We discuss the problem of canonical quantization of a free real massive
scalar field in the Schwarzschild space-time. It is shown that a consistent
procedure of canonical quantization of the field can be carried out without
taking into account the black hole interior, so that in the resulting theory
the canonical commutation relations are satisfied exactly, and the Hamiltonian
has the standard form. However, unlike some papers, in which the expansion of
the quantum field in spherical harmonics was used, here we use an expansion in
scattering-like states for energies larger than the mass of the field. This
reveals a strange property of the resulting quantum field theory – doubling of
the quantum states, which look as having the same asymptotic momentum to an
observer located far away from the black hole. This purely topological effect
cannot be eliminated by moving away from the black hole.

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