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A Note on the Baker–Campbell–Hausdorff Series in Terms of Right-Nested Commutators

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Abstract

We get compact expressions for the Baker–Campbell–Hausdorff series \(Z = \log (\mathrm {e}^X \, \mathrm {e}^Y)\) in terms of right-nested commutators. The reduction in the number of terms originates from two facts: (i) we use as a starting point an explicit expression directly involving independent commutators and (ii) we derive a complete set of identities arising among right-nested commutators. The procedure allows us to obtain the series with fewer terms than when expressed in the classical Hall basis at least up to terms of grade 10.

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References

  1. http://www.gicas.uji.es/Research/bch.html

  2. Arnal, A., Casas, F., Chiralt, C.: A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators. J. Phys. Commun. 2, 035024 (2018)

    Article  Google Scholar 

  3. Blanes, S., Casas, F.: A Concise Introduction to Geometric Numerical Integration. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  4. Bonfiglioli, A.: An ODE’s version of the formula of Baker, Campbell, Dynkin and Hausdorff and the construction of Lie groups with prescribed Lie algebra. Mediterr. J. Math. 7, 387–414 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bonfiglioli, A., Fulci, R.: Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin, vol. 2034 of Lecture Notes in Mathematics. Springer, Berlin (2012)

    MATH  Google Scholar 

  6. Burgunder, E.: Eulerian idempotent and Kashiwara-Vergne conjecture. Ann. Inst. Fourier 58, 1153–1184 (2008)

    Article  MathSciNet  Google Scholar 

  7. Casas, F., Murua, A.: An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications. J. Math. Phys. 50, 033513 (2009)

    Article  MathSciNet  Google Scholar 

  8. Dragt, A., Forest, E.: Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods. J. Math. Phys. 24, 2734–2744 (1983)

    Article  MathSciNet  Google Scholar 

  9. Dynkin, E.: Evaluation of the coefficients of the Campbell–Hausdorff formula. Dokl. Akad. Nauk. SSSR 57, 323–326 (1947)

    Google Scholar 

  10. Dynkin, E.: On the representation by means of commutators of the series \(\log (e^x e^y)\) for noncommutative \(x\) and \(y\). Mat. Sb. (N.S.) 25(67), 155–162 (1949). (in Russian)

    Google Scholar 

  11. Dynkin, E.: Calculation of the coefficients in the Campbell–Hausdorff series. In: Dynkin, E., Yushkevich, A., Seitz, G., Onishchik, A. (eds.) Selected Papers of E.B. Dynkin with Commentary, pp. 31–35. American Mathematical Society, Providence (2000)

    Google Scholar 

  12. Goldberg, K.: The formal power series for \(\log (\text{ e}^x \text{ e}^y)\). Duke Math. J. 23, 13–21 (1956)

    MathSciNet  MATH  Google Scholar 

  13. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag, Berli (2006). Second ed.

    MATH  Google Scholar 

  14. Helmstetter, J.: Série de Hausdorff, d’une algèbre de Lie et projections canoniques de l’algèbre enveloppante. J. Algebra 120, 170–199 (1989)

    Article  MathSciNet  Google Scholar 

  15. Kobayashi, H., Hatano, N., Suzuki, M.: Goldberg’s theorem and the Baker–Campbell–Hausdorff formula. Physica A 250, 535–548 (1998)

    Article  MathSciNet  Google Scholar 

  16. Kolsrud, M.: Maximal reductions in the Baker–Hausdorff formula. J. Math. Phys. 34, 270–285 (1993)

    Article  MathSciNet  Google Scholar 

  17. Li, R.-C. Raising the order of unconventional schemes for ordinary differential equations. PhD thesis, University of California at Berkeley, (1995)

  18. Loday, J.-L.: Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math. 96, 205–230 (1989)

    Article  MathSciNet  Google Scholar 

  19. Loday, J.-L.: Série de Hausdorff, idempotents Eulériens et algèbres de Hopf. Expo. Math. 12, 165–178 (1994)

    MATH  Google Scholar 

  20. McLachlan, R. and Murua, A., The Lie algebra of classical mechanics, tech. rep., (2019) arXiv:1905.07554

  21. Müger, M. Notes on the theorem of Baker–Campbell–Hausdorff–Dynkin, https://www.math.ru.nl/~mueger/PDF/BCHD.pdf

  22. Oteo, J.: The Baker–Campbell–Hausdorff formula and nested commutator identities. J. Math. Phys. 32, 419–424 (1991)

    Article  MathSciNet  Google Scholar 

  23. Reutenauer, C.: Free Lie Algebras, vol. 7. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  24. Solomon, L.: On the Poincaré-Birkhoff-Witt theorem. J. Comb. Theory 4, 363–375 (1968)

