Baker–Campbell–Hausdorff formula


In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation
for possibly noncommutative and 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 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.

History

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 ; elaborated by Henri Poincaré and Baker ; and systematized geometrically, and linked to the Jacobi identity by Hausdorff. The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin. The history of the formula is described in detail in the article of Achilles and Bonfiglioli and in the book of Bonfiglioli and Fulci.

Explicit forms

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. 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 in this section.

Dynkin's formula

Let G be a Lie group with Lie algebra. Let
be the exponential map.
The following general combinatorial formula was introduced by Eugene Dynkin,
where the sum is performed over all nonnegative values of and, and the following notation has been used:
The series is not convergent in general; it is convergent for all sufficiently small and.
Since, the term is zero if or if and.
The first few terms are well-known, with all higher-order terms involving and commutator nestings thereof :
The above lists all summands of order 5 or lower. The X ↔ Y /symmetry in alternating orders of the expansion, follows from. A complete elementary proof of this formula can be found here.

An integral formula

There are numerous other expressions for, many of which are used in the physics literature. A popular integral formula is
involving the generating function for the Bernoulli numbers,
utilized by Poincaré and Hausdorff.

Matrix Lie group illustration

For a matrix Lie group the Lie algebra is the tangent space of the identity I, and the commutator is simply = XY − YX; the exponential map is the standard exponential map of matrices,
When one solves for Z in
using the series expansions for and 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 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.

Questions of convergence

Suppose and are the following matrices in the Lie algebra :
Then
It is then not hard to show that there does not exist a matrix in with.
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, 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.
Concretely, if working with a matrix Lie algebra and is a given submultiplicative matrix norm, convergence is guaranteed if

Special cases

If ' and ' commute, that is, the Baker–Campbell–Hausdorff formula reduces to.
Another case assumes that commutes with both and, as for the nilpotent Heisenberg 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.
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. We also obtain a simple "braiding identity":
which may be written as an adjoint dilation:

Existence results

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. 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. 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 of all non-commuting formal power series with real coefficients in the non-commuting variables X and Y. There is a ring homomorphism from S to the tensor product of S with S over R,
called the coproduct, such that
One can then verify the following properties:
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.

Zassenhaus formula

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, in, follow recursively by application of the above BCH expansion.
As a corollary of this, the Suzuki–Trotter decomposition follows.

An important lemma and its application to a special case of the Baker–Campbell–Hausdorff formula

The identity

Let be a matrix Lie group and its corresponding Lie algebra. Let be the linear operator on defined by for some fixed. Denote with for fixed the linear transformation of given by.
A standard combinatorial lemma which is utilized in producing the above explicit expansions is given by
so, explicitly,
This formula can be proved by evaluation of the derivative with respect to of, solution of the resulting differential equation and evaluation at s = 1,
or

An application of the identity

For central, i.e., commuting with both and,
Consequently, for, it follows that
whose solution is
Taking gives one of the special cases of the Baker–Campbell–Hausdorff formula described above:
More generally, for non-central , the following braiding identity further follows readily,

Application in quantum mechanics

A special case 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 Lie algebra. Specifically, the position and momentum operators in quantum mechanics, usually denoted and, satisfy the canonical commutation relation:
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, we would conclude that
This "exponentiated commutation relation" does indeed hold, and forms the basis of the Stone–von Neumann theorem.
A related application is the annihilation and creation operators, and. Their commutator is central, that is, it commutes with both and. As indicated above, the expansion then collapses to the semi-trivial degenerate form:
where is just a complex number.
This example illustrates the resolution of the displacement operator,, 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, with the resultant displacement equal to the sum of the two displacements,
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.