Quadratic operators and the free-term correction

Identify the product Fock space with ${\Gamma }_{\mathrm{s}}({\mathfrak{h}}_g\oplus {\mathfrak{h}}_{\phi })$. Put

and let $G_R$ be the symmetric tensor obtained by placing $P_{2,R}$ in the two cross-species slots and adding its flip. Thus $\Vert G_R\Vert =\sqrt{2}{\Vert P_{2,R}\Vert }_{\mathrm{HS}}$. Define the symmetric quadratic

With $\mathsf{N}={\mathsf{N}}_g+{\mathsf{N}}_{\phi }$ and ${\Pi }_M=1_{\{\mathsf{N}\le M\}}$, sector estimates give

For completeness, on the $n$-particle sector, $\mathrm{d}\Gamma (T)$ has norm at most $n\Vert T\Vert$, while pair annihilation has norm at most $\sqrt{n(n-1)}\Vert G\Vert$. Pair creation is its adjoint. Summing the orthogonal output sectors proves (eq:B2graph); compression and $\Vert \mathrm{Re}C\Vert \le \Vert C\Vert$ prove (eq:B2compression).

Define

Each generator is bounded self-adjoint. Its free scalar term is compressed by ${\mathsf{N}}_{\phi }$, not by total number.