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The Hopf Algebra of Feynman Diagrams

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Graphs in Perturbation Theory

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Abstract

The content of this chapter is partially based on the author’s article (Borinsky, Lett Math Phys 106(7):879–911, 2016) [1].

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Notes

  1. 1.

    Reprinted by permission from Springer Nature, Letters in Mathematical Physics, 106, 7, Algebraic Lattices in QFT Renormalization by Michael Borinsky, Copyright 2016.

  2. 2.

    Because the coproduct is not of the form \(\Delta \Gamma = \mathbbm {1}\otimes \Gamma + \Gamma \otimes \mathbbm {1}+ \widetilde{\Delta }\), the elements in the kernel of \(\widetilde{\Delta }\) are also called skewprimitive. As we can always divide out the ideal which sets all residues to \(\mathbbm {1}\), we will not treat this case differently.

  3. 3.

    I wish to thank Erik Panzer for quickly coming up with the explicit counterexample in Fig. 6.3a.

  4. 4.

    Note that all residues \(r \in \mathcal {R}^*\) map to \(\mathbbm {1}_\mathcal {H}^\text {P}\) under \(\chi _D\), \(\chi _D(r) = \mathbbm {1}_\mathcal {H}^\text {P}\).

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Authors and Affiliations

  1. Departments of Physics and of Mathematics, Humboldt-Universität zu Berlin, Berlin, Germany

    Michael Borinsky

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  1. Michael Borinsky
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Correspondence to Michael Borinsky .

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Borinsky, M. (2018). The Hopf Algebra of Feynman Diagrams. In: Graphs in Perturbation Theory. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-03541-9_6

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