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Theoretical and Mathematical Physics

, Volume 154, Issue 2, pp 270–293 | Cite as

Nonlinear algebra and Bogoliubov’s recursion

  • A. Yu. Morozov
  • M. N. Serbyn
Article

Abstract

We give many examples of applying Bogoliubov’s forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory.

Keywords

quantum field theory renormalization nonlinear algebra 

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References

  1. 1.
    N. N. Bogoliubow and O. S. Parasiuk, Acta Math., 97, 227–266 (1957); N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields [in Russian], Gostekhizdat, Moscow (1957); English transl., Wiley, New York (1980); B. M. Stepanov and O. I. Zavialov, Yadern. Fiz., 1, 922 (1965); K. Hepp, Comm. Math. Phys., 2, 301–326 (1966); M. Zimmerman, Comm. Math. Phys., 15, 208–234 (1969).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    O. I. Zavialov, Renormalized Feynman Diagrams [in Russian], Nauka, Moscow (1979).Google Scholar
  3. 3.
    M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison Wesley, Reading, Mass. (1995).Google Scholar
  4. 4.
    J. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-Product Expansion, Cambridge Univ. Press, Cambridge (1984).zbMATHGoogle Scholar
  5. 5.
    A. N. Vasil’ev, Quantum Field Renormgroup in the Theory of Critical Behavior and Stochastic Dynamics [in Russian], Izd. PIYaF, St. Petersburg (1998).Google Scholar
  6. 6.
    A. Connes and D. Kreimer, Comm. Math. Phys., 210, 249–273 (2000); arXiv:hep-th/9912092v1 (1999); Comm. Math. Phys., 216, 215–241 (2001); arXiv:hep-th/0003188v1 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Gerasimov, A. Morozov, and K. Selivanov, Internat. J. Mod. Phys. A, 16, 1531–1558 (1995); arXiv:hep-th/0005053v1 (2000).CrossRefMathSciNetGoogle Scholar
  8. 8.
    D. Kreimer and R. Delbourgo, Phys. Rev. D, 60, 105025 (1999); arXiv:hep-th/9903249v3 (1999); K. Ebrahimi-Fard and D. Kreimer, J. Phys. A, 38, R385–R407 (2005); arXiv:hep-th/0510202v2 (2005); D. Kreimer, “Structures in Feynman graphs: Hopf algebras and symmetries,” in: Graphs and Patterns in Mathematics and Theoretical Physics (Proc. Sympos. Pure Math., Vol. 73), Amer. Math. Soc., Providence, R. I. (2005), p. 43–78; arXiv:hep-th/0202110v3 (2002); Ann. Phys., 321, 2757–2781 (2006); arXiv:hep-th/0509135v3 (2005).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    W. D. van Suijlekom, Lett. Math. Phys., 77, 265–281 (2006); arXiv:hep-th/0602126v2 (2006).zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    R. Wulkenhaar, “Hopf algebras in renormalization and NC geometry,” in: Noncommutative Geometry and the Standard Model of Elementary Particle Physics (Lect. Notes Phys., Vol. 596), Springer, Berlin (2002), p. 313–324; arXiv:hep-th/9912221v1 (1999).CrossRefGoogle Scholar
  11. 11.
    D. V. Malyshev, Theor. Math. Phys., 143, 505–514 (2005); arXiv:hep-th/0408230v1 (2004); D. V. Malyshev, “Non RG logarithms via RG equations,” arXiv:hep-th/0402074v1 (2004); Phys. Lett., 578, 231–234 (2004); arXiv:hep-th/0307301v2 (2003).CrossRefMathSciNetGoogle Scholar
  12. 12.
    D. I. Kazakov and G. S. Vartanov, “Renormalizable expansion for nonrenormalizable theories: I. Scalar higher dimensional theories,” arXiv:hep-th/0607177v2 (2006).Google Scholar
  13. 13.
    D. I. Kazakov and G. S. Vartanov, “Renormalizable expansion for nonrenormalizable theories: II. Gauge higher dimensional theories,” arXiv:hep-th/0702004v1 (2007).Google Scholar
  14. 14.
    I. V. Volovich and D. V. Prokhorenko, Proc. Steklov Inst. Math., 245, 273–280 (2004); arXiv:hep-th/0611178v1 (2006).MathSciNetGoogle Scholar
  15. 15.
    B. Delamotte, Amer. J. Phys., 72, 170–184 (2004); arXiv:hep-th/0212049v3 (2002).CrossRefADSGoogle Scholar
  16. 16.
    V. Dolotin and A. Morozov, Introduction to Non-Linear Algebra, World Scientific, Singapore (2007); arXiv:hep-th/0609022v2 (2006).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”RomeItaly
  3. 3.Moscow Physical Technical InstituteMoscowRussia
  4. 4.Landau Institute for Theoretical PhysicsRASMoscowRussia
  5. 5.Bogolyubov Institute for Theoretical PhysicsNational Academy of Sciences of UkraineKievUkraine

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