Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

A finite-width algebra-level coefficient

Interactions, normalization, and a controlled diagnostic › A finite-width algebra-level coefficient · lines 1970–1992

VII-I@1#eq:finite-coefficient

Let $h (a)$ and $ϕ (u)$ be compact smooth linear fields. For the density-prepared first dressing,

B_{f, ϵ}^{D} = - \int f w_{ϵ}^{μ} [h] \partial_{μ} ϕ .

The free Gaussian coefficient functional $ω_{H} (F) = F (0)$ gives the exact finite expression

ω_{H} (B_{f, ϵ}^{D} ⋆_{H} h (a) ⋆_{H} ϕ (u)) = - ℏ^{2} \int f w_{ϵ}^{μ} [H_{hh} a] \partial_{μ} (H_{ϕϕ} u) d^{4} x .

There is one allowed complete contraction: the metric slot meets $h (a)$ , and the scalar slot meets $ϕ (u)$ .

Displayed formulas in this passage

VII-I@1#eq:finite-coefficient · line 1980
$ω_{H} (B_{f, ϵ}^{D} ⋆_{H} h (a) ⋆_{H} ϕ (u)) = - ℏ^{2} \int f w_{ϵ}^{μ} [H_{hh} a] \partial_{μ} (H_{ϕϕ} u) d^{4} x .$
unlabeled · line 1974
$B_{f, ϵ}^{D} = - \int f w_{ϵ}^{μ} [h] \partial_{μ} ϕ .$