Manuscript XVI-I@1 · XVI-I.tex · proposition · a7046a8d

Proposition: Two differentiated edges

An ambient propagator in the selected normalization › Wavefront relation and punctured products · lines 783–812 · proof included or linked

XVI-I@1#prop:product, XVI-I@1#eq:p

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For arbitrary multiindices $μ, ν$ , the product

p_{μν}^{\circ} (r) = (\partial^{μ} d) (r) (\partial^{ν} d) (r)

exists on $M ∖ {0}$ . Its pullback to

Ω = {(y, w) \in M^{2} : y \neq = w}

exists under $r = y - w$ . Finite tensor contractions and smooth multipliers preserve definedness.

Proof. Differentiation does not enlarge the wavefront set. At each nonzero null point both factors have covectors on the same open half-ray in (eq:WFd). Two such nonzero covectors cannot be negatives of each other. The distributional product criterion therefore applies. The difference map $(y, w) \mapsto y - w$ is a submersion, so its pullback exists.

The tensor matrices are constant, and contraction is a finite component sum. Smooth multiplication is always defined.

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XVI-I@1#eq:p · line 786
$p_{μν}^{\circ} (r) = (\partial^{μ} d) (r) (\partial^{ν} d) (r)$
unlabeled · line 793
$Ω = {(y, w) \in M^{2} : y \neq = w}$

Proposition: Two differentiated edges

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