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Nonlinear algebra and Bogoliubov’s recursion. (English. Russian original) Zbl 1150.81014
Theor. Math. Phys. 154, No. 2, 270-293 (2008); translation from Teor. Mat. Fiz. 154, No. 2, 316-343 (2008).
Summary: 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.

81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81R15 Operator algebra methods applied to problems in quantum theory
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI arXiv
[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). · Zbl 0081.43302
[2] O. I. Zavialov, Renormalized Feynman Diagrams [in Russian], Nauka, Moscow (1979).
[3] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison Wesley, Reading, Mass. (1995).
[4] J. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-Product Expansion, Cambridge Univ. Press, Cambridge (1984). · Zbl 1094.53505
[5] A. N. Vasil’ev, Quantum Field Renormgroup in the Theory of Critical Behavior and Stochastic Dynamics [in Russian], Izd. PIYaF, St. Petersburg (1998).
[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). · Zbl 1032.81026
[7] A. Gerasimov, A. Morozov, and K. Selivanov, Internat. J. Mod. Phys. A, 16, 1531–1558 (1995); arXiv:hep-th/0005053v1 (2000). · Zbl 0984.81086
[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).
[9] W. D. van Suijlekom, Lett. Math. Phys., 77, 265–281 (2006); arXiv:hep-th/0602126v2 (2006). · Zbl 1160.81432
[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). · Zbl 1330.81131
[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). · Zbl 1178.81193
[12] D. I. Kazakov and G. S. Vartanov, ”Renormalizable expansion for nonrenormalizable theories: I. Scalar higher dimensional theories,” arXiv:hep-th/0607177v2 (2006). · Zbl 1098.81068
[13] D. I. Kazakov and G. S. Vartanov, ”Renormalizable expansion for nonrenormalizable theories: II. Gauge higher dimensional theories,” arXiv:hep-th/0702004v1 (2007).
[14] I. V. Volovich and D. V. Prokhorenko, Proc. Steklov Inst. Math., 245, 273–280 (2004); arXiv:hep-th/0611178v1 (2006).
[15] B. Delamotte, Amer. J. Phys., 72, 170–184 (2004); arXiv:hep-th/0212049v3 (2002).
[16] V. Dolotin and A. Morozov, Introduction to Non-Linear Algebra, World Scientific, Singapore (2007); arXiv:hep-th/0609022v2 (2006).
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