  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
Hamiltonian contains no surface term at the internal infinity, for reasons
related to the vanishing of the extremal hole surface gravity, and quantization
yields a vanishing black hole entropy. Second, we give a Hamiltonian
quantization that incorporates extremal black holes as a limiting case of
nonextremal ones, and examine the classical limit in terms of wave packets. The
spreading of the packets, even the ones centered about extremal black holes, is
consistent with continuity of the entropy in the extremal limit, and thus with
the Bekenstein-Hawking entropy even for the extremal holes. The discussion
takes place throughout within Lorentz-signature spacetimes.
