Essential Duality and Quantitative Channel Composability in Algebraic Quantum Field Theory

Abstract

Let \(O\mapsto A(O)\) be a local Haag--Kastler net and let \(A^d(O)=A(O')'\) be its dual net. Normal unital completely positive maps on \(B(H)\) that fix \(A(O')\) pointwise are exactly the channels with Kraus operators in \(A^d(O)\). Thus Haag duality at \(O\) is equivalent to equality with the \(A(O)\)-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators \(a\in M\) and \(b\in N\), the small-parameter order defect has coefficient \(\operatorname{diam}\sigma(-i[a,b])\). Uniform optimization gives \(\Gamma(M,N)=2\Delta_{\mathrm{sa}}(M,N)\), and reversible inner-channel cb balls recover the same invariant through \(\lim_{\varepsilon\downarrow0}\Omega^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=\Gamma(M,N)/4\). The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable \(G\), the mean-input-energy-constrained coefficient \(\Gamma_{G,E}\) has the same zero criterion and increases to \(\Gamma\) as \(E\to\infty\), while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either \(0\) or \(4\); generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.