Canonical quantization of a particle near a black hole ∗

We discuss the quantization of a particle near an extreme Reissner-Nordström black hole in the canonical formalism. This model appears to be described by a Hamiltonian with no well-defined ground state. This problem can be circumvented by a redefinition of the Hamiltonian due to de Alfaro, Fubini and Furlan (DFF). We show that the Hamiltonian with no ground state corresponds to a gauge in which there is an obstruction at the boundary of spacetime requiring a modification of the quantization rules. The redefinition of the Hamiltonianà la DFF corresponds to a different choice of gauge. The latter is a good gauge leading to standard quantization rules. Thus, the DFF trick is a consequence of a standard gauge-fixing procedure in the case of black hole scattering. ‡