Lemma: Exact time-slice identity and derivative filter

The reduced field observable of $\mathsf{D}({\eta }_T{s}_R)$ is $\mathcal{L}(q_R,p_R)$. The coefficient of $F_{R,T}$ is

The time-slice identity is exact for every $T>0$.

Proof. Pair the test with a homogeneous field solution and integrate the ${\eta }_T^{\prime \prime }$ term once by parts:

The integrated symplectic pairing is constant in time. Since $\int {\eta }_T^{\prime }=1$, it equals $\mathcal{L}(q_R,p_R)$.

Alternatively the advanced and retarded solutions of $\mathsf{D}({\eta }_T{s}_R)$ are $({\eta }_T-1)s_R$ and ${\eta }_T{s}_R$, respectively, confirming the sign convention. In (eq:testcoefficient), applying $-{{\partial }_t}^2$ to the test multiplies its coefficient by ${\omega }^2$. This proves (eq:kR).