Lemma: Instantaneous coefficient regularity

The map $t\mapsto k_t$ is smooth in the weighted norm and has compact temporal support. Each ${\Phi }_F(t)=X(k_t)$ is a well-defined essentially self-adjoint field operator on the standard finite-particle core of a quasifree Fock realization.

Proof. The compact-region threshold bound controls the weighted low-frequency integral uniformly on compact time intervals. At high frequency,

Time derivatives introduce finite powers of $\omega$ and derivatives of $f$. Spectral unitarity and the compact spatial smoothness of $f$ give the required bounds by applying sufficiently many powers of $\mathsf{L}$ to those derivatives. Thus all time derivatives exist in the weighted norm.

For clarity, a doubled Fock realization is available. On its one-particle space use the real-linear map

It obeys

The vacuum Weyl characteristic function is the required one. The Segal field $a(Tk)+a(Tk{)}^{†}$ is essentially self-adjoint on finite-particle vectors: the usual creation/annihilation estimates make those vectors analytic for each such field. This realizes $X(k_t)$ and its closure. It does not posit a global stationary density in the physical, undoubled radiation Fock space.