- Source: Derivative of the exponential map
In the theory of Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d/dtexp(X(t)):Tg → TG, where X(t) is a C1 path in the Lie algebra, and a closely related differential dexp:Tg → TG.
The formula for dexp was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula.
The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker–Campbell–Hausdorff formula, and it is used frequently in physics for example in quantum field theory, as in the Magnus expansion in perturbation theory, and in lattice gauge theory.
Throughout, the notations exp(X) and eX will be used interchangeably to denote the exponential given an argument, except when, where as noted, the notations have dedicated distinct meanings. The calculus-style notation is preferred here for better readability in equations. On the other hand, the exp-style is sometimes more convenient for inline equations, and is necessary on the rare occasions when there is a real distinction to be made.
Statement
The derivative of the exponential map is given by
Explanation
To compute the differential dexp of exp at X, dexpX: TgX → TGexp(X), the standard recipe
d
exp
X
Y
=
d
d
t
e
Z
(
t
)
|
t
=
0
,
Z
(
0
)
=
X
,
Z
′
(
0
)
=
Y
{\displaystyle d\exp _{X}Y=\left.{\frac {d}{dt}}e^{Z(t)}\right|_{t=0},Z(0)=X,Z'(0)=Y}
is employed. With Z(t) = X + tY the result
follows immediately from (1). In particular, dexp0:Tg0 → TGexp(0) = TGe is the identity because TgX ≃ g (since g is a vector space) and TGe ≃ g.
Proof
The proof given below assumes a matrix Lie group. This means that the exponential mapping from the Lie algebra to the matrix Lie group is given by the usual power series, i.e. matrix exponentiation. The conclusion of the proof still holds in the general case, provided each occurrence of exp is correctly interpreted. See comments on the general case below.
The outline of proof makes use of the technique of differentiation with respect to s of the parametrized expression
Γ
(
s
,
t
)
=
e
−
s
X
(
t
)
∂
∂
t
e
s
X
(
t
)
{\displaystyle \Gamma (s,t)=e^{-sX(t)}{\frac {\partial }{\partial t}}e^{sX(t)}}
to obtain a first order differential equation for Γ which can then be solved by direct integration in s. The solution is then eX Γ(1, t).
Lemma
Let Ad denote the adjoint action of the group on its Lie algebra. The action is given by AdAX = AXA−1 for A ∈ G, X ∈ g. A frequently useful relationship between Ad and ad is given by
Proof
Using the product rule twice one finds,
∂
Γ
∂
s
=
e
−
s
X
(
−
X
)
∂
∂
t
e
s
X
(
t
)
+
e
−
s
X
∂
∂
t
[
X
(
t
)
e
s
X
(
t
)
]
=
e
−
s
X
d
X
d
t
e
s
X
.
{\displaystyle {\frac {\partial \Gamma }{\partial s}}=e^{-sX}(-X){\frac {\partial }{\partial t}}e^{sX(t)}+e^{-sX}{\frac {\partial }{\partial t}}\left[X(t)e^{sX(t)}\right]=e^{-sX}{\frac {dX}{dt}}e^{sX}.}
Then one observes that
∂
Γ
∂
s
=
A
d
e
−
s
X
X
′
=
e
−
a
d
s
X
X
′
,
{\displaystyle {\frac {\partial \Gamma }{\partial s}}=\mathrm {Ad} _{e^{-sX}}X'=e^{-\mathrm {ad} _{sX}}X',}
by (4) above. Integration yields
Γ
(
1
,
t
)
=
e
−
X
(
t
)
∂
∂
t
e
X
(
t
)
=
∫
0
1
∂
Γ
∂
s
d
s
=
∫
0
1
e
−
a
d
s
X
X
′
d
s
.
{\displaystyle \Gamma (1,t)=e^{-X(t)}{\frac {\partial }{\partial t}}e^{X(t)}=\int _{0}^{1}{\frac {\partial \Gamma }{\partial s}}ds=\int _{0}^{1}e^{-\mathrm {ad} _{sX}}X'ds.}
Using the formal power series to expand the exponential, integrating term by term, and finally recognizing (2),
Γ
(
1
,
t
)
=
∫
0
1
∑
k
=
0
∞
(
−
1
)
k
s
k
k
!
(
a
d
X
)
k
d
X
d
t
d
s
=
∑
k
=
0
∞
(
−
1
)
k
(
k
+
1
)
!
