The free scalar carrier and unbounded moments

The parameter $ϵ$ is a dimensionless effective CCR parameter relative to a fixed reference unit of action. It is neither $\kappa$ nor a smoothing width. The classical symbol is fixed while the representation section in (eq:O) varies.

of unbounded traces. If $\rho ={\sum }_{\ell }{p}_{\ell }|{\psi }_{\ell }⟩⟨{\psi }_{\ell }|$ has finite number expectation, its positively weighted vectors with $p_{\ell }>0$ lie in $\mathrm{Dom}{N}^{1/2}$. We define

The first moment is the absolutely convergent associated sum. These are spectral moments, independent of the decomposition. Notation such as $\mathrm{Tr}(\rho \Phi (v{)}^2)$ below means this quadratic moment; it does not assume that the formal product $\rho \Phi (v{)}^2$ is an already-defined trace-class operator.

Lemma lem:field implies