Advertisement

Letters in Mathematical Physics

, Volume 109, Issue 3, pp 725–746 | Cite as

The Baker–Campbell–Hausdorff formula via mould calculus

  • Yong Li
  • David SauzinEmail author
  • Shanzhong Sun
Article
First Online:
  • 74 Downloads

Abstract

The well-known Baker–Campbell–Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product \(\text {e}^X \text {e}^Y\) can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction to Écalle’s mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin (Dokl Akad Nauk SSSR (NS) 57:323–326, 1947) explicit formula for the logarithm, as well as another formula recently obtained by Kimura (Theor Exp Phys 4:041A03, 2017) for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.

Keywords

Baker-Campbell-Hausdorff theorem Dynkin formula Mould calculus 

Mathematics Subject Classification

22E15 17B05 17B60 17B81 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

D.S. and Y.L. thank the Centro Di Ricerca Matematica Ennio De Giorgi and the Scuola Normale Superiore di Pisa for their kind hospitality, during which this work was completed.

References

  1. 1.
    Baumard, S., Schneps, L.: On the derivation representation of the fundamental Lie algebra of mixed elliptic motives. Annales Mathématiques du Québec 41(1), 43–62 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Behtash, A., Dunne, G.V., Schäfer, T., Sulejmanpasic, T., Ünsal, M.: Complexified path integrals, exact saddles, and supersymmetry. Phys. Rev. Lett. 116, 011601 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonfiglioli, A., Fulci, R.: Topics in Noncommutative Algebra—The Theorem of Campbell, Baker, Hausdorff and Dynkin. Lecture Notes in Mathematics, vol. 2034. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  4. 4.
    Bouillot, O., Écalle, J.: Invariants of identity-tangent diffeomorphisms expanded as series of multitangents and multizetas. In: Resurgence, Physics and Numbers, pp. 109–232, CRM Series, 20, Ed. Norm., Pisa (2017)Google Scholar
  5. 5.
    Couso-Santamaría, R., Schiappa, R., Vaz, R.: On asymptotices and resurgent structures of enumerative Gromov–Witten invariants. Commun. Numer. Theor. Phys. 11, 707–790 (2017). arXiv:1605.07473 CrossRefzbMATHGoogle Scholar
  6. 6.
    Couso-Santamaría, R., Mariño, M., Schiappa, R.: Resurgence matches quantization. J. Phys. A Math. Theor. 50, 145402 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dynkin, E.B.: Calculation of the coefficients in the Campbell-Hausdorff formula (Russian). Dokl. Akad. Nauk SSSR (N.S.) 57, 323–326 (1947)Google Scholar
  8. 8.
    Dillinger, H., Delabaere, E., Pham, F.: Résurgence de Voros et périodes des courbes hyperelliptiques. Annales de l’institut Fourier 43(1), 163–199 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Écalle, J.: Les fonctions résurgentes, Publ. Math. d’Orsay [vol. 1: 81-05, vol. 2: 81-06, vol. 3: 85-05] (1981, 1985)Google Scholar
  10. 10.
    Écalle, J.: Cinq applications des fonctions résurgentes. Publ. Math. d’Orsay, pp. 84–62 (1984)Google Scholar
  11. 11.
    Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actualités Math, Hermann (1992)zbMATHGoogle Scholar
  12. 12.
    Écalle, J.: ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan. Journal de Théorie des Nombres de Bordeaux 15(2), 411–478 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fauvet, F., Foissy, L., Manchon, D.: The Hopf algebra of finite topologies and mould composition. Annales de l’Institut Fourier 67(3), 911–945 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kimura, T.: Explicit description of the Zassenhaus formula. Theor. Exp. Phys. 4, 041A03 (2017)Google Scholar
  16. 16.
    Kontsevich, M.: Resurgence and Quantization. Course given at IHES, Paris in April 2017Google Scholar
  17. 17.
    Matone, M.: An algorithm for the Baker–Campbell–Hausdorff formula. J. High Energy Phys. 05, 113 (2015). arXiv:1502.06589 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Menous, F.: Formal differential equations and renormalization. In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois theories, IRMA Lect. Math. Theor. Phys., vol. 15, pp. 229–246 (2009)Google Scholar
  19. 19.
    Novelli, J.-C., Paul, T., Sauzin, D., Thibon, J.-Y.: Rayleigh-Schrödinger series and Birkhoff decomposition. Lett. Math. Phys. 108, 18 (2018).  https://doi.org/10.1007/s11005-017-1040-1 CrossRefzbMATHGoogle Scholar
  20. 20.
    Paul, T., Sauzin, D.: Normalization in Lie algebras via mould calculus and applications. Regul. Chaotic Dyn. 22(6), 616–649 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Reutenauer, C.: Free Lie Algebras, London Mathematical Society Monographs, vol. 7. Clarendon Press, New York (1993)Google Scholar
  22. 22.
    Sauzin, D.: Initiation to mould calculus through the example of saddle-node singularities. Rev. Semin. Iberoam. Mat. 3(5–6), 147–160 (2008)MathSciNetGoogle Scholar
  23. 23.
    Sauzin, D.: Mould expansions for the saddle-node and resurgence monomials. In: Connes, A., Fauvet, F., Ramis, J.-P. (eds.) Renormalization and Galois Theories. IRMA Lectures in Mathematics and Theoretical Physics, vol. 15, pp. 83–163. European Mathematical Society, Zürich (2009)CrossRefGoogle Scholar
  24. 24.
    Sauzin, D.: Introduction to 1-summability and resurgence. In: Mitschi, C., Sauzin, D. (eds.) Divergent Series, Summability and Resurgence. I. Monodromy and Resurgence. Lecture Notes in Mathematics, vol. 2153. Springer, Berlin (2016)Google Scholar
  25. 25.
    Schneps, L.: Double shuffle and Kashiwara–Vergne Lie algebras. J. Algebra 367, 54–74 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Thibon, J.-Y.: Noncommutative symmetric functions and combinatorial Hopf algebras. In: Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation, vol. I, pp. 219–258. CRM Series, 12, Ed. Norm., Pisa (2011)Google Scholar
  27. 27.
    Voros, A.: The return of the quartic oscillator. The complex WKB method. Annales de l’I. H. P. Section A tome 39(3), 211–338 (1983)MathSciNetzbMATHGoogle Scholar
  28. 28.
    von Waldenfels, W.: Zur Charakterisierung Liescher Elemente in freien Algebren. Arch. Math. (Basel) 17, 44–48 (1966)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsCapital Normal UniversityBeijingPeople’s Republic of China
  2. 2.CNRS UMR 8028 IMCCEParisFrance

Personalised recommendations