Theorem: Switched bilinear and polynomial closure

Suppose an initial time $t_{\mathrm{in}}$ precedes the supports of $g$ and $f$. Quantizing the classical bilinear

by replacing $q,\pi$ with $Q,\Pi$ gives exactly ${\mathcal{R}}_{g,f}$ on ${\mathcal{D}}_{\mathrm{mix}}$.

Writing $N=N_g+N_{\varphi }$, there is $C<\infty$ with

Both ${\mathcal{R}}_{g,f}$ and, for every real $\lambda$,

are essentially self-adjoint on ${\mathcal{D}}_{\mathrm{mix}}$.

Proof. On free scalar matrix elements, ${\mathsf{Q}}_{ga}\varphi =L_{ga}\varphi$. The source is zero before $t_{\mathrm{in}}$, so

Smearing proves the switched identity, including derivatives of $g$.

Creation-annihilation estimates give

The two factors commute and

This proves the bound. Schwartz mode vectors and finite particle shifts prove invariance of the common core. Reality proves symmetry.

Each polynomial changes total particle number by at most two. For a vector supported in sectors up to $M$,

After division by $k!$, the resulting analytic-vector series converges for a positive range of its parameter. The core is dense, so Nelson's theorem Nelson proves essential self-adjointness.