In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation
for possibly noncommutativeX and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in and and iterated commutators thereof. The first few terms of this series are:
where "" indicates terms involving higher commutators of and . If and are sufficiently small elements of the Lie algebra of a Lie group , the series is convergent. Meanwhile, every element sufficiently close to the identity in can be expressed as for a small in . Thus, we can say that near the identity the group multiplication in —written as —can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.
If and are sufficiently small matrices, then can be computed as the logarithm of , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that can be expressed as a series in repeated commutators of and .
Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.
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 history of the formula is described in detail in the article of Achilles and Bonfiglioli and in the book of Bonfiglioli and Fulci.
For many purposes, it is only necessary to know that an expansion for in terms of iterated commutators of and exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, where the precise coefficients play no role in the argument.) A remarkably direct existence proof was given by Martin Eichler, see also the "Existence results" section below.
In other cases, one may need detailed information about and it is therefore desirable to compute as explicitly as possible. Numerous formulas exist; we will describe two of the main ones (Dynkin's formula and the integral formula of Poincaré) in this section.
where the sum is performed over all nonnegative values of and , and the following notation has been used:
with the understanding that [X] := X.
The series is not convergent in general; it is convergent (and the stated formula is valid) for all sufficiently small and .
Since [A, A] = 0, the term is zero if or if and .
The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie algebra):
The above lists all summands of order 6 or lower (i.e. those containing 6 or fewer X's and Y's). The X ↔ Y (anti-)/symmetry in alternating orders of the expansion, follows from Z(Y, X) = −Z(−X, −Y). A complete elementary proof of this formula can be found in the article on the derivative of the exponential map.
using the series expansions for exp and log one obtains a simpler formula:
The first, second, third, and fourth order terms are:
The formulas for the various 's is not the Baker–Campbell–Hausdorff formula. Rather, the Baker–Campbell–Hausdorff formula is one of various expressions for 's in terms of repeated commutators of and . The point is that it is far from obvious that it is possible to express each in terms of commutators. (The reader is invited, for example, to verify by direct computation that is expressible as a linear combination of the two nontrivial third-order commutators of and , namely and .) The general result that each is expressible as a combination of commutators was shown in an elegant, recursive way by Eichler.
A consequence of the Baker–Campbell–Hausdorff formula is the following result about the trace:
That is to say, since each with is expressible as a linear combination of commutators, the trace of each such terms is zero.
Suppose and are the following matrices in the Lie algebra (the space of matrices with trace zero):
It is then not hard to show that there does not exist a matrix in with . (Similar examples may be found in the article of Wei.)
This simple example illustrates that the various versions of the Baker–Campbell–Hausdorff formula, which give expressions for Z in terms of iterated Lie-brackets of X and Y, describe formal power series whose convergence is not guaranteed. Thus, if one wants Z to be an actual element of the Lie algebra containing X and Y (as opposed to a formal power series), one has to assume that X and Y are small. Thus, the conclusion that the product operation on a Lie group is determined by the Lie algebra is only a local statement. Indeed, the result cannot be global, because globally one can have nonisomorphic Lie groups with isomorphic Lie algebras.
If and commute, that is , the Baker–Campbell–Hausdorff formula reduces to .
Another case assumes that commutes with both and , as for the nilpotentHeisenberg group. Then the formula reduces to its first three terms.
Theorem: If and commute with their commutator, , then .
This is the degenerate case used routinely in quantum mechanics, as illustrated below. In this case, there are no smallness restrictions on and . This result is behind the "exponentiated commutation relations" that enter into the Stone–von Neumann theorem. A simple proof of this identity is given below.
Another useful form of the general formula emphasizes expansion in terms of Y and uses the adjoint mapping notation :
which is evident from the integral formula above. (The coefficients of the nested commutators with a single are normalized Bernoulli numbers.)
Now assume that the commutator is a multiple of , so that . Then all iterated commutators will be multiples of , and no quadratic or higher terms in appear. Thus, the term above vanishes and we obtain:
Theorem: If , where is a complex number with for all integers , then we have
Again, in this case there are no smallness restriction on and . The restriction on guarantees that the expression on the right side makes sense. (When we may interpret .) We also obtain a simple "braiding identity":
If and are matrices, one can compute using the power series for the exponential and logarithm, with convergence of the series if and are sufficiently small. It is natural to collect together all terms where the total degree in and equals a fixed number , giving an expression . (See the section "Matrix Lie group illustration" above for formulas for the first several 's.) A remarkably direct and concise, recursive proof that each is expressible in terms of repeated commutators of and was given by Martin Eichler.
Alternatively, we can give an existence argument as follows. The Baker–Campbell–Hausdorff formula implies that if X and Y are in some Lie algebra defined over any field of characteristic 0 like or , then
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.
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.
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.
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.
A standard combinatorial lemma which is utilized in producing the above explicit expansions is given by
The series can be written more compactly (cf. main article) as
with the infinite series
Here, M is a matrix whose matrix elements are .
The usefulness of this expression comes from the fact that the matrix M is a vielbein. Thus, given some map from some manifold N to some manifold G, the metric tensor on the manifold N can be written as the pullback of the metric tensor on the Lie group G,
where is the identity operator. It follows that and commute with their commutator. Thus, if we formally applied a special case of the Baker–Campbell–Hausdorff formula (even though and are unbounded operators and not matrices), we would conclude that
A related application 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 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,
^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.
^Henry Frederick Baker, Proceedings of the London Mathematical Society (1) 34 (1902) 347–360; H. Baker, Proceedings of the London Mathematical Society (1) 35 (1903) 333–374; H. Baker, Proceedings of the London Mathematical Society (Ser 2) 3 (1905) 24–47.
^Felix Hausdorff, "Die symbolische Exponentialformel in der Gruppentheorie", Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.
Achilles, Rüdiger; Bonfiglioli, Andrea (May 2012). "The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin". Archive for History of Exact Sciences. 66 (3): 295–358. doi:10.1007/s00407-012-0095-8. S2CID120032172.