In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation
for possibly noncommutative X 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 . Verfication of this claim 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 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 of the Baker–Campbell–Hausdorff formulaEdit
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 (Dykin's formula and the integral formula of Poincaré) in this section.
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:
It should be emphasized that 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 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.
An integral formulaEdit
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:
It should be emphasized that 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 a 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 convergenceEdit
Suppose and are the following matrices in the Lie algebra (the space of matrices with trace zero):
This simple example illustrates an important point: 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.
Selected special casesEdit
We mention here various useful special cases of the general formula.
Of course, if [X,Y] vanishes, then the above formula reduces to X+Y. A more interesting special case is the one in which commutes with both and ; 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. We state this result formally as a theorem.
Theorem: Suppose and commute with their commutator: . Then we have
In this special 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 proof of this identity is given in a subsequent section.
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.)
When the commutator happens to be , for some non-zero s, all iterated commutators will be multiples of , so that no terms of quadratic or higher order in appear. Thus, the " " term above vanishes and we obtain another special case of the Baker–Campbell–Hausdorff formula:
Theorem: Suppose , where is a complex number that is not of the form for a nonzero integer . Then we have
Again, in this special case, there are no smallness restriction on and . The restriction on guarantees that the expression on the right-hand side of the formula makes sense. (When we interpret as having its limiting value as approaches zero, namely 1.)
In this special case, we also have a simple "braiding identity":
which may be written equivalently as an adjoint dilation:
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.
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.
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 lemma and its application to a special case of the Baker–Campbell–Hausdorff formulaEdit
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,
An application of the identityEdit
For [X,Y] central, i.e., commuting with both X and Y,
Consequently, for g(s) ≡ esX esY, 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 [X,Y] , the following braiding identity further follows readily,
Application in quantum mechanicsEdit
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, we note that 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 (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 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.
- 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
- Achilles 2012
- Bonfiglioli 2012
- Rossmann 2002
- Hall 2015
- Hall 2015
- Martin Eichler (1968). "A new proof of the Baker-Campbell-Hausdorff formula", J. Math. Soc. Japan 20, 23-25. online open access.
- Nathan Jacobson, Lie Algebras, John Wiley & Sons, 1966.
- 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.
- A.A. Sagle & R.E. Walde, "Introduction to Lie Groups and Lie Algebras", Academic Press, New York, 1973. ISBN 0-12-614550-4.
- Magnus, Wilhelm (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.
- Suzuki, Masuo (1985). "Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics". Journal of Mathematical Physics. 26 (4): 601–612. Bibcode:1985JMP....26..601S. doi:10.1063/1.526596.; Veltman, M, 't Hooft, G & de Wit, B (2007), Appendix D.
- W. Miller, Symmetry Groups and their Applications, Academic Press, New York, 1972, pp 159–161. ISBN 0-12-497460-0
- Hall 2015 Theorem 5.3
- Martin Eichler (1968). "A new proof of the Baker-Campbell-Hausdorff formula", J. Math. Soc. Japan 20, 23-25. online open access.
- Hall 2015 Example 3.41
- Wei, James (October 1963). "Note on the Global Validity of the Baker-Hausdorff and Magnus Theorems". Journal of Mathematical Physics. 4 (10): 1337–1341. Bibcode:1963JMP.....4.1337W. doi:10.1063/1.1703910.
- Biagi, Stefano; Bonfiglioli, Andrea; Matone, Marco (2018). "On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues". Linear and Multilinear Algebra: 1–19. arXiv:1805.10089. doi:10.1080/03081087.2018.1540534. ISSN 0308-1087.
- Hall 2015 Theorem 5.1
- Hall 2015 Exercise 5.5
- Martin Eichler (1968). "A new proof of the Baker-Campbell-Hausdorff formula", J. Math. Soc. Japan 20, 23–25. online open access.
- Hall 2015 Section 5.7
- Casas, F.; Murua, A.; Nadinic, M. (2012). "Efficient computation of the Zassenhaus formula". Computer Physics Communications. 183 (11): 2386–2391. arXiv:1204.0389. Bibcode:2012CoPhC.183.2386C. doi:10.1016/j.cpc.2012.06.006.
- Hall 2015 Proposition 3.35
- Rossmann 2002 p. 15
- Hall 2015 Chapter 14
- L. Mandel, E. Wolf Optical Coherence and Quantum Optics (Cambridge 1995).
- Greiner 1996 See pp 27-29 for a detailed proof of the above lemma.
- Achilles, R., & Bonfiglioli, A. (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
- Yu.A. Bakhturin (2001) , "Campbell–Hausdorff formula", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Bonfiglioli, A., Fulci, R. (2012): Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin. Springer
- 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.
- Greiner, W.; Reinhardt, J. (1996), Field Quantization, Springer Publishing, ISBN 978-3-540-59179-5
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3-319-13466-6
- Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 978-0-19-859683-7
- Serre, Jean-Pierre (1965). Lie algebras and Lie groups. Benjamin.
- Schmid, Wilfied (1982). "Poincaré and Lie groups" (PDF). Bulletin of the American Mathematical Society. 6: 175−186.
- Shlomo Sternberg (2004) Lie Algebras, Orange Grove Books, ISBN 978-1616100520 free, online
- Veltman, M, 't Hooft, G & de Wit, B (2007). "Lie Groups in Physics", online lectures.