Quantum Gowdy T3 model:: a uniqueness result

Modulo a homogeneous degree of freedom and a global constraint, the linearly polarized Gowdy T-3 cosmologies are equivalent to a free scalar field propagating in a fixed nonstationary background. Recently, a new field parametrization was proposed for the metric of the Gowdy spacetimes such that the associated scalar field evolves in a flat background in (1+1) dimensions with the spatial topology of S-1, although subject to a time-dependent potential. Introducing a suitable Fock quantization for this scalar field, a quantum theory was constructed for the Gowdy model in which the dynamics is implemented as a unitary transformation. A question that was left open is whether one might adopt a different, nonequivalent Fock representation by selecting a distinct complex structure. The present work proves that the chosen Fock quantization is in fact unique (up to unitary equivalence) if one demands unitary implementation of the dynamics and invariance under the group of S-1 translations. These translations are precisely those generated by the global constraint that remains on the Gowdy model. It is also shown that the proof of uniqueness in the choice of the complex structure can be applied to more general field dynamics than that corresponding to the Gowdy cosmologies.