  In curved spacetimes, the lack of criteria for the construction of a unique
quantization is a fundamental problem undermining the significance of the
predictions of quantum field theory. Inequivalent quantizations lead to
different physics. Recently, however, some uniqueness results have been
obtained for fields in non-stationary settings. In particular, for vacua that
are invariant under the background symmetries, a unitary implementation of the
classical evolution suffices to pick up a unique Fock quantization in the case
of Klein-Gordon fields with time-dependent mass, propagating in a static
spacetime whose spatial sections are three-spheres. In fact, the field equation
can be reinterpreted as describing the propagation in a
Friedmann-Robertson-Walker spacetime after a suitable scaling of the field by a
function of time. For this class of fields, we prove here an even stronger
result about the Fock quantization: the uniqueness persists when one allows for
linear time-dependent transformations of the field in order to account for a
scaling by background functions. In total, paying attention to the dynamics,
there exists a preferred choice of quantum field, and only one
$SO(4)$-invariant Fock representation for it that respects the standard
probabilistic interpretation along the evolution. The result has relevant
implications e.g. in cosmology.
