For free scalar fields in any Gaussian state, the modular Hamiltonian restricted to a region is determined by the equal-time two-point functions X and Π via M=Π^{1/2}B^{-1}arcoth(2B)Π^{1/2}, N=Π^{-1/2}B arcoth(2B)Π^{-1/2}, B=sqrt(Π^{1/2}XΠ^{1/2}).
Global Properties of Vacuum States in de Sitter Space
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Starting from the assumption that vacuum states in de Sitter space look for any geodesic observer like equilibrium states with some a priori arbitrary temperature, an analysis of their global properties is carried out in the algebraic framework of local quantum physics. It is shown that these states have the Reeh-Schlieder property and that any primary vacuum state is also pure and weakly mixing. Moreover, the geodesic temperature of vacuum states has to be equal to the Gibbons-Hawking temperature and this fact is closely related to the existence of a discrete PCT-like symmetry. It is also shown that the global algebras of observables in vacuum sectors have the same structure as their counterparts in Minkowski space theories.
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Relating the modular Hamiltonian to two-point functions
For free scalar fields in any Gaussian state, the modular Hamiltonian restricted to a region is determined by the equal-time two-point functions X and Π via M=Π^{1/2}B^{-1}arcoth(2B)Π^{1/2}, N=Π^{-1/2}B arcoth(2B)Π^{-1/2}, B=sqrt(Π^{1/2}XΠ^{1/2}).