Baker–Campbell–Hausdorff formula

In mathematics, the Baker–Campbell–Hausdorff formula is the solution to the equation

Z=\log {\big (}e^{X}e^{Y}{\big )}

for possibly noncommutative $X$ and $Y$ in the Lie algebra of a Lie group. This formula tightly links Lie groups to Lie algebras by expressing the logarithm of the product of two Lie group elements as a Lie algebra element using only Lie algebraic operations. The solution on this form, whenever defined, means that multiplication in the group can be expressed entirely in Lie algebraic terms. The solution on commutative forms is obtained by substituting the power series for $exp$ and $log$ in the equation and rearranging.^[1] The point is to express the solution in Lie algebraic terms. This 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 ^[2] where a convergent power series is given, with terms recursively defined.^[3] 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^[4] (1897); elaborated by Henri Poincaré^[5] (1899) and Baker (1902);^[6] and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906).^[7] The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947).^[8]

The Baker–Campbell–Hausdorff formula: existence

The Baker–Campbell–Hausdorff formula implies that if X and Y are in some Lie algebra $\mathfrak g,$ defined over any field of characteristic 0 like $ℝ$ or $ℂ$ , then

Z = log(exp(X) exp(Y))

,

can formally be written as an infinite sum of elements of ${\mathfrak {g}}$ . [This infinite series may or may not converge, so it need not define an actual element Z in ${\mathfrak {g}}$ .] 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^[9] 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]]

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,

Δ : S \to S \otimes S

,

called the coproduct, such that

Δ(X) = X \otimes1 + 1\otimes X

and

Δ(Y) = Y \otimes1 + 1\otimes 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 $Δ(r) = r \otimes r$ ) if and only if s is primitive (this means $Δ(s) = s \otimes1 + 1\otimes s$ ).
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^[10]^[11])

The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows:^[11] The elements X and Y are primitive, so exp(X) and exp(Y) are grouplike; so their product $exp(X)exp(Y)$ is also grouplike; so its logarithm $log(exp(X)exp(Y))$ 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.

An alternative, remarkably direct and concise, recursive proof that all homogeneous polynomials in $Z$ are in the Lie algebra is due to Eichler.^[12]

An explicit Baker–Campbell–Hausdorff formula

Specifically, let G be a Lie group with Lie algebra ${\mathfrak {g}}$ . Let

\exp :{\mathfrak {g}}\rightarrow G

be the exponential map. The following general combinatorial formula was introduced by Eugene Dynkin (1947),^[11]^[13]

\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}}]}{(\sum _{j=1}^{n}(r_{j}+s_{j}))\cdot \prod _{i=1}^{n}r_{i}!s_{i}!}},

where the sum is performed over all nonegative values of $s_{i}$ and $r_{i}$ , and the following notation has been used:

[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 ]].

Since $[A, A] = 0$ , the term is zero if $s_n > 1$ or if $s_n = 0$ and $r_n > 1$ .^[14]

The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie algebra):

{\begin{aligned}Z(X,Y)&{}=\log(\exp X\exp Y)\\&{}=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}\left([X,[X,Y]]+[Y,[Y,X]]\right)\\&{}\quad -{\frac {1}{24}}[Y,[X,[X,Y]]]\\&{}\quad -{\frac {1}{720}}\left([Y,[Y,[Y,[Y,X]]]]+[X,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{360}}\left([X,[Y,[Y,[Y,X]]]]+[Y,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{120}}\left([Y,[X,[Y,[X,Y]]]]+[X,[Y,[X,[Y,X]]]]\right)+\cdots \end{aligned}}

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.

Selected tractable cases

There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications.

For example, if $[X, Y]$ vanishes, then the above formula reduces to $X + Y$ . If the commutator $[X, Y]$ is a scalar (central, 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.^[15] This is the degenerate case utilized routinely in quantum mechanics, as illustrated below.

