Skip to main content
SpringerLink
Log in

A Lie Theoretic Approach to Renormalization

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson F.V. (1963). Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7: 1–30

    Article  MATH  MathSciNet  Google Scholar 

  2. Bergbauer C. and Kreimer D. (2005). The Hopf algebra of rooted trees in Epstein-Glaser renormalization. Ann. Henri Poincaré 6: 343–367

    Article  MATH  MathSciNet  Google Scholar 

  3. 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

  4. Blaer A.S. and Young K. (1974). Field theory renormalization using the Callan–Symanzik equation. Nucl. Phys. B 83: 493–514

    Article  ADS  Google Scholar 

  5. Blessenohl D. and Schocker M. (2005). Noncommutative character theory of the symmetric group. World Scientific, Singapore

    MATH  Google Scholar 

  6. Bloch S., Esnault H. and Kreimer D. (2006). Motives associated to graph polynomials. Commun. Math. Phys. 267: 181–225

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. 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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. 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

    Article  MATH  ADS  Google Scholar 

  9. Bourbaki, N.: Elements of Mathematics. In: Lie groups and Lie algebras. Chapters 1–3. Berlin: Springer (1989)

  10. Callan C.G. (1970). Broken scale invariance in scalar field theory. Phys. Rev. D 2: 1541–1547

    Article  ADS  Google Scholar 

  11. 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

  12. Cartier, P.: A primer on Hopf algebras. IHES preprint, August 2006, available at http://www.ihes.fr/ PREPRINTS/2006/M/M-06-40.pdf

  13. Caswell W.E. and Kennedy A.D. (1982). Simple approach to renormalization theory. Phys. Rev. D 25: 392–408

    Article  ADS  MathSciNet  Google Scholar 

  14. Collins J.C. (1974). Structure of the counterterms in dimensional regularization. Nucl. Phys. B 80: 341–348

    Article  ADS  Google Scholar 

  15. Collins J.C. (1984). Renormalization. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  16. Connes A. and Moscovici H. (1995). The local index formula in noncommutative geometry. Geom. Func. Anal. 5: 174–243

    Article  MATH  MathSciNet  Google Scholar 

  17. Connes A. and Kreimer D. (1998). Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199: 203–242

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. 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

    MATH  MathSciNet  Google Scholar 

  19. 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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Connes A. and Marcolli M. (2004). Renormalization and motivic Galois theory. Internat. Math. Res. Notices 2004(76): 4073–4091

    Article  MATH  MathSciNet  Google Scholar 

  21. 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

  22. Connes A. and Marcolli M. (2006). Quantum Fields and Motives. J. Geom. Phys. 56: 55–85

    Article  MATH  MathSciNet  Google Scholar 

  23. 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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Ebrahimi-Fard K., Guo L. and Kreimer D. (2005). Integrable Renormalization II: the General case. Ann. H. Poincaré 6: 369–395

    Article  MATH  MathSciNet  Google Scholar 

  25. Ebrahimi-Fard K. and Kreimer D. (2005). Hopf algebra approach to Feynman diagram calculations. J. Phys. A 38: R385–R406

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. 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

    Article  ADS  MathSciNet  Google Scholar 

  27. Ebrahimi-Fard K., Guo L. and Manchon D. (2006). Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion. Commun. Math. Phys. 267: 821–845

    Article  ADS  MathSciNet  Google Scholar 

  28. Ebrahimi-Fard K. and Manchon D. (2006). On matrix differential equations in the Hopf algebra of renormalization. Adv. Theor. Math. Phys. 10: 879–913

    MATH  MathSciNet  Google Scholar 

  29. Falk, S.: Doktor der Naturwissenschaften Dissertation, Mainz, 2005

  30. Figueroa H. and Gracia-Bondía J.M. (2005). Combinatorial Hopf algebras in quantum field theory I. Rev. of Math. Phys. 17: 881–976

