Lemma: Domains and self-adjointness

For every $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on the finite-particle domain ${\mathcal{D}}_{\mathrm{fin}}$. Coherent vectors have finite moments of every order of its self-adjoint closure.

Proof. On vectors with at most $r$ particles, the creation and annihilation estimates give

The analytic-vector series converges for every fixed sufficiently small positive argument, indeed for every argument. The dense finite-particle domain therefore consists of analytic vectors. The analytic-vector criterion gives essential self-adjointness Nelson.

For completeness, if

the exponential-vector expansion yields the unitary displacement identity

The vacuum has finite moments of every order, by the same finite-particle estimates. Applying (eq:displacement) to the vacuum proves the coherent-vector assertion.