Manuscript XVI-H@1 · XVI-H.tex · section · af895076

Diagonal response and completion

Setting and inherited interfaces › Diagonal response and completion · lines 422–461

XVI-H@1#eq:CD, XVI-H@1#eq:Aone

The stated diagonal completion is

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

It is not defined by independently choosing a source pairing.

Its displayed coefficient maps are

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,

and its two retarded legs are

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

The divergence retains derivatives of $g$ , and the mass term is retained. No factor $g$ is inserted into the coordinate forcing.

Displayed formulas in this passage

XVI-H@1#eq:CD · line 425
$C_{3, D} (f, g) [γ, ϕ] = A_{1} [f, g] (u_{D} [γ]; ϕ) .$
XVI-H@1#eq:Aone · line 440
$X_{1}^{μ} [r] V_{1, g} [r] ϕ ϕ_{1} [r; ϕ] A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{α} (σ a_{1}^{α μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} (V_{1, g} [r] ϕ), = \int_{M_{T}} f (ϕ_{1} [r; ϕ] - X_{1}^{μ} [r] \partial_{μ} ϕ) d^{4} x .$
unlabeled · line 433
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,$