Complex structures for Klein–Gordon theory on globally hyperbolic spacetimes

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.