    Article  Google Scholar 

  25. Sornborger, A., Stewart, E.: Higher-order methods for simulations on quantum computers. Phys. Rev. A 60, 1956–1965 (1999)

    Article  Google Scholar 

  26. Strichartz, R.S.: The Campbell-Baker–Hausdorff–Dynkin formula and solutions of differential equations. J. Funct. Anal. 72, 320–345 (1987)

    Article  MathSciNet  Google Scholar 

  27. Varadarajan, V.: Lie Groups, Lie Algebras, and Their Representations. Springer-Verlag, Berlin (1984)

    Book  Google Scholar 

  28. Wilcox, R.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962–982 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

FC would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Geometry, compatibility and structure preservation in computational differential equations”, when work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1 and by Ministerio de Economía y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE). We are also very grateful to Prof. M. Müger for bringing reference [10] to our attention and for providing us his lecture notes [21].

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  1. Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Departament de Matemàtiques, Universitat Jaume I, 12071, Castellón, Spain

    Ana Arnal, Fernando Casas & Cristina Chiralt

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  1. Ana Arnal
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  2. Fernando Casas
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Appendices

A Appendix

The following code in Mathematica implements (perhaps not in the most efficient way) formula (7):

figure a

Next, we compute \(\Phi _m(X_1, \ldots , X_n)\) with the explicit expression (9) for \(\varphi _m \) in terms of non-commutative products Mm. This is contained in PhiMm[m][{\(X_1, \ldots , X_n\)}]:

figure b

Then, we express \(\varphi _n(X_1, \ldots , X_n)\) in terms of commutators, Eq. (13), and finally \(\Phi _m\) by computing PhiCmt[m][{\(X_1, \ldots , X_n\)}]:

figure c

The first block defines the commutator (just the linearity property and the antisymmetry) with the correct format for output if necessary. Linearity properties in an analogous way should be implemented for the non-commutative product Mm. The second block defines the basis and the generic term \(\varphi _n(X_1,\ldots , X_n)\). Finally, let us remark that since the number of variables is free, the same code allows one to compute both the BCH PhiCmt[m][{X, Y}] and the symmetric BCH series, PhiCmt[m][{\(\frac{1}{2} X,Y, \frac{1}{2} X\)}].

B Appendix

The algorithm we have applied to generate the identities among commutators and a basis of the homogeneous subspace \(\mathcal {L}_m(X,Y)\) formed by right-nested commutators is a generalization of a procedure proposed in [17], and can be summarized as follows:

For each \(j=2, \ldots , m\), do:

  1. 1.

    Generate all possible right-nested commutators \(C_i\) involving j operators X and \(m-j\) operators Y. For example, with \(m=4\):

    $$\begin{aligned} \mathcal {B}_ 4=\{ [X,[X,[X,Y]]], [Y,[X,[X,Y]]], [X,[Y,[X,Y]]], [Y,[Y,[X,Y]]]\}. \end{aligned}$$
  2. 2.

    Generate the corresponding element \(\langle C_i \rangle \) in the homogeneous subspace \(\mathcal {U}_m(X,Y)\) of the universal enveloping algebra associated with \(\mathcal {L}(X,Y)\). This is done by expanding each commutator \([A,B] = A B - B A\). For example, on the previous list, for \(C_1=[X,[X,[X,Y]]]\):

    $$\begin{aligned} <C_1> = XXXY-3XXYX+3XYXX-YXXX. \end{aligned}$$
  3. 3.

    The element \(\langle C_i \rangle \) is then a linear combination of words \(X_{i_1} X_{i_2} \cdots X_{i_{m}}\), where \(X_{i_j}\) is either X of Y. The total number of words is \(\left( {\begin{array}{c}m\\ j\end{array}}\right) \). Once all these words are arranged in a prescribed order, the element commutator \(C_i\) can be identified with the vector \((a_1, a_2, \ldots , a_p)\) formed by the linear combination. Then, \(C_1\) would be \(C_1\equiv (1,-3,0,3,0,0,-1,0,0,0,0,0)\).

