Uniqueness of the Fock quantization of the Gowdy $T^3$ model

After its reduction by a gauge-fixing procedure, the family of linearly polarized Gowdy $T^3$ cosmologies admits a scalar field description whose evolution is governed by a Klein-Gordon type equation in a flat background in $1+1$ dimensions with the spatial topology of $S^1$, though in the presence of a time-dependent potential. The model is still subject to a homogeneous constraint, which generates $S^1$-translations. Recently, a Fock quantization of this scalar field was introduced and shown to be unique under the requirements of unitarity of the dynamics and invariance under the gauge group of $S^1$-translations. In this work, we extend and complete this uniqueness result by considering other possible scalar field descriptions, resulting from reasonable field reparametrizations of the induced metric of the reduced model. In the reduced phase space, these alternate descriptions can be obtained by means of a time-dependent scaling of the field, the inverse scaling of its canonical momentum, and the possible addition of a time-dependent, linear contribution of the field to this momentum. Demanding again unitarity of the field dynamics and invariance under the gauge group, we prove that the alternate canonical pairs of fieldlike variables admit a Fock representation if and only if the scaling of the field is constant in time. In this case, there exists essentially a unique Fock representation, provided by the quantization constructed by Corichi, Cortez, and Mena Marugán. In particular, our analysis shows that the scalar field description proposed by Pierri does not admit a Fock quantization with the above unitarity and invariance properties.