Manuscript XV-A@1 · XV-A.tex · section · f53f3515

The two internal placements

Internal kernels are not the joint quantum operation › The two internal placements · lines 874–910

XV-A@1#eq:K5, XV-A@1#eq:internalproducts

XIV-H gives the copied formula

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)] .

Here $R$ , $L$ , and $A$ are the inherited retarded kernel, coordinate-response kernel, and tensor map. Their complete definitions are not recreated by this audit.

The source proposition says that each placement is a finite differential expression, of total kernel derivative order at most four, in

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

That derivative-order statement is a copied input Hkernel.

The elementary existence mechanism for the displayed products can be checked independently.

Displayed formulas in this passage

XV-A@1#eq:K5 · line 877
$K_{5}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z) + f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XV-A@1#eq:internalproducts · line 901
$R (b - a) R (a - z), R (b - z) R (z - a) .$

The two internal placements

Displayed formulas in this passage

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