In mathematics, the Baker–Campbell–Hausdorff formula gives a value of Z {\displaystyle Z} that solves the equation
e X e Y = e Z {\displaystyle e^{X}e^{Y}=e^{Z}}
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 Z {\displaystyle Z} in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X {\displaystyle X} and Y {\displaystyle Y} and iterated commutators thereof. The first few terms of this series are:
Z = X + Y + 1 2 [ X , Y ] + 1 12 [ X , [ X , Y ] ] + 1 12 [ Y , [ Y , X ] ] + ⋯ , {\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+{\frac {1}{12}}[Y,[Y,X]]+\cdots \,,}
where " ⋯ {\displaystyle \cdots } " indicates terms involving higher commutators of X {\displaystyle X} and Y {\displaystyle Y} . If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small elements of the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} , the series is convergent. Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {\displaystyle G} —written as e X e Y = e Z {\displaystyle e^{X}e^{Y}=e^{Z}} —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 X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y {\displaystyle e^{X}e^{Y}} , 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 Z := log ( e X e Y ) {\displaystyle Z:=\log \left(e^{X}e^{Y}\right)} can be expressed as a series in repeated commutators of X {\displaystyle X} and Y {\displaystyle Y} . 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 (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 For many purposes, it is only necessary to know that an expansion for Z {\displaystyle Z} in terms of iterated commutators of X {\displaystyle X} and Y {\displaystyle Y} 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 Z {\displaystyle Z} and it is therefore desirable to compute Z {\displaystyle Z} 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.
Dynkin formula Let G be a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} . Let
exp : g → G {\displaystyle \exp :{\mathfrak {g}}\to G}
be the exponential map. The following general combinatorial formula was introduced by Eugene Dynkin (1947),
log ( exp X exp Y ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n ∑ r 1 + s 1 > 0 ⋮ r n + s n > 0 [ X r 1 Y s 1 X r 2 Y s 2 ⋯ X r n Y s n ] ( ∑ j = 1 n ( r j + s j ) ) ⋅ ∏ i = 1 n r i ! s i ! , {\displaystyle \log(\exp X\exp Y)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sum _{\begin{smallmatrix}r_{1}+s_{1}>0\\\vdots \\r_{n}+s_{n}>0\end{smallmatrix}}{\frac {[X^{r_{1}}Y^{s_{1}}X^{r_{2}}Y^{s_{2}}\dotsm X^{r_{n}}Y^{s_{n}}]}{\left(\sum _{j=1}^{n}(r_{j}+s_{j})\right)\cdot \prod _{i=1}^{n}r_{i}!s_{i}!}},}
where the sum is performed over all nonnegative values of s i {\displaystyle s_{i}} and r i {\displaystyle r_{i}} , and the following notation has been used:
[ X r 1 Y s 1 ⋯ X r n Y s n ] = [ X , [ X , ⋯ [ X ⏟ r 1 , [ Y , [ Y , ⋯ [ Y ⏟ s 1 , ⋯ [ X , [ X , ⋯ [ X ⏟ r n , [ Y , [ Y , ⋯ Y ⏟ s n ] ] ⋯ ] ] {\displaystyle [X^{r_{1}}Y^{s_{1}}\dotsm X^{r_{n}}Y^{s_{n}}]=[\underbrace {X,[X,\dotsm [X} _{r_{1}},[\underbrace {Y,[Y,\dotsm [Y} _{s_{1}},\,\dotsm \,[\underbrace {X,[X,\dotsm [X} _{r_{n}},[\underbrace {Y,[Y,\dotsm Y} _{s_{n}}]]\dotsm ]]}
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 X {\displaystyle X} and Y {\displaystyle Y} . Since [A, A] = 0, the term is zero if s n > 1 {\displaystyle s_{n}>1} or if s n = 0 {\displaystyle s_{n}=0} and r n > 1 {\displaystyle r_{n}>1} . 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.
