard's.  --MOBle (talk) 16:11, 10 August 2011 (UTC)
 * It is a simple lemma not necessarily linked to CBH. I grew up with that rubric, and I cite at least one such informal usage in print, on my talk page. So, if you were researching the propriety of the labelling, I doubt Hadamard discovered it; but I stick to popular usage, e.g. like the "Runge" eponymy in the Runge-Lenz vector... Supplant a better name, if available, but I have not come across one. It is a terrible idea, however, to link CBH's label to it...     Cuzkatzimhut (talk) 12:24, 23 August 2011 (UTC)
 * I edited the lemma today, reassuring the reader that Campbell proved it in print in 1897, as I just confirmed; however, this does not preclude prior implicit usage by somebody else, likely Lie... Cuzkatzimhut (talk) 21:41, 11 October 2022 (UTC)

A good reference: LNM 2034
Happy new year.

BONFIGLIOLI and FULCI : topics in Noncommutative Algebra (Lecture Notes in Mathematics 2034),539p, SpV 2012, ISBN 978-3-642-22596-3

this book is well documented about CBH and Free Lie Algebra

--Guerinsylvie (talk) 09:41, 3 January 2012 (UTC)

Better formula for first few terms
, first few terms are wtitten in the following formula:

$$\begin{align} Z(X,Y)&{}=\log(\exp X\exp Y) \\ &{}= X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] \\ &{}\quad - \frac {1}{24}[Y,[X,[X,Y]]] \\ &{}\quad - \frac{1}{720}([[[[X,Y],Y],Y],Y] +[[[[Y,X],X],X],X]) \\ &{}\quad +\frac{1}{360}([[[[X,Y],Y],Y],X]+[[[[Y,X],X],X],Y])\\ &{}\quad + \frac{1}{120}([[[[Y,X],Y],X],Y] +[[[[X,Y],X],Y],X]) + \cdots \end{align}$$

It has inconsistent format in a sense than in some terms commutators are nested towards the right, while in other terms they are nested towards the left. Also, innermost commutator is sometimes $$[X,Y]$$ and sometimes $$[Y,X]$$. So I propose it to be replaced by:

$$\begin{align} Z(X,Y)&{}=\log(\exp X\exp Y) \\ &{}= X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] \\ &{}\quad - \frac {1}{24}[Y,[X,[X,Y]]] \\ &{}\quad + \frac{1}{720}([Y,[Y,[Y,[X,Y]]]] -[X,[X,[X,[X,Y]]]]) \\ &{}\quad -\frac{1}{360}([X,[Y,[Y,[X,Y]]]]-[Y,[X,[X,[X,Y]]]])\\ &{}\quad + \frac{1}{120}([Y,[X,[Y,[X,Y]]]] -[X,[Y,[X,[X,Y]]]]) + \cdots \end{align}$$

But I'm not sure if I've got all the signs right because there is a lot of it. 193.198.162.13 (talk) 09:07, 25 July 2014 (UTC)


 * Well, the most important feature, beyond subjective aesthetic criteria, is that the Y - X symmetries are manifest and natural, and do not need tweaking to jump at you. So, if anything, it is the 3rd order terms that could use tweaking, and certainly not the higher, 5th, order ones! Your fearing sign mistakes is an ipso facto illustration of the point . You might look at the following sections to appreciate the point and perhaps leave good enough alone? Why don't you get an account? Cuzkatzimhut (talk) 10:46, 25 July 2014 (UTC)  OK, rearranged cubics as per your point. Cuzkatzimhut (talk) 15:35, 25 July 2014 (UTC)

Any field
The beginning of the first section at the moment talks about $$X,Y$$ being small enough, while claiming to be working in an arbitrary field of characteristic zero. What norm is intended here?

