Manuscript Em01-F@3 · Em01-F-v3.tex · proposition · 225e59af

Proposition: Smooth-slot existence and regrouping

The independently defined marked insertion › Marked observable · lines 1885–1942 · proof included or linked

Em01-F@3#prop:regroup-internal, Em01-F@3#eq:combinedB, Em01-F@3#eq:internalregroup

The insertion is a continuous trilinear functional on fixed compact smooth input stages.

Let

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} + D_{dens}^{α μ} (G, h) + a_{1}^{α μ} [r [h, G]] .

Then, exactly,

B_{f, g} (h, G, ψ) = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} b + \int f (b) [R * \partial_{α} B^{α μ} [G; h]] (b) \partial_{μ} ψ (b) d^{4} b + \int g (b) [a_{1}^{μν} [r [h, G]] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r [h, G]] F ψ] (b) d^{4} b .

Proof. All compact-source retarded responses are smooth. The local products involving $h, G$ are compact smooth sources, while the nested response is well defined by causal properness. Final integrations are compact because of $f, g$ . Finite-order differential formulas and compact-local Green estimates prove continuity.

By definition,

- X_{r [h, G]}^{μ} = R * \partial_{α} a_{1}^{α μ} [r [h, G]] .

Linearity of retarded propagation combines the second, third, and fourth lines of (eq:insertion), giving (eq:internalregroup). No crossed-line composition or asymptotic approximation is used.

Displayed formulas in this passage

Em01-F@3#eq:combinedB · line 1897
$B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + D_{dens}^{α μ} (G, h) + a_{1}^{α μ} [r [h, G]] .$
Em01-F@3#eq:internalregroup · line 1907
$B_{f, g} (h, G, ψ) = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} b + \int f (b) [R * \partial_{α} B^{α μ} [G; h]] (b) \partial_{μ} ψ (b) d^{4} b + \int g (b) [a_{1}^{μν} [r [h, G]] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r [h, G]] F ψ] (b) d^{4} b .$
unlabeled · line 1891
$C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),$
unlabeled · line 1933
$- X_{r [h, G]}^{μ} = R * \partial_{α} a_{1}^{α μ} [r [h, G]] .$

Proposition: Smooth-slot existence and regrouping

Displayed formulas in this passage

Nearby in this section