C2 and C3: maps, transposition, signs, and polarization

The advanced operator in these formulas is the transpose of retarded propagation of the scalar. Thus the terminology “retarded map” is consistent with backward causal support of its test.

The inner test ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is an actual smooth compact test. Iteration may be written as applying the first-map rule to $F_{-{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}$, but it must not be written as applying that rule to an unspecified representative of its on-shell class. Equation (eq:first-map) explains why: changing an insertion by $F_{P_{m}v}$ can change its response.