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The Fock quantization of fields propagating in cosmological spacetimes is notuniquely determined because of several reasons. Apart from the ambiguity in thechoice of the quantum representation of the canonical commutation relations,there also exists certain freedom in the choice of field: one can scale itarbitrarily absorbing background functions, which are spatially homogeneous butdepend on time. Each nontrivial scaling turns out into a different dynamicsand, in general, into an inequivalent quantum field theory. In this work weanalyze this freedom at the quantum level for a scalar field in anonstationary, homogeneous spacetime whose spatial sections have $S^3$topology. A scaling of the configuration variable is introduced as part of alinear, time dependent canonical transformation in phase space. In thiscontext, we prove in full detail a uniqueness result about the Fockquantization requiring that the dynamics be unitary and the spatial symmetriesof the field equations have a natural unitary implementation. The mainconclusion is that, with those requirements, only one particular canonicaltransformation is allowed, and thus only one choice of field-momentum pair (upto irrelevant constant scalings). This complements another previous uniquenessresult for scalar fields with a time varying mass on $S^3$, which selects aspecific equivalence class of Fock representations of the canonical commutationrelations under the conditions of a unitary evolution and the invariance of thevacuum under the background symmetries. In total, the combination of these twodifferent statements of uniqueness picks up a unique Fock quantization for thesystem. We also extend our proof of uniqueness to other compact topologies andspacetime dimensions.

Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes