Manuscript X-B-ST@1 · X-B-ST.tex · lemma · d03c150c

Lemma: Pure-gauge Euler identity

First-order background identity and retained defects › An off-shell operator calculation · lines 1126–1157 · proof included or linked

X-B-ST@1#lem:Q, X-B-ST@1#eq:Qidentity

For every smooth vector field $ξ$ ,

Q_{K ξ} = D_{ξ} P_{m} - P_{m} X_{ξ} .

This is an identity on arbitrary smooth scalars.

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 .

In particular,

\partial_{μ} a_{1}^{μν} [K ξ] = - □ ξ^{ν} .

Expansion of $Q_{K ξ} u$ gives

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

Expanding $D_{ξ} P_{m} u - P_{m} (ξ^{ν} \partial_{ν} u)$ gives the same expression.

Displayed formulas in this passage

X-B-ST@1#eq:Qidentity · line 1129
$Q_{K ξ} = D_{ξ} P_{m} - P_{m} X_{ξ} .$
unlabeled · line 1138
$a_{1}^{μν} [K ξ] = (\partial \cdot ξ) η^{μν} - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}, b_{1} [K ξ] = \partial \cdot ξ .$
unlabeled · line 1146
$\partial_{μ} a_{1}^{μν} [K ξ] = - □ ξ^{ν} .$
unlabeled · line 1150
$(\partial \cdot ξ) P_{m} u - (□ ξ^{ν}) \partial_{ν} u - 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u .$