Abstract
In this chapter, we address some applications of the theory of Hopf algebras that have been popular recently, starting essentially in the late 1990s and early 2000s with the works of Alain Connes and Dirk Kreimer on renormalization in perturbative quantum field theory (pQFT).
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Notes
- 1.
See K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti, Hopf algebraic deformations of products and Wick polynomials. International Mathematics Research Notices, rny269, 2018.
- 2.
For details and more advanced results on the subject, see, e.g., G. Peccati and M. S. Taqqu. Wiener chaos: moments, cumulants, and diagrams. Springer and Bocconi University Press, Milan, 2011.
- 3.
Our definition and the proof of the next Proposition follow D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries. Proc. Sympos. Pure Math. 73 (2001), 43-80. The difficult part in handling Feynman graphs is to take into account of isomorphisms and the combinatorial coefficients they create (the symmetry factors). The combinatorics of graph insertions is carefully studied in W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach. Communications in mathematical physics, 276(3), (2007) 773-798. The reader is referred to this article for details and the graph-theoretic framework surrounding these constructions.
- 4.
The dualization has to take into account combinatorial symmetry factors, see again van Suijlekom, op. cit.
- 5.
The content of this section is based on K. Ebrahimi-Fard and F. Patras, Exponential renormalization. Ann. Henri Poincaré 11(5), 411–433 (2002).
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Cartier, P., Patras, F. (2021). Renormalization. In: Classical Hopf Algebras and Their Applications. Algebra and Applications, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-030-77845-3_10
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