Baker–Campbell–Hausdorff formula

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

Z = log(e^X e^Y)

for possibly noncommutative X and Y in the Lie algebra of a Lie group. This formula tightly links Lie groups to Lie algebras by expressing the logarithm of the product of two Lie group elements as a Lie algebra element using only Lie algebraic operations. The solution on this form, whenever defined, means that multiplication in the group can be expressed entirely in Lie algebraic terms. The solution on another form is straightforward to obtain; one just substitutes the power series for exp and log in the equation and rearranges. The point is to express the solution in Lie algebraic terms. This occupied the time of several prominent mathematicians.

The formula is named after Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff who discovered its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of the Lie correspondence and in quantum field theory. It was first noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902); and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947).

The Campbell–Baker–Hausdorff formula: existence

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

Z = log(exp(X) exp(Y)),

can, possibly with conditions on X, Y, and Z, 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 merely assurance of its existence, like, for instance, in this construction of a Lie group representation from a Lie algebra representation. Existence 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

Δ(X) = X⊗1 + 1⊗X

and likewise for Y. (The definition of the coproduct is extended recursively by the rule Δ(XY) = Δ(X)Δ(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. (Friedrichs' theorem )

The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows: The elements X and Y are primitive, so exp(X) and exp(Y) are group like; so their product exp(X)exp(Y) is also group like; 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.

An alternative, remarkably direct and concise, recursive proof that all homogeneous polynomials in Z are in the Lie algebra is due to Eichler.

An explicit Baker–Campbell–Hausdorff formula

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

$\exp : \mathfrak g\rightarrow G$

be the exponential map. The following general combinatorial formula was introduced by Eugene Dynkin (1947),

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

where $s_n$ and $r_n$ are non-negative integers, and the following notation has been used:

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

This term is zero if $s_n > 1$ or if $s_n = 0$ and $r_n > 1$.

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

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

The above lists all summands of order 5 or lower (i.e. those containing 5 or fewer X's and Y's). Note the X ↔ Y (anti-)/symmetry in alternating orders of the expansion, since Z(Y, X) = −Z(−X,−Y). A complete elementary proof of this formula can be found here.

Selected tractable cases

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, if [X,Y] vanishes, then the above formula reduces to X+Y. If the commutator [X,Y] is a scalar (central, cf. the nilpotent Heisenberg group), then all but the first three terms on the right-hand side of the above vanish. This is the degenerate case utilized routinely in quantum mechanics, as illustrated below.

Other forms of the Baker–Campbell–Hausdorff formula, emphasizing expansion in terms of the element Y (and utilizing the linear adjoint endomorphism notation, ad_X Y ≡ [X,Y]), might serve well:

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

as is evident from the integral formula below. (The coefficients of the nested commutators linear in Y are normalized Bernoulli numbers, outlined below.)

Thus, when the commutator happens to be [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

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

or adjoint dilation,

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

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

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

involving the generating function for the Bernoulli numbers,

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

utilized by Poincaré and Hausdorff.

Matrix Lie group illustration

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,

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

When one solves for Z in

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

using the series expansions for exp and log one obtains a simpler formula:

$Z = \sum_{n>0} \frac{(-1)^{n-1}}{n} \sum_{\begin{smallmatrix} r_i+s_i>0\, \\ 1\le i\le n\end{smallmatrix}} \frac{X^{r_1}Y^{s_1}\cdots X^{r_n}Y^{s_n}}{r_1!s_1!\cdots r_n!s_n!}, \quad ||X|| + ||Y|| < \log 2, ||Z|| < \log 2.$

The first, second, third, and fourth order terms are:

• $z_1 = X + Y\,\!$

• $z_2 = \frac{1}{2} (XY - YX)$

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

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

The main significance of the Baker–Campbell–Hausdorff formula formula in this case is that each of the terms of degree two and higher can be expressed as a linear combination of brackets. It is not at all obvious, if one simply computes everything by means of power series, that each term in the expansion is expressible as a combination of brackets.

The Zassenhaus formula

A related combinatoric expansion that is useful in dual applications is

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

where the exponents of higher order in t are likewise nested commutators, i.e., homogeneous Lie polynomials. These exponents, C_n in exp(–tX) exp(t(X+Y)) = Π_n exp(tⁿ C_n), follow recursively by application of the above BCH expansion.

