Modular flows preserve causal spacetime ordering inside causal diamonds, but a bilocal modular Hamiltonian can violate local commutativity at spacelike distances.
Modular conjugation for multicomponent regions
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abstract
We consider a massless Dirac field in $1+1$ dimensions, and compute the Tomita-Takesaki modular conjugation corresponding to the vacuum state and a generic multicomponent spacetime region. We do it by analytic continuation from the modular flow, which was computed recently. We use our result to discuss the validity of Haag duality in this model.
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Modular evolutions and causality in two-dimensional conformal field theory
Modular flows preserve causal spacetime ordering inside causal diamonds, but a bilocal modular Hamiltonian can violate local commutativity at spacelike distances.