Linear coordinate two-point pairings

For each fixed pair of coordinate tests $a,a^{\prime }$, the preceding construction also gives well-defined free on-shell pairings

where $t$ is a smooth compactly supported graviton test tensor. These expressions are independent of the time cutoff used to construct the smooth representatives.

Indeed, two such representatives have a smooth compactly supported difference $d$ with $Ed=0$. Then

and $E_{\mathrm{R}}d=E_{\mathrm{A}}d$ is smooth and compactly supported. The bisolution property of ${\mathcal{H}}_{hh}$ annihilates this difference in either argument.

Equations (eq:linear-wh)–(eq:linear-ww) are statements about smooth-test pairings of free linear observables. They are not a licence to multiply the corresponding unsmeared kernels with scalar propagators, or to pull them back to the coincidences required by a composite insertion.