Required inverses and contraction data

A Q1 completion must supply a specified realization $\tilde{P}$ of the coupled gauge-fixed operator, its domain, and distributions

with the actual left and right inverse, support, normalization, and field-equation identities. The graded ghost and auxiliary blocks must be included.

A Q2 completion must additionally specify a compatible two-point matrix or subtraction parametrix. In a bosonic physical sector, a state-dependent two-point function $W$ must satisfy the appropriate field equations, Hermiticity, positivity, and

under its declared component pairing; factors of $\hslash$ are then restored in the field expectation.

The Hadamard wavefront condition uses the positive-frequency null relation

Its scalar characterization and appropriate vector-valued extensions have precise hypotheses Radzikowski,SV. They are not claims of positivity for the entire ghost-extended field space.