  We study the Fock quantization of scalar fields of Klein-Gordon type in
nonstationary scenarios propagating in spacetimes with compact spatial
sections, allowing for different field descriptions that are related by means
of certain nonlocal linear canonical transformations that depend on time. More
specifically, we consider transformations that do not mix eigenmodes of the
Laplace-Beltrami operator, which are supposed to be dynamically decoupled. In
addition, we assume that the canonical transformations admit an asymptotic
expansion for large eigenvalues (in norm) of the Laplace-Beltrami operator in
the form of a series of half integer powers. Canonical transformations of this
kind are found in the study of scalar perturbations in inflationary
cosmologies, relating for instance the physical degrees of freedom of these
perturbations after gauge fixing with gauge invariant canonical pairs of
Bardeen quantities. We characterize all possible transformations of this type
and show that, independently of the initial field description, the combined
criterion of requiring (i) invariance of the vacuum under the spatial
symmetries and (ii) a unitary implementation of the dynamics, leads to a unique
equivalence class of Fock quantizations, all of them related by unitary
transformations. This conclusion provides even further robustness to the
validity of the proposed criterion, completing the results that have already
appeared in the literature about the uniqueness of the Fock quantization under
changes of field description when one permits exclusively local time dependent
canonical transformations that scale the field configuration.
