Baker-Campbell-Hausdorff formula

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In mathematics, the Baker-Campbell-Hausdorff formula is the solution to

$Z = \log(e^X e^Y)\,$

for non-commuting X and Y. It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff. It was first noted in print by Campbell, elaborated by Henri Poincaré and Baker, and systematized by Hausdorff. The formula below was introduced by Eugene Dynkin.

1 The Baker-Campbell-Hausdorff formula: existence
2 An explicit Baker-Campbell-Hausdorff formula
- 2.1 The Zassenhaus formula
- 2.2 The Hadamard lemma
3 See also
4 References
5 External links

[edit] The Baker-Campbell-Hausdorff formula: existence

The Baker-Campbell-Hausdorff formula implies that if X and Y are in some Lie algebra $\mathfrak g,$ defined over any field of characteristic 0, then

$\log(\exp(X)\exp(Y)) \,$

can be written as a formal infinite sum of elements of $\mathfrak g$ . For many applications one does not need an explicit expression for this infinite sum but just its existence, and this can be seen as follows. The ring

S = R〈X,Y〉

of all non-commuting formal power series in non-commuting variables X and Y has a ring homomorphism Δ from S to the completion of

S⊗S,

called the coproduct, such that

S(X) = X⊗1 + 1⊗X

and similarly for Y. This has the following properties:

exp is an isomorphism (of sets) from the elements of S with constant term 0 to the elements with constant term 1, with inverse log
r=exp(s) is grouplike (this means Δ(r)=r⊗r) if and only if s is primitive (this means Δ(s)=s⊗1+1⊗s).
The grouplike elements form a group under multiplication.
The primitive elements are exactly the formal infinite sums of elements of the Lie algebra generated by X and Y.

The existence of the Baker-Campbell-Hausdorff formula can now be seen as follows: The elements X and Y are primitive, so exp(X) and exp(Y) are grouplike, so their product exp(X)exp(Y) is also grouplike, so its logarithm log(exp(X)exp(Y)) is primitive, and hence can be written as an infinite sum of elements of the Lie algebra generated by X and Y.

The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring S used above is just a completion of this Hopf algebra.

[edit] An explicit Baker-Campbell-Hausdorff formula

Specifically, let G be a simply-connected Lie group with Lie algebra $\mathfrak g$ . Let

$\exp : \mathfrak g\rightarrow G$

be the exponential map. The general formula is given by:

$\log(\exp X\exp Y) = \sum_{n>0}\frac {(-1)^{n-1}}{n} \sum_{ \begin{smallmatrix} {r_i + s_i > 0} \\ {1\le i \le n} \end{smallmatrix}} \frac{(\sum_{i=1}^n (r_i+s_i))^{-1}}{r_1!s_1!\cdots r_n!s_n!} [ X^{r_1} Y^{s_1} X^{r_2} Y^{s_2} \ldots X^{r_n} Y^{s_n} ],$

which uses the notation

$[ X^{r_1} Y^{s_1} \ldots X^{r_n} Y^{s_n} ] = [ \underbrace{X,[X,\ldots[X}_{r_1} ,[ \underbrace{Y,[Y,\ldots[Y}_{s_1} ,\,\ldots\, [ \underbrace{X,[X,\ldots[X}_{r_n} ,[ \underbrace{Y,[Y,\ldots Y}_{s_n} ]]\ldots]].$

This term is zero if $s n > 1$ or if $s n = 0$ and $r n > 1$ (Sagle & Walde 1973, pp. 134+135).

The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie algebra):

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

Note the X-Y (anti-)/symmetry in alternating orders of the expansion, since $Z (Y, X) = - Z ( - X, - Y)$ .

There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications.

For example, note that if [X,Y] vanishes, then the above formula manifestly reduces to X + Y. If the commutator [X,Y] is a constant (central), then all but the first three terms on the right-hand side of the above vanish.

If one of the Lie algebra elements X maps the kernel of ad Y into itself, other forms of the Campbell-Baker-Hausdorff formula might serve well:

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

as is evident from the integral formula below. So, if the commutator is [X, Y] = sY, for some non-zero s, this formula reduces to just Z = X + sY / (1 − exp(-s)), which then leads to braiding identities such as

$e^{X} e^{Y} = e^{\exp (s) Y} e^{X}.\,$

There are numerous such well-known expressions applied routinely in physics (cf. Magnus). A popular integral formula is

$\log(\exp X\exp Y) = X + \left ( \int^1_0 \psi \left ( e^{\text{ad} X} e^{t \,\text{ad} Y}\right ) \, dt \right) \, Y,$

involving a generating function for the Bernoulli numbers,

$\psi(x) \equiv \frac{x \ln x}{x-1}.$

For a matrix Lie group $G \sub \mbox{GL}(n,\mathbb{R})$ the Lie algebra is the tangent space of the identity I, and the commutator is simply [X, Y] = XY − YX; the exponential map is the standard exponential map of matrices,

$\mbox{exp}\ X = e^X = \sum_{n=0}^{\infty}{\frac{X^n}{n!}}.$

When we solve for Z in

$e^Z = e^X e^Y,\,\!$

we obtain a simpler formula:

$Z = \sum_{n>0} \frac{(-1)^{n-1}}{n} \sum_{\begin{smallmatrix} r_i+s_i>0\, \\ 1\le i\le n\end{smallmatrix}} \frac{X^{r_1}Y^{s_1}\cdots X^{r_n}Y^{s_n}}{r_1!s_1!\cdots r_n!s_n!}.$

We note that the first, second, third and fourth order terms are:

$z_1 = X + Y\,\!$

$z_2 = \frac{1}{2} (XY - YX)$

$z_3 = \frac{1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY)$

$z_4 = \frac{1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX).$

[edit] The Zassenhaus formula

A related combinatoric expansion, useful in dual applications is

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

[edit] The Hadamard lemma

A standard combinatoric lemma utilized, among others, in the above explicit expansions is

$e^{X}Y e^{-X} =Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\cdots ,$

easily provable by parametric induction.

[edit] See also

Dyson series

[edit] References

H. Baker, Proc Lond Math Soc (1) 34 (1902) 347–360; ibid (1) 35 (1903) 333–374; ibid (Ser 2) 3 (1905) 24–47.
Yu.A. Bakhturin (2001), “Campbell-Hausdorff formula”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
J. Campbell, Proc Lond Math Soc 28 (1897) 381–390; ibid 29 (1898) 14–32.
L. Corwin & F.P Greenleaf, Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples, Cambridge University Press, New York, 1990, ISBN 0-521-36034-X.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9
F. Hausdorff, Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.
W. Miller, "Symmetry Groups and their Applications", Academic Press, New York, 1972, pp 159–161.
W. Magnus, Comm Pur Appl Math VII (1954) 649–673.
H. Poincaré, Compt Rend Acad Sci Paris 128 (1899) 1065–1069; Camb Philos Trans 18 (1899) 220–255.
M.W. Reinsch, "A simple expression for the terms in the Baker-Campbell-Hausdorff series". Journal of Mathematical Physics, 41(4):2434–2442, April 2000. doi:10.1063/1.533250 (arXiv preprint)
W. Rossmann, Lie Groups: An Introduction through Linear Groups. Oxford University Press, 2002.
A.A. Sagle & R.E. Walde, "Introduction to Lie Groups and Lie Algebras", Academic Press, New York, 1973. ISBN 0-12-614550-4.
J.-P. Serre, Lie algebras and Lie groups , Benjamin (1965)