  We consider the quantum dynamics of a massless scalar field in de Sitter
space-time. 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 show also 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.
