Existence of advanced and retarded Green operators

Statement

Every normally hyperbolic operator on a vector bundle over a globally hyperbolic spacetime has unique future and past Green operators $G^+$ and $G^-$ .

Operator identities and propagator

For each compactly supported source $f$ , the sections $G^+f$ and $G^-f$ solve $Pu=f$ with supports in $J^+(\operatorname{supp}f)$ and $J^-(\operatorname{supp}f)$ , respectively. The uniqueness is as Green operators satisfying both-sided identities on compactly supported sections; the one-sided equation and support condition alone do not state operator-level uniqueness. Their difference

G=G^+-G^-

solves the homogeneous equation after applying $P$ and is called the causal propagator.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Corollary 3.4.3.