As a corollary of this, the Suzuki–Trotter decomposition follows directly.

An important lemma

Let G be a matrix Lie group and g its corresponding Lie algebra. Let ad_X be the linear operator on g defined by ad_X Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with Ad_A for fixed A ∈ G the linear transformation of g given by Ad_AY = AYA⁻¹.

A standard combinatorial lemma which is utilized in producing the above explicit expansions is given by

$\operatorname{Ad}_{e^X} = e^{\operatorname{ad}_X},$

so, explicitly,

$\operatorname{Ad}_{e^X}Y = e^{X}Y e^{-X} = e^{\operatorname{ad} _X} Y =Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots.$

This formula can be proved by evaluation of the derivative with respect to s of f (s)Y ≡ e^sX Y e^−sX, solution of the resulting differential equation and evaluation at s = 1,

$\frac{d}{ds}f(s)Y = \frac{d}{ds} \left (e^{sX}Ye^{-sX} \right )= Xe^{sX}Ye^{-sX} - e^{sX}Ye^{-sX}X = \operatorname{ad}_X (e^{sX}Ye^{-sX})$

or

$f'(s) = \operatorname{ad}_Xf(s), \qquad f(0) = 1 \qquad \Longrightarrow \qquad f(s) = e^{s \operatorname{ad}_X}.$

A direct application of this identity

For [X,Y] central, i.e., commuting with both X and Y,

$e^{sX} Y e^{-sX} = Y + s [ X, Y ] ~.$

Consequently, for g(s) ≡ e^sX e^sY, it follows that

$\frac{dg}{ds}=\Bigl( X+ e^{sX} Y e^{-sX}\Bigr) g(s) =(X+ Y + s [ X, Y ]) ~g(s) ~,$

whose solution is

$g(s)= e^{s(X+Y) +\frac{s^2}{2} [ X, Y ] } ~,$

hence the degenerate form already covered above,

$e^X e^Y= e^{X+Y +\frac{1}{2} [ X, Y] } ~.$

More generally, for non-central [X,Y] , the following braiding identity further follows readily,

$e^{X} e^{Y}= e^{(Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots)} ~e^X.$

Application in quantum mechanics

A degenerate form of the Baker–Campbell–Hausdorff formula is useful in Quantum mechanics and especially quantum optics, where X and Y are Hilbert space operators, generating the Heisenberg group.

A typical example is the annihilation and creation operators, â and â^†. Their commutator [â^†,â]= −I is central, that is, it commutes with both â and â^†. As indicated above, the expansion then collapses to the semi-trivial degenerate form:

$e^{v\hat{a}^\dagger - v^*\hat{a}} = e^{v\hat{a}^\dagger} e^{-v^*\hat{a}} e^{-|v|^{2}/2} ,$

where v is just a complex number.

This example illustrates the resolution of the displacement operator, exp(vâ^†−v^*â), into exponentials of annihilation and creation operators and scalars.

This degenerate Baker–Campbell–Hausdorff formula then displays the product of two displacement operators as another displacement operator (up to a phase factor), with the resultant displacement equal to the sum of the two displacements,

$e^{v\hat{a}^\dagger - v^*\hat{a}} e^{u\hat{a}^\dagger - u^*\hat{a}} = e^{(v+u)\hat{a}^\dagger -(v^*+u^*)\hat{a}} e^{(vu^*-uv^*)/2},$

since the Heisenberg group they provide a representation of is nilpotent. The degenerate Baker–Campbell–Hausdorff formula is frequently used in quantum field theory as well.

See also

• Magnus expansion
• Stone–von Neumann theorem
• Logarithm of a matrix
• Matrix exponential
• Golden–Thompson inequality
• Lie product formula (Trotter product formula)
• Derivative of the exponential map

Bibliography

• 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.
• J.-P. Serre, Lie algebras and Lie groups, Benjamin, 1965.
• Wilfried Schmid (1982). "Poincaré and Lie groups", Bull. Amer. Math. Soc. 6 175−186.
• Shlomo Sternberg (2004) Lie Algebras, Orange Grove Books, ISBN 978-1616100520 free, online

External links

• C.K. Zachos, Crib Notes on CBH expansions doi:10.13140/2.1.3090.2409
• MathWorld page