The uses of Connes and Kreimer's algebraic formulation of renormalization theory

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.

[1]  Le Bois-Marie,et al.  Integrable Renormalization II : the general case , 2004 .

[2]  D. Kreimer.  New mathematical structures in renormalizable quantum field theories , 2002, hep-th/0211136.

[3]  J. Gracia-Bondía.  Improved Epstein–Glaser Renormalization in Coordinate Space I. Euclidean Framework , 2002, hep-th/0202023.

[4]  H. Quevedo,et al.  Normal Coordinates and Primitive Elements¶in the Hopf Algebra of Renormalization , 2001, hep-th/0105259.

[5]  R. Longo.  Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects , 2001 .

[6]  J. C. Várilly.  Hopf Algebras in Noncommutative Geometry , 2001, hep-th/0109077.

[7]  D. Kastler.  Connes-Moscovici-Kreimer Hopf Algebras , 2001, math-ph/0104017.

[8]  J. Gracia-Bond́ıa,et al.  ON THE ANTIPODE OF KREIMER'S HOPF ALGEBRA , 1999, hep-th/9912170.

[9]  Hagen Kleinert,et al.  Critical properties of φ4-theories , 2001 .

[10]  G. Pinter.  The Hopf Algebra Structure of Connes and Kreimer in Epstein–Glaser Renormalization , 2000, hep-th/0012057.

[11]  J. Gracia-Bond́ıa,et al.  Elements of Noncommutative Geometry , 2000 .

[12]  D. Kreimer.  Knots and Feynman Diagrams , 2000 .

[13]  J. M. Gracia-Bondía,et al.  Connes-Kreimer-Epstein-Glaser Renormalization , 2000, hep-th/0006106.

[14]  D. Broadhurst,et al.  Towards Cohomology of Renormalization: Bigrading the Combinatorial Hopf Algebra of Rooted Trees , 2000, hep-th/0001202.

[15]  A. Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 1999, hep-th/9912092.

[16]  A. Connes,et al.  Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group , 1999, hep-th/9909126.

[17]  Karl H. Hofmann,et al.  The Structure of Compact Groups , 2020 .

[18]  A. Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998, hep-th/9808042.

[19]  D. Kreimer.  On the Hopf algebra structure of perturbative quantum field theories , 1997, q-alg/9707029.

[20]  M. Takeuchi.  Book Review: Hopf algebras and their actions on rings , 1995 .

[21]  Susan Montgomery,et al.  Hopf algebras and their actions on rings , 1993 .

[22]  Gian-Carlo Rota,et al.  Coalgebras and Bialgebras in Combinatorics , 1979 .

[23]  P. M. Cohn.  GROUPES ET ALGÉBRES DE LIE , 1977 .

[24]  W. Zimmermann.  Remark on Equivalent Formulations for Bogoliubov’s Method of Renormalization , 1976 .

[25]  W. Zimmermann.  Convergence of Bogoliubov's method of renormalization in momentum space , 1969 .