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For Klein-Gordon fields, it is well known that there exist an infinite numberof nonequivalent Fock representations of the canonical commutation relationsand, therefore, of inequivalent quantum theories. A context in which this kindof ambiguities arises and prevents the derivation of robust results is, e.g.,in the quantum analysis of cosmological perturbations. In these situations,typically, a suitable scaling of the field by a time dependent function leadsto a description in an auxiliary static background, though the nonstationaritystill shows up in a time dependent mass. For such a field description, andassuming the compactness of the spatial sections, we recently proved in threeor less spatial dimensions that the criteria of a natural implementation of thespatial symmetries and of a unitary time evolution are able to select a uniqueclass of unitarily equivalent vacua, and hence of Fock representations. In thiswork, we succeed to extend our uniqueness result to the consideration of allpossible field descriptions that can be reached by a time dependent canonicaltransformation which, in particular, involves a scaling of the field by afunction of time. This kind of canonical transformations modify the dynamics ofthe system and introduce a further ambiguity in its quantum description,exceeding the choice of a Fock representation. Remarkably, for any compactspatial manifold in less than four dimensions, we show that our criteriaeliminate any possible nontrivial scaling of the field other than that leadingto the description in an auxiliary static background. Besides, we show thateither no time dependent redefinition of the field momentum is allowed or, ifthis may happen, the redefinition does not introduce any Fock representationthat cannot be obtained by a unitary transformation.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Criteria for the determination of time dependent scalings in the Fock  quantization of scalar fields with a time dependent mass in ultrastatic  spacetimes