Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

The translation distribution

The complete contraction as a translation distribution › The translation distribution · lines 1159–1185

Em01-C@3#eq:Omega

Let

d (x, v) = x - v, d : R^{8} \times R^{8} ⟶ R^{8},

and define

Ω_{Φ} = I, μ, ν \sum d_{*} (Λ_{I, μν} [Φ] \otimes χ C_{I}^{μν}) .

All factors are compactly supported. Thus $Ω_{Φ}$ is a compact distribution formed by defined tensor products and pushforward, not by an undeclared coincident-point product.

A compact distribution acts on smooth functions by insertion of a cutoff equal to one near its support. This makes the pairing with $ρ_{δ} (x - v)$ precise even though that function need not be compactly supported on $R^{16}$ .

Displayed formulas in this passage

Em01-C@3#eq:Omega · line 1167
$Ω_{Φ} = I, μ, ν \sum d_{*} (Λ_{I, μν} [Φ] \otimes χ C_{I}^{μν}) .$
unlabeled · line 1162
$d (x, v) = x - v, d : R^{8} \times R^{8} ⟶ R^{8},$

The translation distribution

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