We discuss the branching structure of the quantum-gravitational wave functionthat describes the evaporation of a black hole. A global wave function thatinitially describes a classical Schwarzschild geometry is continually decoheredinto distinct semiclassical branches by the emission of Hawking radiation. Thelaws of quantum mechanics dictate that the wave function evolves unitarily, butthis unitary evolution is only manifest when considering the global descriptionof the wave function: it is not implemented by time evolution on a singlesemiclassical branch. Conversely, geometric notions like the position orsmoothness of a horizon only make sense on the level of individual branches. Weconsider the implications of this picture for probes of black holes byclassical observers in definite geometries, like those involved in the AMPSconstruction. We argue that individual branches can describe semiclassicalgeometries free of firewalls, even as the global wave function evolvesunitarily. We show that the pointer states of infalling detectors that arerobust under Hamiltonian evolution are distinct from, and incompatible with,those of exterior detectors stationary with respect to the black hole horizon,in the sense that the pointer bases are related to each other via nontrivialtransformations that mix system, apparatus, and environment. This resultdescribes a Hilbert-space version of black hole complementarity.
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