Theorem: Linear Yang–Feldman characterization

The following statements hold modulo ${\kappa }^3$:

1. The test operator is the unique formal series with leading term $1$ satisfying

   $${\mathcal{R}}_q=1-{\mathcal{R}}_{q}L[q]{\mathsf{E}}_{\mathrm{A}}.$$

   Equivalently,

   $${\mathbf{\varphi }}_q(c)=\mathbf{\varphi }(c)-{\mathbf{\varphi }}_q(L[q]{\mathsf{E}}_{\mathrm{A}}c).$$

2. The scalar map in (eq:scalar-map) obeys

   $$(P_m+L[q])r_{q}u=P_{m}u$$

   for arbitrary smooth $u$, and is the unique solution with the same zeroth-order data and zero positive-order data before the interaction.

3. For every compact $a$,

   $${\mathbf{\varphi }}_q(E_g[q]a)=0.$$

4. If $\mathrm{supp}g\cap J_{\eta }^{-}(\mathrm{supp}c)=\varnothing$, then ${\mathbf{\varphi }}_q(c)=\mathbf{\varphi }(c)$.

5. Its algebraic commutator is

   $$[{\mathbf{\varphi }}_q(c),{\mathbf{\varphi }}_q(d)]=i\hslash ⟨c,({{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}})d⟩1.$$

Proof. The formal inverse identity

proves (i). Coefficient comparison fixes the first operator as $-L_{1,g}{\mathsf{E}}_{\mathrm{A}}$ and the second as $L_{1,g}{\mathsf{E}}_{\mathrm{A}}{L}_{1,g}{\mathsf{E}}_{\mathrm{A}}-L_{2,g}{\mathsf{E}}_{\mathrm{A}}$.

The scalar-side recursion is

Applying $P_m$ gives (ii). Conversely, its positive-order equations are ordinary free inhomogeneous Cauchy problems with smooth compact sources, so free Cauchy uniqueness determines them.

For compact $a$, ${\mathsf{E}}_{\mathrm{A}}{P}_{m}a=a$, and hence

Equation (eq:free) proves (iii). The support condition in (iv) makes $L[q]{\mathsf{E}}_{\mathrm{A}}c=0$, proving that assertion.

For (v), formal self-adjointness gives

Using the inherited free commutator, its kernel between the original tests is

The advanced contribution is

because ${\mathsf{E}}_{\mathrm{A}}(1-L[q]{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}})={{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}$. The retarded contribution is

because $(1-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}L[q]){\mathsf{E}}_{\mathrm{R}}={{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}$. Their difference is ${{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}$. Every composition is well defined on the stated compact tests.