Two probes with complete coverage inside that class

Choose a nonnegative $f\in {C_c}^{\infty }(M)$, with $\int f=1$, supported in a small double cone. Choose a smooth free solution $u$ equal to $x^0$ on a neighborhood of that cone, with smooth compactly supported Cauchy data.

Such a solution is obtained by taking zero initial field and a smooth velocity cutoff equal to one on a sufficiently large ball. Finite propagation speed makes it equal to $x^0$ on the desired cone. Its data have a finite massless one-particle norm, so its Weyl displacement defines a coherent state ${\omega }_u$.

For vacuum-normal-ordered fields, coherent displacement gives

This follows by shifting the creation and annihilation operators by their displacement constants.

Define