  Two important problems in studying the quantum black hole, namely the
construction of the Hilbert space and the definition of the time evolution
operator on such Hilbert space, are discussed using the de Sitter background
field method for an observer far from the black hole. This is achieved through
the ambient space formalism. Remarkably, in this approximation (distant
observer), the theory preserves unitarity and analyticity, it is free from any
infrared divergence, and it renders a quantum black hole entropy that turns out
to be finite.
