Letters in Mathematical Physics

, Volume 48, Issue 1, pp 85–96

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

  • Alain Connes
  • Dirk Kreimer
Article

Abstract

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.

quantum field theory noncommutative geometry renormalization Hopf algebras foliations ODE 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bogoliubov, N. N. and Shirkov, D.V.: Introduction to theTheory of Quantized Fields, 3rd edn, Wiley, New York, 1980; Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem of renormalization, Comm. Math. Phys. 2 (1966), 301–326; Zimmermann, W.: Convergence of Bogoliubov's method of renormalization in momentum space, Comm. Math. Phys. 15 (1969), 208–234.Google Scholar
  2. 2.
    Bjorken, J.D. and Drell, S.D.: RelativisticQuantumFields, McGraw-Hill, Englewood Cliffs, 1965.Google Scholar
  3. 3.
    Broadhurst, D. J.: Massive 3-loop Feynman Diagrams reducible to SC* primitives of algebras at the sixth root of unity, to appear in Eur. Phys. J. C, hep-th/9803091.Google Scholar
  4. 4.
    Broadhurst, D. J. and Kreimer, D.: Association ofmultiple zetavalues with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393 (1997) 403–412, hep-th/9609128.Google Scholar
  5. 5.
    Broadhurst, D. J. and Kreimer, D.: Renormalization automated by Hopf algebra, to appear in J. Symbolic. Comput., hep-th/9810087.Google Scholar
  6. 6.
    Brouder, C.: Runge Kutta methods and renormalization, hep-th/9904014.Google Scholar
  7. 7.
    Butcher, J. C.: An algebraic theory of integration methods, Math.Comp. 26 (1972), 79–106.Google Scholar
  8. 8.
    Cayley, A.: On the theory of the analytical forms called trees, Phil.Mag. XIII (1857), 172–176.Google Scholar
  9. 9.
    Chen, Kuo-Tsai, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831–879.Google Scholar
  10. 10.
    Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155–176, hep-th/9603053.Google Scholar
  11. 11.
    Connes, A.: Noncommutative Geometry, Academic Press, San Diego, CA, 1994.Google Scholar
  12. 12.
    Connes, A., Douglas, M. and Schwarz, A.: Noncommutative geometry and matrix theory: compactification on tori, JHEP 9802:003 (1998), hep-th/9711162.Google Scholar
  13. 13.
    Connes, A. and Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203–242, hep-th/9808042.Google Scholar
  14. 14.
    Connes, A. and Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), 199–246, math.DG/9806109.Google Scholar
  15. 15.
    Connes, A. and Moscovici, H.: Cyclic cohomology and Hopf algebras, Lett.Math. Phys. 48 (1999), 97–108 (this issue).Google Scholar
  16. 16.
    Epstein, H. and Glaser, V.: The role of locality in perturbation theory, Ann. Inst. H. Poincarŕ A 19 (1973), 211–295.Google Scholar
  17. 17.
    Fulton, W. and MacPherson, R.: A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), 183–225.Google Scholar
  18. 18.
    Goncharov, A.: Polylogarithms in arithmetic and geometry, Proc. ICM-94 (Zürich),Vol. 1,2, Birkhäuser, Basel, 1995, pp. 374–387.Google Scholar
  19. 19.
    Kontsevich, M.: Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35–72.Google Scholar
  20. 20.
    Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories, Adv. Math. Phys. 2 (1998), 303–334, q-alg/9707029.Google Scholar
  21. 21.
    Kreimer, D.: On Overlapping Divergences, to appear in Comm.Math. Phys., hep-th/9810022.Google Scholar
  22. 22.
    Kreimer, D.: Chen's iterated integral represents the operator product expansion, hep-th/9901099.Google Scholar
  23. 23.
    Kreimer, D. and Delbourgo, R.: Using the Hopf algebra of QFT in calculations, hep-th/9903249.Google Scholar
  24. 24.
    Wulkenhaar, R.: On the Connes-Moscovici Hopf algebra associated to the diffeomorphisms of a manifold, Preprint CPT Luminy April 1999. hep-th/9904009.Google Scholar
  25. 25.
    Zagier, D.: Values of zeta functions and their applications, First European Congr.Math.,Vol. II, Birkhäuser, Boston, 1994, pp. 497–512.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Alain Connes
    • 1
  • Dirk Kreimer
    • 1
  1. 1.Institut des Hautes Études Scientifiques, Le Bois-MarieBures-sur-YvetteFrance

Personalised recommendations