The $L_{\infty }$-algebra of the S-matrix

Abstract

We point out that the one-particle-irreducible vacuum correlation functions of a QFT are the structure constants of an $L_{\infty }$-algebra, whose Jacobi identities hold whenever there are no local gauge anomalies. The LSZ prescription for S-matrix elements is identified as an instance of the ``minimal model theorem'' of $L_{\infty }$-algebras. This generalises the algebraic structure of closed string field theory to arbitrary QFTs with a mass gap and leads to recursion relations for amplitudes (albeit ones only immediately useful at tree-level, where they reduce to Berends-Giele-style relations as shown in recent work).

32 pages; References and clarifications added (including fixing some questionable analytic points...) v3: minor typos fixed & presentation changes

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