(
a
d
X
)
k
d
X
d
t
=
1
−
e
−
a
d
X
a
d
X
d
X
d
t
,
{\displaystyle \Gamma (1,t)=\int _{0}^{1}\sum _{k=0}^{\infty }{\frac {(-1)^{k}s^{k}}{k!}}(\mathrm {ad} _{X})^{k}{\frac {dX}{dt}}ds=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(k+1)!}}(\mathrm {ad} _{X})^{k}{\frac {dX}{dt}}={\frac {1-e^{-\mathrm {ad} _{X}}}{\mathrm {ad} _{X}}}{\frac {dX}{dt}},}
and the result follows. The proof, as presented here, is essentially the one given in Rossmann (2002). A proof with a more algebraic touch can be found in Hall (2015).
= Comments on the general case
=
The formula in the general case is given by
d
d
t
exp
(
C
(
t
)
)
=
exp
(
C
)
ϕ
(
−
a
d
(
C
)
)
C
′
,
{\displaystyle {\frac {d}{dt}}\exp(C(t))=\exp(C)\phi (-\mathrm {ad} (C))C~',}
where
ϕ
(
z
)
=
e
z
−
1
z
=
1
+
1
2
!
z
+
1
3
!
z
2
+
⋯
,
{\displaystyle \phi (z)={\frac {e^{z}-1}{z}}=1+{\frac {1}{2!}}z+{\frac {1}{3!}}z^{2}+\cdots ,}
which formally reduces to
d
d
t
exp
(
C
(
t
)
)
=
exp
(
C
)
1
−
e
−
a
d
C
a
d
C
d
C
(
t
)
d
t
.
{\displaystyle {\frac {d}{dt}}\exp(C(t))=\exp(C){\frac {1-e^{-\mathrm {ad} _{C}}}{\mathrm {ad} _{C}}}{\frac {dC(t)}{dt}}.}
Here the exp-notation is used for the exponential mapping of the Lie algebra and the calculus-style notation in the fraction indicates the usual formal series expansion. For more information and two full proofs in the general case, see the freely available Sternberg (2004) reference.
= A direct formal argument
=
An immediate way to see what the answer must be, provided it exists is the following. Existence needs to be proved separately in each case. By direct differentiation of the standard limit definition of the exponential, and exchanging the order of differentiation and limit,
d
d
t
e
X
(
t
)
=
lim
N
→
∞
d
d
t
(
1
+
X
(
t
)
N
)
N
=
lim
N
→
∞
∑
k
=
1
N
(
1
+
X
(
t
)
N
)
N
−
k
1
N
d
X
(
t
)
d
t
(
1
+
X
(
t
)
N
)
k
−
1
,
{\displaystyle {\begin{aligned}{\frac {d}{dt}}e^{X(t)}&=\lim _{N\to \infty }{\frac {d}{dt}}\left(1+{\frac {X(t)}{N}}\right)^{N}\\&=\lim _{N\to \infty }\sum _{k=1}^{N}\left(1+{\frac {X(t)}{N}}\right)^{N-k}{\frac {1}{N}}{\frac {dX(t)}{dt}}\left(1+{\frac {X(t)}{N}}\right)^{k-1}~,\end{aligned}}}
where each factor owes its place to the non-commutativity of X(t) and X´(t).
Dividing the unit interval into N sections Δs = Δk/N (Δk = 1 since the sum indices are integers) and letting N → ∞, Δk → dk, k/N → s, Σ → ∫ yields
d
d
t
e
X
(
t
)
=
∫
0
1
e
(
1
−
s
)
X
X
′
e
s
X
d
s
=
e
X
∫
0
1
A
d
e
−
s
X
X
′
d
s
=
e
X
∫
0
1
e
−
a
d
s
X
d
s
X
′
=
e
X
1
−
e
−
a
d
X
a
d
X
d
X
d
t
.
