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

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.