Manuscript XVI-H@1 · XVI-H.tex · section · af895076

Source normal form

Construction of the compact three-slot kernel › Source normal form · lines 880–923

XVI-H@1#eq:j-delta, XVI-H@1#eq:j-id, XVI-H@1#eq:c-sym

Choose a global component frame for the input tensor bundle, with indices $I, J$ . The local polynomial structure of the source yields a finite normal form for its polarized kernel:

j_{D}^{A}_{I J} (y; y_{1}, y_{2}) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum c^{A}_{I J; α β} (y) \partial_{y_{1}}^{α} \partial_{y_{2}}^{β} (δ (y_{1} - y) δ (y_{2} - y)) .

The coefficient tensors are smooth and ambient. The source identity is

⟨ j_{D}^{A} (y; \cdot, \cdot), a \otimes c ⟩ = J_{D}^{A} (a, c) (y) .

This is the source-kernel structure attributed to XII-D in the supplied XVI-H-v3. It also follows abstractly from the finite local quadratic-polynomial premise: a monomial $q (y) (\partial^{α} a_{I}) (\partial^{β} c_{J})$ is represented by the corresponding differentiated deltas with coefficient $(- 1)^{∣ α ∣ + ∣ β ∣} q (y)$ .

The normal form may be symmetrized without changing (eq:j-id), so we choose

c^{A}_{I J; α β} = c^{A}_{J I; β α} .

Its coefficient functions can be chosen to vanish for $y^{0} \leq 0$ , since every source contribution contains $σ$ or a positive-order derivative of $σ$ . No coefficient values of the unexpanded $J_{3}$ are fabricated.

Displayed formulas in this passage

XVI-H@1#eq:j-delta · line 886
$j_{D}^{A}_{I J} (y; y_{1}, y_{2}) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum c^{A}_{I J; α β} (y) \partial_{y_{1}}^{α} \partial_{y_{2}}^{β} (δ (y_{1} - y) δ (y_{2} - y)) .$
XVI-H@1#eq:j-id · line 897
$⟨ j_{D}^{A} (y; \cdot, \cdot), a \otimes c ⟩ = J_{D}^{A} (a, c) (y) .$
XVI-H@1#eq:c-sym · line 914
$c^{A}_{I J; α β} = c^{A}_{J I; β α} .$