  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).
