Abstract
Motivated by recent work of Connes and Marcolli, based on the Connes–Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.
Similar content being viewed by others
References
Atkinson F.V. (1963). Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7: 1–30
Bergbauer C. and Kreimer D. (2005). The Hopf algebra of rooted trees in Epstein-Glaser renormalization. Ann. Henri Poincaré 6: 343–367
Bergbauer, C., Kreimer, D.: Hopf Algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology, In: IRMA lectures in Mathematics and Theoretical Physics, Vol. 10, eds. V. Turaev, L. Nyssen, Berlin: European Mathematical Society, 2006, pp. 133–164
Blaer A.S. and Young K. (1974). Field theory renormalization using the Callan–Symanzik equation. Nucl. Phys. B 83: 493–514
Blessenohl D. and Schocker M. (2005). Noncommutative character theory of the symmetric group. World Scientific, Singapore
Bloch S., Esnault H. and Kreimer D. (2006). Motives associated to graph polynomials. Commun. Math. Phys. 267: 181–225
Broadhurst D.J. and Kreimer D. (2000). Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees. Commun. Math. Phys. 215: 217–236
Broadhurst D.J. and Kreimer D. (2001). Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality. Nucl. Phys. B 600: 403–422
Bourbaki, N.: Elements of Mathematics. In: Lie groups and Lie algebras. Chapters 1–3. Berlin: Springer (1989)
Callan C.G. (1970). Broken scale invariance in scalar field theory. Phys. Rev. D 2: 1541–1547
Callan C.G., Introduction to renormalization theory. In: Methods in Field Theory, (Les Houches 1975), Balian, J., Zinn-Justin, J., (eds.), Amsterdam: North–Holland, 1976
Cartier, P.: A primer on Hopf algebras. IHES preprint, August 2006, available at http://www.ihes.fr/ PREPRINTS/2006/M/M-06-40.pdf
Caswell W.E. and Kennedy A.D. (1982). Simple approach to renormalization theory. Phys. Rev. D 25: 392–408
Collins J.C. (1974). Structure of the counterterms in dimensional regularization. Nucl. Phys. B 80: 341–348
Collins J.C. (1984). Renormalization. Cambridge University Press, Cambridge
Connes A. and Moscovici H. (1995). The local index formula in noncommutative geometry. Geom. Func. Anal. 5: 174–243
Connes A. and Kreimer D. (1998). Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199: 203–242
Connes A. and Kreimer D. (2000). 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: 249–273
Connes A. and Kreimer D. (2001). Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216: 215–241
Connes A. and Marcolli M. (2004). Renormalization and motivic Galois theory. Internat. Math. Res. Notices 2004(76): 4073–4091
Connes, A., Marcolli, M.: From Physics to Number Theory via Noncommutative Geometry II: Renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory. In: Frontiers in Number Theory, Physics and Geometry. Berlin Heidelberg-New York: Springer, 2006, p. 269
Connes A. and Marcolli M. (2006). Quantum Fields and Motives. J. Geom. Phys. 56: 55–85
Ebrahimi-Fard K., Guo L. and Kreimer D. (2004). Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J. Phys. A 37: 11037–11052
Ebrahimi-Fard K., Guo L. and Kreimer D. (2005). Integrable Renormalization II: the General case. Ann. H. Poincaré 6: 369–395
Ebrahimi-Fard K. and Kreimer D. (2005). Hopf algebra approach to Feynman diagram calculations. J. Phys. A 38: R385–R406
Ebrahimi-Fard K., Gracia-Bondía J.M., Guo L. and Várilly J.C. (2006). Combinatorics of renormalization as matrix calculus. Phys. Lett. B 632: 552–558