  4. 4.

    Define a matrix A whose rows are formed by these coefficient vectors and augment it to the right with the \(m \times m\) identity matrix, forming an block matrix \(\left( A | I\right) \). Apply Gauss–Jordan elimination and get the block matrix \(\left( M | P\right) \). For \(m=4\), we have:

    $$\begin{aligned} \left( A | I\right) =\left( \begin{array}{cccccccccccc|cccc} 1 &{} -3 &{} 0 &{} 3 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 &{} 2 &{} 0 &{} 0 &{} -2 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 &{} 2 &{} 0 &{} 0 &{} -2 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} -3 &{} 0 &{} 3 &{} -1 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{array} \right) \end{aligned}$$

    and

    $$\begin{aligned} \left( M | P\right) =\left( \begin{array}{cccccccccccc|cccc} 1 &{} -3 &{} 0 &{} 3 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} -2 &{} 0 &{} 0 &{} 2 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} -3 &{} 0 &{} 3 &{} -1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 \\ \end{array} \right) . \end{aligned}$$
  5. 5.

    Now, the non-vanishing rows on M give the commutators of the basis. The identities we want to find out are obtained after making equal to zero the linear combinations on \(P\cdot \mathcal {B}_ n\) corresponding to the vanishing rows. In our example, since there is one null row on M we get one Grade 4 identity, we equal to zero the last element on the product \(P\cdot \mathcal {B}_ 4\), that is:

    $$\begin{aligned} 0=[Y,[X,[X,Y]]]-[X,[Y,[X,Y]]]. \end{aligned}$$

C Appendix

For completeness, we include in the sequel the explicit compact expression we have obtained for \(\Phi _m(X,Y)\), \(m=1,\ldots , 10\). Here \([ X^{p_1} Y^{q_1} \ldots X^{p_m} Y^{q_m}]\) denotes the right-nested commutator based on the word \( X^{p_1} Y^{q_1} \ldots X^{p_m} Y^{q_m}\). Thus, in particular:

