Boundary conditions and the corresponding states of a quantum field theory depend on how the horizons are taken into account. There is an ambiguity as to which method is appropriate because different ways of incorporating the horizons lead to different results. We propose that a natural way of including the horizons is to first consider the Kruskal extension and then define the quantum field theory on the Euclidean section. Boundary conditions emerge naturally as consistency conditions of the Kruskal extension. We carry out the proposal for the explicit case of the Schwarzschild-de Sitter manifold with two horizons. The required period β is the interesting condition that it is the lowest common multiple of 2π divided by the surface gravity of both horizons. Restricting the ratio of the surface gravity of the horizons to rational numbers yields finite β. The example also highlights some of the difficulties of the off-shell approach with conical singularities in the multi-horizon scenario and serves to illustrate the much richer interplay that can occur among horizons, quantum field theory and topology when the cosmological constant is not neglected in black-hole processes.
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