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The bosonic signature operator is defined for Klein-Gordon fields andmassless scalar fields on globally hyperbolic Lorentzian manifolds of infinitelifetime. The construction is based on an analysis of families of solutions ofthe Klein-Gordon equation with a varying mass parameter. It makes use of theso-called bosonic mass oscillation property which states that integrating overthe mass parameter generates decay of the field at infinity. We derive acanonical decomposition of the solution space of the Klein-Gordon equation intotwo subspaces, independent of observers or the choice of coordinates. Thisdecomposition endows the solution space with a canonical complex structure. Italso gives rise to a distinguished quasi-free state. Taking a suitable limitwhere the mass tends to zero, we obtain corresponding results for masslessfields. Our constructions and results are illustrated in the examples ofMinkowski space and ultrastatic spacetimes.

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