Skip to main content
SpringerLink
Log in

Lax Pair Equations and Connes-Kreimer Renormalization

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We find a Lax pair equation corresponding to the Connes-Kreimer Birkhoff factorization of the character group of a Hopf algebra. This flow preserves the locality of counterterms. In particular, we obtain a flow for the character given by Feynman rules, and relate this flow to the Renormalization Group Flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, M., van Moerbeke, P., Vanhaecke, P.: Algebraic integrability, Painlevé geometry and Lie algebras. Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, Vol. 47, Berlin: Springer-Verlag, 2004

  2. 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, 249–273 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Connes A., Kreimer D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216(1), 215–241 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives. American Mathematical Society Colloquium Publications, Vol. 55, Providence, RI: Amer. Math. Soc., 2008

  5. Delbourgo R., Kreimer D.: Using the Hopf algebra structure of QFT in calculations. Phys. Rev. D 60, 105025 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  6. Ebrahimi-Fard K., Guo L., Kreimer D.: Integrable renormalization II: the general case. Ann. Inst. H. Poincaré 6, 369–395 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ebrahimi-Fard K., Manchon D.: On matrix differential equations in the Hopf algebra of renormalization. Adv. Theor. Math. Phys. 10, 879–913 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276(2), 519–549 (2007)

    Article  MATH  ADS  Google Scholar 

  9. Guest, M.: Harmonic Maps, Loop Groups, and Integrable Systems. LMS Student Texts, Vol. 38, Cambridge: Cambridge U. Press, 1997

  10. Kosmann-Schwarzbach, Y.: Lie bialgebras, Poisson Lie groups and dressing transformations. In: Integrability of nonlinear systems (Pondicherry, 1996), Lecture Notes in Phys., Vol. 495, Berlin: Springer, 1997, pp. 104–170

  11. Kreimer D.: Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3(3), 627–670 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Manchon, D.: Hopf algebras, from basics to applications to renormalization, Comptes-rendus des Rencontres mathématiques de Glanon 2001, Published 2003, available at http://arxiv.org/abs/math/0408405v2, 2006

  13. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Integrable Systems II: Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems. In: Dynamical systems. VII, Encyclopaedia of Mathematical Sciences, Vol. 16, Berlin: Springer-Verlag, 1994

  14. Suris, Y.B.: The problem of integrable discretization: Hamiltonian approach. Progress in Mathematics, Vol. 219, Basel: Birkhäuser Verlag, 2003

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Gabriel Bădiţoiu

  2. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215, USA

    Steven Rosenberg

Authors
  1. Gabriel Bădiţoiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven Rosenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steven Rosenberg.

Additional information

Communicated by M. Aizenman

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bădiţoiu, G., Rosenberg, S. Lax Pair Equations and Connes-Kreimer Renormalization. Commun. Math. Phys. 296, 655–680 (2010). https://doi.org/10.1007/s00220-010-1034-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1034-7

Keywords

Search

Navigation