193.11.79.122 (talk) 08:27, 13 October 2014 (UTC)


 * Good point. The statement "small enough" is independent of norms and fields, but the note box refers, of course, to $R$ and $C$ with the Hilbert-Schmidt norm for matrix groups. YohanN7 (talk) 09:33, 13 October 2014 (UTC)

Generating function for Bernoulli numbers

 * $$ \psi(x) ~\stackrel{\text{def}}{=} ~ \frac{x \log x}{x-1}= 1- \sum^\infty_{n=1}

{(1-x)^n \over n (n+1)} ~, $$ is not the generating function for Bernoulli numbers — Preceding unsigned comment added by 173.48.171.226 (talk) 19:03, 18 January 2015 (UTC)
 * Please sign your posts--or, better, get a WP account. Which part of footnote 2, nb2, where the Generating function involved is stated, are you disagreeing with? ψ(x) is not, but ψ(exp(y)) is, as stated. Are you conflating involvement with identity?Cuzkatzimhut (talk) 21:04, 18 January 2015 (UTC)

possible mistake In "An important lemma"
In "An important lemma" --- A direct application of this idenity

I asume there is something wrong
 * $$e^X e^Y= e^{X+Y +\frac{1}{2} [ X, Y] } ~. $$

should be:
 * $$e^X e^Y= e^{X+Y -\frac{1}{2} [ X, Y] } ~. $$

Because The Zassenhaus formula says:
 * $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~

e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \cdots$$

with t=0 and
 * $$ [X [ X, Y] =0 . $$

... — Preceding unsigned comment added by ChilesheAT (talk • contribs) 12:34, 6 March 2016 (UTC)


 * There is no mistake, as you may verify by consistently expanding everything in powers of X and Y... Applying the correct formula to the Zassenhaus formula to quadratic order in t you get consistency. What exactly is your argument? You must have misunderstood something, but it is not easy to reconstruct your wrong steps. Cuzkatzimhut (talk) 14:15, 6 March 2016 (UTC)

Assessment comment
Substituted at 19:47, 1 May 2016 (UTC)

supersymmetric variant?
So, am I being a complete idiot, or is is the odd part of the supersymmetric variant given by the generation function $$x/(1+e^x)$$ -- i.e. Fermi-Dirac statistics, instead of Bose-Einstein statistics? The basic argument would be that Bose-Einstein -> Fock space (which is the same thing as the tensor algebra) but is for bosons, so its the symmetric algebra while for fermions it would be the exterior algebra so we have to track that extra sign through everything, and doing so would flip it to $$x/(1+e^x)$$ -- finally one reinterprets x to be -E/kT to get the statistics. I guess I should ask on mathoverflow. Or maybe just google it. 67.198.37.16 (talk) 04:12, 25 September 2016 (UTC)

Historical remarks
To prevent quasi-revelatory tweaks on the history of Lie's third fundamental theorem being presented as the CBH formula, I'd defer to the definitive chronicler of the subject, Wilfried Schmid, in Bulletin of the American Mathematical Society  6 (2) March 1982, pp 175-186. Schur's (note, not Issai S!) role and his checkered history with Lie and Engel w.r.t. Lie's third fundamental theorem are detailed, and are adhered to in the NPOV placement in the lede. Cuzkatzimhut (talk) 14:58, 26 September 2016 (UTC)

Error in the explicit formula
I don't think the explicit formula for X\ast Y written is correct. For instance the term [X, Y] has contributions both from n=1, r_1=s_1=1 and n=2, r_1=s_2=1, s_1=r_2=0, giving a coefficient of 1/2-1/4=1/4 instead of the supposed 1/2. Am I missing something? A (correct) version of the formula can be found in https://terrytao.wordpress.com/2011/09/01/254a-notes-1-lie-groups-lie-algebras-and-the-baker-campbell-hausdorff-formula/, exercise 17. — Preceding unsigned comment added by 193.136.196.3 (talk) 17:53, 31 January 2018 (UTC)


 * Please sign your message. Yes, you are badly misunderstanding the notation, and proposing partitions expressly excluded by it. Make a bona-fide effort to actually understand the notation, and, if need be, going to the original reference quoted from. This is not a forum. Cuzkatzimhut (talk) 18:55, 31 January 2018 (UTC)

Error in Footnote
Footnote 31 is wrong. Hall has no chapter 14. Please correct.
 * Apparently, what was meant is chapter 14 in Brian C. Hall, "Quantum Theorty for Mathematicians", not the (not existing) chapter 14 in Brian C. Hall "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction". Can somebody please check? – Tea2min (talk) 08:32, 29 May 2020 (UTC)