Theorem: Two-leg root identity

The root is smooth, linear in $\varphi$, and has support in a fixed causal compact set determined by $f$ and the specified coefficients. It satisfies

For

choose a smooth time collar $\vartheta$ equal to one on the nonnegative-time causal support relevant to the source, and zero sufficiently far in the past. Then

is compactly supported and

The insertion is collar independent. The effective roots ${\sigma }_{\epsilon }{a}_{f,g}$, ${\sigma }_{\epsilon }^{\prime }{a}_{f,g}$, and ${\sigma }_{\epsilon }^{\prime \prime }{a}_{f,g}$ are supported in nonnegative time.

Proof. Transpose the retarded Green operators and the finite differential operators in (eq:Aone). This gives (eq:alpha) and the first identity. Advanced support, compact $f$, and the coefficient support restrictions give a common compact causal support for $\alpha$. All operations are continuous on the corresponding fixed smooth support stages.

The advanced image $\beta$ may extend to the past. A lower time collar makes it compact because the advanced causal past of the fixed compact set, truncated below in time, is compact. Transposing $u_{\mathrm{D}}=-\mathcal{P}{E_0}^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$ gives ${\mathcal{C}}_{3,\mathrm{D}}=\int \beta \cdot {\mathcal{J}}_{\mathrm{D}}$. The source vanishes in negative time and the advanced root vanishes above the observation, so multiplication by the chosen collar does not change this pairing. Ramp factors and their derivatives vanish at nonpositive times.