Baker–Campbell–Hausdorff formula

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

Z = log(eX eY)

for noncommutative X and Y. This formula links Lie groups to Lie algebras by expressing the logarithm of the product of two Lie group elements as a Lie algebra element in canonical coordinates, a significant guiding connection appreciated (Hausdorff 1906)[1] before the full development of the theory.

It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff. It was first noted in print by Campbell[2] (1897); elaborated by Henri Poincaré[3] (1899) and Baker (1902);[4] and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906).[1]

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

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

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 merely assurance of 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


called the coproduct, such that

Δ(X) = X⊗1 + 1⊗X

and similarly 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 [5])

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 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.

An explicit Baker–Campbell–Hausdorff formula[edit]

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 following general combinatoric formula was introduced by Eugene Dynkin (1947):[6]

\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.[7]

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

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 )  \\
- \frac {1}{24}[Y,[X,[X,Y]]]  \\
- \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])\\
+ \frac{1}{120}([[[[Y,X],Y],X],Y] +[[[[X,Y],X],Y],X])
+ \cdots

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

Selected tractable cases[edit]

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, adX 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.[8] A popular integral formula is[9]

\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. Recall

\psi(e^y)=\sum_{n=0}^\infty B_n ~ y^n/n! ,

for the Bernoulli numbers, B0 = 1, B1 = 1/2, B2 = 1/6, B4 = −1/30, ...

Matrix Lie group illustration[edit]

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 [XY] = 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,\,\!

one obtains a simpler formula:

 Z =
\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!}.

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 Zassenhaus formula[edit]

A related combinatoric expansion that is useful in dual[8] 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 exponents of higher order in t are likewise nested commutators.[10]

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

An important lemma[edit]

Let Ln be the space of all complex n×n matrices, and let adX be the linear operator defined by adX Y = [X,Y] for some fixed XLn. A standard combinatorial lemma which is utilized[9] in producing the above explicit expansions is

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 parametric induction: evaluation of the derivative with respect to s of f (s) ≡ esX Y esX; recursive determination of the Taylor expansion coefficients around s=0, in terms of nested commutators; and evaluation at s = 1, namely f (1).

A direct application of this identity for [X,Y] central, i.e., commuting with both X and Y is

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

Consequently, for g(s) ≡ esX esY, 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[edit]

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(v*â), into exponentials of annihilation and creation operators and scalars.[11]

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.

See also[edit]


  1. ^ a b F. Hausdorff, "Die symbolische Exponentialformel in der Gruppentheorie", Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.
  2. ^ J. Campbell, Proc Lond Math Soc 28 (1897) 381–390; ibid 29 (1898) 14–32.
  3. ^ H. Poincaré, Compt Rend Acad Sci Paris 128 (1899) 1065–1069; Camb Philos Trans 18 (1899) 220–255.
  4. ^ H. Baker, Proc Lond Math Soc (1) 34 (1902) 347–360; ibid (1) 35 (1903) 333–374; ibid (Ser 2) 3 (1905) 24–47.
  5. ^ N. Jacobson, Lie Algebras, John Wiley & Sons, 1966.
  6. ^ Dynkin, Eugene Borisovich (1947). "Вычисление коэффициентов в формуле Campbell–Hausdorff" [Calculation of the coefficients in the Campbell–Hausdorff formula]. Doklady Akademii Nauk SSSR (in Russian) 57: 323–326. 
  7. ^ A.A. Sagle & R.E. Walde, "Introduction to Lie Groups and Lie Algebras", Academic Press, New York, 1973. ISBN 0-12-614550-4.
  8. ^ a b Magnus, W. (1954). "On the exponential solution of differential equations for a linear operator". Communications on Pure and Applied Mathematics 7 (4): 649–673. doi:10.1002/cpa.3160070404. 
  9. ^ a b W. Miller, Symmetry Groups and their Applications, Academic Press, New York, 1972, pp 159–161. ISBN 0-12-497460-0
  10. ^ Casas, F.; Murua, A.; Nadinic, M. (2012). "Efficient computation of the Zassenhaus formula". Computer Physics Communications 183 (11): 2386. arXiv:1204.0389. Bibcode:2012CoPhC.183.2386C. doi:10.1016/j.cpc.2012.06.006. 
  11. ^ L. Mandel, E. Wolf Optical Coherence and Quantum Optics (Cambridge 1995).


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