Complex Structures for Klein-Gordon Theory on Globally Hyperbolic 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.
Cite
@article{arxiv.1812.00926,
title = {Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes},
author = {Albert Much and Robert Oeckl},
journal= {arXiv preprint arXiv:1812.00926},
year = {2022}
}
Comments
32 pages; v2: Section 5 expanded with new results, minor corrections; v3: restructure of Section 5 with further new results, minor improvements; v4: corrected and streamlined proof of conservation equations, added result on Gelfand-Dikki equation; v5: final corrections