Other forms of the Baker–Campbell–Hausdorff formula, emphasizing expansion in terms of the element $Y$ (and utilizing the linear adjoint endomorphism notation, $ad X Y \equiv [X, Y]$ ), might serve well:

\log(\exp X\exp Y) = X + \frac{\text{ad} _X ~ e^{\operatorname{ad} _X}}{e^{\operatorname{ad} _X}-1} ~ Y + O(Y^2),

as is evident from the integral formula below. (The coefficients of the nested commutators linear in $Y$ are normalized Bernoulli numbers, outlined below.)

Thus, when the commutator happens to be $[X, Y]= sY$ , for some non-zero s, this formula reduces to just $Z = X + sY / (1 - exp(- s))$ ,^[16] which then leads to braiding identities such as

e^{X} e^{Y} = e^{\exp (s) ~Y} e^{X}~,

or adjoint dilation,

e^{X} e^{Y} e^{-X} = e^{\exp (s) ~Y} ~.

There are numerous such well-known expressions applied routinely in physics.^[10]^[17] A popular integral formula is^[18]^[19]

\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,

involving the generating function for the Bernoulli numbers,

\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.^{[nb 1]}

Matrix Lie group illustration

For a matrix Lie group $G \sub \mbox{GL}(n,\mathbb{R})$ 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 = \sum_{n=0}^\infty {\frac{X^n}{n!}}.

When one solves for Z in

e^{Z}=e^{X}e^{Y},

using the series expansions for $exp$ and $log$ one obtains a simpler formula:

Z=\sum _{n>0}{\frac {(-1)^{n-1}}{n}}\sum _{\begin{smallmatrix}r_{i}+s_{i}>0\,\\1\leq i\leq n\end{smallmatrix}}{\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.

^{[nb 2]}

The first, second, third, and fourth order terms are:

$z_{1}=X+Y$
$z_2 = \frac{1}{2} (XY - YX)$
$z_3 = \frac{1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY)$
$z_4 = \frac{1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX).$

The main significance of the Baker–Campbell–Hausdorff formula is the non-obvious fact that each of the terms of degree two and higher can be expressed as a linear combination of iterated brackets of X and Y, with rational coefficients, as proven compactly in the above-mentioned Eichler note. As an immediate consequence one obtains the following result about the trace:

\operatorname {tr} \ln(e^{X}e^{Y})=\operatorname {tr} X+\operatorname {tr} Y

.

Questions of convergence

There exist^[20] real 2-by-2 matrices X and Y such that there is no real 2-by-2 matrix Z with

e^{Z}=e^{X}e^{Y}.

An example is

X={\begin{bmatrix}0&-4\\4&0\end{bmatrix}}\ \ {\text{ and }}\ \ Y={\begin{bmatrix}0&1\\0&0\end{bmatrix}}

The formulas above that promise to give expressions for Z in terms of iterated Lie-brackets of X and Y therefore describe formal power series whose convergence is not guaranteed; they are guaranteed to be globally valid only if working in the ring of formal power series in the non-commuting variables X and Y, with the commutator as Lie-bracket.

However, one can show that if we work in a Lie algebra over the field R or C and the elements X and Y are close enough to 0, then the Dynkin formula given above converges absolutely to a Lie algebra element Z with

e^{Z}=e^{X}e^{Y}.

Concretely, if working with a matrix Lie algebra and $\|.\|$ is a given submultiplicative matrix norm, then this will happen^[13] if

\|X\|+\|Y\|<\ln(2)/2

.

The Zassenhaus formula

A related combinatoric expansion that is useful in dual^[10] applications is

e^{t(X+Y)}=e^{tX}~e^{tY}~e^{-{\frac {t^{2}}{2}}[X,Y]}~e^{{\frac {t^{3}}{6}}(2[Y,[X,Y]]+[X,[X,Y]])}~e^{{\frac {-t^{4}}{24}}([[[X,Y],X],X]+3[[[X,Y],X],Y]+3[[[X,Y],Y],Y])}\cdots

where the exponents of higher order in t are likewise nested commutators, i.e., homogeneous Lie polynomials.^[21] These exponents, $C n$ in $exp(- tX) exp(t (X+Y)) = Π n exp(t n C n)$ , follow recursively by application of the above BCH expansion.

