Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Completion formulas and the detailed domain check

Completion formulas and the detailed domain check · lines 938–1008

XVI-D@1#eq:coefficients, XVI-D@1#eq:completion-legs, XVI-D@1#eq:Aone, XVI-D@1#eq:CD

Fix

m \geq 0, f \in C_{c}^{\infty} (M_{T}^{\circ}; R), g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]),

and let $ϕ$ be a real smooth scalar. No scalar free equation is required for the existence checks in this section.

The supplied completion coefficients are

a^{μν, ab} a_{1}^{μν} [r] b_{1} [r] = \frac{1}{2} η^{μν} η^{ab} - η^{μ (a} η^{b) ν}, = a^{μν, ab} r_{ab} = - (P r)^{μν}, = \frac{1}{2} η^{ab} r_{ab} .

The equality with trace reversal is a pointwise calculation.

The inherited legs are written

X_{1}^{μ} [r] V_{1, g} [r] ϕ ϕ_{1} [r; ϕ] = - E_{0}^{R} \partial_{α} (σ a_{1}^{α μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} (V_{1, g} [r] ϕ) .

These are inherited definitions on a domain where their operator applications are justified XIID.

The corresponding functional is

A_{1} [f, g] (r; ϕ) = \int_{M_{T}} f (x) (ϕ_{1} [r; ϕ] (x) - X_{1}^{μ} [r] (x) \partial_{μ} ϕ (x)) d^{4} x .

The diagonal completed insertion is

C_{3, D} (f, g) [γ, ϕ] = A_{1} [f, g] (u_{D} [γ]; ϕ)

on its inherited domain.

A direct polarized-response reading would be

C_{D} (f, g; γ_{1}, γ_{2}, ϕ) = A_{1} [f, g] (U_{D} [γ_{1}, γ_{2}]; ϕ),

but only after both legs have been justified on that response. Its witness value is denoted $A_{D} (λ; f, g, ϕ)$ . This is not a definition of a compact class-kernel pairing.

Displayed formulas in this passage

XVI-D@1#eq:coefficients · line 951
$a^{μν, ab} a_{1}^{μν} [r] b_{1} [r] = \frac{1}{2} η^{μν} η^{ab} - η^{μ (a} η^{b) ν}, = a^{μν, ab} r_{ab} = - (P r)^{μν}, = \frac{1}{2} η^{ab} r_{ab} .$
XVI-D@1#eq:completion-legs · line 966
$X_{1}^{μ} [r] V_{1, g} [r] ϕ ϕ_{1} [r; ϕ] = - E_{0}^{R} \partial_{α} (σ a_{1}^{α μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} (V_{1, g} [r] ϕ) .$
XVI-D@1#eq:Aone · line 984
$A_{1} [f, g] (r; ϕ) = \int_{M_{T}} f (x) (ϕ_{1} [r; ϕ] (x) - X_{1}^{μ} [r] (x) \partial_{μ} ϕ (x)) d^{4} x .$
XVI-D@1#eq:CD · line 992
$C_{3, D} (f, g) [γ, ϕ] = A_{1} [f, g] (u_{D} [γ]; ϕ)$
unlabeled · line 941
$m \geq 0, f \in C_{c}^{\infty} (M_{T}^{\circ}; R), g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]),$
unlabeled · line 1000
$C_{D} (f, g; γ_{1}, γ_{2}, ϕ) = A_{1} [f, g] (U_{D} [γ_{1}, γ_{2}]; ϕ),$