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

Lemma: Exact transposed vertex

The complete contraction as a translation distribution › Transposition of the complete vertex · lines 1121–1157 · proof included or linked

Em01-C@3#lem:transpose, Em01-C@3#eq:transpose, Em01-C@3#eq:Lambda-order

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For every smooth symmetric array $H$ ,

⟨ i V (K, H), Φ ⟩ = I, μ, ν \sum ⟨ Λ_{I, μν} [Φ], H_{I}^{μν} ⟩ .

If $K$ has local order at most four, then for $supp Φ \subset L ⋐ U$ ,

∣ Λ_{I, μν} [Φ] (ψ) ∣ \leq C_{L} ∥Φ ∥_{5} ∥ ψ ∥_{5}

on a fixed compact neighborhood of its relative support.

Proof. The kinetic part of the test action is

- i I, μ, ν \sum a_{I, ν} [Φ] (\partial_{2, μ} H_{I}^{μν}) .

Transposition gives $i \partial_{2, μ} a_{I, ν}$ . Symmetry of $H$ gives the symmetrization in (eq:Lambda). The trace-kinetic coefficient $i /2$ becomes $- i /2$ , while the mass coefficient remains $- i m^{2} /2$ . This proves the identity.

The order-four estimate acts on a product of $g$ , a derivative of $Φ$ , and a pulled-back relative test. It uses at most five derivatives of $Φ$ . The additional relative derivative uses at most five derivatives of $ψ$ .

Displayed formulas in this passage

Em01-C@3#eq:transpose · line 1124
$⟨ i V (K, H), Φ ⟩ = I, μ, ν \sum ⟨ Λ_{I, μν} [Φ], H_{I}^{μν} ⟩ .$
Em01-C@3#eq:Lambda-order · line 1133
$∣ Λ_{I, μν} [Φ] (ψ) ∣ \leq C_{L} ∥Φ ∥_{5} ∥ ψ ∥_{5}$
unlabeled · line 1143
$- i I, μ, ν \sum a_{I, ν} [Φ] (\partial_{2, μ} H_{I}^{μν}) .$

Nearby in this section

section: Transposition of the complete vertex