Abstract
We discuss the quantization of two-Killing symmetry reductions of general relativity with cosmological interpretation. In particular we will focus on the obtention of unitary evolution operators in closed form for the class of time-dependent quadratic Hamiltonians that appear in this context. This will be done by extending to field theoretical models the techniques developed for the exact quantization of the harmonic oscillator with time-dependent frequency. The abstract operators that we will derive are useful both to discuss different field representations and the problem of unitary implementation of dynamics. Finally, we consider in some detail the applications of the formalism developed to the Gowdy and Schmidt cosmological models.
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