### 3.6 Thermality of the reduced density matrix of a Killing horizon

The quantum state defined by Eq. (53) 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
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 , which is the
generator of rotations with respect to the angular coordinate defined above, one finds that
, i.e., the density matrix is thermal with respect to the Hamiltonian
with inverse temperature . This formal proof in Minkowski space was outlined in [152]. The
appropriate Euclidean Hamiltonian is then the Rindler Hamiltonian, which generates Lorentz
boosts in a direction orthogonal to the surface . In [142] the proof was generalized to the
case of generic static spacetimes with bifurcate Killing horizons admitting a regular Euclidean
section.