{\displaystyle {\begin{aligned}{\frac {d}{dt}}e^{X(t)}&=\int _{0}^{1}e^{(1-s)X}X'e^{sX}ds=e^{X}\int _{0}^{1}\mathrm {Ad} _{e^{-sX}}X'ds\\&=e^{X}\int _{0}^{1}e^{-\mathrm {ad} _{sX}}dsX'=e^{X}{\frac {1-e^{-\mathrm {ad} _{X}}}{\mathrm {ad} _{X}}}{\frac {dX}{dt}}~.\end{aligned}}}
Applications
=
The inverse function theorem together with the derivative of the exponential map provides information about the local behavior of exp. Any Ck, 0 ≤ k ≤ ∞, ω map f between vector spaces (here first considering matrix Lie groups) has a Ck inverse such that f is a Ck bijection in an open set around a point x in the domain provided dfx is invertible. From (3) it follows that this will happen precisely when
1
−
e
a
d
X
a
d
X
{\displaystyle {\frac {1-e^{\mathrm {ad_{X}} }}{\mathrm {ad} _{X}}}}
is invertible. This, in turn, happens when the eigenvalues of this operator are all nonzero. The eigenvalues of 1 − exp(−adX)/adX are related to those of adX as follows. If g is an analytic function of a complex variable expressed in a power series such that g(U) for a matrix U converges, then the eigenvalues of g(U) will be g(λij), where λij are the eigenvalues of U, the double subscript is made clear below. In the present case with g(U) = 1 − exp(−U)/U and U = adX, the eigenvalues of 1 − exp(−adX)/adX are
1
−
e
−
λ
i
j
λ
i
j
,
{\displaystyle {\frac {1-e^{-\lambda _{ij}}}{\lambda _{ij}}},}
where the λij are the eigenvalues of adX. Putting 1 − exp(−λij)/λij = 0 one sees that dexp is invertible precisely when
λ
i
j
≠
k
2
π
i
,
k
=
±
1
,
±
2
,
…
.
{\displaystyle \lambda _{ij}\neq k2\pi i,k=\pm 1,\pm 2,\ldots .}
The eigenvalues of adX are, in turn, related to those of X. Let the eigenvalues of X be λi. Fix an ordered basis ei of the underlying vector space V such that X is lower triangular. Then
X
e
i
=
λ
i
e
i
+
⋯
,
{\displaystyle Xe_{i}=\lambda _{i}e_{i}+\cdots ,}
with the remaining terms multiples of en with n > i. Let Eij be the corresponding basis for matrix space, i.e. (Eij)kl = δikδjl. Order this basis such that Eij < Enm if i − j < n − m. One checks that the action of adX is given by
a
d
X
E
i
j
=
(
λ
i
−
λ
j
)
E
i
j
+
⋯
≡
λ
i
j
E
i
j
+
⋯
,
{\displaystyle \mathrm {ad} _{X}E_{ij}=(\lambda _{i}-\lambda _{j})E_{ij}+\cdots \equiv \lambda _{ij}E_{ij}+\cdots ,}
with the remaining terms multiples of Emn > Eij. This means that adX is lower triangular with its eigenvalues λij = λi − λj on the diagonal. The conclusion is that dexpX is invertible, hence exp is a local bianalytical bijection around X, when the eigenvalues of X satisfy
λ
i
−
λ
j
≠
k
2
π
i
,
k
=
±
1
,
±
2
,
…
,
1
≤
i
,
j
≤
n
=
dim
V
.
{\displaystyle \lambda _{i}-\lambda _{j}\neq k2\pi i,\quad k=\pm 1,\pm 2,\ldots ,\quad 1\leq i,j\leq n=\dim V.}
In particular, in the case of matrix Lie groups, it follows, since dexp0 is invertible, by the inverse function theorem that exp is a bi-analytic bijection in a neighborhood of 0 ∈ g in matrix space. Furthermore, exp, is a bi-analytic bijection from a neighborhood of 0 ∈ g in g to a neighborhood of e ∈ G. The same conclusion holds for general Lie groups using the manifold version of the inverse function theorem.
It also follows from the implicit function theorem that dexpξ itself is invertible for ξ sufficiently small.
= Derivation of a Baker–Campbell–Hausdorff formula
=
If Z(t) is defined such that
e
Z
(
t
)
=
e
X
e
t
Y
,
{\displaystyle e^{Z(t)}=e^{X}e^{tY},}
an expression for Z(1) = log( exp X exp Y ), the Baker–Campbell–Hausdorff formula, can be derived from the above formula,
exp
(
−
Z
(
t
)
)
d
d
t
exp
(
Z
(
t
)
)
=
1
−
e
−
a
d
Z
a
d
Z
Z
′
(
t
)
.
