Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: Density-valued scalar identity

Exact first-order repair and the larger-support defect › Two linear identities · lines 932–969 · proof included or linked

VII-A@1#lem:QK, VII-A@1#eq:QK, VII-A@1#eq:QKon

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For every smooth scalar $ψ$ ,

Q_{K ξ}^{(m)} ψ = D_{ξ} P_{m} ψ - P_{m} X_{ξ} ψ .

In particular, if $P_{m} ϕ_{0} = 0$ , then

Q_{K ξ}^{(m)} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .

Proof. The unswitched density-valued operator

D [g] ψ = \partial_{μ} (ρ_{g} g^{μν} \partial_{ν} ψ) - m^{2} ρ_{g} ψ

satisfies the change-of-variables identity

D [F^{*} g] (F^{*} ψ) = F_{dens}^{*} (D [g] ψ) .

Differentiate at $g = η$ along the flow of $ξ$ . The variation of the operator is $Q_{K ξ}^{(m)}$ , the variation of the scalar is $X_{ξ} ψ$ , and the variation of the output density is $D_{ξ} P_{m} ψ$ . Rearrangement gives (eq:QK).

Displayed formulas in this passage

VII-A@1#eq:QK · line 935
$Q_{K ξ}^{(m)} ψ = D_{ξ} P_{m} ψ - P_{m} X_{ξ} ψ .$
VII-A@1#eq:QKon · line 942
$Q_{K ξ}^{(m)} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .$
unlabeled · line 952
$D [g] ψ = \partial_{μ} (ρ_{g} g^{μν} \partial_{ν} ψ) - m^{2} ρ_{g} ψ$
unlabeled · line 960
$D [F^{*} g] (F^{*} ψ) = F_{dens}^{*} (D [g] ψ) .$

Lemma: Density-valued scalar identity

Displayed formulas in this passage

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