Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

Regulated quadratic generator

Model 3: a specified sourced form and an exterior pairing › Regulated quadratic generator · lines 1288–1317

IV-B@1#eq:B3, IV-B@1#eq:coherent3

Define on $Dom N$

B_{Λ} (J) = d Γ (h_{Λ}) + \frac{1}{2} (a^{*} (g_{Λ}) + a (g_{Λ})) .

The sector estimates already used give

∥ B_{Λ} (J) Ψ ∥ \leq C_{Λ} ∥ (N + 2) Ψ ∥, B_{Λ, M} (J) = Π_{M} B_{Λ} (J) Π_{M} bounded self-adjoint .

Neither $Π_{M}$ nor the bounded momentum cutoff is asserted to have finite rank.

Normal ordering gives the exact coherent identity

⟨ Coh (z), B_{Λ} (J) Coh (z)⟩ = \int J^{μν} (R^{NW} T (φ_{χ_{Λ} z}))_{μν} d^{4} x .

It follows by replacing the normal-ordered field factors by their coherent one-point functions. The graph bound justifies the expectation.

Displayed formulas in this passage

IV-B@1#eq:B3 · line 1296
$∥ B_{Λ} (J) Ψ ∥ \leq C_{Λ} ∥ (N + 2) Ψ ∥, B_{Λ, M} (J) = Π_{M} B_{Λ} (J) Π_{M} bounded self-adjoint .$
IV-B@1#eq:coherent3 · line 1308
$⟨ Coh (z), B_{Λ} (J) Coh (z)⟩ = \int J^{μν} (R^{NW} T (φ_{χ_{Λ} z}))_{μν} d^{4} x .$
unlabeled · line 1291
$B_{Λ} (J) = d Γ (h_{Λ}) + \frac{1}{2} (a^{*} (g_{Λ}) + a (g_{Λ})) .$