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We consider the quantum dynamics of a minimally coupled massless scalar field in de Sitter spacetime. The classical evolution is represented by a canonical transformation on the phase space for the field theory. By studying the corresponding Bogoliubov transformations, we show that the symplectic map that encodes the evolution between two instants of time cannot be unitarily implemented on any Fock space built from a SO(4)-symmetric complex structure. We will also show that, in contrast with some effectively lower dimensional examples arising from quantum general relativity such as Gowdy models, it is impossible to find a time-dependent conformal redefinition of the massless scalar field leading to a quantum unitary dynamics.