Letters in Mathematical Physics

, Volume 103, Issue 3, pp 331–350 | Cite as

Time-Ordering and a Generalized Magnus Expansion

  • Michel Bauer
  • Raphael Chetrite
  • Kurusch Ebrahimi-FardEmail author
  • 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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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Michel Bauer
    • 1
  • Raphael Chetrite
    • 2
  • Kurusch Ebrahimi-Fard
    • 3
    Email author
  • 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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