# Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes

## TL;DR

This paper introduces a rigorous method to parametrize and construct complex structures for Klein-Gordon theory on globally hyperbolic spacetimes, ensuring unitary evolution and natural quantizations.

## Contribution

It provides a systematic approach using operator differential equations to define complex structures that are conserved under evolution and applicable to various spacetime geometries.

## Key findings

- Constructed complex structures for Klein-Gordon theory on specific spacetimes.
- Proved the differential equation for complex structures is given by the Gelfand-Dikki equation.
- Applied the method to static, expanding, and Friedmann-Robertson-Walker spacetimes.

## Abstract

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.