Manuscript XVI-J@1 · XVI-J.tex · proposition · 6c1f306d

Proposition: The numerical kernel

Ambient propagator and the punctured numerical square › Off-origin product · lines 506–534 · proof included or linked

XVI-J@1#prop:q, XVI-J@1#eq:q

The distribution

q^{\circ} (z) = S (z)^{2} = d (z)^{2}, z \neq = 0,

exists on $R^{4} ∖ {0}$ , including nonzero null points. It is homogeneous of degree $- 4$ , has scaling degree four at the origin, and is nonzero.

Proof. The single conormal orientation just described satisfies Hörmander's product criterion. At non-null points both factors are smooth. The product is therefore defined on the entire punctured domain.

The massless $d$ is homogeneous of degree $- 2$ , so its defined punctured square is homogeneous of degree $- 4$ . It is not zero: at a spacelike point,

q^{\circ} (z) = \frac{1}{16 π ^{4} ( ∣ z ∣ ^{2} - ( z ^{0} ) ^{2} ) ^{2}} .

A nonzero homogeneous distribution of degree $- 4$ has scaling degree four.

Displayed formulas in this passage

XVI-J@1#eq:q · line 509
$q^{\circ} (z) = S (z)^{2} = d (z)^{2}, z \neq = 0,$
unlabeled · line 528
$q^{\circ} (z) = \frac{1}{16 π ^{4} ( ∣ z ∣ ^{2} - ( z ^{0} ) ^{2} ) ^{2}} .$

Proposition: The numerical kernel

Displayed formulas in this passage

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