Fock quantization of a scalar field with time dependent mass on the
  three-sphere: unitarity and uniqueness
  We study the Fock description of a quantum free field on the three-sphere
with a mass that depends explicitly on time, also interpretable as an
explicitly time dependent quadratic potential. We show that, under quite mild
restrictions on the time dependence of the mass, the specific Fock
representation of the canonical commutation relations which is naturally
associated with a massless free field provides a unitary dynamics even when the
time varying mass is present. Moreover, we demonstrate that this Fock
representation is the only acceptable one, up to unitary equivalence, if the
vacuum has to be SO(4)-invariant (i.e., invariant under the symmetries of the
field equation) and the dynamics is required to be unitary. In particular, the
analysis and uniqueness of the quantization can be applied to the treatment of
cosmological perturbations around Friedmann-Robertson-Walker spacetimes with
the spatial topology of the three-sphere, like e.g. for gravitational waves
(tensor perturbations). In addition, we analyze the extension of our results to
free fields with a time dependent mass defined on other compact spatial
manifolds. We prove the uniqueness of the Fock representation in the case of a
two-sphere as well, and discuss the case of a three-torus.
