A New Look at Completeness and Generalized Symmetries
We describe a general principle of completeness in QFT. It asserts that the physical observable algebras produced by local degrees of freedom are the maximal ones compatible with causality. We elaborate on equivalent statements to this principle such as the non-existence of generalized symmetries and the uniqueness of the net of algebras. For non-complete theories, we explain how the existence of generalized symmetries is unavoidable, and further, that they always come in dual pairs with precisely the same "size", measured by the Jones or algebraic index. Entropic order/disorder parameters can be defined that sense the dual pairs of generalized symmetries and satisfy a relation dubbed the certainty principle, due to its connection with the uncertainty principle in quantum mechanics. Using this new understanding we prove a recent conjecture by Harlow and Ooguri concerning a universal formula for the charged density of states in QFT, comment on relations with the Weinberg-Witten theorem, and argue against the existence of generalized symmetries in the bulk of holographic theories.