Advertisement

Letters in Mathematical Physics

, Volume 93, Issue 2, pp 187–201 | Cite as

A Perspective on Regularization and Curvature

  • Susama Agarwala
Article

Abstract

A global connection on the Connes Marcolli renormalization bundle relates β-functions of a class of regularization schemes by gauge transformations, as well as local solutions to β-functions over curved space–time.

Mathematics Subject Classification (2010)

81R99 

Keywords

quantum field theory equisingular connection β-function ζ-function regularization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwala, S.: The β-function over curved space-time under ζ-function regularization (2009). arXiv:0909.4122Google Scholar
  2. 2.
    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, pp. 133–164 (2006). arXiv:hep-th/0506190v2Google Scholar
  3. 3.
    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. In: Communications in Mathematical Physics, vol. 210, pp. 249–273 (2001). arXiv:hep-th/9912092v1Google Scholar
  4. 4.
    Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem II: the β function, diffeomorphisms and renormalization group. In: Communications in Mathematical Physics, vol. 216, pp. 215–241 (2001). arXiv:hep-th/0003188v1Google Scholar
  5. 5.
    Connes A., Marcolli M.: Noncommutative geometry, quantum fields, and motives. American Mathematical Society, Providence (2008)Google Scholar
  6. 6.
    Ebrahimi-Fard, K., Manchon, D.: On matrix differential equations in the Hopf algebra of renormalization. In: Advances in Theoretical and Mathematical Physics, vol. 10, pp. 879–913 (2006). arXiv:math-ph/0606039v2Google Scholar
  7. 7.
    Ebrahimi-Fard K., Kreimer D.: Hopf algebra approach to Feynman diagram calculations. J. Phys. A 38, R385–R406 (2006) arXiv:hep-th/0510202v2CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gross D.: Renormalization groups. In: Deligne, P., Kazhkan, D., Etingof, P., Morgan, J.W., Freed, D.S., Morrison, D.R., Jeffrey, L.C., Witten, E. (eds) Quantum Fields and Strings: A Course for Mathematicians, vol. 1, pp. 551–596. American Mathematical Society, Providence (1999)Google Scholar
  9. 9.
    Manchon D.: Hopf algebras and renormalization. In: Hazelwinkel, M. (eds) Handbook of Algebra, vol. 4, pp. 365–427. Elevier, Oxford (2008)CrossRefGoogle Scholar
  10. 10.
    Mencattini, I.: The structures of insertion elimination Lie algebra, Ph.D. thesis, Boston University (2005)Google Scholar
  11. 11.
    Patras, F., Ebrahimi-Fard, K., Garcia-Bondiá, J.M.: A Lie theoretic approach to renormalizaton. In: Communications in Mathematical Physics, vol. 276, pp. 519–549 (2001). arXiv:hep-th/0609035Google Scholar
  12. 12.
    Ryder L.H.: Quantum Field Theory. Cambridge University Press, New York (1985)MATHGoogle Scholar
  13. 13.
    Speer E.R.: Analytic renormalization. J. Math. Phys. 9, 1404–1411 (1968)CrossRefADSGoogle Scholar
  14. 14.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, 3rd edn, vol. 2. Publish or Perish, Houston (1999)Google Scholar
  15. 15.
    Ticciati R.: Quantum Field Theory for Mathematicians. Cambridge University Press, New York (1999)MATHCrossRefGoogle Scholar
  16. 16.
    Witten E.: Perturbative quantum field theory. In: Deligne, P., Kazhkan, D., Etingof, P., Morgan, J.W., Freed, D.S., Morrison, D.R., Jeffrey, L.C., Witten, E. (eds) Quantum Fields and Strings: A Course for Mathematicians. vol. 1, pp. 419–473. American Mathematical Society, Providence (1999)Google Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA

Personalised recommendations