  The Fock quantization of fields propagating in cosmological spacetimes is not
uniquely determined because of several reasons. Apart from the ambiguity in the
choice of the quantum representation of the canonical commutation relations,
there also exists certain freedom in the choice of field: one can scale it
arbitrarily absorbing background functions, which are spatially homogeneous but
depend on time. Each nontrivial scaling turns out into a different dynamics
and, in general, into an inequivalent quantum field theory. In this work we
analyze this freedom at the quantum level for a scalar field in a
nonstationary, homogeneous spacetime whose spatial sections have $S^3$
topology. A scaling of the configuration variable is introduced as part of a
linear, time dependent canonical transformation in phase space. In this
context, we prove in full detail a uniqueness result about the Fock
quantization requiring that the dynamics be unitary and the spatial symmetries
of the field equations have a natural unitary implementation. The main
conclusion is that, with those requirements, only one particular canonical
transformation is allowed, and thus only one choice of field-momentum pair (up
to irrelevant constant scalings). This complements another previous uniqueness
result for scalar fields with a time varying mass on $S^3$, which selects a
specific equivalence class of Fock representations of the canonical commutation
relations under the conditions of a unitary evolution and the invariance of the
vacuum under the background symmetries. In total, the combination of these two
different statements of uniqueness picks up a unique Fock quantization for the
system. We also extend our proof of uniqueness to other compact topologies and
spacetime dimensions.
