Manuscript XIV-J@1 · XIV-J.tex · input · 1c0e72ca

Input statement: XIV-E regrouping and complete root

J0: copied premises and their domains › Earlier analytic handoffs · lines 588–640

XIV-J@1#input:E, XIV-J@1#eq:regroup, XIV-J@1#eq:root, XIV-J@1#eq:zerolocus

For smooth directions,

M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} \partial_{α} (σ_{ε} B^{α μ} [G; h]) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,

where

C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),

B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}], r_{h} = P E_{0}^{R} (J_{D}^{'} [G] h) .

For $F = E_{m}^{A} f$ ,

\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .

These are exact tree-kernel identities before characteristic composition.

On the retained chronological neighborhood,

q_{R} = q_{1} = q_{2} = 0 ⟺ (b, a, y) \in N^{\circ} .

The elementary common product phase was not clean XIVE.

Displayed formulas in this passage

XIV-J@1#eq:regroup · line 591
$M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} \partial_{α} (σ_{ε} B^{α μ} [G; h]) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,$
XIV-J@1#eq:root · line 621
$\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .$
XIV-J@1#eq:zerolocus · line 632
$q_{R} = q_{1} = q_{2} = 0 ⟺ (b, a, y) \in N^{\circ} .$
unlabeled · line 604
$C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),$
unlabeled · line 609
$B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}], r_{h} = P E_{0}^{R} (J_{D}^{'} [G] h) .$