  Usually quantum theory is formulated in terms of the evolution of states
through spacelike surfaces. However, a generalization of this formulation is
needed for field theory in spacetimes not foliable by spacelike surfaces, or in
quantum gravity where geometry is not definite but a quantum variable. In
particular, a generalization of usual quantum theory is needed for field theory
in the spacetimes that model the process of black hole evaporation. This paper
discusses a spacetime generalization of usual quantum theory that is applicable
to evaporating black hole spacetimes. In this generalization, information is
not lost in the process of evaporation. Rather, complete information is
distributed about four-dimensional spacetime. Black hole evaporation is thus
not in conflict with the principles of quantum theory when suitably generally
stated. (Talk presented at the 49th Yamada Conference: Black Holes and High
Energy Astrophysics, Kyoto Japan, April 7-10, 1998.)