    Article  MATH  Google Scholar 

  31. Gelfand I.M., Krob D., Lascoux A., Leclerc B., Retakh V. and Thibon J.-Y. (1995). Noncommutative symmetric functions. Adv. Math. 112: 218–348

    Article  MATH  MathSciNet  Google Scholar 

  32. Gracia-Bondía, J.M., Lazzarini, S.: Connes–Kreimer–Epstein–Glaser renormalization. http://arxive.org/ list/hep-th/0006106, 2006

  33. Gracia-Bondía J.M. (2003). Improved Epstein–Glaser renormalization in coordinate space I. Euclidean framework. Math. Phys. Anal. Geom. 6: 59–88

    Article  MATH  Google Scholar 

  34. Hazewinkel, M.: Hopf algebras of endomorphisms of Hopf algebras. http://arxive.org/list/math.QA/ 0410364, 2004

  35. Kleinert H., Schulte-Frohlinde V. (2001) Critical Properties of ϕ4-theories. Singapore, World Scientific

    Google Scholar 

  36. Kreimer D. (1998). On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2: 303–334

    MATH  MathSciNet  Google Scholar 

  37. Kreimer D. (2006). Anatomy of a gauge theory. Annals Phys. 321: 2757–2781

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. 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

  39. Kreimer D. (2007). Dyson–Schwinger Equations: from Hopf algebras to number theory. Fields Institute communications 50: 225–248

    MathSciNet  Google Scholar 

  40. Kreimer D., Yeats K.: An Étude in non-linear Dyson–Schwinger Equations. Nucl. Phys. Proc. Suppl. 160, 116–121, 2006. hep-th/0605096

  41. 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

  42. 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

  43. Patras F. (1993). La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier 43: 1067–1087

    MATH  MathSciNet  Google Scholar 

  44. Patras F. (1994). L’algèbre des descentes d’une bigèbre graduée. J. Alg. 170: 547–566

    Article  MATH  MathSciNet  Google Scholar 

  45. Patras F. and Reutenauer C. (2002). On Dynkin and Klyachko idempotents in graded bialgebras. Adv. Appl. Math. 28: 560–579

    Article  MATH  MathSciNet  Google Scholar 

  46. Piguet O. and Sorella S.P. (1995). Algebraic renormalization. Springer, Berlin

    MATH  Google Scholar 

  47. Reutenauer C. (1993). Free Lie algebras. Oxford University Press, Oxford

    MATH  Google Scholar 

  48. 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

  49. Smirnov V.A. (1991). Renormalization and Asymptotic Expansions. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  50. Solomon L. (1976). A Mackey formula in the group ring of a Coxeter group. J. Alg. 41: 255–268

    Article  MATH  Google Scholar 

  51. ’t Hooft G. (1973). Dimensional regularization and the renormalization group. Nucl. Phys. B 41: 455–468

    Article  MathSciNet  Google Scholar 

  52. Vasilev A.N. (2004). The field theoretic renormalization group in critical behavior theory and stochastic dynamics. Boca Raton, FL, Chapman & Hall/CRC

    Google Scholar 

  53. Zimmermann W. (1969). Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15: 208–234

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I.H.É.S., Le Bois-Marie, 35, Route de Chartres, F-91440, Bures-sur-Yvette, France

    Kurusch Ebrahimi-Fard

  2. Departamento de Física Teórica I, Universidad Complutense, Madrid, 28040, Spain

    José M. Gracia-Bondía

  3. Departamento de Física, Universidad de Costa Rica, San Pedro, 2060, Costa Rica

    José M. Gracia-Bondía

  4. Laboratoire J.-A. Dieudonné UMR 6621, CNRS, Parc Valrose, 06108, Nice Cedex 02, France

    Frédéric Patras

Authors
  1. Kurusch Ebrahimi-Fard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José M. Gracia-Bondía
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frédéric Patras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kurusch Ebrahimi-Fard.

Additional information

Communicated by A. Connes

Rights and permissions

Reprints 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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0346-8

Keywords

Search

Navigation