{\displaystyle \exp(-Z(t)){\frac {d}{dt}}\exp(Z(t))={\frac {1-e^{-\mathrm {ad} _{Z}}}{\mathrm {ad} _{Z}}}Z'(t).}
Its left-hand side is easy to see to equal Y. Thus,
Y
=
1
−
e
−
a
d
Z
a
d
Z
Z
′
(
t
)
,
{\displaystyle Y={\frac {1-e^{-\mathrm {ad} _{Z}}}{\mathrm {ad} _{Z}}}Z'(t),}
and hence, formally,
Z
′
(
t
)
=
a
d
Z
1
−
e
−
a
d
Z
Y
≡
ψ
(
e
a
d
Z
)
Y
,
ψ
(
w
)
=
w
log
w
w
−
1
=
1
+
∑
m
=
1
∞
(
−
1
)
m
+
1
m
(
m
+
1
)
(
w
−
1
)
m
,
‖
w
‖
<
1.
{\displaystyle Z'(t)={\frac {\mathrm {ad} _{Z}}{1-e^{-\mathrm {ad} _{Z}}}}Y\equiv \psi \left(e^{\mathrm {ad} _{Z}}\right)Y,\quad \psi (w)={\frac {w\log w}{w-1}}=1+\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m(m+1)}}(w-1)^{m},\|w\|<1.}
However, using the relationship between Ad and ad given by (4), it is straightforward to further see that
e
a
d
Z
=
e
a
d
X
e
t
a
d
Y
{\displaystyle e^{\mathrm {ad} _{Z}}=e^{\mathrm {ad} _{X}}e^{t\mathrm {ad} _{Y}}}
and hence
Z
′
(
t
)
=
ψ
(
e
a
d
X
e
t
a
d
Y
)
Y
.
{\displaystyle Z'(t)=\psi \left(e^{\mathrm {ad} _{X}}e^{t\mathrm {ad} _{Y}}\right)Y.}
Putting this into the form of an integral in t from 0 to 1 yields,
Z
(
1
)
=
log
(
exp
X
exp
Y
)
=
X
+
(
∫
0
1
ψ
(
e
ad
X
e
t
ad
Y
)
d
t
)
Y
,
{\displaystyle Z(1)=\log(\exp X\exp Y)=X+\left(\int _{0}^{1}\psi \left(e^{\operatorname {ad} _{X}}~e^{t\,{\text{ad}}_{Y}}\right)\,dt\right)\,Y,}
an integral formula for Z(1) that is more tractable in practice than the explicit Dynkin's series formula due to the simplicity of the series expansion of ψ. Note this expression consists of X+Y and nested commutators thereof with X or Y. A textbook proof along these lines can be found in Hall (2015) and Miller (1972).
= Derivation of Dynkin's series formula
=
Dynkin's formula mentioned may also be derived analogously, starting from the parametric extension
e
Z
(
t
)
=
e
t
X
e
t
Y
,
{\displaystyle e^{Z(t)}=e^{tX}e^{tY},}
whence
e
−
Z
(
t
)
d
e
Z
(
t
)
d
t
=
e
−
t
a
d
Y
X
+
Y
,
{\displaystyle e^{-Z(t)}{\frac {de^{Z(t)}}{dt}}=e^{-t\,\mathrm {ad} _{Y}}X+Y~,}
so that, using the above general formula,
Z
′
=
a
d
Z
1
−
e
−
a
d
Z
(
e
−
t
a
d
Y
X
+
Y
)
=
a
d
Z
e
a
d
Z
−
1
(
X
+
e
t
a
d
X
Y
)
.
