Perturbative Quantum Field Theory and Homotopy Algebras

Abstract

We review the homotopy algebraic perspective on perturbative quantum field theory: classical field theories correspond to homotopy algebras such as $A_{\infty }$- and $L_{\infty }$-algebras. Furthermore, their scattering amplitudes are encoded in minimal models of these homotopy algebras at tree level and their quantum relatives at loop level. The translation between Lagrangian field theories and homotopy algebras is provided by the Batalin--Vilkovisky formalism. The minimal models are computed recursively using the homological perturbation lemma, which induces useful recursion relations for the computation of scattering amplitudes. After explaining how the homological perturbation lemma produces the usual Feynman diagram expansion, we use our techniques to verify an identity for the Berends--Giele currents which implies the Kleiss--Kuijf relations.

arXiv commentv2: 25 pages, contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity", typos fixed, published version

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• Field Theory Equivalences as Spans of $L_{\infty }$-algebras
• Generalised wavefunction coefficients and acyclonesto-cosmohedra
• Full S-matrices and Witten diagrams with (relative) L-infinity algebras
• Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes
• Extended Field Theories as higher Kaluza-Klein theories