Proposition: A nonzero classical target

Choose

Let $a\in {C_c}^{\infty }((t_1,t_2))$ and $F\in {C_c}^{\infty }((t_3,t_4))$ be nonnegative and nonzero. Let $\varrho \in {C_c}^{\infty }({\mathbb{R}}^3)$ be nonnegative with integral one, and set

Choose the compact switch equal to one wherever this coefficient contributes to the causal dependence region of $f$.

Choose compact smooth scalar Cauchy data equal to $(1,0)$ on a ball containing that dependence region at $t=0$. Their amplitude $z$ belongs to $\mathcal{S}$, and

Consequently $v(f,h,\chi )\ne 0$.

Proof. Finite propagation gives ${\varphi }_z=\cos (mt)$ throughout the observation dependence region. Direct calculation yields

On the free scalar shell there,

Hence

Subtracting $w[h{]}^0{\partial }_t{\varphi }_z$, integrating by parts in $s$, and using

gives the integrand in (eq:nonzero). Integration against $f$ proves the formula.

On the supports of $a(s)$ and $F(t)$, both trigonometric factors are strictly positive. Thus the value is negative. The symbol formula (eq:classical-symbol) excludes $v=0$.

Finally, compact smooth real Cauchy data have amplitude

For $m>0$ the displayed multipliers preserve Schwartz decay.