{\displaystyle Z'={\frac {\mathrm {ad} _{Z}}{1-e^{-\mathrm {ad} _{Z}}}}~\left(e^{-t\,\mathrm {ad} _{Y}}X+Y\right)={\frac {\mathrm {ad} _{Z}}{e^{\mathrm {ad} _{Z}}-1}}~\left(X+e^{t\,\mathrm {ad} _{X}}Y\right).}
Since, however,
a
d
Z
=
log
(
exp
(
a
d
Z
)
)
=
log
(
1
+
(
exp
(
a
d
Z
)
−
1
)
)
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
(
exp
(
a
d
Z
)
−
1
)
n
,
‖
a
d
Z
‖
<
log
2
,
{\displaystyle {\begin{aligned}\mathrm {ad_{Z}} &=\log \left(\exp \left(\mathrm {ad} _{Z}\right)\right)=\log \left(1+\left(\exp \left(\mathrm {ad} _{Z}\right)-1\right)\right)\\&=\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}(\exp(\mathrm {ad} _{Z})-1)^{n}~,\quad \|\mathrm {ad} _{Z}\|<\log 2~~,\end{aligned}}}
the last step by virtue of the Mercator series expansion, it follows that
and, thus, integrating,
Z
(
1
)
=
∫
0
1
d
t
d
Z
(
t
)
d
t
=
∑
n
=
1
∞
(
−
1
)
n
−
1
n
∫
0
1
d
t
(
e
t
a
d
X
e
t
a
d
Y
−
1
)
n
−
1
(
X
+
e
t
a
d
X
Y
)
.
{\displaystyle Z(1)=\int _{0}^{1}dt~{\frac {dZ(t)}{dt}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\int _{0}^{1}dt~\left(e^{t\,\mathrm {ad} _{X}}e^{t\mathrm {ad} _{Y}}-1\right)^{n-1}~\left(X+e^{t\,\mathrm {ad} _{X}}Y\right).}
It is at this point evident that the qualitative statement of the BCH formula holds, namely Z lies in the Lie algebra generated by X, Y and is expressible as a series in repeated brackets (A). For each k, terms for each partition thereof are organized inside the integral ∫dt tk−1. The resulting Dynkin's formula is then
For a similar proof with detailed series expansions, see Rossmann (2002).
= Combinatoric details
=
Change the summation index in (5) to k = n − 1 and expand
in a power series. To handle the series expansions simply, consider first Z = log(eXeY). The log-series and the exp-series are given by
log
(
A
)
=
∑
k
=
1
∞
(
−
1
)
k
+
1
k
(
A
−
I
)
k
,
and
e
X
=
∑
k
=
0
∞
X
k
k
!
{\displaystyle \log(A)=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}{(A-I)}^{k},\quad {\text{and}}\quad e^{X}=\sum _{k=0}^{\infty }{\frac {X^{k}}{k!}}}
respectively. Combining these one obtains
This becomes
where Sk is the set of all sequences s = (i1, j1, ..., ik, jk) of length 2k subject to the conditions in (99).
Now substitute (eXeY − 1) for (eadtXeadtY − 1) in the LHS of (98). Equation (99) then gives
d
Z
d
t
=
∑
k
=
0
∞
(
−
1
)
k
k
+
1
∑
s
∈
S
k
,
i
k
+
1
≥
0
t
i
1
+
j
1
+
⋯
+
i
k
+
j
k
a
d
X
i
1
a
d
Y
j
1
⋯
a
d
X
i
k
a
d
Y
j
k
i
1
!
j
1
!
⋯
i
k
!
j
k
!
X
+
t
i
1
+
j
1
+
⋯
+
i
k
+
j
k
+
i
k
+
1
a
d
X
i
1
a
d
Y
j
1
⋯
a
d
X
i
k
a
d
Y
j
k
X
i
k
+
1
i
1
!
j
1
!
⋯
i
k
!
j
k
!
i
k
+
1
!
Y
,
i
r
,
j
r
≥
0
,
i
r
+
j
r
>
0
,
1
≤
r
≤
k
,
{\displaystyle {\begin{aligned}{\frac {dZ}{dt}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\sum _{s\in S_{k},i_{k+1}\geq 0}&t^{i_{1}+j_{1}+\cdots +i_{k}+j_{k}}{\frac {{\mathrm {ad} _{X}}^{i_{1}}{\mathrm {ad} _{Y}}^{j_{1}}\cdots {\mathrm {ad} _{X}}^{i_{k}}{\mathrm {ad} _{Y}}^{j_{k}}}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!}}X\\{}+{}&t^{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+i_{k+1}}{\frac {{\mathrm {ad} _{X}}^{i_{1}}{\mathrm {ad} _{Y}}^{j_{1}}\cdots {\mathrm {ad} _{X}}^{i_{k}}{\mathrm {ad} _{Y}}^{j_{k}}X^{i_{k+1}}}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!}}Y,\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k,\end{aligned}}}
or, with a switch of notation, see An explicit Baker–Campbell–Hausdorff formula,
d
Z
d
t
=
∑
k
=
0
∞
(
−
1
)
k
k
+
1
∑
s
∈
S
k
,
i
k
+
1
≥
0
t
i
1
+
j
1
+
⋯
+
i
k
+
j
k
[
X
(
i
1
)
Y
(
j
1
)
⋯
X
(
i
k
)
Y
(
j
k
)
X
]
i
1
!
j
1
!
