  We study quantum cosmological models for certain classes of bang/crunch
singularities, using the duality between expanding bubbles in AdS with a FRW
interior cosmology and perturbed CFTs on de Sitter space-time. It is pointed
out that horizon complementarity in the AdS bulk geometries is realized as a
conformal transformation in the dual deformed CFT. The quantum version of this
map is described in full detail in a toy model involving conformal quantum
mechanics. In this system the complementarity map acts as an exact duality
between eternal and apocalyptic Hamiltonian evolutions. We calculate the
commutation relation between the Hamiltonians corresponding to the different
frames. It vanishes only on scale invariant states.

High Energy Physics - Theory;General Relativity and Quantum Cosmology