Theorem: External first retarded coefficient

For real compact $g,f$, and real external $h\in {\mathcal{V}}_{\mathrm{fin}}$,

The test $q_{g,h,f}$ is real and compact smooth. The unsmeared coefficient is

On the TT plane,

Proof. In the subtraction (eq:Rdef), the normal three-scalar block cancels exactly. The contraction difference is $i{G}_R(x-y)$; the additional factor $i$ in the definition produces $-G_R(x-y)$. Contracting with $g{h}^{\mu \nu }/2$ and integrating by parts in the compact vertex variable gives $-G_R{\mathsf{Q}}_{gh}\varphi$.

Smearing with $f$ gives

using formal self-adjointness and compact coefficients. This proves the sign and compact-test formula. Applying $P$ gives the equation in (eq:YangFeldman); (eq:QL) uses $P\varphi =0$.