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

The divergence term in a density transformation

Sign, density, and switch audit › The divergence term in a density transformation · lines 2064–2083

For an orientation-preserving diffeomorphism,

F_{dens}^{*} r = (det D F) (r \circ F) .

For the flow of $ξ$ ,

\frac{d}{d s}_{s = 0} F_{s, dens}^{*} r = ξ^{μ} \partial_{μ} r + (\partial_{μ} ξ^{μ}) r .

This proves (eq:densityLie). Omitting the divergence term would replace the off-shell identity (eq:QK) by an incorrect operator commutator. Its contribution disappears on a free scalar only after imposing $P_{m} ϕ_{0} = 0$ .

Displayed formulas in this passage

unlabeled · line 2067
$F_{dens}^{*} r = (det D F) (r \circ F) .$
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$\frac{d}{d s}_{s = 0} F_{s, dens}^{*} r = ξ^{μ} \partial_{μ} r + (\partial_{μ} ξ^{μ}) r .$