Essential Duality and Maximal Non-signalling Extensions in Algebraic Quantum Field Theory
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| Format: | Preprint |
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2026
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| _version_ | 1866915971637182464 |
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| author | Nasreddine, Hassan |
| author_facet | Nasreddine, Hassan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_00075 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Essential Duality and Maximal Non-signalling Extensions in Algebraic Quantum Field Theory Nasreddine, Hassan Quantum Physics Mathematical Physics 46L60 (Primary), 81T05, 46L30 (Secondary) 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. |
| title | Essential Duality and Maximal Non-signalling Extensions in Algebraic Quantum Field Theory |
| topic | Quantum Physics Mathematical Physics 46L60 (Primary), 81T05, 46L30 (Secondary) |
| url | https://arxiv.org/abs/2605.00075 |