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

DOI: 10.1023/A:1007523409317

Cite this article as:
Connes, A. & Kreimer, D. Letters in Mathematical Physics (1999) 48: 85. doi:10.1023/A:1007523409317


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 theorynoncommutative geometryrenormalizationHopf algebrasfoliationsODE

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