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}).
Adiabatic Ground States in Non-Smooth Spacetimes
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Ground states are a well-known class of Hadamard states in smooth spacetimes. In this paper we show that the ground state of the Klein-Gordon field in a non-smooth ultrastatic spacetime is an adiabatic state. The order of the state depends linearly on the regularity of the metric. We obtain the result by combining microlocal estimates for the causal propagator, propagation of singularities results for non-smooth pseudodifferential operators, and eigenvalue asymptotics for elliptic operators of low regularity.
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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}).