Letters in Mathematical Physics

, Volume 103, Issue 3, pp 331–350

Time-Ordering and a Generalized Magnus Expansion

  • Michel Bauer
  • Raphael Chetrite
  • Kurusch Ebrahimi-Fard
  • Frédéric Patras


Both the classical time-ordering and the Magnus expansion are well known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including, however, directly a non-trivial initial condition. As a by-product we recover a variant of the time-ordering operation, known as \({\mathsf{T}^\ast}\)-ordering. Eventually, placing our results in the general context of Rota–Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations.

Mathematics Subject Classification (2010)

16R60 34L99 


\({\mathsf{T}}\)-ordered products Magnus expansion Rota–Baxter relation Atkinson recursion 


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  1. 1.
    Aczél, J.: Lectures on functional equations and their applications. In: Mathematics in Science and Engineering, vol. 19, Academic Press, Dublin (1966)Google Scholar
  2. 2.
    Aczél, J., Dhombres, J.: Functional Equations in Several Variables: With Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge (1989)Google Scholar
  3. 3.
    Atkinson F.V.: Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7, 1 (1963)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bachmann S., Graf G.M., Lesovik G.B.: Time ordering and counting statistics. J. Stat. Phys. 138, 333 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Baxter G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731 (1960)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Blanes S., Casas F., Oteo J.A., Ros J.: Magnus expansion: mathematical study and physical applications. Phys. Rep. 470, 151 (2009)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Cartier P.: On the structure of free Baxter algebras. Adv. Math. 9, 253 (1972)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chapoton, F., Patras, F.: Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula (2012, preprint). arXiv:1201.2159v1 [math.QA]Google Scholar
  9. 9.
    Connes A., Kreimer D.: Renormalization in quantum field theory and the Riemann–Hilbert problem I: The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Ebrahimi-Fard K., Gracia-Bondía J., Patras F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81, 61 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Ebrahimi-Fard K., Manchon D.: The combinatorics of Bogoliubov’s recursion in renormalisation. in ‘Renormalization and Galois theories’. IRMA Lect. Math. Theor. Phys. 15, 179 (2009)MathSciNetGoogle Scholar
  12. 12.
    Ebrahimi-Fard K., Manchon D.: A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. 9, 295 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ebrahimi-Fard K., Manchon D.: Dendriform equations. J. Algebra 322, 4053 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ebrahimi-Fard K., Manchon D., Patras F.: A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion. J. Noncommut. Geom. 3, 181 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fried H.: Green’s Functions and Ordered Exponentials. Cambridge University Press, Cambridge (2005)Google Scholar
  16. 16.
    Gelfand I.M., Krob D., Lascoux A., Leclerc B., Retakh V., Thibon J.Y.: Noncommutative symmetric functions. Adv. Math. 112, 218 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A.: Lie-group methods. Acta Numerica 9, 215 (2000)CrossRefGoogle Scholar
  18. 18.
    Kingman J.F.C.: Spitzer’s identity and its use in probability theory. J. Lond. Math. Soc. 37, 309 (1962)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649 (1954)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mielnik B., Plebański J.: Combinatorial approach to Baker–Campbell–Hausdorff exponents. Ann. Inst. Henri Poincaré A XII, 215 (1970)Google Scholar
  21. 21.
    Murua A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6, 387 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Rota G.C.: Baxter algebras and combinatorial identities I, II. Bull. Am. Math. Soc. 75, 325 (1969)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rota G.C.: Ten mathematics problems I will never solve. DMV Mitteilungen 2, 45 (1998)MathSciNetGoogle Scholar
  24. 24.
    Rota G.C., Smith D.A.: Fluctuation theory and Baxter algebras. Istituto Nazionale di Alta Matematica IX, 179 (1972)MathSciNetGoogle Scholar
  25. 25.
    Spitzer F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82, 323 (1956)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Strichartz R.S.: The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations. J. Funct. Anal. 72, 320 (1987)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Talkner P., Lutz E., Hanggi P.: Fluctuation theorems: work is not an observable. Phys. Rev. E (Rapid Communication) 75, 050102 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    Vogel W.: Die kombinatorische Lösung einer Operator-Gleichung. Z. Wahrscheinlichkeitstheorie 2, 122 (1963)MATHCrossRefGoogle Scholar
  29. 29.
    Wilcox R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962 (1967)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Michel Bauer
    • 1
  • Raphael Chetrite
    • 2
  • Kurusch Ebrahimi-Fard
    • 3
  • Frédéric Patras
    • 2
  1. 1.Institut de Physique Théorique de SaclayCEA-SaclayGif-sur-YvetteFrance
  2. 2.Laboratoire J.-A. Dieudonné UMR 7351, CNRSNice Cedex 02France
  3. 3.ICMATMadridSpain

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