In mathematics, the Baker–Campbell–Hausdorff formula is the solution to the equation
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 commutative forms is obtained by substituting the power series for exp and log in the equation and rearranging. 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 stated its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. An earlier statement of the form was adumbrated by Friedrich Schur in 1890  where a convergent power series is given, with terms recursively defined. 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. Following Schur, it was 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 Baker–Campbell–Hausdorff formula: existenceEdit
can formally be written as an infinite sum of elements of . [This infinite series may or may not converge, so it need not define an actual element Z in .] For many applications, the mere assurance of the existence of this formal expression is sufficient, and an explicit expression for this infinite sum is not needed. This is for instance the case in the Lorentzian construction of a Lie group representation from a Lie algebra representation. Existence can be seen as follows.
We consider the ring
- S = R[[X,Y]]
- Δ : S → S⊗S,
called the coproduct, such that
- Δ(X) = X⊗1 + 1⊗X and Δ(Y) = Y⊗1 + 1⊗Y
(The definition of Δ is extended to the other elements of S by requiring R-linearity, multiplicativity and infinite additivity.)
One can then verify the following properties:
- The map exp, defined by its standard Taylor series, is a bijection between the set of elements of S with constant term 0 and the set of elements of S with constant term 1; the inverse of exp is 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, where the Lie bracket is given by the commutator [U,V]=UV-VU. (Friedrichs' theorem)
The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows: The elements X and Y are primitive, so and are grouplike; so their product is also grouplike; so its logarithm 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 formulaEdit
Specifically, let G be a Lie group with Lie algebra . Let
where the sum is performed over all nonegative values of and , and the following notation has been used:
Since [A, A] = 0, the term is zero if or if and .
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 casesEdit
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 special case may be established directly. 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:
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
or adjoint dilation,
involving the generating function for the Bernoulli numbers,
utilized by Poincaré and Hausdorff.[nb 1]
Matrix Lie group illustrationEdit
For a matrix Lie group 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,
When one solves for Z in
using the series expansions for exp and log one obtains a simpler formula:
The first, second, third, and fourth order terms are:
The main significance of the Baker–Campbell–Hausdorff formula is the non-obvious fact that each of the terms of degree two and higher can be expressed as a linear combination of iterated brackets of X and Y, with rational coefficients, as proven compactly in the above-mentioned Eichler note. As an immediate consequence one obtains the following result about the trace:
Questions of convergenceEdit
There exist real 2-by-2 matrices X and Y such that there is no real 2-by-2 matrix Z with
An example is
The formulas above that promise to give expressions for Z in terms of iterated Lie-brackets of X and Y therefore describe formal power series whose convergence is not guaranteed; they are guaranteed to be globally valid only if working in the ring of formal power series in the non-commuting variables X and Y, with the commutator as Lie-bracket.
However, one can show that if we work in a Lie algebra over the field R or C and the elements X and Y are close enough to 0, then the Dynkin formula given above converges absolutely to a Lie algebra element Z with
The Zassenhaus formulaEdit
A related combinatoric expansion that is useful in dual applications is
where the exponents of higher order in t are likewise nested commutators, i.e., homogeneous Lie polynomials. These exponents, Cn in exp(–tX) exp(t(X+Y)) = Πn exp(tn Cn), follow recursively by application of the above BCH expansion.
As a corollary of this, the Suzuki–Trotter decomposition follows.
An important lemmaEdit
Let G be a matrix Lie group and g its corresponding Lie algebra. Let adX be the linear operator on g defined by adX Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with AdA for fixed A ∈ G the linear transformation of g given by AdAY = AYA−1.
This formula can be proved by evaluation of the derivative with respect to s of f (s)Y ≡ esX Y e−sX, solution of the resulting differential equation and evaluation at s = 1,
- A direct application of this identity
For [X,Y] central, i.e., commuting with both X and Y,
Consequently, for g(s) ≡ esX esY, it follows that
whose solution is
hence the degenerate form already covered above,
More generally, for non-central [X,Y] , the following braiding identity further follows readily,
Application in quantum mechanicsEdit
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:
where v is just a complex number.
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,
- Rossmann 2002 Equation (2) Section 1.3. For matrix Lie algebras over the fields R and C, the convergence criterion is that the log series converges for both sides of eZ = eXeY. This is guaranteed whenever ||X|| + ||Y|| < log 2, ||Z|| < log 2 in the Hilbert–Schmidt norm. Convergence may occur on a larger domain. See Rossmann 2002 p. 24.
- Rossmann 2002 See equation (2) in section 1.3.
- F. Schur (1890), “Neue Begruendung der Theorie der endlichen Transformationsgruppen,” Mathematische Annalen, 35 (1890), 161-197. online copy
- see, e.g., Shlomo Sternberg, Lie Algebras (2004) Harvard University. (See page 10.)
- J. Campbell, Proc Lond Math Soc 28 (1897) 381–390; J. Campbell, Proc Lond Math Soc 29 (1898) 14–32.
- H. Poincaré, Compt Rend Acad Sci Paris 128 (1899) 1065–1069; Camb Philos Trans 18 (1899) 220–255.
- H. Baker, Proc Lond Math Soc (1) 34 (1902) 347–360; H. Baker, Proc Lond Math Soc (1) 35 (1903) 333–374; H. Baker, Proc Lond Math Soc (Ser 2) 3 (1905) 24–47.
- F. Hausdorff, "Die symbolische Exponentialformel in der Gruppentheorie", Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.
- Rossmann 2002 p. 23
- Hall 2015 Section 5.7
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- 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.
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- Hall 2015 Theorem 5.1
- Hall 2015 Exercise 5.5
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- Hall 2015 Theorem 5.3
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- Rossmann 2002 p. 15
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