  We discuss the canonical quantization of non-unitary time evolution in
inflating Universe. We consider gravitational wave modes in the FRW metrics in
a de Sitter phase and show that the vacuum is a two-mode SU(1,1) squeezed state
of thermo field dynamics, thus exhibiting the link between inflationary
evolution and thermal properties. In particular we discuss the entropy and the
free energy of the system. The state space splits into many unitarily
inequivalent representations of the canonical commutation relations
parametrized by time $t$ and non-unitary time evolution is described as a
trajectory in the space of the representations: the system evolves in time by
running over unitarily inequivalent representations. The generator of time
evolution is related to the entropy operator. A central ingredient in our
discussion is the doubling of the degrees of freedom which turns out to be the
bridge to the unified picture of non-unitary time evolution, squeezing and
thermal properties in inflating metrics.
