Manuscript Oai282-B@1 · Oai282-B.tex · section · f2956de5

The quadratic kernel

Physical membership of the improvement scalar › The quadratic kernel · lines 498–518

Oai282-B@1#eq:Qfock

Let $Q = : ϕ^{2} :$ . For real $f \in S (M)$ , its quadratic Fock decomposition is

Q (f) = ℏ (a^{*} a^{*} (F_{f}) + aa (F_{f}) + 2 d Γ (B_{f})),

where, with the kernel convention of Section sec:model,

F_{f} (p, q) = f (p + q), B_{f} (p, q) = f (p - q) .

The pair kernel is symmetric, and $B_{f}$ is self-adjoint for real $f$ .

Displayed formulas in this passage

Oai282-B@1#eq:Qfock · line 502
$Q (f) = ℏ (a^{*} a^{*} (F_{f}) + aa (F_{f}) + 2 d Γ (B_{f})),$
unlabeled · line 511
$F_{f} (p, q) = f (p + q), B_{f} (p, q) = f (p - q) .$