3.6 Thermality of the reduced density matrix of a Killing horizon

The quantum state defined by Eq. (53View Equation) is the Hartle–Hawking vacuum [126]. The Green’s function in this state is defined by analytic continuation from the Euclidean Green’s function. The periodicity t → t + iβ H is thus inherent in this state. This periodicity indicates that the correlation functions computed in this state are in fact thermal correlation functions when continued to the Lorentzian section. This fact generalizes to an arbitrary interacting quantum field as shown in [121]. On the other hand, being globally defined, the Hartle–Hawking state is a pure state, which involves correlations between modes localized on different sides of the horizon. However, this state is described by a thermal density matrix if reduced to modes defined on one side of the horizon as was shown by Israel [138]. That the reduced density matrix obtained by tracing over modes inside the horizon is thermal can be formally seen by using angular quantization. Introducing the Euclidean Hamiltonian HE, which is the generator of rotations with respect to the angular coordinate ϕ defined above, one finds that ρ(ψ1 ,ψ2 ) = < ψ1|e−2πHE |ψ2 > + + + +, i.e., the density matrix is thermal with respect to the Hamiltonian HE with inverse temperature 2π. This formal proof in Minkowski space was outlined in [152Jump To The Next Citation Point]. The appropriate Euclidean Hamiltonian is then the Rindler Hamiltonian, which generates Lorentz boosts in a direction orthogonal to the surface Σ. In [142] the proof was generalized to the case of generic static spacetimes with bifurcate Killing horizons admitting a regular Euclidean section.
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