An integral formula There are numerous other expressions for Z {\displaystyle Z} , many of which are used in the physics literature. A popular integral formula is
log ( e X e Y ) = X + ( ∫ 0 1 ψ ( e ad X e t ad Y ) d t ) Y , {\displaystyle \log \left(e^{X}e^{Y}\right)=X+\left(\int _{0}^{1}\psi \left(e^{\operatorname {ad} _{X}}~e^{t\operatorname {ad} _{Y}}\right)dt\right)Y,}
involving the generating function for the Bernoulli numbers,
ψ ( x ) = def x log x x − 1 = 1 − ∑ n = 1 ∞ ( 1 − x ) n n ( n + 1 ) , {\displaystyle \psi (x)~{\stackrel {\text{def}}{=}}~{\frac {x\log x}{x-1}}=1-\sum _{n=1}^{\infty }{(1-x)^{n} \over n(n+1)}~,}
utilized by Poincaré and Hausdorff.
Matrix Lie group illustration For a matrix Lie group G ⊂ GL ( n , R ) {\displaystyle G\subset {\mbox{GL}}(n,\mathbb {R} )} 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,
exp X = e X = ∑ n = 0 ∞ X n n ! . {\displaystyle \exp X=e^{X}=\sum _{n=0}^{\infty }{\frac {X^{n}}{n!}}.}
When one solves for Z in
e Z = e X e Y , {\displaystyle e^{Z}=e^{X}e^{Y},}
using the series expansions for exp and log one obtains a simpler formula:
Z = ∑ n > 0 ( − 1 ) n − 1 n ∑ 1 ≤ i ≤ n r i + s i > 0 X r 1 Y s 1 ⋯ X r n Y s n r 1 ! s 1 ! ⋯ r n ! s n ! , ‖ X ‖ + ‖ Y ‖ < log 2 , ‖ Z ‖ < log 2. {\displaystyle Z=\sum _{n>0}{\frac {(-1)^{n-1}}{n}}\sum _{\stackrel {r_{i}+s_{i}>0}{1\leq i\leq n}}{\frac {X^{r_{1}}Y^{s_{1}}\cdots X^{r_{n}}Y^{s_{n}}}{r_{1}!s_{1}!\cdots r_{n}!s_{n}!}},\quad \|X\|+\|Y\|<\log 2,\|Z\|<\log 2.}
The first, second, third, and fourth order terms are:
z 1 = X + Y {\displaystyle z_{1}=X+Y}
z 2 = 1 2 ( X Y − Y X ) {\displaystyle z_{2}={\frac {1}{2}}(XY-YX)}
z 3 = 1 12 ( X 2 Y + X Y 2 − 2 X Y X + Y 2 X + Y X 2 − 2 Y X Y ) {\displaystyle z_{3}={\frac {1}{12}}\left(X^{2}Y+XY^{2}-2XYX+Y^{2}X+YX^{2}-2YXY\right)}
z 4 = 1 24 ( X 2 Y 2 − 2 X Y X Y − Y 2 X 2 + 2 Y X Y X ) . {\displaystyle z_{4}={\frac {1}{24}}\left(X^{2}Y^{2}-2XYXY-Y^{2}X^{2}+2YXYX\right).}
The formulas for the various z j {\displaystyle z_{j}} 's is not the Baker–Campbell–Hausdorff formula. Rather, the Baker–Campbell–Hausdorff formula is one of various expressions for z j {\displaystyle z_{j}} 's in terms of repeated commutators of X {\displaystyle X} and Y {\displaystyle Y} . The point is that it is far from obvious that it is possible to express each z j {\displaystyle z_{j}} in terms of commutators. (The reader is invited, for example, to verify by direct computation that z 3 {\displaystyle z_{3}} is expressible as a linear combination of the two nontrivial third-order commutators of X {\displaystyle X} and Y {\displaystyle Y} , namely [ X , [ X , Y ] ] {\displaystyle [X,[X,Y]]} and [ Y , [ X , Y ] ] {\displaystyle [Y,[X,Y]]} .) The general result that each z j {\displaystyle z_{j}} 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:
tr log ( e X e Y ) = tr X + tr Y . {\displaystyle \operatorname {tr} \log \left(e^{X}e^{Y}\right)=\operatorname {tr} X+\operatorname {tr} Y.}
That is to say, since each z j {\displaystyle z_{j}} with j ≥ 2 {\displaystyle j\geq 2} is expressible as a linear combination of commutators, the trace of each such term is zero.
