Geodesic
Nonlinear algebra and Bogoliubov's recursion
Teoretičeskaâ i matematičeskaâ fizika, Tome 154 (2008) no. 2, pp. 316-343 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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A. Yu. Morozov; M. N. Serbin. Nonlinear algebra and Bogoliubov's recursion. Teoretičeskaâ i matematičeskaâ fizika, Tome 154 (2008) no. 2, pp. 316-343. http://geodesic.mathdoc.fr/item/TMF_2008_154_2_a9/

[1] N. N. Bogoliubow, O. S. Parasiuk, Acta Math., 97 (1957), 227–266 ; Н. Н. Боголюбов, Д. В. Ширков,, Введение в теорию квантованных полей, Гостехиздат, М., 1957 ; Б. М. Степанов, О. И. Завьялов, ЯФ, 1 (1965), 922; K. Hepp, Comm. Math. Phys., 2 (1966), 301–326 ; M. Zimmerman, Comm. Math. Phys., 15 (1969), 208–234 | DOI | MR | Zbl | MR | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[2] O. I. Zavyalov, Perenormirovannye diagrammy Feimana, Nauka, M., 1979 | MR

[3] M. Peskin, D. Shreder, Vvedenie v kvantovuyu teoriyu polya, RKhD, M.–Izhevsk, 2001 | MR

[4] Dzh. Kollinz, Perenormirovka, Mir, M., 1988 | MR | MR | Zbl

[5] A. N. Vasilev, Kvantovopolevaya renormgruppa v teorii kriticheskogo povedeniya i stokhasticheskoi dinamike, Izd-vo PIYaF, SPb., 1998

[6] A. Connes, D. Kreimer, Comm. Math. Phys., 210 (2000), 249–273 ; ; Comm. Math. Phys., 216 (2001), 215–241 ; arXiv: hep-th/9912092arXiv: hep-th/0003188 | DOI | MR | Zbl | DOI | MR | Zbl

[7] A. Gerasimov, A. Morozov, K. Selivanov, Internat. J. Modern Phys. A, 16 (2001), 1531–1558 ; arXiv: hep-th/0005053 | DOI | MR | Zbl

[8] D. Kreimer, R. Delbourgo, Phys. Rev. D (3), 60 (1999), 105025 ; ; K. Ebrahimi-Fard, D. Kreimer, J. Phys. A, 38 (2005), R385–R407 ; ; D. Kreimer, “Structures in Feynman graphs: Hopf algebras and symmetries”, Graphs and Patterns in Mathematics and Theoretical Physics, Proc. Sympos. Pure Math., 73, AMS, Providence, RI, 2005, 43–78 ; ; Ann. Phys., 321 (2006), 2757–2781 ; arXiv: hep-th/9903249arXiv: hep-th/0510202arXiv: hep-th/0202110arXiv: hep-th/0509135 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[9] W. D. van Suijlekom, Lett. Math. Phys., 77 (2006), 265–281 ; arXiv: hep-th/0602126 | DOI | MR | Zbl

[10] R. Wulkenhaar, “Hopf algebras in renormalization and NC geometry”, Noncommutative geometry and the standard model of elementary particle physics (Hesselberg, Germany, 1999), Lecture Notes in Phys., 596, Springer, Berlin, 2002, 313–324 ; arXiv: hep-th/9912221 | DOI | MR | Zbl

[11] D. V. Malyshev, TMF, 143:1 (2005), 22–32 ; ; D. V. Malyshev, Non RG logarithms via RG equations, ; Phys. Lett. B, 578 (2004), 231–234 ; arXiv: hep-th/0408230arXiv: hep-th/0402074arXiv: hep-th/0307301 | DOI | MR | DOI | MR | Zbl

[12] D. I. Kazakov, G. S. Vartanov, Renormalizable expansion for nonrenormalizable theories: I. Scalar higher dimensional theories, arXiv: hep-th/0607177

[13] D. I. Kazakov, G. S. Vartanov, Renormalizable expansion for nonrenormalizable theories: II. Gauge higher dimensional theories, arXiv: hep-th/0702004

[14] I. V. Volovich, D. V. Prokhorenko, Tr. MIAN, 245 (2004), 288–295 ; arXiv: hep-th/0611178 | MR | Zbl

[15] B. Delamotte, Amer. J. Phys., 72:2 (2004), 170–184 ; arXiv: hep-th/0212049 | DOI

[16] V. Dolotin, A. Morozov, Introduction to Non-Linear Algebra, World Sci., Singapore, 2007 ; arXiv: hep-th/0609022 | MR | Zbl