  In this article, we formulate the study of the unitary time evolution of
systems consisting of an infinite number of uncoupled time-dependent harmonic
oscillators in mathematically rigorous terms. We base this analysis on the
theory of a single one-dimensional time-dependent oscillator, for which we
first summarize some basic results concerning the unitary implementability of
the dynamics. This is done by employing techniques different from those used so
far to derive the Feynman propagator. In particular, we calculate the
transition amplitudes for the usual harmonic oscillator eigenstates and define
suitable semiclassical states for some physically relevant models. We then
explore the possible extension of this study to infinite dimensional dynamical
systems. Specifically, we construct Schroedinger functional representations in
terms of appropriate probability spaces, analyze the unitarity of the time
evolution, and probe the existence of semiclassical states for a wide range of
physical systems, particularly, the well-known Minkowskian free scalar fields
and Gowdy cosmological models.
