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.

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

References in corpus (4)

• States of Low Energy on Robertson-Walker Spacetimes
• Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes
• Ultrastatic spacetimes
• Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

Cited by in corpus (6)

• Bosonic and fermionic Gaussian states from Kähler structures
• Stationary Spacetimes and Self-Adjointness in Klein-Gordon Theory
• Entanglement and correlations between local observables in de Sitter spacetime
• Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution
• A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes
• Global Hyperbolicity and Self-adjointness