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).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Brouder Ch., Fauser B., Frabetti A., Krattenthaler Ch.: Non-commutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200, 479 (2006)
Brown, L. (ed.): Renormalization: From Lorentz to Landau (and Beyond). Springer, New York (1993)
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)
Caswell W.E., Kennedy A.D.: A simple approach to renormalization theory. Phys. Rev. D 25, 392 (1982)
Collins J.: Renormalization. Cambridge monographs in mathematical physics, Cambridge (1984)
Connes A., Kreimer D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203 (1998)
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)
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)
Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008)
Delamotte B.: A hint of renormalization. Am. J. Phys. 72, 170 (2004)
Dyson F.: The S matrix in quantum electrodynamics. Phys. Rev. 75, 1736 (1949)
Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276, 519 (2007)
Ebrahimi-Fard K., Gracia-Bondía J.M., Patras F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81(1), 61 (2007)
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)
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 2007
Figueroa H., Gracia-Bondía J.M.: Combinatorial Hopf algebras in quantum field theory I. Rev. Math. Phys. 17, 881 (2005)
Itzykson C., Zuber J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)
Joni S.A., Rota G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61, 93 (1979)
Kreimer D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757 (2006)
Kreimer D.: Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3, 627–670 (1999)
Manchon, D.: Hopf algebras in renormalisation. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 5, pp. 365–427. Elsevier, Oxford (2008)
Reutenauer C.: Free Lie Algebras. Oxford University Press, Oxford (1993)
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)
van Suijlekom W.: Renormalization of gauge fields: a Hopf algebra approach. Commun. Math. Phys. 276, 773 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vincent Rivasseau.
K. Ebrahimi-Fard is on leave from Univ. de Haute Alsace, Mulhouse, France.
Rights and permissions
About this article
Cite this article
Ebrahimi-Fard, K., Patras, F. Exponential Renormalization. Ann. Henri Poincaré 11, 943–971 (2010). https://doi.org/10.1007/s00023-010-0050-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-010-0050-7