As a corollary of this, the Suzuki–Trotter decomposition follows.

An important lemma

Let $G$ be a matrix Lie group and $g$ its corresponding Lie algebra. Let $ad X$ be the linear operator on $g$ defined by $ad X Y = [X, Y] = XY - YX$ for some fixed $X \in g$ . (The adjoint endomorphism encountered above.) Denote with $Ad A$ for fixed $A \in G$ the linear transformation of $g$ given by $Ad A Y = AYA -1$ .

A standard combinatorial lemma which is utilized^[18] in producing the above explicit expansions is given by^[22]

\operatorname{Ad}_{e^X} = e^{\operatorname{ad}_X},

so, explicitly,

\operatorname{Ad}_{e^X}Y = e^{X}Y e^{-X} = e^{\operatorname{ad} _X} Y =Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots.

This formula can be proved by evaluation of the derivative with respect to $s$ of $f (s) Y \equiv e sX Y e - sX$ , solution of the resulting differential equation and evaluation at s = 1,

\frac{d}{ds}f(s)Y = \frac{d}{ds} \left (e^{sX}Ye^{-sX} \right )= Xe^{sX}Ye^{-sX} - e^{sX}Ye^{-sX}X = \operatorname{ad}_X (e^{sX}Ye^{-sX})

or

f'(s)=\operatorname {ad} _{X}f(s),\qquad f(0)=1\qquad \Longrightarrow \qquad f(s)=e^{s\operatorname {ad} _{X}}.

^[23]

A direct application of this identity

For $[X,Y]$ central, i.e., commuting with both $X$ and $Y$ ,

e^{sX} Y e^{-sX} = Y + s [ X, Y ] ~.

Consequently, for $g(s) \equiv e sX e sY$ , it follows that

{\frac {dg}{ds}}={\Bigl (}X+e^{sX}Ye^{-sX}{\Bigr )}g(s)=(X+Y+s[X,Y])~g(s)~,

whose solution is

g(s)= e^{s(X+Y) +\frac{s^2}{2} [ X, Y ] } ~,

hence the degenerate form already covered above,

e^X e^Y= e^{X+Y +\frac{1}{2} [ X, Y] } ~.

More generally, for non-central $[X,Y]$ , the following braiding identity further follows readily,

e^{X} e^{Y}= e^{(Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots)} ~e^X.

Application in quantum mechanics

A degenerate form 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 group.

A typical example 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:

e^{v\hat{a}^\dagger - v^*\hat{a}} = e^{v\hat{a}^\dagger} e^{-v^*\hat{a}} e^{-|v|^{2}/2} ,

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.^[24]

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,

e^{v\hat{a}^\dagger - v^*\hat{a}} e^{u\hat{a}^\dagger - u^*\hat{a}} = e^{(v+u)\hat{a}^\dagger -(v^*+u^*)\hat{a}} e^{(vu^*-uv^*)/2},

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.^[25]

Notes

↑ Recall
$\psi(e^y)=\sum_{n=0}^\infty B_n ~ y^n/n!$ ,
for the Bernoulli numbers, B₀ = 1, B₁ = 1/2, B₂ = 1/6, B₄ = −1/30, ...
↑ 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 $e Z = e X e Y$ . 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.

