Disjoint additivity and local quantum physics

Quantum systems of physical interest are often local, but there are at least three competing perspectives on how "locality" should be formalized: an algebraic framework, a path-integral framework, and a lattice framework. One puzzle in this competition is that systems with higher-form symmetries, which are perfectly local from the path-integral and lattice perspectives, can violate the algebraic principle of "additivity". In this paper, we propose a resolution to this puzzle by introducing a weaker locality principle, "disjoint additivity", which together with Haag duality should always be satisfied in local quantum systems. As evidence, we give examples in which disjoint additivity is preserved when ordinary additivity is violated; we show that Haag duality and disjoint additivity are satisfied in rather general lattice systems with local symmetry constraints; we give examples of nonlocal theories in which either disjoint additivity or Haag duality is violated; and finally we give examples of systems with nonlocal symmetry constraints in which disjoint additivity is violated, but can be restored by passing to a local "SymTFT" system in one higher dimension.