• Corpus ID: 221819469

Species-Theoretic Foundations of Perturbative Quantum Field Theory

@article{Norledge2020SpeciesTheoreticFO,
  title={Species-Theoretic Foundations of Perturbative Quantum Field Theory},
  author={William Norledge},
  journal={arXiv: Mathematical Physics},
  year={2020}
}

• William Norledge
• Published 21 September 2020
• Mathematics
• arXiv: Mathematical Physics

We develop an algebraic formalism for perturbative (local) quantum field theory (pQFT) which is based on Joyal's combinatorial species. We show that certain fundamental structures of pQFT are correctly viewed as algebraic structures internal to species. Aspects of this formalism have appeared in the physics literature, particularly in the work of Bogoliubov-Shirkov, Steinmann, Ruelle, and Epstein-Glaser-Stora. In this paper, we give a fully explicit account in terms of Aguiar-Mahajan's modern theory of the structures involved. We describe the central construction of causal perturbation theory as a homomorphism from the Hopf monoid of set compositions, decorated with local observables, into microcausal polynomial observables. The various operator-valued distributions, e.g. time-ordered products, retarded products, generalized retarded products, are obtained as images of fundamental elements of this Hopf monoid under the (curried) homomorphism. The perturbative \hbox{S-matrix} scheme corresponds to the so-called universal series, and the property of causal factorization is naturally expressed in terms of the action of the Hopf monoid on itself by Hopf powers. Given a system of (re)normalized time-ordered products, the perturbative construction of the corresponding interacting products is via an up biderivation of the Hopf monoid, which recovers Bogoliubov's formula.