Manuscript Em02-E@1 · Em02-E-v1.tex · proposition · f3795050

Proposition: A diagonal correction with a non-diagonal image

Transport of internal counterterms · lines 1274–1314 · proof included or linked

Em02-E@1#prop:image, Em02-E@1#eq:image

Let $c \in C_{c}^{\infty} (R^{3})$ be nonzero and define

C_{c} (y, z) = c (y) δ^{(3)} (y - z) .

Applying a spatial Green kernel in the first slot gives the well-defined distribution

Π C_{c} (x, z) = G_{Δ} (x - z) c (z) .

For every $z$ where $c (z) \neq = 0$ , it is nonzero on open sets with $x \neq = z$ . Its support is therefore not contained in the output diagonal.

Proof. The right side is locally integrable because $∣ x - z ∣^{- 1}$ is locally integrable in three relative spatial dimensions. For a compact smooth test $Φ (x, z)$ , its action is

\int c (z) [\int G_{Δ} (x - z) Φ (x, z) d^{3} x] d^{3} z .

This is also the distributional evaluation of

\int G_{Δ} (x - y) c (y) δ^{(3)} (y - z) d^{3} y .

At separated points the kernel is smooth and nonzero, proving the support assertion.

Displayed formulas in this passage

Em02-E@1#eq:image · line 1284
$Π C_{c} (x, z) = G_{Δ} (x - z) c (z) .$
unlabeled · line 1277
$C_{c} (y, z) = c (y) δ^{(3)} (y - z) .$
unlabeled · line 1300
$\int c (z) [\int G_{Δ} (x - z) Φ (x, z) d^{3} x] d^{3} z .$
unlabeled · line 1307
$\int G_{Δ} (x - y) c (y) δ^{(3)} (y - z) d^{3} y .$

Proposition: A diagonal correction with a non-diagonal image

Displayed formulas in this passage

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