Essential Duality and Maximal Non-signalling Extensions in Algebraic Quantum Field Theory

Abstract

We show that, under additivity, the maximal von Neumann algebra extension of $\mathcal{A}(O)$ inside $B(\mathcal{H})$ whose inner automorphisms are non-signalling with respect to all spacelike-separated regions is $\mathcal{A}(O')'$. Consequently, $\mathcal{A}(O)$ is maximal with respect to this property if and only if essential duality holds. The proof is purely algebraic. When essential duality fails, we construct a proper extension all of whose inner automorphisms, and more generally all normal completely positive maps admitting Kraus operators in the algebra, are non-signalling. Under essential duality, any proper extension necessarily admits a signalling operation. An entropic formulation using Araki relative entropy provides a quantitative diagnostic of signalling, though it is not used in the proof. Additional structural results include the wedge-intersection identity $\mathcal{A}(O')' = \bigcap_{W \supset O}\mathcal{A}(W)$ and equivalent characterisations of essential duality. These results identify essential duality as an operational maximality condition within the given representation.