Manuscript X-B@3 · X-B-v3.tex · lemma · 2d415d58

Lemma: Pure-gauge Euler coefficient

B3: first-order Ward identity and the second-order boundary › The three-term variation formula · lines 1103–1130 · proof included or linked

X-B@3#lem:Q, X-B@3#eq:Qidentity

For every smooth vector field,

Q_{K ξ} = D_{ξ} P_{m} - P_{m} X_{ξ} .

Proof. The density coefficients are

a_{1}^{μν} [K ξ] = (\partial \cdot ξ) η^{μν} - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}, b_{1} [K ξ] = \partial \cdot ξ .

Expanding $Q_{K ξ} u$ , the derivatives of $\partial \cdot ξ$ cancel, leaving

(\partial \cdot ξ) P_{m} u - (□ ξ^{ν}) \partial_{ν} u - 2 \partial^{μ} ξ^{ν} \partial_{μ} \partial_{ν} u .

Expanding the right-hand side of (eq:Qidentity) gives the same expression.

Displayed formulas in this passage

X-B@3#eq:Qidentity · line 1106
$Q_{K ξ} = D_{ξ} P_{m} - P_{m} X_{ξ} .$
unlabeled · line 1114
$a_{1}^{μν} [K ξ] = (\partial \cdot ξ) η^{μν} - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}, b_{1} [K ξ] = \partial \cdot ξ .$
unlabeled · line 1123
$(\partial \cdot ξ) P_{m} u - (□ ξ^{ν}) \partial_{ν} u - 2 \partial^{μ} ξ^{ν} \partial_{μ} \partial_{ν} u .$