Ebrahimi-Fard K., Guo L. and Manchon D. (2006). Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion. Commun. Math. Phys. 267: 821–845
Ebrahimi-Fard K. and Manchon D. (2006). On matrix differential equations in the Hopf algebra of renormalization. Adv. Theor. Math. Phys. 10: 879–913
Falk, S.: Doktor der Naturwissenschaften Dissertation, Mainz, 2005
Figueroa H. and Gracia-Bondía J.M. (2005). Combinatorial Hopf algebras in quantum field theory I. Rev. of Math. Phys. 17: 881–976
Gelfand I.M., Krob D., Lascoux A., Leclerc B., Retakh V. and Thibon J.-Y. (1995). Noncommutative symmetric functions. Adv. Math. 112: 218–348
Gracia-Bondía, J.M., Lazzarini, S.: Connes–Kreimer–Epstein–Glaser renormalization. http://arxive.org/ list/hep-th/0006106, 2006
Gracia-Bondía J.M. (2003). Improved Epstein–Glaser renormalization in coordinate space I. Euclidean framework. Math. Phys. Anal. Geom. 6: 59–88
Hazewinkel, M.: Hopf algebras of endomorphisms of Hopf algebras. http://arxive.org/list/math.QA/ 0410364, 2004
Kleinert H., Schulte-Frohlinde V. (2001) Critical Properties of ϕ4-theories. Singapore, World Scientific
Kreimer D. (1998). On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2: 303–334
Kreimer D. (2006). Anatomy of a gauge theory. Annals Phys. 321: 2757–2781
Kreimer, D.: \’Etude for linear Dyson–Schwinger Equations. IHES preprint, March 2006, available at http://www.hes.fr/PREPRINTS/2006/p/p-06-23.pdf
Kreimer D. (2007). Dyson–Schwinger Equations: from Hopf algebras to number theory. Fields Institute communications 50: 225–248
Kreimer D., Yeats K.: An Étude in non-linear Dyson–Schwinger Equations. Nucl. Phys. Proc. Suppl. 160, 116–121, 2006. hep-th/0605096
Lowenstein, J.H.: BPHZ renormalization. In: Renormalization theory (Proceedings NATO Advanced Study Institute, Erice, 1975), NATO Advanced Study Institute Series C: Math. and Phys. Sci., Vol. 23, Dordrecht:Reidel, 1976
Manchon, D.: Hopf algebras, from basics to applications to renormalization. Comptes-rendus des Rencontres mathématiques de Glanon 2001, http://arxive.org/list/math.QA/0408405, 2009
Patras F. (1993). La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier 43: 1067–1087
Patras F. (1994). L’algèbre des descentes d’une bigèbre graduée. J. Alg. 170: 547–566
Patras F. and Reutenauer C. (2002). On Dynkin and Klyachko idempotents in graded bialgebras. Adv. Appl. Math. 28: 560–579
Piguet O. and Sorella S.P. (1995). Algebraic renormalization. Springer, Berlin
Reutenauer C. (1993). Free Lie algebras. Oxford University Press, Oxford
Shnider, S., Sternberg, S.: Quantum groups. From coalgebras to Drinfel’d algebras. A guided tour, Graduate Texts in Mathematical Physics, II. Cambridge MA: International Press, 1993
Smirnov V.A. (1991). Renormalization and Asymptotic Expansions. Birkhäuser Verlag, Basel
Solomon L. (1976). A Mackey formula in the group ring of a Coxeter group. J. Alg. 41: 255–268
’t Hooft G. (1973). Dimensional regularization and the renormalization group. Nucl. Phys. B 41: 455–468
Vasilev A.N. (2004). The field theoretic renormalization group in critical behavior theory and stochastic dynamics. Boca Raton, FL, Chapman & Hall/CRC
Zimmermann W. (1969). Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15: 208–234
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
Ebrahimi-Fard, K., Gracia-Bondía, J.M. & Patras, F. A Lie Theoretic Approach to Renormalization. Commun. Math. Phys. 276, 519–549 (2007). https://doi.org/10.1007/s00220-007-0346-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0346-8