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We study the Fock quantization of scalar fields with a time dependent mass incosmological scenarios with flat compact spatial sections. This frameworkdescribes physically interesting situations like, e.g., cosmologicalperturbations in flat Friedmann-Robertson-Walker spacetimes, generallyincluding a suitable scaling of them by a background function. We prove thatthe requirements of vacuum invariance under the spatial isometries and of aunitary quantum dynamics select (a) a unique canonical pair of field variablesamong all those related by time dependent canonical transformations which scalethe field configurations, and (b) a unique Fock representation for thecanonical commutation relations of this pair of variables. Though the proof isgeneralizable to other compact spatial topologies in three or less dimensions,we focus on the case of the three-torus owing to its relevance in cosmology,paying a especial attention to the role played by the spatial isometries in thedetermination of the representation.

Unitary evolution and uniqueness of the Fock quantization in flat cosmologies with compact spatial sections