Abstract
Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization. Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counter-factors and of order n bare coupling constants).
Keywords
Hopf Algebra Formal Power Series Feynman Rule Feynman Graph Subtraction Scheme
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References
- 1.Bellon M., Schaposnik F.: Renormalization group functions for the Wess-Zumino model: up to 200 loops through Hopf algebras. Nuclear Phys. B 800, 517 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 2.Brouder Ch., Fauser B., Frabetti A., Krattenthaler Ch.: Non-commutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200, 479 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
- 3.Brown, L. (ed.): Renormalization: From Lorentz to Landau (and Beyond). Springer, New York (1993)Google Scholar
- 4.Cartier, P.: Hyperalgèbres et groupes de Lie formels. In: Séminaire “Sophus Lie” de la Faculté des Sciences de Paris, 1955–56. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 61 pp (1957)Google Scholar
- 5.Caswell W.E., Kennedy A.D.: A simple approach to renormalization theory. Phys. Rev. D 25, 392 (1982)CrossRefMathSciNetADSGoogle Scholar
- 6.Collins J.: Renormalization. Cambridge monographs in mathematical physics, Cambridge (1984)zbMATHCrossRefGoogle Scholar
- 7.Connes A., Kreimer D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 8.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 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 9.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, 215 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 10.Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008)Google Scholar
- 11.Delamotte B.: A hint of renormalization. Am. J. Phys. 72, 170 (2004)CrossRefADSGoogle Scholar
- 12.Dyson F.: The S matrix in quantum electrodynamics. Phys. Rev. 75, 1736 (1949)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 13.Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276, 519 (2007)zbMATHCrossRefADSGoogle Scholar
- 14.Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81(1), 61 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 15.Ebrahimi-Fard K., Manchon D., Patras F.: A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion. J. Noncommutative Geom. 3(2), 181 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
- 16.Ebrahimi-Fard, K., Patras, F.: A Zassenhaus-type algorithm solves the Bogoliubov recursion. In: Doebner, H.-D., Dobrev, V.K. (eds.) Proceedings of VII International Workshop“Lie Theory and Its Applications in Physics”, Varna, June 2007Google Scholar
- 17.Figueroa H., Gracia-Bondía J.M.: Combinatorial Hopf algebras in quantum field theory I. Rev. Math. Phys. 17, 881 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
- 18.Itzykson C., Zuber J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)Google Scholar
- 19.Joni S.A., Rota G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61, 93 (1979)zbMATHMathSciNetGoogle Scholar
- 20.Kreimer D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
- 21.Kreimer D.: Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3, 627–670 (1999)zbMATHMathSciNetGoogle Scholar
- 22.Manchon, D.: Hopf algebras in renormalisation. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 5, pp. 365–427. Elsevier, Oxford (2008)Google Scholar
- 23.Reutenauer C.: Free Lie Algebras. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
- 24.van Suijlekom W.: Multiplicative renormalization and Hopf algebras. In: Ceyhan, O., Manin, Yu.-I., Marcolli, M. (eds) Arithmetic and Geometry Around Quantization, Birkhäuser, Basel (2008)Google Scholar
- 25.van Suijlekom W.: Renormalization of gauge fields: a Hopf algebra approach. Commun. Math. Phys. 276, 773 (2007)zbMATHCrossRefADSGoogle Scholar
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