Doubling of physical states in the quantum scalar field theory for a remote observer in the Schwarzschild spacetime

We discuss the problem of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime. 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 is used, here we use an expansion in scatteringlike 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.