Fock conventions, domains, and number moments

Let $\mathfrak{h}$ be a separable complex Hilbert space whose inner product is linear in the second argument. Let

On the finite-particle domain ${\mathcal{D}}_{\mathrm{fin}}$, use

The map $v\mapsto \Phi (v)$ is real linear. Define

These conventions give

Standard Fock and Weyl constructions are discussed in Derezinski; the domain argument used below invokes the analytic-vector theorem Nelson.

A normal state is represented by a positive trace-class operator $\rho$ with $\mathrm{Tr}\rho =1$. Its residual number moment is

For an unbounded self-adjoint $B$, $\mathrm{Tr}(\rho B^2)$ means the spectral quadratic moment, which may be finite even when an operator product is not trace class in a naive sense.