⋯
i
k
!
j
k
!
+
t
i
1
+
j
1
+
⋯
+
i
k
+
j
k
+
i
k
+
1
[
X
(
i
1
)
Y
(
j
1
)
⋯
X
(
i
k
)
Y
(
j
k
)
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{\displaystyle {\begin{aligned}{\frac {dZ}{dt}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\sum _{s\in S_{k},i_{k+1}\geq 0}&t^{i_{1}+j_{1}+\cdots +i_{k}+j_{k}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!}}\\{}+{}&t^{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+i_{k+1}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!}},\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k\end{aligned}}.}
Note that the summation index for the rightmost eadtX in the second term in (97) is denoted ik + 1, but is not an element of a sequence s ∈ Sk. Now integrate Z = Z(1) = ∫dZ/dtdt, using Z(0) = 0,
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{\displaystyle {\begin{aligned}Z=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\sum _{s\in S_{k},i_{k+1}\geq 0}&{\frac {1}{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+1}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!}}\\{}+{}&{\frac {1}{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+i_{k+1}+1}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!}},\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k\end{aligned}}.}
Write this as
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{\displaystyle {\begin{aligned}Z=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\sum _{s\in S_{k},i_{k+1}\geq 0}&{\frac {1}{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+(i_{k+1}=1)+(j_{k+1}=0)}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1}=1)}Y^{(j_{k+1}=0)}\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!(i_{k+1}=1)!(j_{k+1}=0)!}}\\{}+{}&{\frac {1}{i_{1}+j_{1}+\cdots +i_{k}+j_{k}+i_{k+1}+(j_{k+1}=1)}}{\frac {\left[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y^{(j_{k+1}=1)}\right]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!(j_{k+1}=1)!}},\\\\&(i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k).\end{aligned}}}
This amounts to
where
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{\displaystyle i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k+1,}
using the simple observation that [T, T] = 0 for all T. That is, in (100), the leading term vanishes unless jk + 1 equals 0 or 1, corresponding to the first and second terms in the equation before it. In case jk + 1 = 0, ik + 1 must equal 1, else the term vanishes for the same reason (ik + 1 = 0 is not allowed). Finally, shift the index, k → k − 1,
This is Dynkin's formula. The striking similarity with (99) is not accidental: It reflects the Dynkin–Specht–Wever map, underpinning the original, different, derivation of the formula. Namely, if
X
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{\displaystyle X^{i_{1}}Y^{j_{1}}\cdots X^{i_{k}}Y^{j_{k}}}
is expressible as a bracket series, then necessarily
Putting observation (A) and theorem (B) together yields a concise proof of the explicit BCH formula.
See also
Matrix logarithm
Remarks
Notes
References
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 ; translation from Google books.
Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
Miller, Wllard (1972), Symmetry Groups and their Applications, Academic Press, ISBN 0-12-497460-0
Poincaré, H. (1899), "Sur les groupes continus", Cambridge Philos. Trans., 18: 220–55
Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
Schur, F. (1891), "Zur Theorie der endlichen Transformationsgruppen", Abh. Math. Sem. Univ. Hamburg, 4: 15–32
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.
Tuynman (1995), "The derivation of the exponential map of matrices", Amer. Math. Monthly, 102 (9): 818–819, doi:10.2307/2974511, JSTOR 2974511
Veltman, M, 't Hooft, G & de Wit, B (2007). "Lie Groups in Physics", online lectures.
Wilcox, R. M. (1967). "Exponential Operators and Parameter Differentiation in Quantum Physics". Journal of Mathematical Physics. 8 (4): 962–982. Bibcode:1967JMP.....8..962W. doi:10.1063/1.1705306.
External links
Sternberg, Shlomo (2004), Lie Algebras (PDF)
Schmid, Wilfried (1982), "Poincaré and Lie groups" (PDF), Bull. Amer. Math. Soc., 6 (2): 175–186, doi:10.1090/s0273-0979-1982-14972-2
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