$$\begin{aligned}{}[Y^2 \, X^4 \, Y] \equiv [Y, [Y, [X, [X, [X, [X, Y]]]]]]. \end{aligned}$$
$$\begin{aligned} \Phi _1(X,Y)&= X + Y \\ \Phi _2(X,Y)&= \frac{1}{2} [X \, Y] \\ \Phi _3(X,Y)&=\frac{1}{12} [X^2 \, Y] - \frac{1}{12} [Y X Y] \\ \Phi _4(X,Y)&=-\frac{1}{24} [X Y X Y]\\ \Phi _5(X,Y)&= -\frac{1}{720} [X^4 \, Y] - \frac{1}{120} [X Y X^2 Y] - \frac{1}{360} [X Y^2 X Y]+ \frac{1}{360} [Y X^3 Y] \\&\qquad \,\, + \frac{1}{120} [Y^2 X^2 Y] + \frac{1}{720} [Y^3 X Y] \\ \Phi _6(X,Y)&= -\frac{1}{720} [X^2 Y^2 X Y] + \frac{1}{240} [X Y^2 X^2 Y] + \frac{1}{1440} [X Y^3 X Y] + \frac{1}{1440}[Y X^4 Y] \\ \Phi _7(X,Y)&= \frac{1}{30240}[X^6 Y]+ \frac{1}{5040}[X^2 Y X^3 Y] - \frac{1}{10080}[X^2 Y^2 X^2 Y] \\&\qquad \,\, + \frac{1}{10080}[X Y X^4 Y] + \frac{1}{1008}[X Y X Y X^2 Y] + \frac{1}{5040}[X Y X Y^2 X Y] \\&\qquad \,\,- \frac{1}{7560}[X Y^2 X^3 Y] + \frac{1}{3360}[X Y^3 X^2 Y] + \frac{1}{10080}[X Y^4 X Y] \\&\qquad \,\, - \frac{1}{10080}[Y X^5 Y] - \frac{1}{1260} [Y X Y X^3 Y]- \frac{1}{1680}[Y X Y^2 X^2 Y] \nonumber \\&\qquad \,\,+ \frac{1}{3360} [Y^2 X^4 Y]- \frac{1}{3360}[Y^2 X Y X^2 Y] - \frac{1}{2520}[Y^2 X Y^2 X Y] \\&\qquad \,\, + \frac{1}{7560}[Y^3 X^3 Y] + \frac{1}{10080}[Y^4 X^2 Y] - \frac{1}{30240}[Y^5 X Y] \end{aligned}$$
$$\begin{aligned} \Phi _8&(X,Y) = -\frac{5}{24192} [X^3 Y^3 X Y] +\frac{1}{2520} [X^2 Y X Y^2 X Y] +\frac{1}{20160} [X^2 Y^4 X Y] \\&\qquad \qquad + \frac{1}{15120}[X Y X^2 Y^2 X Y] -\frac{1}{2016} [X Y X Y^2 X^2 Y] - \frac{1}{20160}[X Y X Y^3 X Y] \\&\qquad \qquad +\frac{1}{20160} [X Y^2 X Y X^2 Y] -\frac{1}{10080} [X Y^2 X Y^2 X Y] -\frac{1}{60480} [X Y^5 X Y] \\&\qquad \qquad - \frac{1}{60480}[Y X^6 Y] + \frac{1}{20160} [Y X^3 Y X^2 Y] -\frac{1}{ 5040} [Y X^2 Y X^3 Y] +\frac{1}{20160} [Y^2 X^5 Y] \end{aligned}$$
$$\begin{aligned} \Phi _9&(X,Y) = - \frac{1}{302400} [X^5 Y X^2 Y] +\frac{1}{113400} [X^5 Y^2 X, Y] -\frac{1}{40320} [X^4 Y^2 X^2 Y] \\&-\frac{1}{120960} [X^4 Y^3 X Y]+\frac{13}{604800} [X^3 Y^4 X Y] + \frac{1}{24192}[X^2 Y X^2 Y X^2 Y] \\&\qquad \qquad +\frac{1}{30240} [X^2 Y X^2 Y^2 X Y] -\frac{1}{60480} [X^2 Y X Y X^3 Y] - \frac{1}{20160} [X^2 Y X Y^3 X Y] \\&\qquad \qquad - \frac{11}{604800} [X^2 Y^5 X Y] -\frac{1}{151200} [X Y X^6 Y] -\frac{1}{20160} [X Y X^3 Y X^2 Y] \\&\qquad \qquad -\frac{1}{10080} [X Y X^2 Y^2 X^2 Y] + \frac{1}{40320} [X Y X^2 Y^3 X Y] +\frac{1}{18900} [X Y X Y^4 X Y] \\&\qquad \qquad -\frac{1}{20160} [X Y^2 X Y^2 X^2 Y] -\frac{1}{17280} [X Y^2 X Y^3 X Y] - \frac{1}{120960} [X Y^4 X^3 Y] \\&\qquad \qquad -\frac{1}{302400} [X Y^6 X Y] +\frac{1}{302400} [Y X^7 Y] +\frac{1}{50400} [Y X^4 Y X^2 Y] \\&\qquad \qquad +\frac{1}{120960} [Y X^4 Y^2 X Y] +\frac{1}{20160} [Y X^2 Y X Y X^2 Y]- \frac{1}{40320} [Y X Y^2 X^4 Y] \\&\qquad \qquad +\frac{1}{10080} [Y X Y^2 X Y X^2 Y] + \frac{1}{30240} [Y X Y^2 X Y^2 