Theorem: Exact induced Weyl effect

The local scattering morphism obeys

For a centered quasifree probe state with variance convention

the induced effect is

Its expectation in every system state is

The induced system test is compactly supported in $K$.

Proof. For future-supported free test vectors $H$, comparison of advanced coupled and uncoupled solutions gives the test transformation

Equivalently, pair the coupled solution with its advanced test solution and use Green's identity in an early Cauchy region, where both theories agree. The test classes are those modulo the respective wave operators.

The transformation preserves the symplectic form: compare the conserved coupled symplectic current on past and future Cauchy surfaces outside $K$. It therefore gives the Weyl scattering morphism ${\Theta }_{ϵ}(W(H))=W({\theta }_{ϵ}H)$. This is the local linear scattering construction stated in Section sec:scope.

Applying it to $H=(0,g/ϵ)$ yields (eq:exactscatteredtests). System and probe Weyl operators belong to separate tensor factors, so no additional mixed phase appears. Apply $\mathrm{id}\otimes \sigma$, then take the real part. Since $c_{ϵ}$ is real and nonnegative, this gives (eq:inducedeffect). Multiplication by $\rho$ gives the support assertion.