Questions of convergence Suppose X {\displaystyle X} and Y {\displaystyle Y} are the following matrices in the Lie algebra s l ( 2 ; C ) {\displaystyle {\mathfrak {sl}}(2;\mathbb {C} )} (the space of 2 × 2 {\displaystyle 2\times 2} matrices with trace zero):
X = ( 0 i π i π 0 ) ; Y = ( 0 1 0 0 ) . {\displaystyle X={\begin{pmatrix}0&i\pi \\i\pi &0\end{pmatrix}};\quad Y={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}
Then
e X e Y = ( − 1 0 0 − 1 ) ( 1 1 0 1 ) = ( − 1 − 1 0 − 1 ) . {\displaystyle e^{X}e^{Y}={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}={\begin{pmatrix}-1&-1\\0&-1\end{pmatrix}}.}
It is then not hard to show that there does not exist a matrix Z {\displaystyle Z} in sl ( 2 ; C ) {\displaystyle \operatorname {sl} (2;\mathbb {C} )} with e X e Y = e Z {\displaystyle e^{X}e^{Y}=e^{Z}} . (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. Concretely, if working with a matrix Lie algebra and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is a given submultiplicative matrix norm, convergence is guaranteed if
‖ X ‖ + ‖ Y ‖ < ln 2 2 . {\displaystyle \|X\|+\|Y\|<{\frac {\ln 2}{2}}.}
Special cases If X {\displaystyle X} and Y {\displaystyle Y} commute, that is [ X , Y ] = 0 {\displaystyle [X,Y]=0} , the Baker–Campbell–Hausdorff formula reduces to e X e Y = e X + Y {\displaystyle e^{X}e^{Y}=e^{X+Y}} . Another case assumes that [ X , Y ] {\displaystyle [X,Y]} commutes with both X {\displaystyle X} and Y {\displaystyle Y} , as for the nilpotent Heisenberg group. Then the formula reduces to its first three terms.
This is the degenerate case used routinely in quantum mechanics, as illustrated below and is sometimes known as the disentangling theorem. In this case, there are no smallness restrictions on X {\displaystyle X} and Y {\displaystyle Y} . 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 ad X ( Y ) = [ X , Y ] {\displaystyle \operatorname {ad} _{X}(Y)=[X,Y]} :
log ( exp X exp Y ) = X + ad X 1 − e − ad X Y + O ( Y 2 ) = X + ad X / 2 ( 1 + coth ad X / 2 ) Y + O ( Y 2 ) , {\displaystyle \log(\exp X\exp Y)=X+{\frac {\operatorname {ad} _{X}}{1-e^{-\operatorname {ad} _{X}}}}~Y+O\left(Y^{2}\right)=X+\operatorname {ad} _{X/2}(1+\coth \operatorname {ad} _{X/2})~Y+O\left(Y^{2}\right),}
which is evident from the integral formula above. (The coefficients of the nested commutators with a single Y {\displaystyle Y} are normalized Bernoulli numbers.) Now assume that the commutator is a multiple of Y {\displaystyle Y} , so that [ X , Y ] = s Y {\displaystyle [X,Y]=sY} . Then all iterated commutators will be multiples of Y {\displaystyle Y} , and no quadratic or higher terms in Y {\displaystyle Y} appear. Thus, the O ( Y 2 ) {\displaystyle O\left(Y^{2}\right)} term above vanishes and we obtain:
Again, in this case there are no smallness restriction on X {\displaystyle X} and Y {\displaystyle Y} . The restriction on s {\displaystyle s} guarantees that the expression on the right side makes sense. (When s = 0 {\displaystyle s=0} we may interpret lim s → 0 s / ( 1 − e − s ) = 1 {\textstyle \lim _{s\to 0}s/(1-e^{-s})=1} .) We also obtain a simple "braiding identity":
e X e Y = e exp ( s ) Y e X , {\displaystyle e^{X}e^{Y}=e^{\exp(s)Y}e^{X},}
which may be written as an adjoint dilation:
e X e Y e − X = e exp ( s ) Y . {\displaystyle e^{X}e^{Y}e^{-X}=e^{\exp(s)\,Y}.}
Existence results If X {\displaystyle X} and Y {\displaystyle Y} are matrices, one can compute Z := log ( e X e Y ) {\displaystyle Z:=\log \left(e^{X}e^{Y}\right)} using the power series for the exponential and logarithm, with convergence of the series if X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small. It is natural to collect together all terms where the total degree in X {\displaystyle X} and Y