We study the Fock quantization of scalar fields with a time dependent mass incosmological scenarios with flat compact spatial sections. This framework describes physicallyinteresting situations like, e.g., cosmological perturbations in flat Friedmann-Robertson-Walkerspacetimes, generally including a suitable scaling of them by a background function. Weprove that the requirements of vacuum invariance under the spatial isometries and of a unitaryquantum dynamics select (a) a unique canonical pair of field variables among all those related bytime dependent canonical transformations which scale the field configurations, and (b) a uniqueFock representation for the canonical commutation relations of this pair of variables. Though theproof is generalizable to other compact spatial topologies in three or less dimensions, we focuson the case of the three-torus owing to its relevance in cosmology, paying a especial attention tothe role played by the spatial isometries in the determination of the representation.
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