Proposition: Outgoing packet expectation

The definition (eq:quantumscatter) gives canonical outgoing commutators. Its number covariance is (eq:numberdensity), and $N_h$ has the finite expectation (eq:packetlaw).

Proof. The two incoming channels commute, and unitarity gives

Hence the outgoing commutator is canonical. The input state is a product state with vanishing means and no cross-channel covariance. Lemma lem:horizonoccupation therefore gives

This is (eq:numberdensity). Smearing proves (eq:packetlaw). The integral is finite because ${\Gamma }_{\ell }\le 1$, $\omega \ge {\omega }_{-}>0$ on the support, and $h\in L^2$.