Hamiltonian evolution and quantization for extremal black holes

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein -Maxwell theory First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only The Hamilton ian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yie lds a vanishing black hole entropy. Second, we give a Hamiltonian quantizat ion that incorporates extremal black holes as a limiting case of nonextrema l ones, and examine the classical limit in terms of wave packets. The sprea ding of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus w ith the Bekenstein-Hawking entropy even for the extremal holes. The discuss ion takes place throughout within Lorentz-signature spacetimes.