Lemma: Self-adjoint fields and the number estimate

For every $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its closure, denoted by the same symbol, satisfies

and the quadratic-form inequality

Proof. The creation–annihilation bounds are

Using $\Vert x+y{\Vert }^2\le 2\Vert x{\Vert }^2+2\Vert y{\Vert }^2$ proves (eq:number-bound) on ${\mathcal{D}}_{\mathrm{fin}}$.

For a vector supported in particle sectors at most $r$,

Its analytic-vector series converges. The analytic-vector criterion Nelson gives essential self-adjointness. Particle truncation converges in the graph norm of $N^{1/2}$; the displayed bound and closedness then extend the estimate and domain inclusion.