  We develop a rigorous method to parametrize complex structures for
Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a
completeness condition. The complex structures are conserved under
time-evolution and implement unitary quantizations. They can be interpreted as
corresponding to global choices of vacuum. The main ingredient in our
construction is a system of operator differential equations. We provide a
number of theorems ensuring that all ingredients and steps in the construction
are well-defined. We apply the method to exhibit natural quantizations for
certain classes of globally hyperbolic spacetimes. In particular, we consider
static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a
huge class of spacetimes we prove that the differential equation for the
complex structure is given by the Gelfand-Dikki equation.
