Renormalization as a functor on bialgebras

Abstract

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)^+), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of B are not renormalised, i.e. when Feynman diagrams containing one single vertex are not renormalised. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as the standard renormalisation procedure for the scalar field.

24 pages, no figure. Several changes in the connection with standard renormalization

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Cited by in corpus (10)

• Combinatorial Hopf algebras
• Quantum field theory and Hopf algebra cohomology
• A Hopf laboratory for symmetric functions
• Quantum field theory meets Hopf algebra
• The renormalisation bialgebra and operads
• Combinatorial Hopf algebras from renormalization
• The Dirichlet Hopf algebra of arithmetics
• Renormalization : A number theoretical model
• On the algebra of quasi-shuffles
• Renormalization groupoids in algebraic topology