References

↑ Rossmann 2002 See equation (2) in section 1.3.
↑ 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
↑ Hall 2015 Section 5.7
1 2 3 Magnus, W. (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.
1 2 3 N. Jacobson, Lie Algebras, John Wiley & Sons, 1966.
↑ Eichler, M. (1968). "A new proof of the Baker-Campbell-Hausdorff formula", J. Math. Soc. Japan 20, 23-25. online open access.
1 2 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.
↑ Hall 2015 Theorem 5.1
↑ Hall 2015 Exercise 5.5
↑ 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. Bibcode:1985JMP....26..601S. doi:10.1063/1.526596. ; Veltman, M, 't Hooft, G & de Wit, B (2007), Appendix D.
1 2 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
↑ 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.
↑ Casas, F.; Murua, A.; Nadinic, M. (2012). "Efficient computation of the Zassenhaus formula". Computer Physics Communications. 183 (11): 2386. arXiv:1204.0389. Bibcode:2012CoPhC.183.2386C. doi:10.1016/j.cpc.2012.06.006.
↑ Hall 2015 Proposition 3.35
↑ Rossmann 2002 p. 15
↑ 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.

Bibliography

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) [1994], "Campbell–Hausdorff formula", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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 3-540-59179-6
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 0-19-859683-9
J.-P. Serre, Lie algebras and Lie groups, Benjamin, 1965.
Wilfried Schmid (1982). "Poincaré and Lie groups", Bull. Amer. Math. Soc. 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.

External links

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[20] Recall
$\psi(e^y)=\sum_{n=0}^\infty B_n ~ y^n/n!$ ,
for the Bernoulli numbers, B₀ = 1, B₁ = 1/2, B₂ = 1/6, B₄ = −1/30, ...

[21] 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 $e Z = e X e Y$ . 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.

[1] Rossmann 2002 See equation (2) in section 1.3.

[2] F. Schur (1890), “Neue Begruendung der Theorie der endlichen Transformationsgruppen,” Mathematische Annalen, 35 (1890), 161-197. online copy

[3] see, e.g., Shlomo Sternberg, Lie Algebras (2004) Harvard University. (See page 10.)

[4] J. Campbell, Proc Lond Math Soc 28 (1897) 381–390; J. Campbell, Proc Lond Math Soc 29 (1898) 14–32.

[5] H. Poincaré, Compt Rend Acad Sci Paris 128 (1899) 1065–1069; Camb Philos Trans 18 (1899) 220–255.

[6] 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.

[7] F. Hausdorff, "Die symbolische Exponentialformel in der Gruppentheorie", Ber Verh Saechs Akad Wiss Leipzig 58 (1906) 19–48.

[8] Rossmann 2002 p. 23

[9] Hall 2015 Section 5.7

[magnus-10] 1 2 3 Magnus, W. (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.

[Jacobson-11] 1 2 3 N. Jacobson, Lie Algebras, John Wiley & Sons, 1966.

[12] Eichler, M. (1968). "A new proof of the Baker-Campbell-Hausdorff formula", J. Math. Soc. Japan 20, 23-25. online open access.

[dynkin-13] 1 2 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.

[14] A.A. Sagle & R.E. Walde, "Introduction to Lie Groups and Lie Algebras", Academic Press, New York, 1973. ISBN 0-12-614550-4.

[15] Hall 2015 Theorem 5.1

[16] Hall 2015 Exercise 5.5

[17] 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. Bibcode:1985JMP....26..601S. doi:10.1063/1.526596. ; Veltman, M, 't Hooft, G & de Wit, B (2007), Appendix D.

[miller-18] 1 2 W. Miller, Symmetry Groups and their Applications, Academic Press, New York, 1972, pp 159–161. ISBN 0-12-497460-0

[19] Hall 2015 Theorem 5.3

[22] 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.

[23] Casas, F.; Murua, A.; Nadinic, M. (2012). "Efficient computation of the Zassenhaus formula". Computer Physics Communications. 183 (11): 2386. arXiv:1204.0389. Bibcode:2012CoPhC.183.2386C. doi:10.1016/j.cpc.2012.06.006.

[24] Hall 2015 Proposition 3.35

[25] Rossmann 2002 p. 15

[26] L. Mandel, E. Wolf Optical Coherence and Quantum Optics (Cambridge 1995).

[27] Greiner 1996 See pp 27-29 for a detailed proof of the above lemma.