A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes

TL;DR

This paper constructs a canonical complex structure and a bosonic signature operator for scalar fields in globally hyperbolic spacetimes, providing a new framework for analyzing solutions of the Klein-Gordon equation.

Contribution

It introduces a canonical decomposition of the solution space and a complex structure based on the bosonic mass oscillation property, applicable to both massive and massless fields.

Findings

Derived a canonical solution space decomposition independent of observers.

Established a bosonic signature operator using mass oscillation properties.

Illustrated the framework with Minkowski and ultrastatic spacetime examples.

Abstract

The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.