Quantization of maximally - charged slowly - moving black holes ∗

We discuss the quantization of a system of slowly-moving extreme Reissner-Nordström black holes. In the near-horizon limit, this system has been shown to possess an SL(2, R) conformal symmetry. However, the Hamiltonian appears to have 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 apply the Faddeev-Popov quantization procedure to show that the Hamiltonian with no ground state corresponds to a gauge in which there is an obstruction at the singularities of moduli space 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. ‡