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Time-Ordering and a Generalized Magnus Expansion

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Abstract

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.

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References

  1. Aczél, J.: Lectures on functional equations and their applications. In: Mathematics in Science and Engineering, vol. 19, Academic Press, Dublin (1966)

  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)

  3. Atkinson F.V.: Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7, 1 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bachmann S., Graf G.M., Lesovik G.B.: Time ordering and counting statistics. J. Stat. Phys. 138, 333 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Baxter G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blanes S., Casas F., Oteo J.A., Ros J.: Magnus expansion: mathematical study and physical applications. Phys. Rep. 470, 151 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  7. Cartier P.: On the structure of free Baxter algebras. Adv. Math. 9, 253 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chapoton, F., Patras, F.: Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula (2012, preprint). arXiv:1201.2159v1 [math.QA]

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Ebrahimi-Fard K., Gracia-Bondía J., Patras F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81, 61 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  12. Ebrahimi-Fard K., Manchon D.: A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. 9, 295 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ebrahimi-Fard K., Manchon D.: Dendriform equations. J. Algebra 322, 4053 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fried H.: Green’s Functions and Ordered Exponentials. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  16. Gelfand I.M., Krob D., Lascoux A., Leclerc B., Retakh V., Thibon J.Y.: Noncommutative symmetric functions. Adv. Math. 112, 218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A.: Lie-group methods. Acta Numerica 9, 215 (2000)

    Article  Google Scholar 

  18. Kingman J.F.C.: Spitzer’s identity and its use in probability theory. J. Lond. Math. Soc. 37, 309 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  19. Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mielnik B., Plebański J.: Combinatorial approach to Baker–Campbell–Hausdorff exponents. Ann. Inst. Henri Poincaré A XII, 215 (1970)

    Google Scholar 

  21. Murua A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6, 387 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rota G.C.: Baxter algebras and combinatorial identities I, II. Bull. Am. Math. Soc. 75, 325 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rota G.C.: Ten mathematics problems I will never solve. DMV Mitteilungen 2, 45 (1998)

    MathSciNet  Google Scholar 

  24. Rota G.C., Smith D.A.: Fluctuation theory and Baxter algebras. Istituto Nazionale di Alta Matematica IX, 179 (1972)

    MathSciNet  Google Scholar 

  25. Spitzer F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82, 323 (1956)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Talkner P., Lutz E., Hanggi P.: Fluctuation theorems: work is not an observable. Phys. Rev. E (Rapid Communication) 75, 050102 (2007)

    Article  ADS  Google Scholar 

  28. Vogel W.: Die kombinatorische Lösung einer Operator-Gleichung. Z. Wahrscheinlichkeitstheorie 2, 122 (1963)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut de Physique Théorique de Saclay, CEA-Saclay, 91191, Gif-sur-Yvette, France

    Michel Bauer

  2. Laboratoire J.-A. Dieudonné UMR 7351, CNRS, Parc Valrose, 06108, Nice Cedex 02, France

    Raphael Chetrite & Frédéric Patras

  3. ICMAT, C/Nicolás Cabrera, no. 13-15, 28049, Madrid, Spain

    Kurusch Ebrahimi-Fard

Authors
  1. Michel Bauer
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  2. Raphael Chetrite
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  3. Kurusch Ebrahimi-Fard
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  4. Frédéric Patras
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Corresponding author

Correspondence to Kurusch Ebrahimi-Fard.

Additional information

K. Ebrahimi-Fard is on leave from UHA, Mulhouse, France.

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Bauer, M., Chetrite, R., Ebrahimi-Fard, K. et al. Time-Ordering and a Generalized Magnus Expansion. Lett Math Phys 103, 331–350 (2013). https://doi.org/10.1007/s11005-012-0596-z

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  • DOI: https://doi.org/10.1007/s11005-012-0596-z

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