Uniqueness of the Fock quantization of a free scalar field on $S^1$ with time dependent mass

We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time-dependent potential, also interpretable as a time-dependent mass. Classically, the field satisfies a linear wave equation of the form $\ddot{\xi }-{\xi }^{\prime \prime }+f(t)\xi =0$. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field ($f=0$) provides a unitary implementation of the dynamics for sufficiently general mass terms, $f(t)$. Furthermore, this representation is uniquely specified, among the class of representations determined by $S^1$-invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.