X Y] +\frac{1}{151200} [Y X Y^5 X Y] \\&\qquad \qquad +\frac{1}{302400} [Y^2 X^6 Y] + \frac{1}{20160} [Y^2 X Y X^4 Y] - \frac{1}{30240} [Y^2 X Y^2 X^3 Y] \\&\qquad \qquad + \frac{1}{60480} [Y^2 X Y^3 X^2 Y] - \frac{13}{604800} [Y^3 X^5 Y] -\frac{1}{90720} [Y^3 X^2 Y^2 X Y] \\&\qquad \qquad + \frac{1}{120960} [Y^4 X^4 Y] + \frac{1}{453600} [Y^5 X^3 Y] +\frac{1}{302400} [Y^6 X^2 Y] \\&\qquad \qquad +\frac{1}{1209600} [Y^7 X Y] -\frac{1}{1209600} [X^8 Y] \end{aligned}$$
$$\begin{aligned} \Phi _{10}&(X,Y) = - \frac{1}{2419200} [X^4YX^4 Y]- \frac{1}{604800} [X^4Y^2X^3 Y] +\frac{1}{604800} [X^3YX^5 Y] \\&\qquad \qquad +\frac{1}{ 151200} [X^3YXYX^3 Y] +\frac{1}{604800} [X^3Y^2X^4 Y] + \frac{1}{134400} [X^3Y^2XYX^2 Y]\\&\qquad \qquad +\frac{1}{362880} [X^3Y^3X^3 Y] + \frac{1}{134400}[X^3Y^4X^2 Y] - \frac{1}{403200} [X^2 YX^6 Y] \\&\qquad \qquad - \frac{1}{100800}[X^2YX^2YX^3 Y] - \frac{1}{201600} [X^2YXYX^4 Y] - \frac{1}{50400} [X^2YXYXYX^2 Y]\\&\qquad \qquad - \frac{1}{604800}[X^2YXYXY^2X Y] -\frac{1}{302400} [X^2YXY^2X^3 Y] -\frac{1}{80640} [X^2YXY^3X^2 Y] \\&\qquad \qquad - \frac{1}{100800} [X^2Y^2XYX^3 Y]-\frac{1}{201600} [X^2Y^2XY^2X^2 Y] +\frac{1}{403200} [X^2Y^3X^4 Y] \\&\qquad \qquad -\frac{1}{75600} [X^2Y^3XYX^2 Y] - \frac{1}{362880} [X^2Y^3XY^2X Y] + \frac{1}{403200} [X^2Y^4X^3 Y] \\&\qquad \qquad -\frac{1}{134400} [X^2Y^5X^2 Y] - \frac{1}{604800} [X^2Y^6X Y] + \frac{1}{604800} [XYX^7 Y] \\&\qquad \qquad + \frac{1}{40320} [XYX^2YX^4 Y] +\frac{1}{67200} [XYX^2YXYX^2 Y] - \frac{1}{151200} [XYX^2Y^2X^3 Y]\\&\qquad \qquad - \frac{1}{50400} [XYXYX^5 Y] + \frac{1}{25200}[XYXYXYX^3 Y] +\frac{1}{25200} [XYXYXY^2X^2 Y] \\&\qquad \qquad - \frac{1}{100800} [XYXY^2X^4 Y]- \frac{1}{100800}[XYXY^2XYX^2 Y] + \frac{1}{302400} [XYXY^2XY^2X Y]\\&\qquad \qquad +\frac{1}{75600} [XYXY^3X^3 Y] +\frac{1}{50400} [XYXY^4X^2 Y] + \frac{1}{151200}[XY^2X^6 Y] \\&\qquad \qquad - \frac{1}{100800} [X Y^2X^2YX^3 Y] -\frac{1}{50400} [XY^2X^2Y^2X^2 Y] +\frac{1}{403200} [XY^2XYX^4 Y] \\&\qquad \qquad + \frac{1}{25200} [XY^2XYXYX^2 Y]+ \frac{1}{151200} [XY^2XYXY^2X Y]- \frac{1}{43200} [XY^2XY^2X^3 Y] \\&\qquad \qquad -\frac{1}{67200} [XY^2XY^3X^2 Y] +\frac{1}{100800} [XY^2XY^4X Y] - \frac{1}{75600} [XY^3XYX^3 Y] \\&\qquad \qquad -\frac{1}{302400} [XY^3XY^2X^2 Y] + \frac{13}{1209600}[X Y^4 X^4 Y] + \frac{1}{75600} [XY^4XYX^2 Y] \\&\qquad \qquad -\frac{1}{60480} [XY^4XY^2X Y] - \frac{1}{226800}[XY^5X^3 Y] + \frac{1}{86400} [XY^6X^2 Y]\\&\qquad \qquad + \frac{1}{2419200}[X Y^7X Y]. \end{aligned}$$

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Arnal, A., Casas, F. & Chiralt, C. A Note on the Baker–Campbell–Hausdorff Series in Terms of Right-Nested Commutators. Mediterr. J. Math. 18, 53 (2021). https://doi.org/10.1007/s00009-020-01681-6

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