Manuscript XIV-H@1 · XIV-H.tex · proposition · 695c7c3a

Proposition: Exact two-placement kernel formula

H2d: the two internal placements in K 5 › The existing tree maps with z retained · lines 1217–1284 · proof included or linked

XIV-H@1#prop:K5exact, XIV-H@1#eq:K5, XIV-H@1#eq:K5products

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The untested mixed kernel of $B_{5}$ is

K_{5}^{I, J} (a, b, z) = \partial_{b^{μ}} [+ f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z) f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .

Each placement is a finite differential expression, of total kernel derivative order at most four, in the products

R (b - a) R (a - z), R (b - z) R (z - a) .

These products are defined, including at $a = z$ .

Proof. Start with the two terms

\int f (b) R (b - s) \partial_{s^{α}} [σ_{ε} (s) (a_{1}^{α β} [h (s)] \partial_{s^{β}} X_{G}^{μ} (s) + a_{1}^{α β} [G (s)] \partial_{s^{β}} X_{h}^{μ} (s))] \partial_{b^{μ}} ψ (b) d^{4} s d^{4} b .

Transfer the divergence to the incoming retarded kernel and the scalar derivative to the final kernel. The two signs cancel. Insert (eq:L), and use the actual local incidences $s = a$ and $s = z$ . This gives (eq:K5).

For the first product in (eq:K5products), the coordinate map

(a, b, z, y) ⟼ (a, b - a, a - z, y)

is a diffeomorphism onto its image. The product is therefore the pullback of $1 \otimes R \otimes R \otimes 1$ . The second placement uses the analogous diffeomorphism

(a, b, z, y) ⟼ (z, b - z, z - a, y) .

Tensor products of distributions exist, and these pullbacks are defined. Differentiation and multiplication by the fixed smooth weights preserve existence.

Displayed formulas in this passage

XIV-H@1#eq:K5 · line 1220
$K_{5}^{I, J} (a, b, z) = \partial_{b^{μ}} [+ f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z) f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XIV-H@1#eq:K5products · line 1240
$R (b - a) R (a - z), R (b - z) R (z - a) .$
unlabeled · line 1251
$\int f (b) R (b - s) \partial_{s^{α}} [σ_{ε} (s) (a_{1}^{α β} [h (s)] \partial_{s^{β}} X_{G}^{μ} (s) + a_{1}^{α β} [G (s)] \partial_{s^{β}} X_{h}^{μ} (s))] \partial_{b^{μ}} ψ (b) d^{4} s d^{4} b .$
unlabeled · line 1269
$(a, b, z, y) ⟼ (a, b - a, a - z, y)$
unlabeled · line 1277
$(a, b, z, y) ⟼ (z, b - z, z - a, y) .$

Proposition: Exact two-placement kernel formula

Displayed formulas in this passage

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